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Interactive Enzyme and Molecular Regulation
C H A P T E R
10 Interactive Enzyme and Molecular Regulation O U T L I N E 10.1 Protein Oligomerization and Interactive Enzyme 538 10.1.1 Covalently Bound Oligomers 539 10.1.2 Noncovalent Association Oligomerization 544 10.1.3 Domain Swapping Oligomerization 552 10.1.4 Interactive Enzyme Oligomer Mixture Model 557 10.2 Ligand Binding and Cooperativity 565 10.2.1 Single Ligand Binding on Homosteric Enzymes 569 10.2.2 Sequential Single Ligand Binding on Allosteric Enzymes 575 10.2.3 Single-Ligand Binding on Random-Access Allosteric Enzymes 579 10.3 Competitive Multiligand Binding on an Interactive Enzyme 581 10.3.1 Competitive Ligand Binding on a Homosteric Enzyme 581 10.3.2 Site-Sequential Multiligand Binding on Allosteric Enzymes 590 10.3.3 Multiligand Binding on Random-Access Allosteric Enzymes 593
Bioprocess Engineering http://dx.doi.org/10.1016/B978-0-444-63783-3.00010-1
10.3.4 Enzymes With Homosteric Paired Allosteric Sites 10.4 Catalytic Reaction Rate on Interactive Enzymes 10.4.1 Catalytic Reactions on Homosteric Sites 10.4.2 Catalytic Reactions on Site-Sequential Allosteric Sites 10.4.3 Random-Access Allosteric Enzymes 10.5 Kinetics of Polymorphic Catalysis and Allosteric Modulation 10.5.1 Substrate-Free Polymorph Interconversion 10.5.2 Substrate-Inert Polymorph Interconversion 10.5.3 Oligomers of Paired Allosteric Sites
597 598 598
601 603
604 604 607 612
10.6 Influence of a Competitive Effector on Interactive Enzymes
615
10.7 Summary
618
Bibliography
623
Problems
624
535
# 2017 Elsevier B.V. All rights reserved.
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10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
What is an interactive enzyme? How is an interactive enzyme formed? How does an interactive enzyme regulate biotransformations? All chemical reactions necessary for life are inadequately slow and thermodynamically complex to selectively occur spontaneously. One or more unique enzyme catalysts are required to direct and accelerate the reaction. Almost all enzymes are proteins that fold into domains. The majority of enzymes contain one domain in each molecule. In particular, a lot of enzymes contain one active center only that can interact with a specific type of ligand. This is what we call a simple enzyme in Chapter 7. Still many are composed of two or more domains allowing complex interactions between the enzyme and the molecules with which it can interact (for example, multifunctional proteins). An enzyme molecule contain more than one active centers is called an interactive enzyme, or duoweimei (多位酶) or multiple-sited enzyme. Mutual (two-way) interactions can only occur when multiple active centers are available. Most enzymes are designed to function at a constant rate, but interactive enzymes are sensitive to physiological controls, and thereby they adjust their rate and determine the flux through the metabolic pathway that they control. Most simple enzymes abide by the Michaelis-Menten kinetics quantitatively where the highest rate is the saturation rate that can be achieved (or closely approached) at relatively low substrate concentrations as we have learned in Chapter 7. As a result, a relatively constant rate can be delivered by the enzymes in normal conditions to the living organism they support. The critical concentration at which an enzyme approaches saturation rate (or half the saturation rate) is directly related to the affinity of the substrate to the enzyme, and the saturation rate is proportional to the turnaround frequency of the enzyme. The enzyme is regulated by changing the conformal structures (through interactions). Multiple sites or apparent multiple sites can appear because of the “aggregation” or conformal structure of multiple simple enzymes together to form an enzyme complex that becomes interactive. The conformal structural change of a complex enzyme induces the change of its affinity for a substrate and/or the enzyme turnaround frequency. This constitutes an interactive enzyme. In Chapter 7, we have learned the functions and fundamental approaches to the kinetics of enzyme-catalyzed reactions. All enzymes are remarkable for their ability to bind one or more substrates with appropriate specificity, and then facilitate a particular type of chemical reaction, producing one or more new products that are essential for the function of a living cell. For simple enzymes, Michaelis-Menten model can describe the kinetic behavior quite well even though mechanistically not an ideal model as compared with pseudosteady state approximations to the kinetic steps involved. A brief summary of the Michaelis-Menten kinetics is reviewed below that will set the stage for our discussions in this chapter. Michaelis-Menten model can be mechanistically presented as: Km
kc
E + S ! ES ! E + P
(10.1)
where E stands for the enzyme molecule, S stands for the substrate, and P stands for the product. There are two reaction steps represented by Eq. (10.1), the first step being the fast equilibrium step of enzyme (E) binding with the substrate (S) to produce the enzyme-substrate complex: Km ¼
½E½S ½ES
(10.2)
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
537
In Eq. (10.2), [E] is the concentration of the enzyme that is not bound with the substrate S, and [S] is the concentration of the substrate (that is not bound with the enzyme), [ES] is the concentration of enzyme-substrate complex, and Km is the Michaelis-Menten saturation constant. One can observe from Eqs. (10.1) and (10.2) that Km is the inverse of the equilibrium constant of binding, and thus the inverse of affinity of substrate to enzyme. In an enzyme reactive system, the concentration of the enzyme (or more precisely the enzyme loading) is much lower than the concentration of the substrate. Therefore, the effect of the concentration of the substrate by the enzyme-substrate binding is negligible, while substrate-enzyme complex consumes most of the enzyme. That is: E ¼ ½E + ½ES
(10.3)
where E is the total enzyme in the system, which is independent of enzyme-substrate binding. Solving Eqs. (10.2) and (10.3), one can obtain the ratio of enzyme-substrate complex to the total enzyme, θS ¼
½ES ½S ¼ ½ E Km + ½ S
(10.4)
where θS may also be termed as the substrate saturation ratio on the enzyme. It is the fraction of enzyme that binds with the substrate. The second step represented in Eq. (10.1) is the catalytic conversion of [ES] to E + P, or the turnover of the enzyme from the enzyme-substrate complex to free enzyme (again) and the released product. The overall reaction rate is then given by: rP ¼ kc ½ES
(10.5)
where kc is the enzyme turnaround frequency, or the catalytic rate constant. Combining Eqs. (10.2) and (10.3) into (10.4), one obtains the Michaelis-Menten equation: rP ¼
kc E½S rmax ½S ¼ Km + ½S Km + ½S
(10.6)
Enzymes can be amazingly fast. For normal chemical reactions, we have the example of a turnaround frequency (kc) greater than 106 s1 for catalase and for carbonic anhydrase. Enzymes can significantly increase the reaction rate; for example, orotidine monophosphate (OMP) decarboxylase can increase the rate of the decarboxylation of OMP by a factor of 1017 over the spontaneous rate. Over 5000 different enzymes have been characterized, and almost all of these are proteins. A limited number of catalytic reactions have been demonstrated with certain types of RNA molecules, and such catalytic RNAs are now called ribozymes. These first two types of enzymes are normal biological molecules that have evolved to have the features that make them so essential. In Chapter 7, we have learned some characteristics of allosteric enzymes. An allosteric enzyme, as the name implies, has more than one binding site available. In this chapter, we will learn more about how the enzymes have multiple sites and how they catalyze biotransformations.
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10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME Enzymes, or proteins in general, can consist of multiple subunits or oligomers. The presence of multiple active domains within one enzyme molecule or complex is the prefix for the enzyme to be interactive as the multiple domains are interconnected. Homo- or heterounits of proteins are commonly associated with each other through covalent bonds that are almost always irreversibly stable, through often reversible associations mediated by electrostatic and hydrophobic interactions, or through hydrogen bonds. The term “oligomer” is usually restricted to be less than 25 for the number of subunits. Marianayagam et al. (2004) characterized 450 enzymes, of which only about 140 are monomeric. Among the other 310 oligomeric enzymes, 200 are homo-oligomers; specifically, 125 homodimers, 50 homotetramers, and 25 structures larger than tetramers. The remaining about 110 are hetero-oligomers. Oligomers, particularly molecules of varying number of subunits, are common for enzymes. Oligomerization is the process of association or the addition of subunits to a base molecule to form a new molecule(s); therefore, it can be generalized as polymerization. However, the term “oligomerization” usually refers to polymerization with a low degree. Protein oligomerization has also been called various other terms, including assembly and multimerization. There is no clear distinction amongst the use of the terms; thus, they are interchangeable. Protein oligomerization is often accompanied by the onset of bioactivity and thus is a phenomenon crucial in triggering various physiological pathways. Biotransformations are often enabled by oligomeric enzymes that can respond to different/changing environmental conditions. For example, proteins or enzymes found with thermophiles or microbes living in high-temperature environments are often associated together by charge clusters, networks of hydrogen bonds, optimization of packing, and hydrophobic interactions, which is an indication that protein oligomers are more stable and can sustain a severe environment. Indeed, some hyperthermostable proteins have larger oligomeric size when compared to their mesophilic homologues. As such, one can expect that protein oligomerization is important to bioengineering or biotechnology in general. Proteins can self- or cross-associate either naturally or artificially. Protein oligomerization can occur when environmental conditions are changed, or crosslinking chemical reagent(s) are introduced. If a monomeric protein, ie, lacking quaternary structure, is considered as the starting species, we can obtain the first polymeric seed when a dimer, or different dimers, form. Polymerization can then continue towards trimer(s), tetramer (s), pentamer(s), and so on. The smallest subset of different subunits forming an oligomer is the structural unit of an oligomeric protein and is called a protomer. A protomer can be a protein subunit or several different subunits that assemble(s) in a defined stoichiometry to form an oligomer. When the oligomers are made of identical protomers, they are called homo-oligomer complexes; otherwise, they are hetero-oligomers. The heterostructures refer to chains of different sequences, which undergo association in a manner less statistically favorable and easily controllable, either qualitatively or quantitatively, than protein self-association. Protein hetero-oligomerization represents a very important phenomenon in the formation of molecular machines, like for motor proteins (eg, kinesin, microtubules), or alternatively, can be obtained artificially by the use of asymmetric bifunctional reagents.
10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
539
10.1.1 Covalently Bound Oligomers The type of protein oligomerization that first comes to mind is perhaps covalently linking protein residue units, just like well-known polymers from small monomers, such as cellulose (β-1,4-glucopyranose polymer), starch (mostly α-1,4-glucopyranose polymer), etc. The only difference here is that protein oligomers are formed from large monomers: proteins, which are already polymers of amino acids, or polypeptides having a DP or number of amino acid residues usually greater than 50. Protein crosslinking can occur by naturally forming covalently linked species (usually through a nonpeptide bond, for obvious reasons, as a peptide bond is what forms protein in the first place) that display quaternary structures, or active covalent complexes starting from inactive protomer precursors. Post-transduction modifications, photochemical process(es), or coenzyme binding (ie, going from apo- to halo-forms) can also induce protein self- or crosslinking. Crosslinking can sometimes occur in nature through free cysteines (ie, proteins containing thiol or –SH groups) of two different subunits that can couple to form intermolecular disulfides: R SH + HS R0 ! R S S R0
(10.7)
Disulfide bond is the common way to crosslinking proteins: A well-known example of oligomerization through disulfide bonding is represented by the antibodies Immunoglobulin M (IgM) and Immunoglobulin A (IgA). Antibodies are heavy (150 kDa) globular plasma proteins. Antibodies have sugar chains added to some of their amino acid residues; in other words, antibodies are glycoproteins. The basic functional unit of each antibody is an immunoglobulin (Ig) monomer (containing only one Ig unit); secreted antibodies can also be dimeric with two Ig units such as IgA, tetrameric with four Ig units like teleost fish IgM, or pentameric with five Ig units, like mammalian IgM. IgA is known to be able to form dimer, trimer, and tetramer. Fig. 10.1 shows a schematic of an IgM FIG. Light chains
Ig M monomer unit
Heavy chains J chain
Disulfide bonds
10.1 Immunoglobulin M (IgM) pentameric antibody molecule structure. The region enclosed by the dashed line represents a monomer that resembles that in Fig. 2.16. With permission from Liu, S., 2015. IJPEM, 16(13), 2731-2760.
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10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
pentamer. In fact, an antibody monomer itself may be regarded as a heterotetramer. The Ig monomer is a “Y”-shaped molecule that consists of four polypeptide chains, two identical heavy chains (450–550 amino acid residues each), and two identical light chains (211–217 amino acid residues each) connected by disulfide bonds. Each chain is composed of structural domains called immunoglobulin domains. These domains contain about 70–110 amino acid residues each and are classified into different categories (eg, variable or IgV, and constant or IgC) according to their size and function. They have a characteristic immunoglobulin fold in which two beta sheets create a “sandwich” shape, held together by interactions between conserved cysteines and other charged amino acids. To induce multiple functionalities, heterodimeric antibodies can be artificially made. Heterodimeric antibodies, which are also asymmetrical antibodies, allow for greater flexibility and new formats for attaching a variety of drugs to the antibody arms. One of the general formats for a heterodimeric antibody is the “knobs-into-holes” format. This format is specific to the heavy chain part of the constant region in antibodies. The “knobs” part is engineered by replacing a small amino acid with a larger one. It fits into the “hole,” which is engineered by replacing a large amino acid with a smaller one. What connects the “knobs” to the “holes” are the disulfide bonds between each chain. The “knobs-into-holes” shape facilitates antibodydependent cell-mediated cytotoxicity. Single chain variable fragments are connected to the variable domain of the heavy and light chain via a short linker peptide. The linker is rich in glycine, which gives it more flexibility, and serine/threonine, which gives it specificity. Two different single-chain variable fragments can be connected together via a hinge region to the constant domain of the heavy chain or the constant domain of the light chain. This gives the antibody bispecificity, allowing for the binding specificities of two different antigens. The “knobs-into-holes” format enhances heterodimer formation but doesn’t suppress homodimer formation. To further improve the function of heterodimeric antibodies, artificial antibodies have been constructed with largely diverse protein motifs that use the functional strategy of the antibody molecule, but aren’t limited by the structural constraints of the natural antibody. Being able to control the combinational design of the sequence and three-dimensional space could transcend the natural design and allow for the attachment of different combinations of drugs to the arms. Heterodimeric antibodies have a greater range in shapes they can take, and the drugs that are attached to the arms don’t have to be the same on each arm, allowing for different combinations of drugs to be used in cancer treatment. Pharmaceuticals are able to produce highly functional bispecific and even multispecific antibodies. Bovine seminal ribonuclease (BS-RNase) is another example that natural homooligomerization can occur through disulfide bonding. BS-RNase is the unique member of the large pancreatic-type secretory ribonuclease superfamily, which is dimeric in nature. Other proteins that can form covalent supramolecular structures are structural proteins such as collagen or elastin. The rubber properties of elastin, for example, are caused by desmosine (Fig. 10.2). Desmosine may appear to be a crosslinking agent, even though oligomerization was the cause of desmosine’s appearance, which resulted from the joining of lysine residues of the protomers (proteins). Elastic stress or relaxation can cause desmosine bridge formation or disruption in elastin in tissues or vessels. The J chain in IgA and IgM oligomers facilitates protein oligomerization through covalent binding to protein oligomers. Agents like the J chain that are not part of the original protomers
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10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
NH2
O
NH2
O
OH
OH
HO
HO
O
NH2 N+
NH2
HO
O
O
NH2
NH2
OH
HO
desmosine
(A)
O
N+
NH2
NH2
O
O
OH
isodesmosine
(B)
FIG. 10.2 Isoforms of desmosine, apparent covalent crosslinker found in elastin.
are crosslinking agents. As we have alluded to, protein oligomerization can be induced artificially. Chemical crosslinking can be and has been extensively employed to make protein oligomers artificially through additional dehydrating molecules, such as 1-ethyl-3-(3-dimethylaminopropyl)carbodiimide (EDC) or carbodiimides in general, or through the use of several bifunctional reagents, such as dialdehydes or diimidoesters, shown in Fig. 10.3. The two chemicals shown in Fig. 10.3, or more precisely crosslinkers, display two terminal reactive
−
O
O
Cl+NH2
(A) Glutaraldehyde
[
−Cl+NH 2
]n
O
O
(B) Diiimidoesters: n = 2, dimethyl-adipimidate; n = 3, -pimelimidate; n = 4, -suberimidate R-NH2 O
[
]4
NH2+Cl−
NH2+Cl−
R-NH
R-NH
[
R-NH2+ O
O
]4
NH2+Cl−
R-NH - 2 CH3OH
O
O
]4
NH2Cl−
NH2+Cl−
NH2+Cl−
[
+ R-NH2
R-NH
[
]4
NH3+Cl−
O NH3+Cl−
(C) Cross-linking reaction of dimethylsuberimidate FIG. 10.3 Glutaraldehyde and diimidoesters, and the crosslinking of proteins (in particular with a lysine residue) by dimethylsuberimidate.
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10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
groups separated by a variable number of unreactive spacers, such as methylenes. They are often symmetric, but asymmetric ones have also been employed for special properties. Some of these crosslinking reagents were extensively used to produce protein oligomers that have been useful, after limited proteolysis, to study protein primary and tertiary structures and, thus, protein conformations. Covalently linked oligomers can have increased activity, higher stability against proteases, and so on. Both disulfide bonding and dehydration crosslinkers react mainly with lysine residues, which act as nucleophiles towards the aldehydic or imidic carbon of the crosslinker under slightly basic conditions (pH ranging from 7.5 to 9). Fig. 10.3 shows two types of crosslinking reagents and an illustration of dehydration crosslinking by imidate, where R-NH2 stands for a protein (N-terminal), especially of that from a lysine residue. The yield of dimers is the highest, which can be more than 20%, while decreasing amounts of higher DP oligomers can be obtained: trimers, tetramers, and traces of even higher DP oligomers. The advantages of protein crosslinker diimidoesters over dialdehydes are that diimidoesters are nontoxic, less reactive and consequently inducing less undesired changes in the desired protein product. For example, dialdehydes can result in side intramolecular reactions potentially inactivating the protein. Diimidoester reactivity can be controlled and lysines may be modified allowing the overall charge of the protein to remain unmodified. Despite a lower oligomerization yield, the products obtained from diimidoesters are more specific and active. The longer the spacer, the higher the intermolecular yield and the less rigid the oligomeric product. Thus, dimethyl-suberimidate is more useful to artificially oligomerize a protein than the shorter diimidoesters. The symmetrical bifunctional N-substituted maleimide derivatives as shown in Fig. 10.4 are useful in protein oligomerization. The reaction occurs between the maleimide and the free R-SH
O
O
O
O
R-S N
N
N
Spacer
O
Spacer
N
O
O
O + R′-SH
Spacer O
O
R-S S N
S
Spacer
N S-R′
O O
O
O
FIG. 10.4 Reaction of a bismaleimide derivative with sulfhydryl groups. The spacer can be a benzene residue or other groups, such as those shown in the insert.
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10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
sulfhydryl group of a cysteine of the protein as an adduction, which saturates the double link of the maleimide conjugated with two carbonyls. The spacer can be benzene, N,N-1,2- or 1,3- or 1,4 phenyldimaleimide (ortho-, meta-, or para-PDM), or derivatives displaying spacers of varying length between the two reactive maleimides. Protein crosslinking is irreversible unless the spacers contain disulfide bonds (Fig. 10.4 with a disulfide spacer). Disulfide bonds can be broken and as a result, species can be drawn inside cells as heterodimers and then released as dissociated monomers exploiting the reducing environment of the cytosol. Thus, spacers containing disulfides can be applied to all bifunctional reagents, if reversibility is desired. There are spacers other than those shown in Fig. 10.4 that can lead to asymmetric structures. Asymmetric bifunctional crosslinking reagents can provide special properties, especially for covalently linking antibodies (or part of the light/heavy chains) with proteins, protein domains, or toxins. They can be a combination of maleimides or succinimides with or without dithio-derivatives in the spacers, and coupled, in the second terminus, with imidoesters, diones, thiones, 2-iminothiolane, etc. Almost all of these bifunctional crosslinking agents cause a two-step reaction, in which one of the partners is modified and activated (for example, with 2-iminothiolane) before it can react with the appropriate partner and form the new heterodimeric adduct. Some of the most important reactants used for artificial heterodimerization are shown in Fig. 10.5.
O
O
O
O S
O
N
N
O
N
S
O
N O
O
O
(B) SPDP, or N- succinimidyl-3-(2-pyridyldithio)
(A) MBS, or 1-(3-((2,5-dioxopyrrolidinyl)
propionate
oxy-carbonyl)phenyl)-1H-pyrrole-2,5dione O
O
O
N
O
SSO3−
O
N
O
S
S N
O
(C) SMBT, or S-4-succinimidyloxycarbonyla-methyl benzyl thiosulfate
O
(D) SMPT, or 4-succinimidyloxycarbonyl-amethyl-a(2pyridyldithio)toluene
FIG. 10.5 Examples of asymmetric bifunctional protein crosslinking reagents.
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10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
O
NO2
F
F
NO2
S O
(A) Divinyl sulfone (DVS)
FIG. 10.6
(B) Dinitrodifluorobenzene (DFDNB)
Chemical structures of short bifunctional reagents (A) DVS and (B) DFDNB.
Carbodiimides (R1 N ¼ C ¼ C R2), such as EDC, are bifunctional crosslinkers, which forms new isopeptide bonds between the side chains of lysine and glutamic acid or aspartic acid residues of proteins, producing “zero-length” crosslinks. They can be applied to covalently fix existing oligomeric protein aggregates avoiding undesired insertions of group(s) or net charge modifications in the protein complex. Fig. 10.6 shows the chemical structures of two “short” bifunctional reagents. They lack spacers and stabilize preformed structures without affecting oligomers’ conformations. DVS is specific for histidines, while DFDNB reacts with lysines. The limited dimensions of both molecules allow crosslinking only between residues that are very close to each other. The crosslinking can produce intra- or intermolecular adducts, and can thus chromatographically or electrophoretically reveal if the protein was monomeric or already oligomeric before covalent stabilization. Covalent protein oligomerization can also be carried out without the additional crosslinking agent by sealing a lyophilized protein in a vacuum at high temperature (<85°C) for 24–96 h. Dimers, trimers, and traces of tetramers of ribonuclease A (RNase A) and lysozyme can be made without introducing chemical groups that could negatively affect the protein residues. In fact, the heat vacuum treatment of proteins induces the dehydration of some of the lysine and aspartic acid or glutamic acid sidechains, producing newly formed intermolecular isopeptide bonds. This reaction, thought to be specific to a single couple of lysine and glutamic acid residues of RNase A, was found to involve more than one of these acid or basic residues, producing a mixture of heterogeneous products. Crosslinking can also be achieved through UV and/or microwave photochemistry. UV irradiation of a monomer or pre-formed non-covalent oligomers is known to produce covalently stabilized oligomers. However, irradiation treatments can inactivate the native protein or possibly its preformed oligomers.
10.1.2 Noncovalent Association Oligomerization While covalently bound oligomers are intuitive to understand, noncovalent protein oligomers are commonly involved in biotransformations. Protein association can occur often naturally without covalent modification(s), through a homo- or heteroassociation mediated by a weak bond network formed by electrostatic-hydrophobic interactions, and/or specific hydrogen bond(s). If the interface between monomers or protomers is large, almost all types of interactions can occur and are more frequently conserved. These interactions can be crucial for the active forms of several classes of proteins, such as enzymes and transporters. The
10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
545
interaction(s)/association(s) may occur naturally because of the sequence and structural features of the subunits, which build the oligomeric complex(es). The noncovalent associations can also result due to environmental changes, such as pH or ionic strength, or as a consequence of the increase of the monomers’ local concentration. These interactions allow the protein to overpass its dimerization dissociation constant(s) KD: E + E ! E2 KD ¼
(10.8)
½E2 ½E2
(10.9)
A further increase in the concentration can augment the degree of polymerization (DP) with the formation of trimers (E3, DP ¼ 3), tetramers (E4, DP ¼ 4), and larger oligomers (En, DP ¼ n > 4), if the entropy cost is balanced by favorable interface interaction(s): E + E2 ! E3
(10.10)
E + E3 ! E4
(10.11)
E + En1 ! En
(10.12)
or stoichiometrically, the oligomerization in general can be expressed as: nE ! En
(10.13)
The free energy of oligomerization can be represented by: ΔG ¼ ΔG0 + RT ln
½ En ½ E n
(10.14)
in which [E] is the concentration of protein segment(s) exposed, or protein interfaces that are able to interact with other protomers. The simplistic stoichiometry representation in Eqs. (10.8) through (10.14) does not mean that oligomerization can be arbitrary, nor that n can be large. For example, the DNA-binding protein Arc repressor is predominantly monomeric and unfolded at low concentrations, while mostly dimeric at high concentrations in 10 mM KH2PO4/K2HPO4 and 100 mM KCl with pH 7.3. Higher and/or different oligomers are not found unless the structure is altered chemically. There are many factors that can determine the number of oligomers and/or the number of subunits/protomer residues in an oligomer. The protomer concentration, temperature, and solvent are the key parameters governing oligomerization. Protomer chemical properties are key in deciding whether oligomerization could occur, and/or how many protomers could be associated together in a certain fashion. Chemical bonding can only occur between specific groups. The oxygen transporter hemoglobin (Hb) (Fig. 10.7) is an example of a natural oligomeric protein. Structurally, Hb is consisted of mostly α-helices (illustrated as ribbons in Fig. 10.7). Hb is functionally active only as an α2β2 tetramer (or, in the fetus, α2γ2, endowed with a greater affinity to oxygen), with its α and β subunits being associated through salt bridges and other
546
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION hems a1
b2
N
N Fe2+ N
N
a2
b1
O OH
(A) A schematic of human hemoglobin (hHb)
O
OH
(B) The chemical structure of hem
FIG. 10.7 Structure of hemoglobin and its oxygen-binding hem. The functional protein is an oligomer containing two α subunits, two β subunits, and an iron-containing heme group for each protein subunit. Both α and β subunits are consisted of mostly α-helices. With permission from Liu, S., 2015. IJPEM, 16(13), 2731–2760. O OH
OH O
P HO O
N
FIG. 10.8 Pyridoxal 5-phosphate (PLP).
weak interactions. The loss of these interactions leads Hb to switch its conformation from the deoxygenated to the oxygenated form; this modification is due to the cooperative binding interactions triggered by the first oxygen bound. Hb, unless it is tetrameric, cannot be active, while myoglobin, the oxygen collector in tissues, is active as a monomer. Pyridoxal 5-phosphate (PLP) enzymes are also typical non-covalent associated proteins. Fig. 10.8 shows the chemical structure of PLP. For example, the aspartic aminotransferase (EC 2.6.1.1, AST), alanine-glyoxylate-transaminase (EC 2.6.1.44, AGT) (Fig. 10.9A), aromatic L-amino acid decarboxylase (EC 4.1.1.28, AADC) (Fig. 10.9B), and cystalisin (E.C. 4.4.1.1) are inactive as monomers and active only when they are in the dimer forms. The PLP-dependent enzyme family is known to have at least 145 distinct enzymes. It can be classified into 5 enzyme categories: 1-oxido-reductases (EC 1; one enzyme), 2-transferases (EC 2; eighty enzymes), 3-hydrolases (EC 3; two enzymes), 4-lyases (EC 4; Forty nine enzymes), and 5-isomerases (EC 5; thirteen enzymes). Basically, all the PLP-dependent-enzymes are mainly involved in amino acid transformations as decarboxylation, transamination, racemization, β,γ-elimination (elimination or truncation from β or γ location), and β,γ-substitution. The multiple functional catalytic modes of PLP-enzymes, according to the position of net reaction, are summarized in Table 10.1, along with the type of the enzyme oligomers. The PLP coenzyme
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10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
Cα
Nα
PLP Cα
O
OH
OH O P
O
HO O
N OH
OH O P
HO
N
O
Nβ
α-Subunit Nβ
Cβ
PLP
Cβ
α-Subunit
FIG. 10.9 Structure of holo natively dimeric PLP enzymes: AGT and AADC.
plays a pivotal role in catalysis of various enzymatic reactions. Structurally, aldehyde groups of PLP are bound covalently as internal aldimine (Schiff base)/imine linkage to the ε-amine group of lysine residues close to the N-terminus, as in Fig. 10.9A. Glycogen phosphorylase (EC 2.4.1.1) has inactive monomer and tetramer forms and an active homodimer form. It has a PLP at each active site. Glycogen phosphorylase is the enzyme responsible for catalyzing the conversion of a terminal glucose residue in linear starch or glycogen to glucose-1-phosphate (G1P). Orthophosphate recognition at allosteric binding sites is a key feature for the regulation of enzyme activity in glycogen (or starch) phosphorylases. Fig. 10.10 shows the recovery of catalytic activity of denatured bacterial starch phosphorylase (from Corynebacterium callunae) CcGlgP when PLP is introduced into the medium with different amounts of orthophosphate addition. One can observe that the addition of PLP can restore catalytic activity in partly inactivated enzyme preparations. Orthophosphate has a positive effect on the catalytic activity of CcGlgP. Another interesting example of an oligomeric protein is phenylalanine hydroxylase (PAH, EC 1.14.16.1). The molar mass of a PAH monomer is 51.9 kDa. Prokaryotic PAH is monomeric, whereas eukaryotic PAH exists in an equilibrium between homotetrameric and homodimeric forms. The catalytic active forms are either homodimer or homotetramer, with a homotetramer being more active. Its activity depends on its tetrameric structure and the presence of tetrahydrobiopterin (BH4) and is cooperatively regulated (also known as allosterically regulated by the substrate, in this case Phe, itself). Some mutations become pathogenic because they destabilize and inactivate the tetramer, which consequently results in the organism to become phenylketonuria (PKU). PAH catalyzes the hydroxylation of the aromatic sidechain of phenylalanine to generate tyrosine: +
NH3
O
+
NH3
O
O2 +
PAH
O–
O–
OH
ð10:15Þ NH
NH2
N
OH
NH OH
N
NH
NH
NH OH
O
NH2
N
OH
OH
O
Oligomeric Type and Functional Catalytic Modes of PLP-Enzymes According to the Position of Net Reaction
Reaction Site
Reaction Type
α
Racemization
Mechanism of Reaction
Example of PLP-Enzyme
R
H
C H
R
NH2
NH2 COOH
CH
CH R
NH2
COOH
COOH
NH2
COOH CH
CH H NH2
NH2 Replacement
COOH
COOH R1 β
O
EC 2.1.2.1 Glycine hydroxylmethyltransferase A dimer
H
R
β
R1
R2 O
NH2
CH
+
CH
R2
EC 2.6.1.5 Tyrosine amino-transferase A dimer
COOH
COOH
CH
+
CH R1
Elimination
+ CO2
R NH2
Transamination
EC 4.1.1.17 Ornithine decarboxylase A homodimer
H
CH α NH2
R2 β
CH α NH2
EC 4.2.1.20 Tryptophan synthase A tetramer (αββα)
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Decarboxylation
EC 5.1.1.1 Alanine racemase A dimer
COOH
COOH C
548
TABLE 10.1
TABLE 10.1 Oligomeric Type and Functional Catalytic Modes of PLP-Enzymes According to the Position of Net Reaction—cont’d Reaction Type
Mechanism of Reaction
Elimination
R β γ
Example of PLP-Enzyme
COOH
COOH
CH α
CH
C O
NH2
Replacement
NH2
COOH
R1
γ
β
Elimination
R
γ
β
EC 4.3.1.17 Serine dehydratase A homodimer
COOH
CH α
R2
γ
β
NH2 COOH
COOH
CH α
CH
NH2
EC 2.5.1.48 Cystathionine γ-synthase A tetramer
COOH CH α NH2
EC 4.4.1.11 Methionine γ-lyase A dimer of dimers
COOH C O
NH2
10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
Reaction Site
549
550
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION N
N C
N
(B) Ebola VP 40 antiparallel dimer
(A) Ebola VP40 protein structure. C-domain is colored green, while N-domain is colored brown. All the α-helics and β-sheets are labeled
N N
(C) Ebola VP 40 parallel dimer
° 80A
(D) Ebola VP 40 heximer
° 20A
(E) Ebola VP 40 octamer
FIG. 10.10
The structures of Ebola matrix protein VP40: (A) detailed monomeric unit structure, (B) an antiparallel dimer, (C) a butterfly-shaped (parallel) dimer structure critical for membrane trafficking, (D) a rearranged hexamer structure essential for building and releasing nascent virions, and (E) an RNA-binding octamer ring that controls transcription in infected cells. Reprinted with permission from Liu, S., 2015. IJPEM, 16(13), 2731–2760.
A natively oligomeric protein can also switch towards higher DP oligomers. For example, the natively homotetrameric L-rhamnulose-1-phosphatase aldolase becomes an octamer only upon a single A88F mutation. The introduction of a single residue displaying large nonpolar sidechains (Phe, Trp) can be sufficient to induce the formation of larger oligomeric complex(es). Protein oligomerization can also occur after its production or purified from fermentation broth. Monomeric proteins can naturally and noncovalently undergo oligomerization during post translation, which can become a switch between active and inactive products. For example, several transmembrane receptors are known to oligomerize during applications, which often display kinase activity. Upon ligand binding, the intracellular domain dimerizes, which triggers (auto)phosphorylation of the intracellular domain undergoing conformational changes able to activate a signal transduction cascade that induces or tunes important physiological phenomena. Examples of families of this type of receptors are growth hormone (GH), interferon (IFN), cytokine and Tyr-kinase, and G- protein-coupled receptor families
10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
551
(GPCRs). For example, one of the Tyr-kinase receptors, insulin receptor is, once released into the bloodstream, insulin can bind to receptors on the surface of cells in muscle or other tissues. The insulin receptor is a protein with an (αβ)2 quaternary structure. The two large α-subunits are extracellular, while the smaller β-subunits have a transmembrane domain as well as extraand intracellular domains. In the absence of insulin, the two intracellular domains of the β subunits are separated. Binding with insulin triggers a conformational change in the receptor that brings them closer together (dimerization). Each β subunit intracellular domain is a tyrosine kinase that phosphorylates its partner in the receptor. A family of proteins involved in apoptosis, the Caspase-3, -7, and -9, are interesting oligomeric proteins. Under normal physiological conditions, Caspase-9 exists as an inactive monomer, forming a 1:1 complex with the Apaf-1 cofactor in the presence of cytochrome c and ATP to produce a heteromultimer. This complex colocalizes with a multiple array of Caspase-9 molecules, which consequently increase their concentration above the Caspase-9 homodimer dissociation constant KD. This process allows the homodimer to be formed through an activation loop, and the active dimer provides the catalytic activity necessary to activate Caspase-3 and-7. The membrane channel-forming tetrameric natural oligomer complexes allow specific ions (Na+ or K+) or water to permeate cells, such as aquaporins or aquaglyceroporins. Aquaporins form tetramers in the cell membrane, with each monomer acting as a water channel. The different aquaporins differ in their peptide sequences, which allows for the size of the pore in the protein to differ between aquaporins. The resultant size of the pore directly affects what molecules are able to pass through the pore, with small pore sizes only allowing small molecules like water to pass through the pore. Aquaporins selectively conduct water molecules in and out of the cell, while preventing the passage of ions and other solutes. As such, aquaporins are integral membrane pore proteins and are also known as water channels. Some of them, known as aquaglyceroporins, also transport other small, uncharged solutes, such as glycerol, CO2, ammonia, and urea across the membrane, depending on the size of the pore. For exam˚ and allows the passage of hydrophilic ple, the aquaporin 3 channel has a pore width of 8–10 A molecules between 150 and 200 Da. However, the water pores are completely impermeable to charged species, such as protons, which is a property critical for the conservation of the membrane’s electrochemical potential difference. Proteins can also form large pathogenic oligomers that can evolve towards pathogenic supramolecular structures. Important examples of these malignant processes are the uncontrolled aggregation of the Glu6Val Hb mutant of Hb (E6V-Hb or HbS) in sickle cell anemia and the formation of amyloid or amyloid-like fibrils, as it occurs with several proteins related to severe neurodegenerative diseases. Viruses can be under tremendous evolutionary pressure for the economy of genomic information. The Ebola virus, which causes hemorrhagic fevers with up to 90% lethality, for example, encodes only seven genes transcripting seven proteins, the protein products of which must achieve all of the different steps of the virus life cycle. As a result of this genomic economy, each of the proteins it encodes is essential and commonly multifunctional. One of these proteins is EBOVE VP40 (shown in Fig. 10.10), the viral structural matrix protein that builds the protein shell underneath the viral membrane to assemble and release progeny viruses from the infected cell. VP40 alone is necessary and sufficient for the assembly and release of Ebola virus-looking particles from transfected cells. The general fold of VP40
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10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
has weakly associated N- and C-terminal domains, and a ring-like arrangement made by the N-terminal domains alone. RNA was observed bound to the ring. Purified VP40 protein is shown to be dimeric, not monomeric. This dimeric VP40 grew multiple crystal forms. In fact, two dimeric structures have been confirmed; both are associated by the N-termini, with one being parallel and the other being antiparallel, as shown in Fig. 10.10. No matter what space group the crystals belonged to or what species of Ebolavirus was analyzed, the VP40 dimers assembled end to end in identical linear filaments. The filaments were assembled by protein-protein interfaces that were distinct from those that assembled the RNA-binding ring. The RNA-binding ring assembly of VP40 was not involved in virus assembly. The protein VP40 is called a “transformer” to reflect its ability to refold its structure in order to achieve new functions: a dimer for membrane trafficking, an RNA-binding ring structure for viral transcription, and a filamentous oligomeric assembly for building and budding new virions. One other transformer has been identified, ie, the transcriptional regulatory protein RfaH.
10.1.3 Domain Swapping Oligomerization A peculiar way of forming protein oligomers is through the exchange of a section of peptide(s) or entire domain(s) among the monomeric subunits. Monomers exploit short, flexible hinge regions present in their sequence to address a given domain (or more) in the corresponding partner subunit that can reciprocally swap an identical domain with the former, as illustrated in Fig. 10.11. Three regions of the monomers can be identified in the swapped horse cytochrome c dimer: the main domain (containing α-helix carboxylic terminal CA or CB); the swapped domain (or secondary domain, containing α-helix amine terminal NA or NB); and the hinge region. This mechanism is called DS, which is also known as threedimensional domain swapping or 3D-DS and schematically shown in Fig. 10.12. Beyond dimers, this mechanism, which is known for diphtheria toxin (DT) as well as for other proteins, can also lead to the formation of higher DP oligomers exploiting small, flexible hinge regions that are able to adopt different conformations within various different environmental conditions.
Hinge region NB CB
CB
CA
+ CA NA
NA
NB A
B
Hinge region Swapped domains
FIG. 10.11 An example of domain swapping: two hose cytochrome c monomers A and B to α-helix amine terminal swapped horse cytochrome c dimer. Redraw based on Deshpande, M.S., et al. 2014. Biochemistry, 53, 4696–4703.
10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
Domain swapping
+
553
FIG. 10.12 Schematic view of the mutually reciprocal domain swapping mechanism. The opening of the hinge region in the “starting” monomer (left) allows the formation of the dimer by recreating the intramolecular interdomain interface present in the monomer in an intermolecular-dimeric interface instead. This subsequently forms a new dimeric interface (between the hinge regions), which is absent in the monomer. Through this mechanism, a protein can form an active dimer while still maintaining functional units (F.U.).
The domain-swapped oligomer reconstitutes the native contacts present in the monomer except the hinge region, while a new interface (between the hinge regions) forms in the oligomer only to stabilize it, as one can infer from Figs. 10.11 and 10.12. One can imagine that a stable oligomer can be formed by mutually (or reciprocally) swapping secondary domains from each protomer. However, this would allow a dimer form only for protomers with singular secondary (swappable) domains. Potential stable oligomers of single-swappable domain protomers with DP greater than 2 are cyclic oligomers. Moving beyond dimers without cyclic swapping will require additional oligomerization means for singularswappable domain protomers. As proteins have two end groups, ie, C-terminus and N-terminus, it is not difficult to imagine that a lot of proteins/enzymes can swap two secondary domains. The capability of swapping two subdomains allows for a high degree of oligomerization, as illustrated in Fig. 10.13. The RNase A oligomers are in general mutually and reciprocally domain-swapped oligomers, as shown in Fig. 10.13. In appropriate conditions, such as 40% ethanol/60% water (v/v) at 60°C, two RNase A monomers (E1) can dimerize by DS the C-terminal β-strand to form Cdimer (DC, or CE2), or their N-terminal α-helix to form N-dimer (DN, or NE2). DN can combine with another monomer to produce a cyclic N-swapped trimer (cNP3). Whereas only DC can combine with another monomer to produce a cyclic C-swapped trimer (cCE3), both dimers can combine with another monomer to produce a linear trimer (NCE3). The addition of another monomer via N-terminal swapping to cCE3 produces the tetramer NcCE4. The addition of a monomer to NCE3 can produce the linear tetramers CNCE4 and NCNE4. NCNE4 is also in equilibrium with its bent form bNCNE4. Alternatively, the NCNE4 and CNCE4 tetramers may also be formed via the combination of two dimers. Isomerization of the protein can lead to various forms of the same stoichiometry. For example, cis $ trans isomerization of the Asn 113-Pro 114 peptide bond in one subunit in the central interface is responsible for the two NCNE4 shown with cis in bNCNE4 and trans in linear NCNE4. The formation of cyclic C-swapped tetramers (cCE4) can start with two C-swapped dimers or a cyclic C-swapper trimer and a monomer. However, rearrangement in the swapping of C-termini amounts to multiple swapping steps. RNase A also forms pentamers (E5), hexamers (E6) and higher DP oligomers (En), up to tetradecamers (E14). The linear oligomers are all mutually domain swapped. However, there are cooperative domain swapped oligomers, such as the cyclic oligomers in Fig. 10.13(v), (vi), (vii), and (xi). Cooperative DS means swapping among multiple protomers,
554 10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
FIG. 10.13
Schematic of RNase A domain-swapped oligomers formations. With permission from Liu, S., 2015. IJPEM, 16(13), 2731–2760.
555
10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
FIG. 10.14 Schematic view of the cooperative domain swapping. The swapping of secondary domain is not mutually reciprocal. E1
14
DC or CE2 DN or NE2
12
Concentration, mM
E3 10
E4
8 6 4 2 0 0
20
40
60 80 Time, min
100
120
140
FIG. 10.15
Ribonuclease A oligomerization in 40% aqueous ethanol at 60°C with 200 mg/mL (14.6 mM ¼ 0.0146 M) proteins. Five species are shown: E1, monomer; DC, C-swapped dimer; DN, N-swapped dimer; E3, trimers; E4, tetramers. Data from Geiger, R., et al. 2011. J. Biol. Chem., 286, 5813–5822.
or else the swapping is not mutually reciprocal, as illustrated in Fig. 10.14. Cooperative swapping can lead to higher degree of oligomerization than mutual swapping. The oligomerization is fast and can occur in minutes. Fig. 10.15 shows the kinetics of oligomerization for RNase A in 40% aqueous ethanol solution at 60°C. After 60 min, all concentrations of the oligomers plateaued. There are multiple oligomer species present at equilibrium. One can observe that there are more monomers (P1) in the solution than dimers (P2), trimers (P3), tetramers (P4), and oligomers of higher DP at equilibrium. Some of the oligomerization rate constants and equilibrium constants are shown in Table 10.2. Beyond RNases, several other proteins involved in important biologic processes form domain-swapped oligomers. One DS-prone protein which is highly structurally versatile is cyanovirin-N, an 11 kDa protein that inhibits HIV. It can be active either as a monomer or as a metastable domain-swapped dimer. Interestingly, some mutants that become active only as domain-swapped dimers were found to form two different relatively stable DS dimeric conformers, one DS trimer, and two DS tetramers. Cystatins, a class of proteins comprised of stefins that inhibit cysteine proteases, can also dimerize through DS. Cytochrome c, which was known to polymerize, was shown to form
556
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
TABLE 10.2 Parameters Describing the Oligomerization of Ribonuclease A Oligomerization Step
Terminus Swapped
Keq
kf (M21 s21)
kb (s21)
E + E ! CD (or CE2)
C
20
1.0 10
E + E ! ND (or NE2)
N
11
4.0 103
3.63 104
CD + E ! cCE3
C
12
2.0 102
1.67 103
CD + E ! NCE3
N
34
8.0 103
2.35 104
ND + E ! NCE3
C
60
2.0 102
3.33 104
! NcCE4
N
23
8.0 103
3.48 104
NCE3 + E
! NCNE4
N
23
8.0 103
3.48 104
NCE3 + E
! CNCE4
N
40
2.0 102
5.0 104
ND + ND ! NCNE4
C
125
2.0 102
1.60 104
CD + CD ! CNCE4
N
68
8.0 103
1.18 104
0.1 s1
0.1 s1
cCE3 + E
NCNE4
! bNCNE4
1
2
5.00 104
Reprinted with permission from Lo´pez-Alonso, J.P. et al., Arch. Biochem. Biophys. 2009, 489(1–2):41–47.
domain-swapped inactive supramolecular structures via a cooperative DS of its C-termini. In particular, domain-swapped dimers, trimers, tetramers, and polymers up to 40-mers have been characterized. The L68Q human cystatin-C (hcC) mutant, in particular, can initialize the cooperative DS, inducing severe massive amyloidosis in brain arteries and lethal cerebral hemorrhages. There are three more important examples of proteins able to form domain-swapped structures. The first is BCL-XL, an anti-apoptotic protein belonging to the BCL-2 family, which can form active C-terminal-swapped dimers when highly concentrated or alternatively when heated up to 50°C. The second is cadherins, which are cell adhesion proteins that dimerize through β-strand swapping to mediate the adhesion itself. It is worth mentioning that several protein cell receptors are known to dimerize to become active, but less is known about the mechanism responsible for the dimerization. Thus, it is possible that some of them could undergo DS. Finally, histones are also known to fold through DS in their evolutionary pathway. While protein oligomerization is generally beneficial in health, uncontrolled oligomerization or polymerization can lead to severe problems or diseases. Oligomerization could be viewed either as beneficial or detrimental depending on the situation or desired application. Protein oligomers can be the first deleterious seed leading to protein fibrilization (polymerization to large/long molecules), a process implicated in several devastating neurodegenerative diseases. Amyloidosis, for example, is a rare disease that results from accumulation of inappropriately folded proteins, or more appropriately, protein oligomers of a high DP. These misfolded proteins or high DP oligomers are called amyloids. Technically, these high DP macromolecular proteins are not oligomers anymore; they are properly termed polymers. Only brief discussion will be made in this paper on the larger than normal oligomers. While proteins are normally soluble in water, amyloids are insoluble and deposit in organs or tissues, disrupting normal
10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
557
function. The type of protein that is misfolded into an amyloid and the organ or tissue in which the amyloids are deposited determine the clinical manifestations of amyloidosis. For example, amyloidosis plays a role in the pathologic process of amyloid plaque formation in Alzheimer’s disease and prion infections such as scrapie and Creutzfeld-Jacob disease. While stable mutual domain-swapped oligomers are common and practical applications abound, there are protomers known to follow cooperative DS as well. Some amyloidogenic proteins form fibrils through cooperative DS as illustrated. For example, the structural similarity existing between the Asn-based open interface of RNase A C-swapped dimer, –CE2– or –CD– and the fibrillogenic nature of poly-Q expansions, which are structured as cross-β-spines. Either prion protein (PrP) or cystatin-C, two amyloidogenic cross-βspine-prone proteins, form DS dimers. The PrP, which is associated with the lethal neurodegenerative Creutzfeld-Jacobs Disease and Scrapie, dimerizes through DS. The cooperative domain-swapped oligomers are formed with open-ended edges “neutralized” or stabilized by intermolecular newly formed disulfide bonds, but do not affect the overall tertiary structure of the globular main domain of PrP. The finding that the prevention of DS effectively inhibits cystatin-C dimerization and oligomerization and studies of the hinge region governing the DS process confirm that cooperative DS plays a key role in cystatin-C fibrillogenesis. Another important amyloidogenic protein able to self-associate through cooperative DS is β2-micro globulin (β2-m), the 10.9 kDa light chain of type-I histocompatibility complex, which can seed as amyloid fibrils during long-term hemodyalisis treatments. Like PrP, β2-m dimerizes to form an open-ended dimer and subsequently propagate to open-ended domainswapped amyloid fibrils stabilized by disulfide bonds. Three more amyloidogenic proteins are known to dimerize and massively aggregate through cooperative DS. The first worth mentioning is the immunoglobulin G-binding B1 domain, which forms DS conformational different dimers and tetramers induced by coredomain mutations before forming fibrils. The second one is T7-endonuclease I, which forms cooperative domain-swapped fibrils stabilized by core-domain intermolecular disulfides. The last one is the cell cycle protein Cks1, which fibrilizes through the preliminary formation of a domain-swapped dimer. Conversely, for another important amyloidogenic protein, transthyretin (TTR), cooperative DS is hypothesized, but the direct stacking model interaction through disulfide bonds between subunits is the one still accepted. Domain-swapped oligomers generally have known monomers of identical primary and secondary domains. In rare occasions, natural protein oligomers resembling domainswapped oligomers have no known protomers, and these multidomain proteins are called quasidomain-swapped oligomers. Examples of quasidomain swapping include crystalline pheromone/odorant binding/transport proteins, rice yellow mottle viral capsid protein, human cystatin C (protease inhibitor), Mycobacterium tuberculosis CarD, etc.
10.1.4 Interactive Enzyme Oligomer Mixture Model When an interactive enzyme can oligomerize while binding ligand(s) or catalyzing biotransformations, we have an interactive mixture of enzymes to deal with that was generally termed allostericity. The allostericity was coined to describe the “different” binding sites.
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10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Naturally, we have been introduced to allostericity in Chapter 7. Early models of enzyme mixture are all restricted to two “distinct oligomers” or molecules, known as allosteric modulators. Monod et al. (1963) proposed their model for the kinetics of allostericity, which is later known as the MWC model, based on a tense oligomer and a relaxed oligomer, with both having the same degree of oligomerization. The following year, Koshland et al. (1966) extended this model to add additional modes for conformational change, leading to the KNF model, as known later. The morpheerin model was proposed by Jaffe and coworkers (eg, Jaffe, 2005) to modify MWC model allowing the two oligomers to have different degrees of oligomerization. A summary of the three models is shown in Fig. 10.16. All three models impose different modes of interaction between the oligomers:
The concerted MWC model
0
(A)
ET
ER
ET + ER
ER + S
ER•S
ERSi + S
ET + S
ET•S
ETSi + S
ER•Si+1 ET•Si+1
ER•S
ERSi + S
ER•Si +1
The sequential KNF model
0
(B)
ET
ER
ET + ER
ER + S
The morpheein model
0
(C)
1M
2M
1M + 2M
iM + S
iM•S
Increasing ligand concentration
FIG. 10.16 Schematics of the ligand binding on a pair of enzyme oligomers as described by three classic models: MWC, KNF and morpheein. The MWC (Monod-Wyman-Changeux) model considers the interconversion between two states or forms of an oligomer (ET and ER of a tetramer in the drawing), while ligands can bind on either states. The KNF (Koshland-Ne´methy-Filmer) model restricts the ligands binding to one state, while the interconversion between the two states or forms is subunit-wise (shown for a tetramer). The morpheein model imposes on strict homogeneity among subunits on each enzyme molecule and no conversion directly between enzyme oligomers (shown a trimer and a tetramer); effectively, each subunit can be treated separately rather than the oligomers themselves, ie, morpheein 1M and 2M.
10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
559
FIG. 10.17
T
KT
KT Ke
R
S KR
Reaction scheme of a MonodWyman-Changeux (MWC) model of a protein (or protein complex) made up of two protomers. The protomer can exist under two states (ET and ER), each with a different affinity for the ligand (S): KT and KR. Ke is the ratio of states (concentrations) in the absence of ligand (or equilibrium constant).
S KR
MWC model explored the “allostericity” based on thermodynamics and three-dimensional conformations. Assuming an oligomeric protein with symmetrically arranged identical subunits, each of which has one active center (ligand binding site); two (or more) interconvertible conformational states of a complex protein were postulated to coexist in a thermodynamic equilibrium. The states, often termed tense (T) and relaxed (R), differ in affinity for the ligand molecule. The distribution between the two states (or allosteric sites) is affected by the binding of ligand molecules, as illustrated in Fig. 10.17. All subunits of an enzyme molecule change states at the same time, a phenomenon known as “concerted transition.” The resulting model is better known today as the MWC model for concerted allosteric transitions. When n ligand molecules can bind with one single protein complex (n ¼ 2 as shown in Fig. 10.17), the fractional occupancy based on MWC model is given by: ½S ½S n1 ½S ½S n1 1+ + αKe 1+α K K K K R R (10.16) θ S ¼ R R n ½S ½S n 1+ + Ke 1 + α KR KR where Ke is the isomerization constant or equilibrium constant between the two states when no ligand molecule is bound and α is the ratio of the dissociation constants for the ligand from the R and T states: Ke ¼
½ET0 ½ER0
(10.17)
KR (10.18) KT The corresponding catalytic reaction rate equation is given by: kT ½S ½S n1 ½S ½S n1 1+ + αKe 1+α k K KR K K n R n R rP ¼ rmax R R (10.19) ½S ½S 1+ + Ke 1 + α KR KR where kT is the catalytic reactivity of T state while kR is the catalytic reactivity of R state. α¼
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10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Since its inception, the MWC framework has been extended and generalized. Variations have been proposed; eg, to cater to proteins with more than two states (Edelstein et al., 1996), proteins that bind to several types of ligands (Mello and Tu, 2005) or several types of allosteric modulators (Najdi et al., 2006), and allosteric proteins (with nonidentical subunits or ligand-binding sites) (Stefan et al., 2009). It became known that protein changes structure upon binding a ligand. Koshland et al. (1966) refined the biochemical explanation of the mechanism described by Pauling (1935) to reflect the conformational structure change of protein. The Koshland-Ne´methy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as “induced fit.” For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula: θS ¼
3 4 3 3 6 KAB ðKS Ke ½SÞ + 3KAB KBB ðKS Ke ½SÞ2 + 3KAB KBB ðKS Ke ½SÞ3 + KBB ðKS Ke ½SÞ4 3 4 K ðK K ½SÞ2 + 4K 3 K 3 ðK K ½SÞ3 + K6 ðK K ½SÞ4 1 + 4KAB ðKS Ke ½SÞ + 6KAB BB S e S e S e BB AB BB
(10.20)
where KS is the association constant for S, Ke is the ratio of the proteins in B and A states in the absence of ligand ("transition"), and KAB and KBB are the relative stabilities of pairs of neighboring subunits relative to a pair where both subunits are in the A state. The KNF model is thus a model more suited for homosteric enzyme binding than polymorphs. While MWC and KNF models took the protein/enzyme modeling to a height, it does not explain the arrival of the protein structure, or assuming the two structures are of the same DP. Jaffe (2005) took the protein/enzyme modeling to the next peak by introducing the morpheein model highlighting the importance of conformational flexibility for protein oligomer functionality. Morpheein is a protein that exists simultaneously in two or more different forms or homo-oligomers (morpheein forms), but the subunits must come apart and change shape to convert between forms as illustrated in Fig. 10.16. The shape of the subunit dictates which oligomer is formed. Each oligomer has a finite number of subunits (stoichiometry). Morpheeins can interconvert between forms under physiological conditions and can exist as an equilibrium of different oligomers. The different oligomers have distinct functionalities. Interconversion of morpheein forms can be a structural basis for “allosteric regulation.” A mutation that shifts the normal equilibrium of morpheein forms can serve as the basis for a conformational disease. Features of morpheeins can be exploited for drug discovery. The one protein that is established to function as a morpheein is porphobilinogen synthase (PBGS). The morpheein model postulates that an enzyme mixture of “noninteractive” enzymes (morpheeins). Equal and independent (noninteractive) sites on each morpheein lead to: θS ¼ rP ¼ k 1 E
f1 ½S ð1 f1 Þ½S + K1 + ½S K2 + ½S
(10.21)
f1 ½S ð1 f1 Þ½S + k2 E K1 + ½S K2 + ½S
(10.22)
where f1 is the fraction of enzyme that belongs to morpheein 1, and K1 and K2 are the Michaelis-Menten saturation constants of morpheein 1 and 2, respectively.
10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
561
(A) 1E2, Hugging hetero-dimer (B) 2E2, Detached hetero dimer
(C) E8, Hetero octamer, or homo-tetramer of hugging heterodimers
FIG. 10.18
(D)
E6, Hetero hexamer, or homo-trimer of detached heterodimers
Structural characteristics of wild-type human porphobilinogen synthase (PBGS). Based on Selwood et al.
(2008).
The morpheein model was applied to PBGS by Jaffe et al., with a basis shown in Fig. 10.18. The basic unit is not a monomer, but a dimer of the PBGS: Fig. 10.19 illustrates a more realistic dynamic relationship among protein oligomers. Polymorphs are allowed, as opposed to the morpheein model, where only morpheeins (or paired forms) are allowed for a protein in a given system. A polymorph is an oligomeric protein that can exist as an ensemble of physiologically significant and functionally distinct alternate quaternary assemblies (monomers and/or oligomers), which can be simplified to a morpheein if it is a paired system. Strictly speaking, morpheein can exist only for proteins that only form dimers and maybe trimers of the smallest stable enzyme BCHOU), from which strictly a
562
(v)
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
30E3
(ii)
(vi)
13E4
(i)
10E2
(x)
00E1
(vii)
(viii)
03E3
(iii)
01E2
04E4
(iv)
(ix)
22E5
(xii)
23E6
11E3
21E4
(xi)
12E4
FIG. 10.19 A schematic of interconversions between enzyme polymorphs. Twelve polymorphs are shown: one monomer (E1), two dimers (E2), three trimers (E3), four tetramers (E4), one pentamer (E5), and one hexamer (E6) are shown. For each enzyme oligomer of a given DP (EDP), there can be multiple folding isomers or conformers, especially for high DP oligoenzymes. Some of these isomers or conformers are distinct enough to be classified as polymorphs. With permission from Liu, S., 2015. IJPEM, 16(13), 2731-2760.
single-step transit to an oligomer is required. As we learned in chemical kinetics that trimolecular reaction is already rare, higher oligomers cannot be formed from a single-step “combination” without intermediates formed. Polymorphs exist in nature and use conformational equilibria between different tertiary structures to form distinct oligomers as a means of regulating their function. In some situations, the transition from one polymorph to another can be important, whereas in other situations, the transition is effectively completed before the cooperative regulatory effects of the polymorph take effect. Alternate polymorph forms are not misfolded forms of a protein; rather, they are differently assembled native states that contain alternate subunit conformations. Transitions between alternate polymorph assemblies involve conformational changes, eg, dissociation, further association, or even oligomer dissociation; conformational change in the dissociated state; and reassembly to a different oligomer. These transitions occur in response to the environment changes, eg, effector molecules or allosteric effects, and represent a new model of cooperative/allosteric regulation. The unique features of polymorphs are being revealed through RNase: Table 10.3 shows a collection of some of the polymorphs known today. Some of these polymorphs were identified as morpheeins (Selwood and Jaffe, 2012). The list is expanded based on the latest known data, especially that which are discussed in this paper. As illustrated in Fig. 10.19, the protein oligomers can interchange, with or without the modulation of monomeric forms. Therefore, all the protein forms, either monomers or oligomers, are polymorphs, so long as they exist in the system. Although dynamic interchange relationships exist among the polymorphs, polymorphs may not share the same physiological and/or kinetic properties. As such, each polymorph needs to be treated differently.
563
10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
TABLE 10.3
Polymorphic Enzymes
Protein
Example Species
E.C. Number
CAS Number
Polymorphsa
Acetyl-CoA carboxylase-1
Gallus domesticus
EC 6.4.1.2
9023-93-2
TE2, RE2,
α-Acetylgalactos-aminidase
Bos taurus
EC 4.3.2.2
9027-81-0
TE1, RE4
S-Adenosyl-L-homocysteine hydrolase
Dictyostelium discoideum
EC 3.3.1.1
9025-54-1
E4, En
Adenylosuccinate lyase
Bacillus subtilis
EC 4.3.2.2
9027-81-0
E1, E2, E3, E4
Alanine-glyoxylate transaminase
Mammalian
EC 2.6.1.44
9015-67-2
E1, E2
D-Amino
Homo sapiens
EC 1.4.3.3
9000-88-8
E1, E2, En
Aristolochene synthase
Penicillium roqueforti
EC 4.2.3.9
94185-89-4
E1, En
Aromatic L-aminoacid decarboxylase
Homo sapiens
EC 4.1.1.28
9042-64-2
E1, E2
L-Asparaginase
Leptosphaeria michotii
EC 3.5.1.1
9015-68-3
E2, E4, TE8
Aspartate kinase
Escherichia coli
EC 2.7.2.4 & EC 1.1.1.3
9012-50-4
E1, E2, E4
Aspartic aminotransferase
Homo sapiens
EC 2.6.1.1
9000-97-9
E1, E2
ATPase of the ABCA1 transporter
Homo sapiens
EC 3.6.1.3
9000-83-3
E2, E4
Biodegrative threonine dehydratase/threonine ammonialyase
Escherichia coli
EC 4.3.1.19
774231-81-1
aE1, bE1, aE4,
Biotin—(acetyl-CoA-carboxylase) ligaseholoenzyme synthetase
Escherichia coli
Caspase-9
Homo sapiens
Chorismate mutase
Escherichia coli
EC 5.4.99.5
9068-30-8
E1, E3, E6
Citrate synthase
Escherichia coli
EC 2.3.3.1
9027-96-7
E1, E2, E3, E4, E5, E6, E12
Cyanovirin-N
Nostoc ellipsosporum
918555-82-5
E1, E2
3-Oxoacid CoA-transferase
Sus scrofa domestica
EC 2.8.3.5
9027-43-4
E2, E4
Cystalysin
Treponema denticola
EC 4.4.1.1
Cystathionine beta-synthase
Homo sapiens
EC 4.2.1.22
9023-99-8
E2, …, E16
Dihydrolipoamide dehydrogenase
Sus scrofa domestica
EC 1.8.1.4
9001-18-7
E1, aE2, bE2, E4
Dopamine beta-monooxygenase
Bos taurus
EC 1.14.17.1
9013-38-1
E2, E4
acid oxidase
bE4
EC 6.3.4.15
37340-95-7
E1, E2 E1, E2
E1, E2
Ebolavirus VP 40 Geranylgeranyl pyrophosphate synthase/farnesyltrans-transferase
En
aE2, bE2,
Homo sapiens
EC 2.5.1.29
9032-58-0
E4, E8
E6, E8 Continued
564
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
TABLE 10.3 Polymorphic Enzymes—cont’d Protein
Example Species
E.C. Number
CAS Number
Polymorphs
GDP-mannose 6-dehydrogenase
Pseudomonas aeruginosa
EC 1.1.1.132
37250-63-8
E3, aE4, bE4, E6
Glucosamine-6-phosphate synthase
Escherichia coli
EC 2.6.1.16
9030-45-9
RE2, TE6
Glutamate dehydrogenase
Bos taurus
EC 1.4.1.2
9001-46-1
TE5, RE6,
Glutamate racemase
Mycobacterium tuberculosis, Escherichia coli, Bacillus subtilis, Aquifex pyrophilus
EC 5.1.1.3
9024-08-02
E1, aE2, bE2, E4
Glyceraldehyde-3-phosphate dehydrogenase
Oryctolagus cuniculus, Sus scrofa domestica
EC 1.2.1.12
9001-50-7
E1, E2, E4
Glycerol kinase
Escherichia coli
EC 2.7.1.30
9030-66-4
E1, aE4, bE4
EC 2.4.1.1
9035-74-9
TE1, RE2, TE4
Glycogen phosphorylase
En
Growth hormone receptor
Homo sapiens
E1, E2
GPCRs
Homo sapiens
E1, E2
HIV-integrase
Human immunodeficiency virus-1
EC 2.7.7.-
HPr-kinase/phosphatase
Bacillus subtilis, Lactobacillus casei, Mycoplasma pneumoniae, Staphylococcus xylosus
EC 2.7.1.EC 3.1.3.-
Interferon γ receptor
Homo sapiens
Lactate dehydrogenase
Bacillus stearothermophilus
EC 1.1.1.27
9001-60-9
aE1, bE1,
Lon protease
Escherichia coli, Mycobacterium smegmatis
EC 3.4.21.53
79818-35-2
E1, E2, E3, E4
Mitochondrial NAD(P) + malic enzyme/malate dehydrogenase (oxaloacetate-decarboxylating) (NADP+)
Homo sapiens
EC 1.1.1.40
9028-47-1
E1, aE2, bE2, E4
Peroxiredoxins
Salmonella typhimurium
EC 1.6.4.-& EC 1.11.1.15
207137-51-7
aE2, bE2,
Phenylalanine hydroxylase
Eukaryotes
EC 1.14.16.1
9029-73-6
E2, E4
Phosphoenol-pyruvate carboxylase
Escherichia coli, Zea mays
EC 4.1.1.31
9067-77-0
TE2, RE4
Phosphofructo-kinase
Bacillus stearothermophilus, Thermus thermophilus
EC 2.7.1.11
9001-80-3
TE2, RE4
Polyphenol oxidase
Agaricus bisporus, Malus domestica, Lactuca sativa L.
EC 1.10.3.1
9002-10-2
E1, E3, E4, E6, E12
Porphobilinogen synthase
Drosophila melanogaster, Danio rerio
EC 4.2.1.24
9036-37-7
E2, E6, E8
E1, E2, E4, En 9026-43-1
E1, E2, E3, E6
E1, E2, E4 E2, E4
E10
565
10.2 LIGAND BINDING AND COOPERATIVITY
TABLE 10.3
Polymorphic Enzymes—cont’d
Protein
Example Species
E.C. Number
CAS Number
Pyruvate kinase
Homo sapiens
EC 2.7.1.40
9001-59-6
E1, TE2, RE2, E3, RE4, E5
Ribonuclease A
Bos taurus
EC 3.1.27.5
9901-99-4
E1, E2, E3, E4, E5, E6, En
Ribonucleotide reductase
Mus musculus
EC 1.17.4.1
9047-64-7
E4, E6
Tyr-kinase receptors
Homo sapiens
EC 2.7.10.1
Tyrosine aminotransferase
Homo sapiens
EC 2.6.1.5
9014-55-5
E1, E2
β-Tryptase
Homo sapiens
EC 3.4.21.59
97501-93-4
TE1, RE1, TE4,
Polymorphs
E1, E2
RE4
Tumor necrosis factor-alpha
Homo sapiens
Uracil phosphoribosyl-transferase
Escherichia coli
EC 2.4.2.9
94948-61-5
E1, E2, E3
9030-24-4
E3, E5
a
Subscripts (right) indicate the number of subunits (DP), where n stands for other or higher oligomers. Left subscripts T and R indicate inactive (or tense) and active (or relaxed); left subscripts a and b indicate different forms. Augmented from Liu, S., 2015. “A review on protein oligomerization process,” IJPEM, 16(13), 2731–2760.
10.2 LIGAND BINDING AND COOPERATIVITY The specificity of enzymes is due to the binding site-specific nature of enzyme. Reviewing the many results with various enzymes, Koshland (1958) proposed the concept of induced fit to explain how enzymes can discriminate for the correct substrate, and explained how this would allow them to ignore smaller, but incorrect analogs. One enzyme unit contains one binding site, which leads to the specific nature of the enzyme kinetics, the Michaelis-Menten equation. However, in many occasions, the Michaelis-Menten equation fails to describe the reaction rate changes with substrate concentration. The single binding site for enzymes is thus too restrictive, leading to the cooperativity of enzymes. For simple enzymes, Eq. (10.4) can be rearranged to give: θS ¼ Km ½S 1 θS
(10.23)
One can notice this is a linearized form of the Michaelis-Menten binding. The linearity of Eq. (10.23) serves as the comparison for complex enzymes. To follow the development of interactive enzyme binding, we start with hemoglobin, which is an enzyme. When plotting the equilibrium hemoglobin fractional occupancy of oxygen (ie, θS) as a function of the partial pressure of oxygen, Christian Bohr in 1904 obtained a sigmoidal (or “S-shaped”) curve as shown in Fig. 10.20. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind, until all binding sites are saturated. In addition, Bohr noticed that increasing CO2 pressure shifted this curve to the right, as shown in Fig. 10.20. One can infer that higher concentrations of CO2 make it more difficult
FIG. 10.20
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Oxygen binding by dog blood as a function of oxygen partial pressure at different partial pressures of carbon dioxide at 37.8°C as shown by Bohr (1904).
100
80 O2 Saturation, %
566
60 pCO2 = 5 Torr
40
pCO2 = 10 Torr pCO2 = 20 Torr
20
pCO2 = 40 Torr pCO2 = 80 Torr
0 0
25
50
75 pO2, Torr
100
125
150
for hemoglobin to bind oxygen (Bohr et al., 1904). This latter phenomenon, together with the observation that hemoglobin’s affinity for oxygen increases with increasing pH, is known as the Bohr effect. The cause of the oxygen binding on hemoglobin differing from the Michaelis-Menten binding behavior lies on the multiple domains of a hemoglobin molecule. However, due to the lack of proper analytical model, earlier studies have focused on the phenomenological observations alone. For example, cooperative binding or cooperativity was coined for this phenomenon. A receptor molecule is said to exhibit cooperative binding if it is binding to ligand scales nonlinearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor’s apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The empirical concept of cooperative binding only applies to molecules or complexes with more than one ligand-binding site. If several ligand-binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be noncooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO2 reduces hemoglobin’s facility to bind oxygen.) Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill (1910) suggested a phenomenological equation that has since been named after him: θS ¼
½Sn n + ½Sn KH
(10.24)
where n is the “Hill coefficient” or cooperativity index, and KH is a microscopic dissociation constant. If n < 1, the system exhibits negative cooperativity, whereas cooperativity is positive
10.2 LIGAND BINDING AND COOPERATIVITY
567
if n > 1. The total number of ligand binding sites is an upper bound for n. Hill equation can be linearized to give: log
θS ¼ n log ½S n log Kd 1 θS
(10.25)
θS versus log[S]. In the case of the Hill equa1 θS tion, it is a line with slope n and intercept log Kd. The Hill equation is an improvement over the Michaelis-Menten expression as Eq. (10.24) can correlate the bulk of the data in Fig. 10.20, ie, a wider range of data than Eq. (10.4) does. Hill equation is extended to catalytic reactions: The “Hill plot” is obtained by plotting log
rP ¼
rmax ½Sn n + ½Sn KH
(10.26)
A combination of experimental observations and theoretical treatments reviewed so far leads from a simple enzyme that has a single binding site to an aggregate or conformal complex structure of enzyme molecules that has multiple binding sites. It is the multiple binding sites that give rise to numerous features of enzymes that enable regulations of reactions in living cells. The Hill empirical equation can correlate some of the experimental data to some degree, but it fails to give an account of the structure of the enzyme, and the parameters have no physical meaning other than the perceived empirical concept of cooperativity. We will examine the treaties for the complex enzymes. KNF model was the first to treat an interactive enzyme from a mechanistic point of view. An enzyme is said to be interactive if the binding of one ligand molecule will affect the further binding of more ligand molecule(s), ie, resulting in the “cooperation” among the bound ligands and enzyme. The binding of ligand on an interactive enzyme progresses one at a time. One sometimes calls this one at a time event as sequential, although sequential could have a more restrictive meaning, as the receptor sites themselves can be selective leading to preferential binding. A more precise term should be “facilitate” or cooperative. When one substrate molecule is bound to one site on an interactive protein/enzyme (of multiple subunits), it facilitates the binding of other substrate molecules on the other n 1 sites of the same protein/ enzyme molecule/complex: K1
En + S ! En S Ki
En Si1 + S ! En Si 81 i n
(10.27) (10.28)
where n is the total number of binding sites available on the protein/enzyme complex/molecule, En denotes for a protein/enzyme complex having n-subunits, S denotes for the binding substrate, and EnSi denotes for one enzyme complex/molecule bound with i substrate molecules. The binding on the first site by the substrate can facilitate further binding on the other sites. In enzyme analysis, we often use Michaelis-Menten kinetic Eq. (10.6). The MichaelisMenten saturation coefficient corresponds to the ratio of the dissociation rate constant over the binding rate constant. The analogy to kinetics on a simple enzyme is applied here for the binding coefficient K’s. The first binding coefficient is denoted as K1 (instead of Km for a
568 Homosteric
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Allosteric sequential
(A)
Allosteric random
(B)
(C) FIG. 10.21 Schematics of ligand binding on an interactive enzyme with (A) homosteric sites, (B) allosteric sites with preference in binding, and (C) allosteric sites without preference in binding. Each enzyme molecule is illustrated with three interactive binding sites. An allosteric enzyme has different sites as illustrated with squares, ovals, and hexagons in (B and C). The binding on an allosteric enzyme can be sequential if there is a preference of the ligand on particular sites. However, a homosteric enzyme contains identical sites as illustrated by the identical shapes. The binding occurs one at a time, but no order on the sites is available. In all the cases, when the concentration of ligand molecules increases (from left to right), increasing number of ligand molecules (shown as yellow circles) bound on the sites of the enzyme molecules. Binding of one ligand molecule will result in the change of conformal structure of the enzyme, and thus the bindings are also color coded to illustrate the interactions or the interactive nature of the enzyme.
simple enzyme) and for the subsequent binding sites, Eq. (10.28) shows that the binding rate constant is Ki for binding the ith substrate on the enzyme En. The interaction of the ligandenzyme-ligand is captured through the variation of binding affinity (or constant Ki’s). Fig. 10.21 shows the cooperative binding schematics of three different types of interactive enzymes: (a) homosteric (or homotropic) enzymes, (b) sequential allosteric (or heterotropic), and (c) random allosteric (or heterotropic). Homosteric enzymes contain multiple identical binding sites (homotropic sites). Each site on a homosteric enzyme molecule interacts with the ligand molecule in the same fashion; thus, the homosteric sites “compete” equally for binding with the ligand molecule. In other words, the complexing of one ligand molecule with any one of the available sites on the enzyme molecule is random. However, when a ligand molecule is bound on the homosteric enzyme molecule, the conformal structure change of the enzyme molecule can be induced; thus, the interaction of the vacant sites with the ligand molecule can be altered, or different from those already bound. As the ligand molecules are bound on the homosteric enzyme, and they possess identical interactions with the enzyme. Therefore, depending on how the ligand can bind on the multiple sites available on an enzyme molecule, the kinetics and equilibrium of the binding itself will be different. There is a marked interaction of the enzyme (with ligands). This is what we call cooperative binding.
569
10.2 LIGAND BINDING AND COOPERATIVITY
Allosteric enzymes, on the other hand, have different types of binding sites (hetero binding sites). The difference in chemical and form/steric properties can present a scaled preference (or sequence), for which the ligand molecule can bind. The preference or sequence in the approach of the ligand can thus be expected, but in some cases, preference may not exist. Binding and reactivity on each site of an allosteric enzyme molecule are also expected to be unique. The binding on an allosteric enzyme can thus be either sequential (or preferential) or random. In either case, binding of one ligand molecule will cause the change of conformal structure of the enzyme, and thus modify the affinity and reactivity afterwards. This cooperativity is clear at least for sites that are randomly accessible. However, for preferential binding, the cooperativity is obscured due to the sequence imposed. We shall discuss the three extreme situations (of interactions) in the following sections.
10.2.1 Single Ligand Binding on Homosteric Enzymes For enzyme homo-oligomers, all the binding sites are equal. Monod et al. (1963) pioneered the analysis on cooperative homosteric binding, while the KNF model was more accurate by inclusion of ligand-enzyme molecular interactions. Fig. 10.22 shows an illustration of the ligand binding on an interactive homosteric enzyme. Since all the binding sites are identical, one can interpret the rates in reactions (10.27) and (10.28) as: En + S ! En S, En S!En + S, En S + S!En S2 , En S2 !En S + S, En Si1 + S!En Si , En Si ! En Si1 + S,
KS
FIG. 10.22
a1KS
r + n1 ¼ nkn1 + ½En ½S
(10.29a)
rn1 ¼ kn1 ½En S
(10.29b)
r + n2 ¼ ðn 1Þkn2 + ½En S½S rn2 ¼ 2kn2 ½En S2
(10.30b)
r + ni ¼ ðn i + 1Þkni + ½En Si1 ½S rn2 ¼ ikni ½En Si
a2KS
(10.30a)
(10.31a) (10.31b)
a3KS
Schematics of ligand binding on a homosteric interactive enzyme with four subunits. The ligand binding occurs one at a time. When the concentration of the ligand molecules increases, an increasing number of ligand molecules bind on the sites of the enzyme molecule. Successive binding of ligand molecules results in the changes in the conformal structure (shape) of the enzyme, and thus the bindings affinity changes. The affinity change factor is denoted as α’s.
570
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
where n is the total number of binding sites available on the protein/enzyme complex/molecule, En denotes for a protein/enzyme complex having n-subunits, S denotes for the binding substrate, EnSi denotes for one complex enzyme complex/molecule bound with i substrate molecules, kni+ is the binding rate constant for substrate molecule on each of the (n i + 1) vacant sites, and kni is the dissociation of a substrate molecule from each of the i bound substrate molecules. The binding of the first substrate molecule can facilitate further binding on the homosteric enzyme molecule. The subsequent binding can have a different affinity as the conformational change could occur because of the binding of the first site. When binding on the first site, there are n equal sites available to bind. and thus the rate of binding on the enzyme is n-times that of each site. When binding on the ith site, there are a total of (n i + 1) equal vacant sites; thus, the binding rate is (n i + 1)-times that of each site. To determine the isotherm of substrate binding on a protein/enzyme, we assume that thermodynamic equilibrium applies to the binding of substrates on all the sites, that is: nkn1 + ½En ½S n½En ½S ¼ kn1 KS Y i1 ðn i + 1Þkni + ½En Si1 ½S ½En ½Si n ½En Si ¼ ¼ αj i ikni KSi ½En S ¼
(10.32)
(10.33)
j¼1
where n ðn 1Þ ðn i + 1Þ n ¼ i 12 i KS ¼
kn1 kn1 +
kni + kn1 + 1 αi1 ¼ kni kn1
(10.34a)
(10.34b)
(10.34c)
where αj is the affinity change factor (or cooperativity factor) due to the binding of the jth ligand molecule. When αi > 1, the affinity is increased due to the binding of the previous i substrates. Otherwise when αi < 1, the affinity is decreased due to the binding of the previous i substrates. The overall enzyme balance: n n Y i1 X ½En ½S X ½Si1 n E n ¼ ½ En + ½En Si ¼ ½En + αj i1 (10.35) KS i¼1 i j¼1 KS i¼1 Leads to: ½En ¼
En i1 i1 n Y ½S αj i1 i KS j¼1
n X
½S 1+ KS i¼1
(10.36)
10.2 LIGAND BINDING AND COOPERATIVITY
Y i1 ½Si n αj i1 i KS j¼1 ½En Si ¼ k1 n k1 X n Y ½S KS + ½S αj k1 k KS j¼1 k¼1
571
En
(10.37)
The total bound substrates on the interactive enzyme of n-binding sites, BnS, can be computed by: n Y i1 X i½Si n En αj i1 i n KS X i¼1 j¼1 i½En Si ¼ (10.38) Bn S ¼ i n i1 X n Y ½S i¼1 KS + αj i1 i KS i¼1 j¼1 which retains the details for thermodynamic properties of each successive binding. In addition, Eq. (10.38) reflects the kinetics of ligand binding where the affinity can vary with the number of sites already bound. When [S] ! 1, Eq. (10.38) is reduced to: B n S 1 ¼ En n
(10.39)
ie, the amount of saturation-bound substrate is identical to the number of binding sites on an enzyme molecule times the amount of enzyme (as each homosteric enzyme molecule/ complex has n binding sites available), which is intuitive. Therefore, Eq. (10.38) can also be written as: n Y i1 1X i½Si n αj i1 n i¼1 i j¼1 KS Bn S ¼ (10.40) θnS ¼ n Y k1 Bn S1 X ½Sk1 n KS + ½S αj k1 k KS j¼1 k¼1 where θnS is the ratio of substrate saturation on the homosteric enzyme of n-binding sites. Before we start to examine the features of the binding characteristics of homosteric enzymes, let us examine the extremes or asymptotic behaviors of Eq. (10.40). When n ¼ 1, Eq. (10.40) is reduced to: θ1S ¼
½S KS + ½ S
(10.41)
which is the identical to Eq. (10.4). When the ligand concentration is low, ie, [S] < KS, Eq. (10.40) is reduced to: 0 1 n Y i½S1 αj 11 n 1 j¼1 KS ½S θnS (10.42) ¼ Y 0 0 + n½S K S ½S n KS + ½S αj 11 1 KS j¼1
572
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Not surprisingly, it is very similar to Eq. (10.4). However, Eq. (10.42) renders a lower saturation than that of Eq. (10.4). When [S] is very high, that is, [S]> > KS, Eq. (10.40) is reduced to: 9 8 nY n1 n1 = 2 1< n Y n½Sn ð n 1 Þ ½ S n n1 αj + αj KS + ½S n1 ; n : n j¼1 KSn1 KSn2 j¼1 α 9 ¼ n1 8 (10.43) θnS n nY < nY 1 2 n½Sn ðn 1Þ½Sn1 = α KS + ½S n n KS + ½ S αj n1 + αj n1 n1 ; : n KS KSn2 j¼1
j¼1
which is, again, strikingly similar to Eq. (10.40). However, the intercept or when [S] is extended to 0, θnS ¼ (n 1)/n 6¼ 0, unless n ¼ 1. We next examine the extreme case of stiff enzyme oligomers. If there is no cooperativity or the further enzyme binding is not affected by the bound substrate, αj 1. In this case, Eq. (10.40) is reduced to: i n 1X n i½S n i¼1 i KSi1 ½S (10.44) ¼ θnS ¼ k1 n X n ½S KS + ½S KS + ½S k KSk1 k¼1 which is identical to the Michaelis-Menten saturation, Eq. (10.4). Therefore, if the binding of the ligand does not affect the enzyme-folding or binding affinity, a homo-oligomer can be treated as multiple simple enzymes. More frequently, the binding of a ligand on an interactive enzyme changes the conformal structure of the enzyme, thus leading to a change in the binding affinity of the remaining binding sites. Without loss of generality, we let: Y i1 n αj (10.45) ani ¼ i j¼1
be the (cumulative) effective affinity modification factor. Then, Eqs. (10.37) and (10.40) are reduced to: ½Si KSi ½En Si ¼ En n X ½Sk 1+ ank k KS k¼1 ani
(10.46)
n X iani ½Si
θnS ¼
1 n
i¼1
1+
KSi
n X ani ½Si i¼1
KSi
(10.47)
10.2 LIGAND BINDING AND COOPERATIVITY
573
While it is plausible to have the affinities vary extensively with the successive binding of each ligand, one may find some situations where the variation of affinity becomes negligible after the binding of a few ligands. If we let: α2 ¼ α3 ¼ … ¼ αi ¼ … ¼ αn
82 i n
(10.48)
That is, the binding of third and subsequent ligands will not further affect the structure of the enzyme, thus binding affinity. In this case: i1 Y αj ¼ α1 α2i2
8i > 1
(10.49a)
when i ¼ 1
(10.49a)
j¼1 i1 Y
αj ¼ 1,
j¼1
n Y i1 X ½Si α 1 ½S n α1 ½S α1 n αj i ¼ 2 1 + α2 2 +n 1 i α2 KS KS α2 KS α 2 i¼1 j¼1
(10.50)
n Y i1 X i½Si α1 ½S ½S n1 ½S α1 n αj i ¼ n 1 + α2 +n 1 i α2 α2 KS KS KS KS i¼1 j¼1
(10.51)
Thus: α1 ½S n1 α1 1 + α2 +1 n X nEn ½S α2 α2 KS n Bn S ¼ i½En Si ¼ KS α1 ½S α1 ½S α1 i¼1 1 + α2 + 1 2 +n 1 α2 KS KS α22 α2
(10.52)
α1 ½S n1 α 1 + α2 + 1 n2 B n S ½S KS α2 αn1 n θnS ¼ ¼ Bn S1 KS α1 ½S α1 ½S αn2 + n 1 + α + 1 1 2 αn1 KS KS α22 α22
(10.53)
Now Eqs. (10.52) and (10.53) are closed form expressions for any n 1. The overwhelmingly large number of studies with hemoglobin has provided ample opportunity to use this molecule to show the cooperative binding behavior. Fig. 10.23 shows the oxygen saturation with horse hemoglobin as a function of the oxygen partial pressure. One can observe that the curve fits with the experimental data (symbols) quite well. The fitting parameters, based on Eq. (10.53), is given by KS ¼ 8.715 Torr-O2, α1 ¼ 0.04349, α2 ¼ 64.56, n ¼ 4. Fig. 10.24 shows the oxygen saturation isotherm of human hemoglobin at 25°C and pH 7.3. The curve shown in Fig. 10.24 has parameters given by KS ¼ 61.634 Torr-O2, α1 ¼ 0.03199, α2 ¼ 88.26, n ¼ 4. One can observe that the model, Eq. (10.53), can describe the human hemoglobin binding on oxygen quite well. There are 4 oxygen-binding sites available on each hemoglobin molecule, as indicated from the model fit.
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
FIG. 10.23 Oxygen saturation isotherm for horse hemoglobin. The symbols are data as calibrated by Pauling (1935) for pH 8.3. The curve is data fitting to Eq. (10.53) with KS ¼ 8.715 Torr-O2, α1 ¼ 0.04349, α2 ¼ 64.56, n ¼ 4.
1.0
0.8 O2 saturation
574
0.6
0.4
0.2
0.0 0
2
4
6
8
10
pO2, Torr
1.0
0.8 O2 saturation
FIG. 10.24 Oxygen saturation isotherm for human hemoglobin at pH 7.4 and 25°C. The symbols are experimental data taken from Imai, K., 1973. Biochemistry 12, 798–808) and the curve is based Eq. (10.53).
0.6
0.4
0.2
0.0 1
10
pO2, Torr
100
The human hemoglobin is known to have four subunits or to be tetramers. Based on the isotherm fitting, there are 4 oxygen-binding sites, which give one oxygen-binding site per subunit. Also, the binding affinity of the first oxygen-binding site on a hemoglobin molecule corresponds to KS ¼ 61.634 Torr, which is lower than for the second binding site as α1 ¼ 0.03199. In effect, the binding coefficient for the first site is equivalent to 61.634/4 Torr-O2 ¼ 15.4085 Torr-O2 because all four sites are equally accessible. The subsequent sites have an oxygen affinity corresponding to KS/α2 ¼ 0.6986 Torr. Therefore, hemoglobin is a remarkable cooperative transporter that has an appropriately low affinity (82.634 Torr) in the lungs, allowing more oxygen to be bound when oxygen is abundant and an appropriately high affinity (0.6986 Torr) in the deoxygenated capillaries, allowing the oxygen to be carried farther.
575
10.2 LIGAND BINDING AND COOPERATIVITY
10.2.2 Sequential Single Ligand Binding on Allosteric Enzymes Fig. 10.25 shows an illustration of the ligand binding on an allosteric enzyme. Adair (1925) and Klotz et al. (1946) pioneered the analysis on allosteric binding. Klotz (2003) likened the ligand binding on allosteric enzymes to the dissociation of multivalent acid or base. The binding and dissociation are strictly sequential and all the binding sites are distinctively different. Following the framework of Klotz et al. (1946), we generalize the sequential binding of substrate for reactions (10.27) and (10.28) by: En + S !En S, En S !En + S, En S + S !En S2 , En S2 !En S + S, En Si1 + S !En Si ,
r + n1 ¼ kn1 + ½En ½S
(10.54a)
rn1 ¼ kn1 ½En S
(10.54b)
r + n2 ¼ kn2 + ½En S½S
(10.54c)
rn2 ¼ kn2 ½En S2
(10.54d)
r + ni ¼ kni + ½En Si1 ½S
(10.54e)
rn2 ¼ kni ½En Si
(10.54f)
En Si !En Si1 + S,
Only when binding occurred on the first site can the substrate further bind on the second site. The bindings on the allosteric sites are expected to have different affinities. To determine the isotherm of substrate binding on protein/enzyme, we assume that thermodynamic equilibrium applies to the binding of substrates on all the sites, ie, the overall rate for each site is zero. One can show that: ½En ½S K1S
(10.55)
½En Si1 ½S ½En ½Si ¼ i Y KiS KjS
(10.56)
½En S ¼ ½En Si ¼
j¼1
where KiS ¼
K1S
K2S
kni kni +
(10.57)
K3S
K4S
FIG. 10.25 Schematics of ligand binding on an enzyme with four allosteric sites. The ligand binding occurs one at a time in an orderly manner. When the concentration of ligand molecules increases, increasing number of ligand molecules bound on the sites of the enzyme molecule. Successive binding of ligand molecules can result in the changes in the conformal structure (as indicated by the green arrow) of the enzyme, and the binding affinity is different for the different (allosteric) sites.
576
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
is the substrate binding constant on the ith allosteric site. With Eq. (10.57), we have included the affinity change factor into the affinity coefficient itself. KiS is a combination of the affinity coefficient and the cooperativity factor due to the binding of the prior substrate molecules. The overall enzyme balance leads to: En ¼ ½ E n +
n X
n X ½Si ½En Si ¼ ½En + ½En i Y i¼1 i¼1 KjS
(10.58)
j¼1
Leads to: ½En ¼
En n X ½Si 1+ i Y i¼1 KjS
(10.59)
j¼1
En ½En Si ¼
½Si i Y KjS j¼1
n X
½Sk 1+ k Y k¼1 KjS
(10.60)
j¼1
The total bound substrates on the allosteric enzyme of n-binding sites, BnS, can be computed by: En Bn S ¼
n X
i½En Si ¼
i¼1
n X i½Si i Y i¼1 KjS j¼1
n X ½Si 1+ i Y i¼1 KjS
(10.61)
j¼1
When n ¼ 1, the substrate binding isotherm Eq. (10.61) is reduced to: B1 S ¼
E1 ½ S K1 + ½S
(10.62)
which is similar to the Langmuir adsorption isotherm of solid adsorbates. When [S] ! 1, Eq. (10.4) is reduced to: Bn S1 ¼ En n
(10.63)
577
10.2 LIGAND BINDING AND COOPERATIVITY
ie, the amount of saturation bound substrate is identical to the number of binding sites on an enzyme molecule times the amount of enzyme (as each allosteric enzyme molecule/complex has n binding sites available), which is intuitive. Therefore, Eq. (10.61) can also be written as: n 1X i½Si i n i¼1 Y KjS
θnS ¼
Bn S ¼ B n S1
j¼1
(10.64)
n X ½Sk 1+ k Y k¼1 KjS j¼1
where θnS is the ratio of substrate saturation on the allosteric enzyme of n-binding sites. The two asymptotes for Eq. (10.64) are similar to those for Eq. (10.40). When [S] is small, ie, [S] < KjS, Eq. (10.64) reduces to its limiting behavior: θnS
½S nðK1S + ½SÞ
(10.65)
and when [S] is high, ie, [S] > > KjS, then: n1 KnS + ½S θnS n KnS + ½S
(10.66)
Both Eqs. (10.65) and (10.66) can be easily confused with the Michaelis-Menten equation (10.6). However, Eq. (10.65) has a lower saturation value, while (10.66) has a nonzero intercept when extended to a zero substrate concentration. Eq. (10.64) can be reduced to Eq. (10.47), if we let: ani ¼
Kni i Y KjS
(10.67a)
j¼1
Or αi ¼
ði + 1ÞnK1S ðn iÞKi + 1S
(10.67b)
TABLE 10.4
Stoichiometric Binding Constants for Binding of Azosulfathiazole by Serum Albumin
I
1
2
3
4
5
6
7
8
9
10
125
45.0
21.6
11.7
67.0
3.99
2.43
1.51
0.95
0.60
1
0.8
0.3830
0.3403
0.3222
0.3420
0.4801
1
1/KiS, mM αi1
Data from Klotz (2003).
0.6481
0.5348
0.4466
578
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
1.0
VC I–II, K1S = 0.0166 mM, K2S = 0.0992 mM VC II, KS = 0.0166 mM
Glycine Saturation
0.8
VC I, KS = 0.0992 mM
0.6 0.4 0.2 0.0 10−8
10−7
10−6
10−5 10−4 10−3 Glycine Concentration (M)
10−2
10−1
FIG. 10.26 Allosteric interaction demonstrated on ligand binding isotherm. A comparison of binding curves for glycine with the normal VC I-II containing both allosteric sites and with the altered VC II containing only one site. VC is an RNA riboswitch. The symbols are data taken from Mandal et al., 2004. The curves are fittings to Eq. (10.64), with parameters shown on the upper left corner.
Experimentally, Eqs. (10.40) and (10.64) cannot be distinguished. The coefficients can be related through ani or redefining αi, (Eqs. 10.45, 10.67a and 10.67b). Therefore, the allosteric enzymes and homosteric enzymes have similar binding behaviors experimentally. Table 10.4 shows the binding coefficients of azosulfathiazole by serum albumin. The total number of binding sites is n ¼ 10, and K10 ¼ 0.08 mM. Both parameters for Eqs. (10.40) and (10.64) are listed. Fig. 10.26 shows the allosteric interaction based on substrate binding isotherms between an allosteric enzyme and a modified enzyme that contains only one of its two allosteric sites. When only a single binding site is available on a protein/enzyme/RNA molecule, the isotherm follows Eq. (10.64) with n ¼ 1. This is clearly shown in Fig. 10.26, where one of the site in the two-sited RNA molecule has a binding affinity equal to that of the single-site binding affinity. One can observe that K2S ¼ 0.0166 mM for VC I-II is the same as the K1S for VC II alone. The binding curve for VC I alone is also plotted in Fig. 10.26. One can observe that the sequential binding of the two allosteric sites falls right between the two binding behaviors of the constitutive sites. The identical affinity of the allosteric binding site on the allosteric enzyme as that when present alone indicates that the binding is strictly sequential on allosteric enzymes and there are no conformational structural changes induced by the binding of the VCI site. This is intuitive, as the sites are not identical and they do not compete in the same way for the ligand molecule. While enzymes are known mostly as proteins, we are aware of a few essential RNA ribozyme activities. Bacteria use many RNA molecules as riboswitches, which upon binding some specific metabolic ligand alter their structure and are then able to bind to a DNA regulatory site to influence the expression of genes coding for enzymes that specifically metabolize the ligand that activates that particular riboswitch. Most such riboswitches appear to have a single-ligand binding site (Mandal et al., 2004). This example demonstrating the effect of cooperativity is taken from Mandal et al. (2004) who demonstrated that a specific RNA
579
10.2 LIGAND BINDING AND COOPERATIVITY
riboswitch from Vibrio cholerae has two aptamers with binding sites for glycine, VC I and VC II. The genes controlled by this riboswitch express enzymes for the catabolism of glycine as an energy source, and ideally should only be turned on when glycine is abundant in order to permit its normal use for protein synthesis. Protein enzymes show cooperativity due to the fact that they have more than one binding site within one oligomer, even though this is, in most cases, proportional to a stoichiometry of one per subunit. While RNAs do not normally form oligomers, a cooperative response is possible with this riboswitch by virtue of the fact that it contains two binding sites on a single RNA chain. Binding of a glycine at the first of these sites then induces a conformational change in the RNA, leading to formation of the adjacent site to enable its binding of glycine, which explains the classic cooperative binding curve shown in Fig. 10.26. The comparison to a single-site binding isotherm is made possible by the authors by making a modified version of the riboswitch, in which site I has been deleted. As shown in Fig. 10.26, the modified RNA (VC II) now has a simple saturation-binding curve, consistent with a single site at a constant affinity. The cooperative response is ideal for a switch, since it greatly narrows the concentration range over which the riboswitch becomes fully active. While it is highly improbable for allosteric enzymes, if the binding affinities are identical for all the allosteric sites, then: KS ¼ K1S ¼ K2S ¼ … ¼ KiS ¼ … ¼ KnS
(10.68)
n + 1 n ½S ½S 1+n ðn + 1 Þ ½S KS KS θnS ¼ KS ½S 2 ½S n n 1 1+ KS KS
(10.69)
Eq. (10.64) is reduced to:
Eq. (10.69) represents the binding isotherm for site-sequential or preferential binding of a single ligand on an allosteric enzyme containing sites of identical affinities.
10.2.3 Single-Ligand Binding on Random-Access Allosteric Enzymes We discuss another case where all the binding sites are equally accessible, but they are all different. This is similar to homosteric enzyme; however, each site is different. Fig. 10.27 shows a schematic of random access of a single ligand on an allosteric enzyme. When first substrate molecule approaches the enzyme molecule: Ki1
En + S ! En Si1
(10.70)
where the subscript i1 stands for the location of the binding site among (1,2, …, n). The subsequent binding can be written as: Ki j
En Si1 …Sij1 + S ! En Si1 …Sij1 Sij
(10.71)
As expected, the affinity will change factor after each successive binding of ligands:
580
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
FIG. 10.27
A schematic of random access of a single ligand on an interactive enzyme with three allosteric sites. The ligand can bind on any one of the three sites at any time, and each successive binding results in a change in the enzyme conformal structure, leading to a change in the affinity of binding on the next available allosteric site.
a1−1K2
K1
−1 K3 a12
a1−1K3 a2−1K1
K2
−1 K2 a13
a2−1K3 a3−1K1 −1 a23 K1
a3−1K2
K3
h i ½En ½Sj En Si1 …Sij1 Sij ¼ α i1 , …, ij1 j Y Kim
(10.72)
m¼1
where α(i1, …, ij1) is the affinity change factor after j 1 sites (i1, …, ij1) are bound by ligand S. In terms of the power law of the substrate, one can obtain: n X n X ⋯ E n Sj ¼ i1 ¼1 i2 ¼1 i2 6¼i1
¼
n X n X i1 ¼1 i2 ¼1 i2 6¼i1
h
n X
En Si1 …Sij1 Sij
i
ij ¼1 ij 6¼i1 , …, ij1
⋯
n X
½En ½Sj α i1 , …, ij1 j Y ij ¼1 ij 6¼i1 , …, ij1 Kim
(10.73)
m¼1
Overall enzyme balance leads to: ½En ¼ 1+
n X
½S
n X n X j
En ⋯
i1 ¼1 i2 ¼1 i2 6¼i1
j¼1
n X
α i1 , …, ij1
ij ¼1 ij 6¼i1 , …, ij1
j Y
(10.74)
Kim
m¼1
½Sk
n X n X i1 ¼1 i2 ¼1 i2 6¼i1
½En Sk ¼ En 1+
n X j¼1
½S
⋯
n X
αði1 , …, ik1 Þ i Y ij ¼1 Kim ik 6¼i1 , …, ik1
n X n X j i1 ¼1 i2 ¼1 i2 6¼i1
⋯
n X ij ¼1 ij 6¼i1 , …, ij1
α i1 , …, ij1
m¼1
j Y m¼1
K im
(10.75)
10.3 COMPETITIVE MULTILIGAND BINDING ON AN INTERACTIVE ENZYME
581
When the affinity change is negligible, ie, α ¼ 1, we can obtain: n n X n n X X 1X 1 j½Sj ⋯ n j n j¼1 Y d Y ½S i1 ¼1 i2 ¼1 ij ¼1 1 + Kim i2 6¼i1 ij 6¼i1 , …, ij1 Kj ½S d½S j¼1 m¼1 ¼ θnS ¼ n n Y Y n ½S ½S 1+ 1+ Kj Kj j¼1 j¼1
¼
(10.76)
n 1 ½S X Kj + ½S n j¼1
Now Eq. (10.76) is binding saturation for a nonsite-sequential allosteric enzyme with negligible affinity change. One can show that Eq. (10.76) is a special case of Eq. (10.47).
10.3 COMPETITIVE MULTILIGAND BINDING ON AN INTERACTIVE ENZYME Although enzymes are very specific in the ligand in that they can interact, still there are ligands that are similar in stereostructural and/or chemical forms. These similar ligands can act on the same enzyme binding sites. When multiple ligands act on an interactive enzyme competitively, the situation is more complicated. We will now have a brief discussion on the competitive ligand binding on interactive enzymes.
10.3.1 Competitive Ligand Binding on a Homosteric Enzyme Fig. 10.28 shows a schematic of two ligands A and B competitively binding on an interactive enzyme of four subunits. One can observe that there is an increasing number of states for the binding over the single ligands. In this section, we shall examine the competitive binding over multiple sites for two-sited, three-sited and four-sited systems. This will allow one to appreciate how the system can be modeled and understand the procedure better. Instead of looking at one species a time (as that shown in Fig. 10.28 in rows), we will examine the competitive binding one site a time sequentially (column-wise from left to right in Fig. 10.28). 10.3.1.1 Two-Site Homosteric Enzyme There are three situations for the binding when focusing on one particular species: (1) one molecule of species j is bound only: [E2(Sj)]; (2) Species j bound on one of the two sites: [E2(Si) (Sj)] + [E2(Sj)(Si)]; and (3) Species j bound on both sites: [E2(Sj)(Sj)]: 2½ E2 S j E2 S j ¼ (10.77) Kj
582
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
E4
Ki1
a −1 i1Ki2 E4Si1
a −1 i1i2Ki3 E4Si1Si2
E4Si1Si2Si3
−1 a i1i2i3 Ki4
E4Si1Si2Si3Si4
A A A
A
A
a−1AAAKA
a−11AAKA
A
A A
KA
a−1AKA
A
A
a−1AAKB
A B
B
a−1AABKA
A
A
A A A
A
a−1AKB
A
a−1AABKB
A A
B A
A
a−1ABKA
a−1AABKB
B
A
A B
A
B A
B
a−1ABKB
B A
a−1ABBKA
A
a−1BKA
B B B B
B
B
a−1ABBKB
A B B
A
KB
B
a−1BKB
B
a−1BBKA
a−1BBBKA
B B
B
a−1BBKB
a−1BBBKB
B B
B
B
B B
B
B
B FIG. 10.28
A schematic of two ligand species A and B competitively binding on an interactive enzyme of four
subunits.
which describes binding on the first site. The binding on the second site is given by: 2 E2 Sj Sj ½ E2 S j ¼ αjð jÞ (10.78) E2 Sj Sj ¼ αjð jÞ 2Kj Kj2 and,
E2 Sj ½Si ½E2 Sj ½Si 0 0 ½E2 Si Sj E2 Sj Si ¼ αið jÞ + αjðiÞ ¼ ðαið jÞ + αjðiÞ Þ Ki Ki K j Kj
(10.79)
10.3 COMPETITIVE MULTILIGAND BINDING ON AN INTERACTIVE ENZYME
583
The concentration or saturation of species j (bound species j) can be computed by
NS X E2 S j + E2 Si Sj + 2 E2 Sj Sj
B2 S j ¼ ½ E2
(10.80)
i¼1 i6¼j
½ E2
NS 2 Sj Sj X Sj ½Si ½Si + 2αjð jÞ + αjðiÞ + αið jÞ ¼2 Kj Kj Ki Ki Kj Kj i¼1 i6¼j
where αi(j) denotes for the affinity change factor for species i after species j is bound on the enzyme. This equation can be reduced to NS B2 S j Sj Sj X ½Si ¼2 + ðαið jÞ + αjðiÞ Þ (10.81) ½ E2 Ki Kj Kj i¼1 The site balance is given by NS X
E2 ¼1+ ½ E2
j¼1
NS X NS NS X X E2 Sj + E2 Si Sj + E2 S j S j j¼1 i¼1 i6¼j
j¼1
½ E2
(10.82)
) ( 2 NS NS NS X X X Sj Sj ½Si 2+ ðαið jÞ + αjðiÞ Þ αjð jÞ 2 ¼1+ Ki Kj Kj j¼1 i¼1 i¼1 The saturation of species j is given by NS Sj Sj X ½Si 2 + ðαið jÞ + αjðiÞ Þ Ki K K j j i¼1 1 B2 Sj 1 ( ) θ2j ¼ ¼ NS NS NS 2 E2 2 X X X ½Sk ½Si ½Si 2 1+ 2+ ðαiðkÞ + αkðiÞ Þ αiðiÞ 2 Kk Ki Ki i¼1 i¼1 k¼1
(10.83)
For noninteracting homosteric enzyme, all the affinity change factors are unity. Eqs. (10.81) and (10.83) are then reduced to NS B2 S j Sj Sj X ½Si ¼2 +2 (10.84) ½E2 Kj Kj i¼1 Ki
θ2j ¼ 1+
Sj Kj NS X ½Si i¼1
Ki
(10.85)
584
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
10.3.1.2 Three-Site Homosteric Enzyme For a three-site homosteric enzyme, there are four situations in binding: (1) Species j bound only: [E3Sj]; (2) Species j bound on one of the three sites: [E3SiSj] + [E3SiSjSk]; (3) Species j bound on two of the three sites: [E3SjSj] + [E3SjSjSk]; and (4) Species j bound on all three sites: [E3SjSjSj]. Binding on the first site, 3½E3 Sj E3 Sj ¼ (10.86) Kj on the second site,
2 2 E3 Sj Sj ½ E3 S j ¼ 3αjð jÞ E3 Sj Sj ¼ αjð jÞ 2Kj Kj2
½E3 Sj ½Si 0 2½E3 Si Sj 0 2 E3 Sj ½Si + αið jÞ ¼ 3ðαið jÞ + αjðiÞ Þ E3 Sj Si ¼ αjðiÞ Kj Ki Ki K j
(10.87)
(10.88)
and on the third site, 3 E 3 Sj Sj Sj ½ E3 S j ¼ αjð j, jÞ αjð jÞ E3 Sj Sj Sj ¼ αjð j, jÞ 3Kj Kj3
E 3 Si Sj Sj E3 Sj Sj ½Si 0 + αið j, jÞ Ki 2Kj 2 2 ½E3 ½Si Sj ½E3 ½Si Sj 3 0 0 ¼ αjði, jÞ ðαið jÞ + αjðiÞ Þ + αið j, jÞ 3αjð jÞ 2 Ki Kj2 Ki Kj2 i ½E ½S S 2 3h 0 3 i j 0 ¼ αjði, jÞ ðαið jÞ + αjðiÞ Þ + 2αið j, jÞ αjð jÞ 2 Ki Kj2 i ½E ½S S 2 3h 3 i j ¼ αjði, jÞ ðαið jÞ + αjðiÞ Þ + 2αið j, jÞ αjð jÞ 4 Ki Kj2 ½E3 Sj ½Si ½Sk E3 Sj Si Sk ¼ αði, j, kÞ Ki Kj Kk
(10.89a)
E3 Sj Sj Si ¼ α0jði, jÞ
(10.89b)
(10.89c)
where αj,i denotes for the affinity change factor after species j and i bound on the enzyme, and αða, b, cÞ ¼ αaðb, cÞ ðαbðcÞ + αcðbÞ Þ + αbðc, aÞ ðαcðaÞ + αaðcÞ Þ + αcða, bÞ ðαaðbÞ + αbðaÞ Þ
(10.90)
The bound species j can be computed by NS NS X NS X X B3 Sj ¼ E3 Sj + E3 Sj Si + 2 E3 Sj Sj + E3 Sj Si Sk i¼1 i6¼j
+2
NS X i¼1 i6¼j
E 3 Sj Si Sj + 3 E 3 Sj Sj Sj
i¼1 k¼1 i6¼j k6¼j
(10.91)
10.3 COMPETITIVE MULTILIGAND BINDING ON AN INTERACTIVE ENZYME
585
Which gives NS NS X NS X 1 Kj B3 Sj ½Si 1 X ½Si ½Sk f1 ¼ ¼1+ ðαið jÞ + αjðiÞ Þ + αði, j, kÞ 3 Sj ½E3 K 3 Ki Kk i i¼1 i¼1 k¼1 i6¼j
(10.92)
i6¼j k6¼j
NS h 2 i ½S Sj X Sj Sj i + αjði, jÞ ðαið jÞ + αjðiÞ Þ + 2αið j, jÞ αjð jÞ + 2αjð jÞ + αjð j, jÞ αjð jÞ 2 Ki 2Kj i¼1 Kj Kj i6¼j
and NS NS NS NS NS X Sj X S j XX E3 ½Si X ½Si ½Sk f0 ¼ ¼1+3 ðαið jÞ + αjðiÞ Þ + αði, j, kÞ ½E3 Ki j¼1 Kj i¼1 k¼1 Ki Kk Kj i¼1 j¼1 i6¼j
i6¼j k6¼j
2 N S h 2 NS NS i ½S X X Sj X Sj i + αjði, jÞ ðαið jÞ + αjðiÞ Þ + 2αið j, jÞ αjð jÞ + αjð jÞ 2 2 K 2Kj i¼1 Kj i j¼1 j¼1 i6¼j
+
NS X
αjð j, jÞ αjð jÞ
j¼1
(10.93)
3 Sj Kj3
The enzyme saturation is then given by
1 B 3 Sj f1 Sj θ3j ¼ ¼ f0 Kj 3 E3
(10.94)
For noninteracting homosteric enzyme, all the α’s are unity. Eqs. (10.91) and (10.94) are then reduced to NS NS X NS X Sj Sj ½Si X Sj ½Si ½Sk B3 S j ¼3 +6 +3 (10.95) ½E3 K Kj K Ki Kj Kk i j i¼1 i¼1 k¼1 NS NS X NS X Sj Sj ½Si X Sj ½Si ½Sk +2 + K Kj K Ki Kj Kk i j i¼1 i¼1 k¼1
θ3j ¼ 1+3
NS X NS NS X NS X NS X ½Sk ½Si X ½Sm ½Si ½Sk +3 + K Ki K Ki Km Kk i k m¼1 i¼1 k¼1 k¼1 i¼1
NS X ½Si i¼1
¼ 1+
Sj Kj NS X ½Si i¼1
(10.96)
Ki
10.3.1.3 Four-Site Homosteric Enzyme For a four-site homosteric enzyme, there are five situations in binding: (1) Species j bound only: [E4Sj]; (2) Species j bound on one of the four sites: [E4Si1Sj] + [E4Si1SjSi2] + [E4Si1SjSi2Si3]; (3) Species j bound on two of the four sites: [E4SjSj] + [E4SjSjSi] + [E4SjSjSi1Si2]; (4) Species j
586
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
bound on three of the four sites: [E4SjSjSj] + [E4SjSjSjSi] + [E4SjSjSjSi1]; and (5) Species j bound on all four sites: [E4SjSjSjSj]. Binding on the first site, 4½E4 Sj (10.97a) E4 Sj ¼ Kj on the second site, 2 2 ½E4 Sj 0 3 E4 Sj Sj 0 ½E4 Sj ¼ 6αjð jÞ ¼ 6αjð jÞ E4 Sj Sj ¼ αjð jÞ 2Kj Kj2 Kj2
3½E4 Si Sj 3 E4 Sj ½Si ½E4 Sj ½Si E4 Sj Si ¼ α0ið jÞ + α0jðiÞ ¼ 6ðαið jÞ + αjðiÞ Þ Kj Ki Ki Kj
(10.98a)
(10.98b)
on the third site, 3 3 2 E4 Sj Sj Sj ½E4 Sj ½E4 Sj 0 0 ¼ 4αjð j, jÞ αjð jÞ ¼ 4αjð j, jÞ αjð jÞ E4 Sj Sj Sj ¼ αjð j, jÞ 3Kj Kj3 Kj3 2 E4 S i S j S j 2 E4 Sj Sj ½Si 0 0 E4 Sj Sj Si ¼ αjði, jÞ + αið j, jÞ Ki 2Kj 2 2 ½E4 ½Si Sj ½E4 ½Si Sj 0 0 ¼ αjði, jÞ 6ðαið jÞ + αjðiÞ Þ + 2αið j, jÞ 6αjð jÞ Ki Kj2 Ki Kj2 h i ½E ½S S 2 4 i j 0 0 ¼ 6 αjði, jÞ ðαið jÞ + αjðiÞ Þ + 2αið j, jÞ αjð jÞ 2 Ki Kj h i ½E ½S S 2 4 i j ¼ 3 αjði, jÞ ðαið jÞ + αjðiÞ Þ + 2αið j, jÞ αjð jÞ Ki Kj2 2½E4 Si1 Si2 Sj 3½E4 Si1 ½Si2 Sj 0 0 0 + ⋯ ¼ 2αjði1, i2Þ αi1ði2Þ +⋯ E4 Sj Si1 Si2 ¼ αjði1,i2Þ Kj Ki2 Kj ½E4 ½Si1 ½Si2 Sj ½E4 ½Si1 ½Si2 Sj + ⋯ ¼ 4αði1,i2, jÞ ¼ 4!α0jði1, i2Þ α0i1ði2Þ Ki1 Ki2 Kj Ki1 Ki2 Kj
(10.99a)
(10.99b)
(10.99c)
and on the fourth site,
E4 Si1 Si2 Sj Sj E4 Si2 Sj Sj ½Si1 0 + αi1ðj, i2, jÞ +⋯ E4 Sj Si1 Si2 Sj ¼ 2Kj Ki1 h i ¼ α0jði1,i2, jÞ 2αði1,i2, jÞ + α0i2ðj, i1, jÞ 3 αjði1, jÞ ðαi1ð jÞ + αjði1Þ Þ + 2αi1ð j, jÞ αjð jÞ
α0jði1,i2, jÞ
+ α0i1ðj,i2, jÞ 3
h i ½E ½S ½S S S 4 i1 i2 j j αjði2, jÞ ðαi2ð jÞ + αjði2Þ Þ + 2αi2ð j, jÞ αjð jÞ Ki1 Ki2 Kj Kj
587
10.3 COMPETITIVE MULTILIGAND BINDING ON AN INTERACTIVE ENZYME
h i 2 αjði1, i2, jÞ αði1,i2, jÞ + αi2ðj,i1, jÞ αjði1, jÞ ðαi1ð jÞ + αjði1Þ Þ + 2αi1ð j, jÞ αjð jÞ 3 h i ½E ½S ½S S S 4 i1 i2 j j (10.100a) + αi1ðj, i2, jÞ αjði2, jÞ ðαi2ð jÞ + αjði2Þ Þ + 2αi2ð j, jÞ αjð jÞ Ki1 Ki2 Kj Kj E4 Si Sj Sj Sj E4 Sj Sj Sj ½Si 0 0 + αið j, j, jÞ +⋯ E4 Sj Si Sj Sj ¼ αjði, j, jÞ 3Kj Ki ¼
n h io ½E ½S S S S 4 i j j j 0 0 ¼ αið j, j, jÞ 4αjð j, jÞ αjð jÞ + αjði, j, jÞ 3 αjði, jÞ ðαið jÞ + αjðiÞ Þ + 2αið j, jÞ αjð jÞ Ki Kj Kj Kj h i ½E ½S S S S 3 4 i j j j ¼ αið j, j, jÞ αjð j, jÞ αjð jÞ + α0jði, j, jÞ αjði, jÞ ðαið jÞ + αjðiÞ Þ + 2αið j, jÞ αjð jÞ KKKK 4 i
½E4 Si1 Si2 Si3 E4 Sj Si1 Si2 Si3 ¼ α0i1, i2,i3 Kj
Sj
j
j
j
(10.100b) +⋯
½E4 ½Si1 ½Si2 ½Si3 ¼ α0jði1,i2,i3Þ 4αði1, i2, i3Þ Ki1 Ki2 Ki3 Kj
Sj
+⋯
(10.100c)
½E4 ½Si1 ½Si2 ½Si3 Sj ¼ αði1,i2, i3, jÞ Ki1 Ki2 Ki3 Kj where αða, b, c, dÞ ¼ αaðb, c, dÞ αðb, c, dÞ + αbða, c, dÞ αða, c, dÞ + αcða, b, dÞ αða, b, dÞ + αdða, b, cÞ αða, b, cÞ
(10.101)
When four different ligands bind on all four homosteric sites, there are four interaction factors assuming there is no difference in the order (or sequence) of binding. The bound species j can be computed by NS NS X X B4 Sj ¼ E4 Sj + 2 E4 Sj Sj + E4 Sj Si + 3 E4 Sj Sj Sj + 2 E4 Sj Si Sj i¼1 ij
+
NS X NS X
i¼1 ij
NS X E4 Sj Si1 Si2 + 4 E4 Sj Sj Sj Sj + 3 E4 Sj Si Sj Sj
i1¼1 i2¼1 i16¼j i26¼j
+2
i¼1 ij
NS X NS NS X NS X NS X X E4 Sj Si1 Si2 Sj + E4 Sj Si1 Si2 Si3 i1¼1 i2¼1 i16¼j i26¼j
i1¼1 i2¼1 i3¼1 i16¼j i26¼j i36¼j
(10.102)
588
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
f1 ¼
B4 Sj Kj ½ E4 4 S j
(10.103)
And overall site balance leads to 8 > NS > NS NS < X X X E4 Sj + E4 Sj Sj + E 4 Sj Si + E 4 Sj Sj Sj + E4 Sj Sj Si2 E 4 ¼ ½ E4 + > j¼1 > i¼1 i2¼1 : i6¼j i26¼j +
NS X NS NS X X E4 Sj Si1 Si2 + E4 Sj Sj Sj Sj + E4 Sj Sj Sj Si3 i1¼1 i2¼1 i16¼j i26¼j
+
NS X NS X i2¼1 i3¼1 i26¼j i36¼j
i3¼1 i36¼j
E4 Sj Sj Si2 Si3 +
NS X NS X NS X
E4 Sj Si1 Si2 Si3
i1¼1 i2¼1 i3¼1 i16¼j i26¼j i36¼j
(10.104)
9 > > =
E4 f0 ¼ > ½ E4 > ;
The enzyme saturation is then given by θ4j ¼
B4 Sj f1 Sj ¼ 4E4 f0 Kj
(10.105)
For noninteracting homosteric enzyme, Eqs. (10.103) and (10.105) are reduced to NS NS X NS NS X NS X NS X X X B4 Sj Sj Sj ½Si ½Si1 ½Si2 Sj ½Si1 ½Si2 ½Si3 Sj ¼4 + 12 + 12 +4 (10.106) ½ E4 Ki Kj Kj Ki1 Ki2 Kj Ki1 Ki2 Ki3 Kj i¼1 i1¼1 i2¼1 i1¼1 i2¼1 i3¼1
θ4j ¼ 1+
Sj Kj NS X ½Si i¼1
(10.107)
Ki
10.3.1.4 n-Site Homosteric Enzyme For the competitive binding of two species (S and P) on an n-site homosteric enzyme, the concentration of a given binding is given by
10.3 COMPETITIVE MULTILIGAND BINDING ON AN INTERACTIVE ENZYME
En Si Pj ¼ αði, jÞ
n! ½Si ½Pj ½E n i j i! j! ðn i jÞ! KP K
589 (10.108)
P
where αði, jÞ ¼ αSði1, jÞ αði1, jÞ + αPði, j1Þ αði, j1Þ
(10.109)
αS ði1, jÞ ¼ αPS 8 i, j > 0
(10.110a)
αPði, j1Þ ¼ αSP 8 i, j > 0
(10.110b)
αð j, 0Þ ¼ αðj1, 0Þ αP0
(10.110c)
αð0, iÞ ¼ αð0, i1Þ αS0
(10.110d)
αð1, 0Þ ¼ αð0, 1Þ ¼ 1
(10.110e)
letting
Site balance leads to n X ni En X n! ½Si ½Pj ¼ αði, jÞ ½En i j ½En i¼0 j¼0 i! j! ðn i jÞ! KS K P
½S ½P n ½S ½P ¼ 1 + αPS + αSP + nð1 αS0 Þ + nð1 αP0 Þ KS KP KS KP ½S n ½S n ½P n ½P n + 1 + αS0 1 + αPS + 1 + αP0 1 + αSP KS KS KP KP
(10.111)
And the S-binding is given by n X ni Bn S 1X n! ½Si ½Pj ¼ i αði, jÞ ½ En i j n½En n i¼1 j¼0 i! j! ðn i jÞ! KS K P
½S ½S ½P n1 ½S ¼ αPS 1 + αPS + αSP + ð1 αS0 Þ KS KS KP KS
(10.112)
½S ½S n1 ½S ½S n1 + αS0 1 + αS0 αPS 1 + αPS KS KS KS KS θnj ¼
Bn Sj nEn
(10.113)
590
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
And when all of the interactive coefficients are unity or are for a noninteracting homosteric enzyme, the binding saturation is reduced to Bn Sj ½S ½S ½P n1 ¼ 1+ + (10.114) θnj ¼ nEn KS KS KP
10.3.2 Site-Sequential Multiligand Binding on Allosteric Enzymes 10.3.2.1 Two-Site Sequential Allosteric Multiple-Ligand Interactive Binding When multiple ligands are acting on the same two-sited allosteric enzyme, there is a combination of randomness (no preference on the ligands) and order (site 1 and site 2). The reaction scheme can be represented by: E2 + Sj ! E2 Sj
(10.115a)
E2 Si + Sj ! E2 Si Sj
(10.115b)
E2 Sj + Si ! E2 Sj Si
(10.115c)
E2 Sj + Sj ! E2 Sj Sj
(10.115d)
where E2Sj stands for the two-site allosteric enzyme, with ligand species j bound on the first site, and j i denotes for the two-site allosteric enzyme, with ligand species j bound on the first site and species i bound on the second (and last) site. Chemical equilibria give rise to: ½E2 Sj E2 Sj ¼ (10.116a) K1j E2 Sj ½Si ½E2 Sj ½Si ¼ αj E2 Sj Si ¼ αj K2i K1j K2i ½E2 Si Sj ½E2 ½Si Sj ¼ αi E2 Si Sj ¼ αi K2j K1i K2j
(10.116b)
(10.116c)
Enzyme balance gives rise to: E 2 ¼ ½ E2 +
NS X j¼1
NS X NS X E2 S j + E2 Sj Si
(10.117)
i¼1 j¼1
Substituting Eqs. (10.116a) and (10.116b) into (10.117), we obtain: NS NS NS X Sj X Sj X ½Si E2 ¼1+ + αj K1j j¼1 K1j i¼1 K2i ½ E2 j¼1
(10.118)
591
10.3 COMPETITIVE MULTILIGAND BINDING ON AN INTERACTIVE ENZYME
The ligand species j binding: NS NS X X E 2 Sj Si + E 2 Si Sj + 2 E 2 Sj Sj B2 S j ¼ E 2 S j + i¼1 i6¼j
(10.119)
i¼1 i6¼j
Substituting Eq. (116) into (10.119), we obtain: NS NS B 2 Sj Sj Sj X ½Si Sj X Sj Sj ½Si ¼ + αj + αi + 2αj ½E2 K1j K1j i¼1 K2j K2j i¼1 K1i K1j K2j i6¼j
i6¼j
(10.120)
NS NS Sj Sj X ½Si Sj X ½Si ¼ + αj + αi K1j K1j i¼1 K2i K2j i¼1 K1i
The saturation of species j on the two-site sequential allosteric enzyme is given by: NS NS Sj Sj X ½Si Sj X ½Si + αj + αi K K1i K K K 1j i¼1 2j 2j i¼1 1 B2 Sj 1 1j θ2j ¼ ¼ (10.121) NS NS NS 2 E2 2 X Sj X Sj X ½Si 1+ + αj K1j j¼1 K1j i¼1 K2i j¼1
10.3.2.2 Three-Site Sequential Allosteric Multiple Ligand Interactive Binding When multiple ligands are acting on the same three-sited allosteric enzyme, the reaction scheme can be represented by: ⎯ ⎯→ E3 + Sj ← ⎯⎯ ⎯ ⎯→ E3Sj + Si1 ← ⎯⎯ ⎯ ⎯→ E3SjSi1 + Si2 ← ⎯⎯ ⎯ ⎯→ E3Si1 + Sj ← ⎯⎯
E3Sj, E3SjSi1, E3SjSi1Si2,
E3Si1Sj, ⎯ ⎯→ E3Si1Sj + Si2 ← E ⎯⎯ 3Si1SjSi2, ⎯ ⎯→ E3Si1Si2Sj, E3Si1Si2 + Sj ← ⎯⎯
ð10:122aÞ ð10:122bÞ ð10:122cÞ ð10:122dÞ ð10:122eÞ ð10:122fÞ
where E2Si1SjSi2 denotes for the three-site allosteric enzyme with ligand species i1 bound on the first site, species j bound on the second site, and species i2 bound on the third site. Chemical equilibria give rise to: ½ E3 S j E3 S j ¼ (10.123a) K1j E3 Sj ½Si1 ½E3 Sj ½Si1 ¼ αj (10.123b) E3 Sj Si1 ¼ αj K2i K1j K2i1
592
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
E3 Si1 Sj ½Si2 ½E3 ½Si1 Sj ½Si2 0 ¼ αi1, j αj K3i2 K1i1 K2j K3i2 ½E3 ½Si1 Sj ½Si2 ¼ αi1, j K1i1 K2j K3i2
E3 Si1 Sj Si2 ¼ α0i1, j
(10.123c)
where αi1,j is the overall affinity change of the enzyme owing to the binding of the first site by species i1 and the second site by species j. Enzyme balance gives rise to: E3 ¼ ½E3 +
NS X
NS X NS NS X NS X NS X X E3 S j + E3 Sj Si1 + E3 Sj Si1 Si2
j¼1
i1¼1 j¼1
(10.124)
i2¼1 i1¼1 j¼1
Substituting Eqs. (10.123) into (10.124), we obtain: ! ! NS NS NS NS NS NS X X E3 ½Si ½Si1 X ½Si2 X ½Si2 X ½Si1 X ½Si3 ¼1+ + αi1 + αi1, i2 ½ E3 K K K K K 1i1 i2¼1 2i2 i2¼1 2i2 i1¼1 1i1 i3¼1 K3i3 i¼1 1i i1¼1
(10.125)
The ligand species j binding: NS X B3 Sj ¼ E3 Sj + E3 Sj Si + E3 Si Sj + 2 E3 Sj Sj i¼1 i6¼j
+
NS X NS X E3 Sj Si1 Si2 + E3 Si1 Sj Si2 + E3 Sj Si1 Si2 i2¼1 i1¼1 i26¼j i16¼j
+2
NS X
(10.126)
E3 Si Sj Sj + ½E3 Sj Si Sj + E3 Sj Sj Si Þ + 3 E3 Sj Sj Sj
i¼1 i6¼j
Substituting Eqs. (10.123) into (10.126), we obtain: ! NS NS NS X B3 S j S j ½Si X ½Si2 X ½Si1 ¼ 1 + αj + αi1 ½E3 K1j K K K1i1 i¼1 2i i2¼1 2i2 i1¼1 ! ! NS NS NS NS NS Sj X ½Si X Sj X ½Si1 X ½Si2 ½S2i2 X ½Si1 + αi + α α + K2j i¼1 K1i i1¼1 i1j K1i1 i2¼1 K3i2 K3j i2¼1 K2i2 i1¼1 i1i2 K1i1
(10.127)
The saturation of species j on the three-site sequential allosteric enzyme is given by: θ3j ¼
1 B3 Sj 3 E3
(10.128)
593
10.3 COMPETITIVE MULTILIGAND BINDING ON AN INTERACTIVE ENZYME
10.3.2.3 n-Site Sequential Allosteric Multiple Ligand Interactive Binding In general, for an n-site allosteric enzyme, competitive ligand binding is given by: 8 NS NS NS n X Sji2 X Sj
i2
i1
NS NS X X S j1 Sji1 + ⋯ αj1 ⋯ji1 j K K j ¼1 1j1 j ¼1 i1ji1 1
i1
NS NS 9 NS NS NS n X X X X X Sj1 Sji1 S ji + 1 Sjm1 X Sjm = + ⋯ ⋯ αj1 ⋯ji1 jji + 1 ⋯jm1 K K K Km1jm1 j ¼1 Kmjm ; m¼i + 2 j ¼1 1j1 j ¼1 i1ji1 j ¼1 i + 1ji + 1 j ¼1 1
i1
m1
i+1
m
(10.129) 0 1 NS NS NS NS NS NS n X X X Sj X Sj X ½Sm X Sji @X Sj1 Sji1 A En ¼1+ (10.130) + αm + ⋯ α ½En K1j j¼1 K2j m¼1 K1m i¼3 j ¼1 Kiji j ¼1 K1j1 j ¼1 j1 ⋯ji1 Ki1ji11 j¼1 1
i
i1
The saturation of species j on the n-site allosteric enzyme is given by: θnj ¼
1 Bn S j n En
(10.131)
One can appreciate that conformal structural change due to different ligand species and the competitive bindings on an allosteric enzyme. For a site-sequential noninteractive allosteric enzyme binding of multiligands, the above equations give: k NS n X k X Sj Y X ½Sl K K k¼1 m¼1 mj i¼1 l¼1 il i6¼m
θnj ¼ 1+
NS n Y k X X k¼1 i¼1
½Sl K l¼1 il
(10.132)
10.3.3 Multiligand Binding on Random-Access Allosteric Enzymes 10.3.3.1 Two-Site Random-Access Allosteric Enzyme Multiple species binding on a two-site random-access allosteric enzyme is the simplest case of multiligand interactive binding on a random access allosteric enzyme. In this, the maximum number of any species that can be bound on a given enzyme molecule is 2. There are three situations for the binding when focusing on one particular species: (1) One molecule of species j is bound only: [E2(Sj)1] + [E2(Sj)2]; (2) Species j bound on one of the two sites: [E2(Si)1(Sj)2] + [E2(Sj)1(Si)2]; and (3) Species j bound on both sites: [E2(Sj)1(Sj)2].
594
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
The concentration or saturation of species j (bound species j) can be computed by: NS X ½E2 Sj 1 + E2 Sj 2 + E2 ðSi Þ1 Sj 2 + ½E2 Sj 1 ðSi Þ2 + 2½E2 Sj 1 Sj 2 i¼1 i6¼j
B2 S j ¼ ½ E2
½ E2 2 h
NS i X αj=1 + αi=2 αj=2 + αi=1 Sj Sj Sj +2 + + Sj ½Si + αj=1 + αj=2 ¼ K1j K2i K2j K1i K1j K2j K1j K2j i¼1 i6¼j
where αj/n denotes for the affinity change factor after species j bound on the site n. This equation can be reduced to:
NS αj=1 + αi=2 αj=2 + αi=1 Sj Sj X B 2 Sj ¼ + + Sj ½Si + ½E2 K1j K1j K2i K2j K1i K2j i¼1
(10.133)
The site balance is given by: E2 ¼½E2 +
NS NS X NS X 1X E2 Sj 1 + ½E2 Sj 2 + E2 ðSi Þ1 Sj 2 + ½E2 Sj 1 ðSi Þ2 2 j¼1 j¼1 i¼1 i6¼j
+
NS X
E 2 Sj 1 Sj 2
j¼1
¼½E2 +
NS NS X NS X 1X E2 Sj 1 + ½E2 Sj 2 + E2 ðSi Þ1 Sj 2 + ½E2 Sj 1 ðSi Þ2 2 j¼1 i¼1 j¼1
which can be yielded to: ) ) ( ( NS NS NS NS X X X X Sj Sj E2 ½Si ½Si ¼1+ 1+ αi=2 1+ αi=1 + ½ E2 K2i K1i K1j K2j j¼1 i¼1 j¼1 i¼1
(10.134)
The saturation of species j is given by:
NS αj=1 + αi=2 αj=2 + αi=1 Sj Sj X + + Sj ½Si + K1j K2i K2j K1i K1j K2j 1 i¼1 ( ) ( ) θ2j ¼ NS NS NS NS 2 X X X X Sj Sj ½Si ½Si + 1+ αi=2 1+ αi=1 1+ K2i K1i K1j K2j j¼1 i¼1 j¼1 i¼1
(10.135)
595
10.3 COMPETITIVE MULTILIGAND BINDING ON AN INTERACTIVE ENZYME
For non-interacting allosteric enzyme, Eqs. (10.133) and (10.135) are reduced to: NS NS B2 S j Sj Sj Sj X ½Si Sj X ½Si ¼ + + + ½E2 K1j K2j K1j i¼1 K2i K2j i¼1 K1i 0 NS NS 1 Sj Sj Sj X ½Si Sj X ½Si Sj Sj + + + C B K2j K1j i¼1 K2i K2j i¼1 K1i 1 B C K K 1 K1j 1j 2j C B ! ! θ2j ¼ + ¼ C B N N N N S S S S 2 2@ X X X X ½Si ½Si ½Si ½Si A 1+ 1+ 1+ 1+ K K K K i¼1 1i i¼1 2i i¼1 1i i¼1 2i
(10.136)
(10.137)
10.3.3.2 Three-Site Allosteric Enzyme For a three-site allosteric enzyme, there are four situations in binding: (1) Species j bound only: [E3(Sj)1] + [E3(Sj)2] + [E3(Sj)3]; (2) Species j bound on one of the three sites: [E3(Si)1(Sj)2] + [E3(Sj)1(Si)2] + [E3(Si)1(Sj)3] + [E3(Sj)1(Si)3] + [E3(Si)3(Sj)2] + [E3(Sj)3(Si)2] + [E3(Si1)1(Sj)2(Si2)3] + [E3(Sj)1(Si1)2(Si2)3] + [E3(Si)1(Sj)3(Sk)2] + [E3(Sj)1(Si1)3(Si2)2] + [E3(Si)3(Sj)2(Sk)1] + [E3(Sj)3(Si)2(Sk)1]; (3) Species j bound on two of the three sites: [E3(Sj)1(Sj)2] + [E3(Sj)1(Sj)3] + [E3(Sj)2(Sj)3] + [E3(Sj)1(Sj)2(Si)3] + [E3(Sj)1(Sj)3(Si)2] + [E3(Sj)2(Sj)3(Si)1] and (4) Species j bound on all three sites: [E3(Sj)1(Sj)2(Sj)3]. The bound species j can be computed by: B3 Sj ¼
3 X
NS X NS X NS X 3 X 3 3 X X E3 S j n + E3 Sj n ðSi Þn1 + E3 Sj n ðSi1 Þn0 ðSi2 Þ6nn0
n¼1
+2
i¼1 n¼1 n1¼1 n16¼n i6¼j 3 h X
i1¼1 i2¼1 n¼1 i16¼j i26¼j
NS X 3 h i h i i X E3 Sj n Sj n0 + 2 E3 ðSi Þn Sj n0 Sj 6nn0 + 3 E3 Sj 1 Sj 2 Sj 3
n¼1
i¼1 n¼1 i6¼j
(10.138) where n0 ¼ n + 1 if n < 3, or n0 ¼ 1. ( n0 ¼
n + 1, 8n < 3 1,
if n ¼ 3
(10.139)
596
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
NS 3 3 3 B3 S j X Sj XX X αj=n + αi=n1 ¼ + Sj ½Si ½E3 n¼1 Knj i¼1 n¼1 n1¼1 Knj Kn1i i6¼j
n16¼n
NS X NS X 3 α 3 α X X ðj=n, i=n0, k=0nÞ::2 j=n + αj=n0 2 + Sj ½Si1 ½Si2 + 2 Sj Knj Kn0i K0nk Knj Kn0j n¼1 i1¼1 i2¼1 n¼1
(10.140)
i16¼j i26¼j
+2
NS X 3 α X αðj=1, j2, j=3Þ::2 3 ðj=n, j=n0,i=0nÞ::2 2 Sj ½Si + 3 Sj Knj Kn0j K0ni K1j K2j K3j i¼1 n¼1 i6¼j
where 0n ¼ 6 n n0, αj/n,i/m denotes for the affinity change factor after species j first bound on the site n and then species i bound on site m, and: αða, b, cÞ::2 ¼ αa,b + αa,c + αb,a + αb,c + αc,a + αc,b
(10.141)
That is there are 6 ¼ 3! binding interactions when all three allosteric sites are bound with three different ligands. Eq. (10.140) can be rearranged to give: 0 1 NS α NS X NS α 3 3 X X X C B3 S j X Sj B j=n + αi=n1 ðj=n, i1=n0, i2=0nÞ::2 B1 + ¼ ½ S + ½Si1 ½Si2 C i @ A ½E3 n¼1 Knj Kn1i Kn0i1 K0ni2 n1¼1 i¼1 i2¼1 i1¼1
(10.142)
n16¼n
And overall site balance leads to: ! NS X NS NS NS NS 3 3 X X Sj E3 1X ½Si1 X ½Si2 X ½Si1 X ½Si2 ¼1 + + α + α ½ E3 2! n¼1 i1¼1 Kn0i1 i2¼1 i2=n Kni2 i1¼1 Kni1 i2¼1 i2=n0 Kn0i2 K j¼1 n¼1 nj NS NS NS 1X ½Si1 X ½Si2 X ½Si3 + α 3! i1¼1 K1i1 i2¼1 K2i2 i3¼1 ði3=1, i2=2,i1=3Þ::2 K3i3
(10.143)
For random binding on a noninteracting three-site allosteric enzyme, Eqs. (10.142) and (10.143) are reduced to: 3 NS 3 NS NS 3 3 Sj X Sj X X ½Si X Sj X ½Si2 X B3 S j X ½Si1 ¼ + + ½E3 n¼1 Knj n¼1 Knj i¼1 n1¼1 Kn1i n¼1 Knj i2¼1 K0ni2 i1¼1 Kn0i1 n16¼n
(10.144)
10.3 COMPETITIVE MULTILIGAND BINDING ON AN INTERACTIVE ENZYME
And overall site balance leads to: 3 NS NS NS NS NS NS 3 X X Sj XX ½Si1 X E3 ½Si2 X ½Si1 X ½Si2 X ½Si3 ¼1 + + + ½ E3 Knj n¼1 i1¼1 Kn0i1 i2¼1 Kni2 i1¼1 K1i1 i2¼1 K2i2 i3¼1 K3i3 n¼1 j¼1 0 1 NS 3 Y X Sj A ¼ @1 + K nj n¼1 j¼1
597
(10.145)
The saturation of species j on a noninteracting random allosteric enzyme is given by: !1 3 NS X Sj X 1 1 B3 S j ½Si ¼ K 1+ (10.146) θ3j ¼ 3 E3 K 3 n¼1 nj i¼1 ni One can appreciate that bindings of multiple ligands are more complicated than that of single ligand. For non-site sequential noninteractive allosteric enzyme binding of multi ligands, we can show the binding saturation as given by: ! NS n X d Y ½Sm !1 1+ n NS K X Sj d Sj i¼1 Sj X 1 ½Sm m¼1 im ! ¼ K 1+ (10.147) θnj ¼ NS n K n n i¼1 ij X Y ½Sm m¼1 im 1+ K m¼1 im i¼1 which turned out to be rather “simple.”
10.3.4 Enzymes With Homosteric Paired Allosteric Sites Fig. 10.29 shows examples of a special class of enzyme oligomers, where each subunit contains a pair of active centers: site type 1 and site type 2. The two site types are distinct and represented by diamonds and triangles. Assuming substrates Si 8i ¼ 1, 2, …, NS can bind on site type 1, while ligands Aj 8j ¼ 1, 2, …, NP can bind on the site type 2. Since the bindings of Si and Aj on the allosteric sites, the bindings can be computed almost separately. Site type 1
(i) Monomer
FIG. 10.29
Site type 2
(ii) Dimer
(iii) Trimer
(iv) Tetramer
Oligomeric enzymes of homosteric paired allosteric sites. Homosteric subunits of the enzyme with each subunit containing a pair of allosteric sites: site type 1 and site type 2. The distinctly different sites allow different ligands to be bound.
598
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Following Eq. (10.108), the binding on homosteric pairs of allosteric sites is as given by: ! ! m l Y Y Ajk ½Sik n n Aj1 …Ajl En Si1 …Sim ¼ (10.148) αði1 , …, im , j1 , …, jl Þ ½En l m KSik KAjk k¼1
"
αði1 , i2 , …, im , j1 , …, jl Þ ¼
k¼1
m 1 X αði , i , …, ik1 , ik + 1 , …, im , j1 , …, jl Þ αSði1 , i2 , …, ik1 , ik + 1 , …, im , j1 , …, jl Þ! + ik m + l k¼1 1 2 # l X αði1 , i2 , …, im , j1 , …, jk1 , jk + 1 , …, jl Þ αAði1 , i2 , …, im , j1 , …, jk1 , jk + 1 , …, jl Þ! + jk + k¼1
(10.149) Enzyme balance: En ¼
n X n X Aj1 …Ajl En Si1 …Sim
(10.150)
m¼0 l¼0
The binding concentration or saturation can be computed for each species. We can see the additional complexity over the homosteric enzyme. However, the equations are quite similar:
10.4 CATALYTIC REACTION RATE ON INTERACTIVE ENZYMES As noted in the previous section, binding on interactive enzymes can be allosteric or homosteric. The enzymatic catalysis of cooperative binding can be summarized following Eqs. (10.27) and (10.28). Again, there are two distinctly different situations: a homosteric enzyme, a site-sequential allosteric enzyme, and a random site allosteric enzyme, as illustrated in Fig. 10.21. To simplify the discussions, we shall restrict our discussions on the initial rates, ie, the effect of product(s) and their bindings are negligible. The binding and catalytic reaction on the first site of an interactive enzyme can be stoichiometrically represented by: Kn
kn
En + S ! En S ! En + P
(10.151)
where Kn is the ratio of binding and dissociation rate constants for the first site of the n-site enzyme En, and kn is the turnaround frequency of the catalytic ability of the first binding site. As we have learned from the above: discussions, the interpretation of the binding equilibrium can be different for different types of binding sites.
10.4.1 Catalytic Reactions on Homosteric Sites For a homosteric enzyme, all the binding sites are equal. The binding and catalytic reaction can be represented by: α1 KS i
ði + 1Þβi kn
En Si + S ! En Si + 1 ! En Si + P
81 i n
(10.152)
10.4 CATALYTIC REACTION RATE ON INTERACTIVE ENZYMES
599
The binding of the first substrate molecule can facilitate the reaction to occur, as well as further binding of additional substrate molecules. When an interactive ligand/substrate is bound on a homosteric enzyme, it can induce a change in the conformal structure of the enzyme as whole and thus immediately change the interactive capability of the vacant binding sites. The subsequent binding of substrate molecules can have different affinities, as the conformational change could occur because of the binding of a substrate molecule. The reactivity can also be changed. The Michaelis-Menten saturation coefficient for the first binding is denoted as KS and the catalytic reaction rate constant is denoted as kn, in Eq. (10.151). For the subsequent binding, Eq. (10.152) shows that the “Michaelis-Menten saturation coefficient” is KS/αi and the catalytic reaction rate constant is βikn. Eq. (10.152) shows that the reactivity for each bound site on a homosteric enzyme is the same when multiple sites are bound. The overall catalytic reaction rate for an enzyme with n-equally reactive binding sites, rPn, as illustrated by Eq. (10.152) is given by: rPn ¼ kn ½En S + kn
n X
iβi1 ½En Si
(10.153)
i¼2
Substituting Eq. (10.39) in (10.160), we obtain the general catalytic rate expression for enzymes with n-equally reactive binding sites: n Y i1 X iβ ½Si1 n n+ αj i1 1 i KS i¼2 j¼1 (10.154) rPn ¼ kn En ½S k1 n k1 X n Y ½S KS + ½S αj k1 k KS j¼1 k¼1 Letting: rmax n ¼ nkn En
(10.155)
Eq. (10.154) is reduced to: n Y i1 1X iβ ½Si1 n αj i1i1 n i¼2 i j¼1 KS n ½S k1 n k1 X n Y ½S KS + ½S αj k1 k KS j¼1 k¼1
1+
rPn ¼ rmax
(10.156)
The two asymptotes are worth mentioning. The first one is at low substrate concentration. When [S] < KS, Eq. (10.156) is reduced to: rPn ¼ rmax n ½S
1 + ðn 1Þα1 β1 KS + n½S
½S KS
(10.157)
which can be confused with the Michaelis-Menten equation (10.6), if we had neglected the second-order term in the numerator. Not to mention that when n ¼ 1, Eq. (10.156) is reduced
600
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
to Eq. (10.6). Another case is at high substrate concentrations, [S] ≫ KS; therefore, Eq. (10.156) is reduced to:
rPn ¼ rmax
βn2 n
n1 KS + βn1 ½S αn1 n KS + ½S αn1
(10.158)
which could be easily confused with the Michaelis-Menten equation. However, the intercept at zero substrate concentration is not zero. One can rewrite Eq. (10.156) without loss of generality to: n 1X iβ ani ½Si1 n i¼2 i1 n ½S n X KS + ½S ani ½Si1
1+
rPn ¼ rmax
(10.159)
k¼1
While Eqs. (10.156) and (10.159) are valid in general, it is inconvenient to use them, as there are multiple terms in the equation, and the number of terms increase with increasing number of binding sites. If the binding affinity and catalytic activity are not altered by the binding of substrate(s), both Eqs. (10.156) and (10.159) are reduced to rPn ¼
rmax n ½S KS + ½S
(10.160)
which is identical to the Michaelis-Menten equation. When the variations of affinity and reactivity become negligible from the third site on: α2 ¼ α3 ¼ ….. ¼ αi ¼ … ¼ αn 82 i n
(10.161)
β2 ¼ β3 ¼ … ¼ βi ¼ … ¼ βn1 82 i n
(10.162)
and:
Eq. (10.158) is reduced to:
rPn ¼ rmax n
½S KS
1
α1 β β2 α1 ½S n1 + ðn 1Þ½Sα1 1 + β2 1 + α2 α2 KS α2 KS α1 α1 ½S n ½S α1 1 2 + 2 1 + α2 +n 1 α2 KS KS α2 α2
(10.163)
Chen et al. (2014) studied the structure and catalytic behaviors of a recombinant Clonorchis sinensis hexokinase (rCsHK). In this case, the enzyme is known to be homo-trimeric. Fig. 10.30 shows the initial catalytic rate variation with substrate concentration for three different substrates: glucose, mannose and fructose. Fitting of the data to Eq. (10.163) gives the kinetic parameters for the different substrate as shown in Table 10.5. One can observe from Fig. 10.30 that Eq. (10.163) can describe the kinetic behavior quite well.
601
10.4 CATALYTIC REACTION RATE ON INTERACTIVE ENZYMES
FIG. 10.30
The catalytic activity of a homotrimeric recombinant Clonorchis sinensis hexokinase (rCsHK) on hexose conversion to hexose 6-phosphate. The symbols are data taken from Chen et al., 2014. The system consists of 3 mM ATP, 100 mM KCl, 15 mM MgCl2, 0.4 mM NADH, 5 mM phosphoenolpyruvate, 5 U/mL of rabbit muscle pyruvate kinase, 10 U/mL rabbit muscle lactate dehydrogenase, and 100 mM Tris-Cl (pH 8.5). The curve is fitted with Eq. (10.163).
5
rP/E, mmol/min/g
4
3 Mannose
Glucose
Fructose
2
1
0 0.01
0.1
1
10
[S], mM
TABLE 10.5
Kinetic Parameters of Recombinant Clonorchis sinensis Hexokinase (rCsHK) Responding to Different Substrates. The Number of Binding Sites per Enzyme Molecule is n ¼ 3
Substrate
rmax/E (mmol/(min g))
Fructose Mannose Glucose
KS (mM)
19.43
α1
6.207
373.9
19.10
1857
99.89
α2
β1
β2
17.09
2.254
0.20212
154.7
4.318
8.21 10
2.080
0.01191
1.48 10
1.816
0.002512
33.13
4 6
10.4.2 Catalytic Reactions on Site-Sequential Allosteric Sites For a site-sequential allosteric enzyme, the ligand has preference on the binding sites. The difference of the binding sites and sequence of the binding are only part of the characteristics of the allosteric enzyme. It is intuitive that the reactivity of the sites are different as well: KiS
kci
En Si1 + S ! En Si ! En Si1 + P
81 i n
(10.164)
The binding on the first site by the substrate can facilitate the reaction to occur, and further binding on the other sites and reactions. The catalytic reaction rate constant is different for different sites. For the ith site, the catalytic rate constant is denoted as kci in Eq. (10.164). The overall catalytic reaction rate for an enzyme with n-allosteric reactive binding sites, rnP, as illustrated by (10.164) is given by: rnP ¼ kc1
n X i¼1
½En Si + kc2
n X i¼2
½En Si + … + kci
n X
½En Si + … + kcn ½En Sn
(10.165)
i¼m
if each site on the allosteric enzyme can catalyze the reaction independently and the reactivity is not altered by the binding of substrate.
602
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
By substituting Eq. (10.62) into (10.165), we obtain the catalytic rate expression for sitesequential allosteric enzymes with n-reactive allosteric sites: i X n X i¼1
kcj ½Si
j¼1 i Y
KjS
j¼1
rPn ¼ En 1+
n X ½Si i Y i¼1 KjS
(10.166)
j¼1
which is a general site-sequential allosteric enzyme catalytic rate expression. If the variations of affinity and reactivity becomes negligible from the third site on, or: K3S ¼ K4S ¼ … ¼ KiS 83 i n
(10.167)
kc3 ¼ kc4 ¼ … ¼ kci ¼ … ¼ βcn 83 i n
(10.168)
and:
Eq. (10.166) is reduced to:
kc1 + rnP ¼ En ½S
½S K2S
n2 ½S 1 kc3 ½S2 K3S + ½S K2S K3S ½S 2 1 1 K3S K3S 8 (10.169) n1 9 > > ½S > > > > 1 = < ½S K3S K1S + ½S 1 + ½S > > K2S > > > > 1 : K3S ;
kc1 + kc2 ½kc1 + kc2 + ðn 2Þkc3
½S K3S
n1
Eq. (10.169) is the catalytic rate expression for random-access allosteric enzymes with variable binding and reactivity on the first three sites. To show how the kinetic model fits with actual experimental data, Fig. 10.31 shows the relative catalytic activity of allosteric enzyme, ie, glucokinase (or hexokinase IV). The fitting of the data showed that glucokinase has 2 allosteric binding sites for glucose (n ¼ 2), and the parameters are shown in Table 10.6. The curves in Fig. 10.31 are based on Eq. (10.169), it fitted the data quite well. When an activator is added, the kinetic parameters changed (for both glucose binding sites). One can observe from Fig. 10.31 that hexokinase IV can be modeled with a 2-site allosteric enzyme.
603
10.4 CATALYTIC REACTION RATE ON INTERACTIVE ENZYMES
FIG. 10.31 Allosteric behaviors
1.6
Relative glucokinase activity
1.4 1.2
No effector addition BAD SAHBA GKA (RO0281675) BAD SAHBA + GKA
of glucokinase (hexokinase IV). The symbols are experimental data take from Szlyk, B., et al. 2014. Nat. Struct. Mol. Biol. 21(17), 36–42. The curves are fittings with Eq. (10.169) with parameters shown in Table 10.6.
1.0 0.8 0.6 0.4 0.2 0.0 10−2
10−1
100
101
102
[Glucose], mM
TABLE 10.6
Kinetic Parameters of Glucokinase With Eq. (10.169) kc1En
Effector addition 1. No effector addition
K1S (mM)
0.02121
2. With BAD SAHBA (S118D)
8.640
34.52
10.09
K2S (mM)
kc2/kc1
N
3.690
4903
2
3.391
3.655
2
3. With GKA (RO0281675)
144.8
5.083
1.299
0
2
4. With both BAD SAHBA and GKA
105.5
4.738
0.6764
0.2069
2
10.4.3 Random-Access Allosteric Enzymes For multiligand binding on a random-access allosteric enzymes, the binding saturation of species j is given by n n X n n X X X α i1 , …, ij1 j ½S ⋯ kcj i Y j¼1 i1 ¼1 i2 ¼1 ij ¼1 Kim i2 6¼i1 ij 6¼i1 , …, ij1 m¼1 (10.170) rPn ¼ En n n X n n X X X α i1 , …, ij1 j 1+ ½S ⋯ j Y j¼1 i1 ¼1 i2 ¼1 ij ¼1 i2 6¼i1 ij 6¼i1 , …, ij1 Kim m¼1
Letting:
α i1 , …, ij1 anj ¼ ⋯ i Y i1 ¼1 i2 ¼1 ij ¼1 Kim i2 6¼i1 ij 6¼i1 , …, ij1 n X n X
n X
m¼1
(10.171)
604
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Eq. (10.170) is reduced to: n X
rPn ¼ En
kci ani ½Si
i¼1
1+
n X
(10.172) ani ½S
i
i¼1
where ani is the cumulative affinity to the order of i, and kci is the effective reactivity of the ith order. Eq. (10.172) is conveniently valid for single species binding on all the interactive enzymes.
10.5 KINETICS OF POLYMORPHIC CATALYSIS AND ALLOSTERIC MODULATION In the above section, we have learned that enzyme catalytic rate deviates from the Michaelis-Menten rate if enzymes aggregate to form complex of multiple binding sites. Still, we have dealt only with enzymes that exist in a single form. When polymorphs all show catalytic activity, the situation is more complex. Both binding isotherm and catalytic rate are affected by polymorphs. For a polymorphic system, the total number of enzyme subunits is distributed among the polymorphs. Again, we shall restrict our discussions to the initial rates. Polymorphs can interconvert. The interconversion of polymorphs can affect the enzymatic catalysis. We must then consider how polymorphs interconvert. For this reason, if equilibrium can be assumed, let: nE1 >j En At thermodynamic equilibrium:
h KjEn ¼
(10.173) i
j En
(10.174)
½E1 n
And the overall enzyme unit balance leads to:
! jn jn N N n h i h i X X X X X i n ¼ n E¼ j E n S + j En j En S i n¼1
j¼1
n¼1
j¼1
(10.175)
i¼1
where jn is the total number of polymorph members that each has n subunits and N is number of subunits of the polymorph that has maximum number of subunits.
10.5.1 Substrate-Free Polymorph Interconversion When enzymes have substrate bound, a polymorph may be stabilized against conversion to different polymorphs. Fig. 10.32 illustrates a substrate-free polymorph. In this case,
605
10.5 KINETICS OF POLYMORPHIC CATALYSIS AND ALLOSTERIC MODULATION
Ligand-Free Polymorph E1
E2
E3
En Ei + Ek
jEi+k
j Ei
+S
jEi
S
jEi
Sm + S
jEi
Sm+1
Increasing ligand concentration
FIG. 10.32 A schematic of ligand-free polymorph system. A polymorph is a mixture of oligomers of enzyme protomer E1. The oligomers or polymorph members can interconvert only when no ligands are bound.
polymorphs can only interconvert when no substrates were bound. Eq. (10.174) can be expressed for ligand-free enzyme polymorphs: h i n (10.176) j En ¼ KjEn pe where jEn is the jth polymorph member with n substrate subunits, KjEn is the equilibrium constant between the singular polymorph member and the jth polymorph member with n subunits, and pe is polymorph enzyme parameter, an analog to the concentration of protomer. Since the interconversion can occur among the polymorphs, the overall enzyme mixture composition is dynamic. Assuming that the interconversion of enzyme polymorphs is fast, we consider the enzyme polymorphs in thermodynamic equilibrium only here. Substituting Eqs. (10.33) and (10.176) into (10.175), one can obtain after rearrangements: ( ) jn N X n Y k1 X X ½Sk n n nKjEn pe 1 + αjnl k (10.177) E¼ k Kjn n¼1 j¼1 k¼1 l¼1 which can be solved for pe, irrespectively of whether any form of the protomer (monomeric enzyme) is present in the system. Here, pe is simply a parameter for the polymorph, having the unit of enzyme concentration.
606
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Substituting Eq. (10.176) into (10.33), one can obtain: h ih ii Y i1 i1 h i Y j En S pne KjEn ½Si n n αjnl ¼ α jnl j En Si ¼ i i i i Kjn Kjn l¼1 l¼1
(10.178)
With Eq. (10.178), one can derive the enzyme saturation and the catalytic reaction rate. The total bound enzyme is given by: jn X jn N X n h N X n Y i1 i X X pne KjEn X i½Si n i j En S i ¼ αjnl i1 (10.179) BS ¼ Kjn i¼1 i l¼1 Kjn n¼1 j¼1 i¼1 n¼1 j¼1 The enzyme saturation is given by: θS ¼
jn N n Y i1 KjEn X BS X pne X i½Si n αjnl i1 ¼ E n¼1 E j¼1 Kjn i¼1 i l¼1 Kjn
(10.180)
The overall catalytic reaction rate for an enzyme with n-equally reactive binding sites of invariable reactivity, rPn, as illustrated by Eq. (10.153) is given by: rPn ¼kn ½En S + kn
n X
iβni1 ½En Si
i¼2
Y n X n i1 pne KEn ½S pn KEn ½Si + kn iβni1 αjnl e i ¼kn n Kn Kn i l¼1 i¼2 Summing up all the polymorphs, one obtains the overall rate: ( ) Y jn N n i1 X X KjEn kjn 1X ½Si1 n n npe 1+ iβ α rP ¼ ½S Kjn n i¼2 jni1 i l¼1 jnl Kni1 n¼1 j¼1
(10.181)
(10.182)
which is the general overall catalytic rate expression for a substrate-free interconvertible polymorph. 10.5.1.1 Substrate-Free Interconvertible Polymorph Of Limiting Structural Changes If Eq. (10.48) holds, or: αjn2 ¼ αjn3 ¼ … ¼ αjni ¼ … ¼ αjnn 82 i n Then the polymorph ratio is given by: ( !) jn N X X αjn1 αjn1 αjn1 ½S n ½S n npe KjEn 1 2 + 2 1 + αjn2 +n 1 E¼ αjn2 Kjn Kjn αjn2 αjn2 n¼1 j¼1
(10.183)
(10.184)
When both Eqs. (10.161) and (10.162) are applicable, Eq. (10.182) is reduced to: ( ) jn N X X αjn1 αjn1 βjn1 βjn2 KjEn kjn ½S n1 n npe 1 βjn2 + ðn 1Þ½Sαjn1 + βjn2 1 + αjn2 rP ¼ ½ S Kjn αjn2 Kjn αjn2 Kjn n¼1 j¼1 (10.185)
607
10.5 KINETICS OF POLYMORPHIC CATALYSIS AND ALLOSTERIC MODULATION
which is the catalytic rate form for a substrate-free interconvertible polymorph of homosteric enzymes, when the substrate-binding affecting affinity and reactivity are only in the first two successive bindings. 10.5.1.2 Substrate-Free Interconvertible Polymorph of No Structural Changes In the extreme case where substrate binding is not affecting the enzyme structure, ie, αjnl 1, Eq. (10.177) is reduced to: E¼
N X
npne
½S n KjEn 1 + Kjn j¼1
jn X
n¼1
(10.186)
Furthermore, when βjnl 1, Eqs. (10.185) or (10.182) is reduced to: rP ¼ ½S
N X
n½E1 n
n¼1
jn X KjEn kjn ½S n1 1+ Kjn Kjn j¼1
For a bimorph having the same number of subunits, Eq. (10.186) is reduced:
½S n ½S n E ¼ npne K1En 1 + + K2En 1 + K1n K2n
(10.187)
(10.188)
which can be solved for the parameter pe: npne ¼
E n ½S ½S n K1En 1 + + K2En 1 + K1n K2n
(10.189)
Substituting Eq. (10.189) into (10.187), we obtain: K1En k1n ½S n1 K2En k2n ½S n1 1+ + 1+ K1n K2n K K2n 1n n rP ¼ E½S ½S ½S n K1En 1 + + K2En 1 + K1n K2n
(10.190)
which is readily reduced to Eq. (10.16). Therefore, the MWC model is applicable only in the case where the interaction of substrate and enzyme is restricted to the form change only. The binding itself does not change the affinity (nor reactivity) of the enzyme molecule. As one substrate molecule is bound on the enzyme, it does not induce any conformal changes that would alter the affinity or the reactivity of the rest of the binding sites.
10.5.2 Substrate-Inert Polymorph Interconversion Fig. 10.33 shows a schematic of a substrate-inert polymorph system. When there are no effects of substrate binding on the polymorph interconversion: j En
¼ KjEn pne
(10.191)
608
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Ligand-Inert Polymorph
• • •
E1
• • •
• • •
• • •
E2
E3 • • •
• • •
En
•••
Ei + Ek
•••
jEi+k
j Ei
+S
Ei S + Ek
•••
j Ei
S
jEi+k
jEi
S
Sm + S
Ei Sm1 + Ek Sm2
j Ei
Sm+1
jEi+k Sm+1
Increasing ligand concentration
FIG. 10.33
A schematic of the ligand-inert polymorph system. A polymorph is a mixture of oligomers of enzyme protomer E1. The oligomers or polymorph members can interconvert without any effects from the ligand binding.
and the enzyme unit balance Eq. (10.175) leads to: jn jn N N X X X X n n npe KjEn E¼ j En ¼ n¼1
j¼1
n¼1
(10.192)
j¼1
The equations with regards to individual polymorphs are still valid. Using Eqs. (10.33) and (10.191), one can derive at the enzyme saturation ratio as: n Y i1 X i½Si n n K p α jEn jnl i1 e jn N X i Kjn BS 1 X i¼1 l¼1 (10.193) θS ¼ ¼ n i 1 X n Y E E n¼1 j¼1 ½Si Kjn + αjnl i1 i Kjn i¼1 l¼1 The overall catalytic reaction rate when all subunits in an enzyme complex are equally reactive can be obtained by summing up all the polymorphs with each giving a rate described by Eq. (10.153). That is: n Y i1 X iβjni1 ½Si1 n n + α jnl jn i1 N i X X Kjn i¼2 l¼1 pne KjEn kjn (10.194) rP ¼ ½S n Y i1 X ½Si1 n n¼1 j¼1 Kjn + ½S αjnl i1 i Kjn k¼1 l¼1
10.5 KINETICS OF POLYMORPHIC CATALYSIS AND ALLOSTERIC MODULATION
609
with the parameter pe determined by Eq. (10.192). When both Eqs. (10.161) and (10.162) are applicable, Eq. (10.194) is reduced to: ½S n1 1 βjn2 + ðn 1Þ½Sαjn1 + βjn2 1 + αjn2 αjn2 Kjn αjn2 Kjn ! n αjn1 αjn1 αjn1 ½S ½S 1 2 + 2 1 + αjn2 +n 1 αjn2 Kjn Kjn αjn2 αjn2 αjn1
rP ¼ ½S
N X n¼1
npne
jn X KjEn kjn j¼1
Kjn
βjn1 βjn2
αjn1
(10.195) When substrate binding does not alter the binding affinity nor the reactivity, ie, αjnl 1 and βjnl 1, Eqs. (10.193) and (10.195) are reduced to: rP ¼ ½S
N X n¼1
npne
jn X KjEn kjn ½S 1 1+ Kjn Kjn j¼1
jn N X KjEn ½S X ½S 1 n np 1+ θS ¼ E n¼1 e j¼1 Kjn Kjn
(10.196)
(10.197)
For a bimorph, Eqs. (10.192) and (10.196) give: n2 E ¼ n1 En1 1 KEn1 + n2 E1 KEn2
1 1 ) K k ½ S K k ½ S En1 n1 En2 n2 1+ + n2 pn2 1+ rP ¼ ½S n1 pn1 e e Kn1 Kn2 Kn1 Kn2
(10.198)
(
(10.199)
letting: f1 ¼
n1 pn1 e KEn1 E
(10.200)
Eq. (10.198) gives: 1 f1 ¼
n2 pn2 e KEn2 E
(10.201)
Substituting Eqs. (10.200) and (10.201) into (10.199), we obtain: (
) f1 kn1 ½S 1 ð1 f1 Þkn2 ½S 1 rP ¼ E½S 1+ + 1+ Kn1 Kn2 Kn1 Kn2
(10.202)
which is readily reduced to Eq. (10.21). Therefore, we have recovered the Morpheein model.
610
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Let us examine the kinetics of human PBGS catalyzed 5-aminolevulinic acid (ALA) to asymmetrically condense to porphibilinogen (PBG): OH
O O
OH
O
ð10:203Þ 2 NH
OH
2
+ 2H 2O
NH 2
O
NH
which was used as a model reaction system for the morpheein model by Selwood et al. (2008). Human PBGS exist as a bimorph, a hexamer, and an octamer, as shown in Fig. 10.15. At neutral pH, human PBGS exist predominantly as the high-activity octamer, while increasing pH will have the octamer converted to the low-activity hexamer. For each polymorph (or bimorph, in this case), the binding and catalytic reaction can be represented by: Kn
En + S ! En S α1 K ni n
iβni kn
! En Si + S En Si + 1 ! En Si1 + PBG + 2H2 O
(10.27) 81 i n
(10.204)
where Kn is the binding saturation coefficient (Michaelis-Menten constant) for the first site of the n-site enzyme En, and 2βn1kn/2 is the turnaround frequency of the catalytic ability of the first pair of the binding sites. Following the derivations above, one can obtain: n Y i1 X iβ ½Si n αnl ni1i Kn kn En i¼2 i l¼1 (10.205) rPn ¼ i n i1 X 2 n Y ½S 1+ αnl i i Kn k¼1 l¼1 and the enzyme unit balance, Eq. (10.192), for the bimorph of 6 and 8 subunits leads to: E ¼ 6p6e KE6 + 8p8e KE8
(10.206)
letting: 6p6e KE6 E
(10.207a)
8p8e KE8 ¼ 1 f6 E
(10.207b)
f6 ¼ Similarly: f8 ¼
The overall initial PBG production rate is given by: rPBG ¼ rP6 + rP8
(10.208)
10.5 KINETICS OF POLYMORPHIC CATALYSIS AND ALLOSTERIC MODULATION
When both Eqs. (10.161) and (10.162) are applicable, Eq. (10.205) gives: ½S n1 β =β 1 1 + α 1 + ðn 1Þ½Sαn2 n1 n2 n2 n nα β KEn pe kn Kn Kn ½S rPn ¼ n1 n2 2αn2 Kn αn1 αn1 ½S n ½S αn1 1 2 + 2 1 + αn2 +n 1 αn2 Kn Kn αn2 αn2 Substituting Eqs. (207) and (10.209) into (10.208), we obtain: ½S 5 β =β 1 1 + α62 1 + 5½Sα62 61 62 rPBG α61 β62 k6 K6 K6 ¼ f6 ½S 6 E 2α62 K6 α61 α61 ½S ½S α61 1 2 + 2 1 + α62 +6 1 K6 K6 α62 α62 α62 7 ½S β =β 1 1 + α82 1 + 7½Sα82 81 82 α81 β82 k8 K8 K8 ½S + f8 2α82 K8 α81 α81 ½S 8 ½S α81 1 2 + 2 1 + α82 +8 1 α82 K8 K8 α82 α82
611
(10.209)
(10.210)
Unlike the rate expressions presented so far, Eq. (10.210) shows the catalytic rate dependence on the substrate concentration to the second order at low substrate concentrations. The reason is that the reaction consumes two moles of the substrate ALA to make one mole of product PBG, as indicated by the reaction (10.203). Fig. 10.34 shows the catalytic behavior of PBGS as affected by the pH, where the lines are based on Eq. (10.210). Assuming that pH only affects the form, we effectively have a polymorph (or bimorph of a hexamer and octamer) that the interconversion is controlled by pH (not the substrate ALA). The binding and reactivity are not affected by pH. The assumptions here may be too strong. In this case, Eq. (10.210) can be applied to fit the experimental data. The kinetic parameters are listed in Table 10.7. FIG. 10.34 Specific production rate of porphobilinogen (PBG) from 5-aminolevulinic acid (ALA), as catalyzed by wild type human PBG synthase. The symbols represent the data are obtained from Selwood et al., 2008. The lines are based on Eq. (10.210) with parameters given in Table 10.7.
40
rPBG/E, mmol/h/g
E = 5 mg/L, pH = 7 30
E = 10 mg/L, pH = 8
20
10 E = 20 mg/L, pH = 8.8 0 0.001
0.01
0.1 1 [ALA], mM
10
100
612
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
TABLE 10.7 Kinetic Parameters Used in Fig 10.30. The Dimers are Assumed to be Inactive in Catalyzing the Reaction, k2 ¼ 0, and the Amount of Hexamer at pH 7 is Negligible n/2fnβn2kn (h21) Polymorph
n
pH 7
pH 8
pH 8.8
Kn (mM)
αn1
αn2
βn1/βn2
Octamer
8
40.01
10.12
2.741
0.3785
294.6
3.130
0.9911
Hexamer
6
14.96
4.416
0.1025
596.3
9.560
0.9856
0
One can observe from Fig. 10.34 that the catalytic rate increases with increasing substrate concentration, decrease with increasing pH except at very low substrate concentrations. The polymorph model can describe the experimental well:
10.5.3 Oligomers of Paired Allosteric Sites Fig. 10.29 shows schematics of some oligomeric enzymes with paired allosteric sites. The allosteric binding of a substrate S and an effector I on an oligomeric enzyme with paired allosteric sites is illustrated in Fig. 10.35. One can derive the kinetic expression straightforwardly by noting that the substrate can bind on one type of sites while the effector always go to the other type of sites. Since the bindings of I and S on the allosteric sites, the bindings can be computed almost separately following Section 10.3.4 as given by: Y i1 ½En ½Si ½En ½Si n n ½En Si ¼ αj ¼ α (10.211a) ð0, iÞ i i KSi KSi j¼1
KS E+S + I
E•S +
+ S
a−1ISKI a−1ISKI
+ I
a −1I0KI I2•E + S
I•E•S2 + S −1 a SIKS I +
+ S I•E•S a−1SIKS + a −1ISKI I
I•E + S
FIG. 10.35
E•S2 + S + a−1S0KS I
I KI
•••
a−1S0KS
a −1ISKI a −1SIKS
I2•E•S +
a−1SIKS
S
a −1SIKS
a−1S0KS
I
a−1ISKI a−1ISKI + S
I•E•S3 + I
a−1SIKS
I2•E•S4 + S
a −1SIKS
I•E•S4 + I
a−1ISKI
a−1ISKI
I2•E•S2 + S
•••
E•S4 +
E•S3 + S + I
I2•E•S3
a−1SIKS •••
An illustration of the kinetics of the effector acting on an oligomeric enzyme of paired homosteric sites. The effector is bound on a different site than where catalytic reaction takes place.
613
10.5 KINETICS OF POLYMORPHIC CATALYSIS AND ALLOSTERIC MODULATION
½En ½Ii n ½Ii En ¼ αði, 0Þ i KIi
(10.211b)
½En ½Si ½Ij n n αð j, iÞ Ij En Si ¼ j j i Ki K
(10.211c)
1 jαðj1, iÞ αIS + iαðj, i1Þ αSI i+j
(10.212a)
S
αð j, iÞ ¼
I
αð j, 0Þ ¼ αðj1, 0Þ αI0
(10.212b)
αð0, iÞ ¼ αð0, i1Þ αS0
(10.212c)
αð1, 0Þ ¼ αð0, 1Þ ¼ 1
(10.212d)
where α(i,j) denotes for the affinity change due to the binding of i-effector molecules (I) and j-substrate molecules (S), αIS stands for the affinity change factor of effector binding when multiple S already bound, and αSI stands for the affinity change factor of substrate (S) binding when multiple effector I molecules already bound. In writing Eqs. (10.211a) through (10.212d), we have assumed that the conformal structural change only occurs once when each effector and substrate molecules are bound, as shown in Fig. 10.35. These equations can be reduced directly from (10.148) and (10.149). The catalytic rate is determined by the rate-limiting steps: iβð j, iÞ kc Ij En Si ! Ij En Si1 + P
rP ¼ kc ½ES + kc
n X i¼2
iβð0, iÞ ½ESi + kc
81 i n n X n X
(10.213)
iβð j, iÞ Ij ESi
(10.214)
j¼1 i¼1
Enzyme balance: En ¼ ½ E n +
n X j¼1
n X n X ½Ii En + En Sj + I i En S j
(10.215)
i¼1 j¼1
Substituting Eqs. (10.211a) through (10.211c) into (10.215), one can obtain: ( ) n n X n X X ½Si ½Ii ½Si ½Ij n n n αð0,i1Þ i + αði1, 0Þ i + ½En (10.216) αð j, iÞ En ¼ ½En + ½En j j i i KS KI Ki K i¼1
i¼1 j¼1
S
Eq. (10.214) can be reduced to: n n X n X i½Si X i½Si ½Ij n n n αð0, iÞ βð0, iÞ i + αð j, iÞ βð j, iÞ j i i KS i¼1 j¼1 j KSi KI i¼1 ( ) rP ¼ k c E n n n X n X X ½Si ½Ii ½Si ½Ij n n n αð0, iÞ i + αði, 0Þ i + 1+ αð j, iÞ j j i i KS KI K i KI i¼1 i¼1 j¼1 S
I
(10.217)
614
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
or: n n X n X i½Si X i½Si ½Ij n n n αð0, iÞ βð0, iÞ i + αð j, iÞ βð j, iÞ j i i KS i¼1 j¼1 j KSi KI i¼1 rmax ( ) rP ¼ n n X n n X X ½Si ½Ii ½Si ½Ij n n n αð0, iÞ i + αði, 0Þ i + 1+ αð j, iÞ j j i i KS KI Ki KI i¼1 i¼1 j¼1
(10.218)
S
where: rmax ¼ nkc En
(10.219)
Solutions of Eqs. (10.212a) through (10.212c) yield αð j, 0Þ ¼ αI0
j1
(10.220a)
i1 αð0, iÞ ¼ αS0
(10.220b)
j1
j1
j1
i1 i1 i1 αð j, iÞ ¼ AαSI αIS + BαS0 αIS + CαSI αI0
8i, j > 0
(10.220c)
With B¼
αSI ðαIS + αSI Þ 2αS0 αIS 6ðαSI αS0 Þ
(10.221a)
C¼
αIS ðαIS + αSI Þ 2αI0 αSI 6ðαIS αI0 Þ
(10.221b)
αIS + αSI BC 2
(10.221c)
βð0, iÞ ¼ β0i1
(10.222a)
A¼ we further let
j
βð j, iÞ ¼ βI β1 β2i1 8i, j > 0
(10.222b)
Substituting Eqs. (10.220a) through (10.222b) into (10.218), one can obtain, rP ¼ rmax where
½Sf1 KS f0
½S n ½I n En 1 1 1 1 ¼ 1 αS0 αI0 + αS0 1 + αS0 + αI0 1 + αI0 f0 ¼ KS ½ En I n K
½S ½I n A + 1 + αSI 1 1 + αIS 1 KS KI αSI αIS
n n B ½S ½I + 1 + αS0 1 1 + αIS 1 αS0 αIS KS KI
C ½S n ½I n + 1 + αSI 1 1 + αI0 1 αSI αI0 KS KI
(10.223)
(10.224a)
615
10.6 INFLUENCE OF A COMPETITIVE EFFECTOR ON INTERACTIVE ENZYMES
0.025 2.5 × 10−4 M 0.020
r, OD291/min
1.0 × 10−4 M 0.015 5.0 × 10−5 M 0.010
2.5 × 10−5 M
FIG. 10.36 Catalytic activity of CTP synthetase as affected by an ignitor: GTP, for the conversion of glutamine. The symbols are data taken from Levitzki, A., Koshland, Jr., D.E., 1969. Negative cooperativity in regulatory enzymes. Proc. Natl. Acad. Sci. U.S.A. 62 (20), 1121–1128. The lines are based on Eq. (10.223) with parameters given below
[GTP] 0.005
0.000 10−2
10−1
100
101
102
[Glutamine], mM
rmax, OD291/ min
KS, mM
KI, mM
αS0
αI0
αSI
αIS
β0
βI
β1
β2
n
14.19
0.0888
29.59
0.1143
141.68
0.22159
163.73
0
0.3605
3.0228
1.048
4
½S n1 AβI ½S n1 ½I n + 1 + αSI β2 1 + αIS β1 1 f1 ¼ 1 + αS0 β0 αIS KS KS KI n1 n
Bβ ½S ½I 1 + αIS β1 1 + I 1 + αS0 β2 αIS KS KI n1 n
Cβ ½S ½I + I 1 + αSI β2 1 + αI0 β1 1 αIS KS KI
(10.224b)
For catalytic enzyme applications, allosteric effector binding can greatly enhance the catalytic activity as the number of catalytic sites is unaffected. Fig. 10.36 shows such a case, where the catalytic rate is increased by the addition of GTP for the conversion of glutamine to CTP by CTP synthetase. The CTP synthetase here is of a tetramer with each monomer unit containing both GTP and Glutamine (different) sites. It is understandable that the great increase of catalytic activity is found for an enzyme with allosteric sites for effector(s). The binding of effector on different (other than substratebinding) sites also made the kinetic equation easier to apply.
10.6 INFLUENCE OF A COMPETITIVE EFFECTOR ON INTERACTIVE ENZYMES We have discussed the competitive binding of ligands on interactive enzymes in Section 10.3. The theory on competitive binding can be applied to examine the influence of a competitive effector on interactive enzymes.
616
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
When competitive binding is examined for interactive enzymes, one may think of homosteric enzymes. As discussed in Section 10.3, competitive binding can occur in both homosteric and allosteric enzymes. In this section, we shall use the example of glucokinase, which is an allosteric enzyme. Glucokinase has two allosteric sites on each molecule and they both can bind glucose and fructose. This is a simple situation of the class of problems we have discussed in Section 10.5.3. Fig. 10.37 shows a schematic of the competitive binding of substrate S and effector I on a two-sited allosteric enzyme. The sequential or preferential competitive effector binding on a site-sequential allosteric enzyme for a two allosteric site enzyme can represented stoichiometrically by: K1S
kc1
K2S
kc2
E + S ! ES ! E + P
(10.225a)
ES + S ! ES2 ! ES + P
(10.225a)
K1I
E + I ! EI
(10.225c)
K2I
EI + I ! EI2 α1 K2I S
kI1
α1 K2S I
kI2
(10.225d)
ES + I ! ESI ! EI + P
(10.225e)
EI + S ! EIS ! EI + P
(10.225f)
The fast equilibrium steps of binding, as shown in reactions (10.225a) through (10.225f), can be written as ½E½S K1S
(10.226a)
½ES½S ½E½S2 ¼ K1S K2S K2S
(10.226b)
½ES ¼ ½ES2 ¼
aSK2S K1S
S
S
aSK2I
S
I
aIK2S
I
S
I
I
S
K1I I
aIK2I
FIG. 10.37
A schematic of competitive sequential binding of S and I on a two-sited allosteric enzyme.
10.6 INFLUENCE OF A COMPETITIVE EFFECTOR ON INTERACTIVE ENZYMES
617
½E½I K1I
(10.226c)
½EI½I ½E½I2 ¼ K1I K2I K2I
(10.226d)
½EIS ¼ αI
½EI½S ½E½I½S ¼ αI K2S K1I K2S
(10.226e)
½ESI ¼ αS
½ES½I ½E½S½I ¼ αS K2I K2I K1S
(10.226f)
½EI ¼ ½EI2 ¼
The enzyme balance leads to: E ¼ ½E + ½ES + ½ES2 + ½EI + ½EI2 + ½EIS + ½ESI
(10.227)
Substituting Eqs. (10.226a) through (10.226f) into (10.227), we obtain: ½ E ¼
E
½S ½S ½I ½I2 αS αI + + + + ½I½S + 1+ K1S K1S K2S K1I K1I K2I K1S K2I K2S K1I 2
(10.228)
The catalytic rate is determined by the rate-limiting steps: rP ¼ kc1 ð½ES + ½ES2 Þ + kc2 ½ES2 + kI1 ½ESI + kI2 ½EIS
(10.229)
which can be reduced to:
kc1 kc1 + kc2 αS kI1 αI kI2 + ½S + ½I + K1S K1S K2S K1S K2I K2S K1I rP ¼ E½S ½S ½S2 ½I ½I2 αS αI + + + + ½I½S + 1+ K1S K2I K2S K1I K1S K1S K2S K1I K1I K2I
(10.230)
When the substrate concentration is fixed while measuring the catalytic rate variation with the competitive effector concentration, one can reduce Eq. (10.230) to: rP ¼ a 0
1 + a1 ½I 1 + b1 ½I + b2 ½I2
(10.231)
with the four constants a0, a1, b1, and b2 defined by: kc1 kc1 + kc2 + ½S K K1S K2S a0 ¼ E½S 1S ½S ½S2 + 1+ K1S K1S K2S
(10.232a)
αS kI1 αI kI2 + K K K2S K1I a1 ¼ 1S 2I kc1 kc1 + kc2 + ½S K1S K1S K2S
(10.232b)
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
FIG. 10.38 Variation of Fructose phosphorylation rate on human hexokinase IV with glucose concentration. While hexokinase IV phosphorylates glucose as well, the data shown is for fructose phosphorylation. The symbols represent data taken from Moukil, M.A., van Schaftingen, E., 2001. J. Biol. Chem. 276(6), 3872– 3878, while the line is drawn based on Eq. (10.231).
50
40 rF6P, mmol/min/g
618
30
20
10
0 0
20
40
60
80
100
[Glucose], mM
1 αS αI + ½S + K1S K2I K2S K1I K b1 ¼ 1I ½S ½S2 + 1+ K1S K1S K2S 1
b2 ¼ K1I K2I
½S ½S2 1+ + K1S K1S K2S
!
(10.232c)
(10.232d)
The kinetic nature of the competitive activation can be demonstrated with human hexokinase IV. Fructose is a poor substrate due to having much lower affinity. Human hexokinase IV is known to be monomeric, but has two active heterosteric sites (allosteric) that are accessible to both fructose and glucose. The addition of low concentrations of glucose produces an almost five-fold increase in the enzyme’s activity with fructose (Fig. 10.38). The line drawn in Fig. 10.38 is based on Eq. (10.231) with a0 ¼ 10.27 mmol-F6P/min/g, a1 ¼ 1.587 mM1, b1 ¼ 0.1172 mM1, b2 ¼ 0.01466 mM2. There is a maximum fructose conversion rate, while the activation of the enzyme catalyzed fructose reaction is less at low and high glucose concentrations.
10.7 SUMMARY An enzyme having more than one active center is called an interactive enzyme, or dueoweimei (多位酶), a multisited enzyme. Enzymes or proteins are commonly present in oligomeric forms for active biological functions. The formation of protein oligomers, either in nature or by artificial means, can take place via covalent bonding or through weak bond network associations. Disulfide bonds are more common in forming covalent protein
619
10.7 SUMMARY
oligomers. Covalent bound protein oligomers usually have additional elements introduced, while proteins associated through weak bond networks usually do not involve any additional bound chemical groups. Proteins are commonly present in stable forms or folds. Oligomerization can occur when stable proteins unfold, creating exposed interfaces for association interactions. One well-known process of weak bond network oligomerization is DS. Separating the two touching/interacting domains creates opportunities for similar interactions with different protein molecules, leading to oligomerization. Both disulfide bonding and DS oligomerization can be dynamic (or reversible), at least at low DPs. The occurrence of oligomerization is usually in response to environmental changes. The dynamic nature of the protein oligomerization is important for bioactivity control. The morpheein model and the polymorph model are thus important in quantifying enzyme-catalyzed biotransformations. Disulfide bonding and DS can act together to form supramolecules. The formation of supramolecules, such as amyloids, in nature can be benign or harmful. The oligomerization of proteins or enzymes allow for multiple active centers or binding sites to be available on one enzyme/protein complex molecule. The multiple active centers provide what is necessary for the regulation of molecule flow or interactions with ligands. Oligomeric enzymes are common interactive enzymes, although interactive enzymes can be monomeric. Oligomeric enzymes provide homosteric active centers (or binding sites). Interactive enzymes can also be allosteric, in that multiple active centers have distinctly different ligand banding affinities and/or catalytic activities. In general, ligand binding on an interactive enzyme results in the interactive enzyme to undergo conformal structural changes. Therefore, even a homosteric interactive enzyme can exhibit variable affinity and/or catalytic activity among different active centers. Treaties have been developed to quantitatively describe the behaviors of interactive enzymes with single and multiple ligands. The complex structure of an interactive enzyme provides opportunity for an effector (or cofactor) or an inhibitor to regulate the biotransformations through either competitive or allosteric regulations. Mechanistic kinetic model expressions have been developed for the complex interactions. The general binding single ligand saturation is given by: ½Si KSi ½ En S i ¼ E n n X ½Sk 1+ ank k KS k¼1 ani
(10.46)
n X iani ½Si
θnS ¼
1 n
i¼1
1+
KSi
n X ani ½Si i¼1
(10.47)
KSi
where En is the total concentration of an interactive enzyme having n-active centers, [EnSi] is the concentration of the interactive enzyme bound with i-substrate S molecules, ani is the effective affinity ratio (or cooperativity) of the ith order, and KS is the characteristic affinity of the substrate on the interactive enzyme.
620
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
When a simple reaction: En
S ! P is catalyzed by an interactive enzyme, the catalytic reaction rate is given by: n X
rPn ¼ En
kci ani ½Si
i¼1
1+
n X
(10.172) ani ½Si
i¼1
For multiple ligands acting on an interactive enzyme, the binding behavior is more complex due to the interactions. For the competitive binding of two species S and P on an n-site homosteric enzyme, the competitive binding is governed by E n S i P j ¼ αð i , j Þ
n! ½Si ½Pj ½En i j i! j! ðn i jÞ! KS K
(10.108)
P
in the case of constant shape change factor, n X ni En X n! ½Si ½Pj ¼ αði, jÞ ½ En i j ½En i¼0 j¼0 i! j! ðn i jÞ! KS KP n ½S ½P ½S ½P ¼ 1 + αPS + αSP + nð1 αS0 Þ + nð1 αP0 Þ KS KP KS KP n n n ½S ½S ½P ½P n + 1 + αS0 1 + αPS + 1 + αP0 1 + αSP KS KS KP KP n X ni Bn S 1X n! ½Si ½Pj ¼ i αði, jÞ ½ En i j n½En n i¼1 j¼0 i! j! ðn i jÞ! KS KP n1 ½S ½S ½P ½S ¼ αPS 1 + αPS + αSP + ð1 αS0 Þ KS KS KP KS ½S ½S n1 ½S ½S n1 + αS0 1 + αS0 αPS 1 + αPS KS KS KS KS
(10.111)
(10.112)
And when all the interactive coefficients are unity or for noninteracting homosteric enzyme, the binding saturation is reduced to Bn Sj ½S ½S ½P n1 ¼ 1+ + (10.114) θnj ¼ nEn KS KS KP For a three-sited homosteric enzyme, the binding of species j in a total of NS ligand species is governed by NS NS NS NS NS X Sj X S j XX E3 ½Si X ½Si ½Sk f0 ¼ ¼1+3 ðαið jÞ + αjðiÞ Þ + αði, j, kÞ ½ E3 Ki j¼1 Kj i¼1 k¼1 Ki Kk Kj i¼1 j¼1 i6¼j
i6¼j k6¼j
621
10.7 SUMMARY
2 N S h 2 NS NS i ½S X X Sj X Sj i + αjði, jÞ ðαið jÞ + αjðiÞ Þ + 2αið j, jÞ αjð jÞ + αjð jÞ 2 2 K 2K Kj i j i¼1 j¼1 j¼1 i6 ¼ j 3 N S X Sj + αjð j, jÞ αjð jÞ 3 K
(10.93)
j
j¼1
NS NS X NS X 1 Kj B3 Sj ½Si 1 X ½Si ½Sk ¼1+ ðαið jÞ + αjðiÞ Þ + αði, j, kÞ f1 ¼ 3 Sj ½E3 K 3 Ki Kk i i¼1 i¼1 k¼1 i6¼j
i6¼j k6¼j
NS h 2 i ½S Sj X Sj Sj i + α ðα + αjðiÞ Þ + 2αið j, jÞ αjð jÞ + 2αjð jÞ + αjð j, jÞ αjð jÞ 2 (10.92) Ki 2Kj i¼1 jði, jÞ ið jÞ Kj Kj i6¼j
1 B 3 Sj f1 Sj θ3j ¼ ¼ f0 Kj 3 E3 For a three-sited sequential allosteric enzyme: ! ! NS NS NS NS NS NS X X E3 ½Si ½Si1 X ½Si2 X ½Si2 X ½Si1 X ½Si3 ¼1+ + αi1 + αi1, i2 ½ E3 K K1i1 i2¼1 K2i2 i2¼1 K2i2 i1¼1 K1i1 i3¼1 K3i3 i¼1 1i i1¼1
! NS NS NS X B3 Sj Sj ½Si X ½Si2 X ½Si1 ¼ 1 + αj + αi1 ½E3 K1j K K K1i1 i¼1 2i i2¼1 2i2 i1¼1 ! ! NS NS NS NS NS Sj X ½Si X Sj X ½Si1 X ½Si2 ½S2i2 X ½Si1 + αi + α α + K2j i¼1 K1i i1¼1 i1j K1i1 i2¼1 K3i2 K3j i2¼1 K2i2 i1¼1 i1i2 K1i1
(10.94)
(10.125)
(10.127)
and for a three-site random accessible allosteric enzyme: ! NS X NS NS NS NS 3 3 X X Sj E3 1X ½Si1 X ½Si2 X ½Si1 X ½Si2 ¼1 + + α + α ½ E3 2! n¼1 i1¼1 Kn0i1 i2¼1 i2=n Kni2 i1¼1 Kni1 i2¼1 i2=n0 Kn0i2 K j¼1 n¼1 nj NS NS NS 1X ½Si1 X ½Si2 X ½Si3 α + 3! i1¼1 K1i1 i2¼1 K2i2 i3¼1 ði3=1, i2=2, i1=3Þ::2 K3i3
0 1 N N N 3 S S S X X αj=n + αi=n1 X X αðj=n, i1=n0, i2=0nÞ::2 C B3 S j Sj B B1 + ¼ ½Si + ½Si1 ½Si2 C @ A ½E3 n¼1 Knj Kn1i Kn0i1 K0ni2 n1¼1 i¼1 i2¼1 i1¼1 3 X
(10.143)
(10.142)
n16¼n
αða, b, cÞ::2 ¼ αa, b + αa, c + αb,a + αb,c + αc,a + αc,b
(10.141)
622
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Interactive enzymes can exist as a mixture of multiple oligomers that can interconvert. This type of interactive enzymes is called a polymorph. Each oligomer is a member of polymorph (or a polymorph). When there are only two members in a polymorph, the polymorph is called a bimorph. For example, morpheein model deals with a bimorph of constant (invariable) affinity and catalytic activity homosteric enzymes that can interconvert without any effect from the substrate binding. MWC model assumes a bimorph of equal number of homosteric units with invariable affinity and catalytic activity so that interbimorph conversion can occur only when substrate is not bound. When the substrate binding stabilizes the polymorph members, the interconversion between polymorphs can only occur if substrate(s) is no bound. For polymorph of substrate-free interconversion, the substrate-free polymorph concentration is given by: h i n (10.176) j En ¼ KjEn pe where jEn is the jth polymorph member with n substrate subunits, KjEn is the equilibrium constant between the singular polymorph member and the jth polymorph member with n subunits, and pe is polymorph enzyme fraction parameter analogic to the concentration of protomer. The enzyme saturation is given by: jn N n Y i1 KjEn X BS X pne X i½Si n αjnl i1 (10.180) θS ¼ ¼ E n¼1 E j¼1 Kjn i¼1 i l¼1 Kjn and the overall catalytic rate is given by: ( ) Y jn i1 N n i1 X X X K k 1 ½ S n jEn jn npne 1+ iβ α rP ¼ ½S Kjn n i¼2 jni1 i l¼1 jnl Kni1 n¼1 j¼1
(10.182)
which is the general overall catalytic rate expression for a substrate-free interconvertible polymorph. When the interconversion amongst the polymorph members is not affected by the substrate, the total concentration of a polymorph member is given by: j En
¼ KjEn pne
(10.191)
The enzyme saturation ratio is given by:
θS ¼
jn N X X
BS 1 ¼ E E n¼1
j¼1
KjEn pne Kjn +
n Y i1 X n
i¼1 n X
i
l¼1
i1 n Y i
i¼1
αjnl
i½Si i1 Kjn
½Si αjnl i1 Kjn l¼1
(10.193)
And the overall catalytic reaction rate is:
rP ¼ ½S
N X n¼1
pne
jn X j¼1
n+ KjEn kjn
n Y i1 X n
i
αjnl
iβjni1 ½Si1
i1 Kjn n i1 X ½Si1 n Y Kjn + ½S αjnl i1 i Kjn i¼2
k¼1
l¼1
l¼1
(10.194)
10.7 SUMMARY
623
The effect of an effector (or cofactor), either competitive or allosteric, can be derived based on the same principles. For an interactive enzyme with n-pair of active sites (n-homosteric sites for substrate S and another n-homosteric sites for effector I), the rate expression is given by: n n X n X i½Si X i½Si ½Ij n n n αð0, iÞ βð0, iÞ i + αð j, iÞ βð j, iÞ j i i KS i¼1 j¼1 j KSi KI i¼1 rmax ( ) rP ¼ (10.223) n n X n X X n ½Si ½Ii ½Si ½Ij n n n αð0, iÞ i + αði, 0Þ i + 1+ αð j, iÞ j j i i KS KI KSi KI i¼1 i¼1 j¼1
Bibliography Adair, G.S., 1925. The hemoglobin system. VI. The oxygen dissociation curve of hemoglobin. J. Biol. Chem. 63, 529–545. Bohr, C., 1904. Die Sauerstoffaufnahme des genuinen Blutfarbstoffes und des aus dem Blute dargestellten Ha¨moglobins. Zentralbl. Physiol. 23, 688–690. Bohr, C., Hasselbalch, K., Krogh, A., 1904. Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensa¨urespannung des Blutes auf dessen Sauerstoffbindung u¨bt 1. Skandinavisches Archiv fu¨r Physiologie 16, 402–412. Chen, T., et al., 2014. Sequence analysis and molecular characterization of clonorchis sinensis hexokinase, an unusual trimeric 50-kDa glucose-6-phosphate-sensitive allosteric enzyme. PLoS One 9 (9), e107940. Edelstein, S.J., Schaad, O., Henry, E., Bertrand, D., et al., 1996. A kinetic mechanism for nicotinic acetylcholine receptors based on multiple allosteric transitions. Biol. Cybern. 75, 361–379. Fersht, A., 1999. Structure and Mechanism in Protein Science: A Guide to Enzyme Catalysis and Protein Folding. W.H. Freeman, San Francisco. Hill, A.V., 1910. The possible effects of the aggregation of the molecules of hæmoglobin on its dissociation curve. J. Physiol. 40, iv–vii. Jaffe, E.K., 2005. Morpheeins—a new structural paradigm for allosteric regulation. Trends Biochem. Sci. 30, 490–497. Klotz, I.M., 2003. Ligand-receptor complexes: origin and development of the concept. J. Biol. Chem. 279 (1), 1–12. Klotz, I.M., Walker, F.M., Pivan, R.B., 1946. The binding of organic ions by proteins. J. Am. Chem. Soc. 68, 1486–1490. Koshland Jr., D.E., 1958. Applications of a theory of enzyme specificity to protein synthesis. Proc. Natl. Acad. Sci. U. S. A. 44, 98–104. Koshland Jr., D.E., Ne´methy, G., Filmer, D., 1966. Comparison of experimental binding data and theoretical models in proteins containing subunits. Biochemistry 5, 365–385. Lawrence, S.H., Jaffe, E.K., 2008. Expanding the concepts in protein structure-function relationships and enzyme kinetics: teaching using morpheein. Biochem. Mol. Biol. Educ. 36 (20), 274–283. Mandal, M., Lee, M., Barrick, J.E., Weinberg, Z., Emilsson, G.M., Ruzzo, W.L., Breaker, R.R., 2004. A glycinedependent riboswitch that uses cooperative binding to control gene expression. Science 306, 275–279. Marianayagam, N.J., Sunde, M., Matthews, J.M., 2004. The power of two: protein dimerization in biology. Trends Biochem. Sci. 29, 618–625. Mello, B.A., Tu, Y., 2005. An allosteric model for heterogeneous receptor complexes: understanding bacterial chemotaxis responses to multiple stimuli. Proc. Natl. Acad. Sci. U. S. A. 102, 17354–17359. Monod, J., Changeux, J.P., Jacob, F., 1963. Allosteric proteins and cellular control systems. J. Mol. Biol. 6, 306–329. Monod, J., Wyman, J., Changeux, J.P., 1965. On the nature of allosteric transitions: a plausible model. J. Mol. Biol. 12, 88–118. Najdi, T.S., Yang, C.R., Shapiro, B.E., Hatfield, G.W., Mjolsness, E.D., 2006. Application of a generalized MWC model for the mathematical simulation of metabolic pathways regulated by allosteric enzymes. J. Bioinf. Comput. Biol. 4 (2), 335–355.
624
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Pauling, L., 1935. The oxygen equilibrium of hemoglobin and its structural interpretation. Proc. Natl. Acad. Sci. U. S. A. 21, 186–191. Selwood, T., Jaffe, E.K., 2012. Dynamic dissociating homo-oligomers and the control of protein function. Arch. Biochem. Biophys. 519, 131–143. Selwood, T., Tang, L., Lawrence, S.H., Anokhina, Y., Jaffe, E.K., 2008. Kinetics and thermodynamics of the interchange of the morpheein forms of human porphobilinogen synthase. Biochemistry 47, 3245–3257. Stefan, M.I., Edelstein, S.J., Nove`re, N.L., 2009. Computing phenomenologic Adair-Klotz constants from microscopic MWC parameters. BMC Syst. Biol. 3, 68.
PROBLEMS 10.1. What is a covalent protein oligomer? List the examples of covalent bonding for protein oligomers. 10.2. Why do one study protein oligomerization? How could protein or enzyme oligomerize if no covalent bonds form? 10.3. What is a PLP enzyme? 10.4. What is domain-swapping oligomerization? 10.5. What is a polymorphic enzyme? 10.6. Experimental data of Bohr (1904) for oxygen binding on normal dog blood at 37.8°C is given in Table P10.6. TABLE P10.6
Oxygen Binding of Normal Dog Hemoglobin at 37.8°C as a Function of Partial Pressures of CO2 and O2 Observed
No
pCO 2 (Torr)
pO 2 (Torr)
O2 Uptake
VI.1
2.3
7.2
35.5
VI.2
8.2
6.7
13.1
48.8
6.8
3.4
IV.1
5.7
12.2
37.2
V.3
10.6
12.2
25.9
V.4
28.3
12
16.6
V.2
54.3
12.1
11.8
IV.2
4.6
25.9
85.8
V.6
8.1
26.7
77.2
I.2
8.6
25.4
68.4
II.2
12.9
25.7
63.2
III.2
12.9
26
57.4
V.7
17
31.7
74.6
VIII.1
625
PROBLEMS
TABLE P10.6
Oxygen Binding of Normal Dog Hemoglobin at 37.8°C as a Function of Partial Pressures of CO2 and O2—cont’d Observed
No
pCO 2 (Torr)
pO 2 (Torr)
O2 Uptake
IV.3
89.2
33.2
35
VI.3
3
55.6
94.9
VI.4
25
55.1
87.1
VI.5
88.4
63.9
72.9
IV.4
0–6
150
VII.1
75.2
152.9
99.6
151.3
155.8
96.2
100
Determine the binary oxygen-carbon dioxide-binding isotherm. 10.7. A “Noninteracting” Complex Enzyme. An enzyme E contains two allosteric nonintereacting sites. The two sites behave independently except when a competing allosteric ligand is bound on the other site. When a competitive allosteric ligand is bound on any of the two sites, it causes the enzyme to undergo conformal structural changes and thus change the binding affinity, as well as the catalytic reactivity towards the competitive substrate. Let the reaction: S!P be catalyzed by such an enzyme. (a) Derive a relationship for the overall enzyme-binding saturation as a function of the substrate concentration, if the substrate is the only molecule that can interact with the enzyme. Assume that the two sites have different affinities. (b) Derive a relationship for the overall enzyme binding saturation by the substrate, if an allosteric ligand I can interact with the enzyme and causing conformal changes to the enzyme. Assume that the two sites have different affinities. (c) If the catalytic reaction can occur, following part b), determine the catalytic reaction rate as a function of substrate and ligand concentrations. 10.8. A Growing Polymorph. An enzyme E is employed to convert substrate S to P. The overall stoichiometry is represented by S!P The enzyme E exists in a monomeric form when S is absent. However, when substrate S is bound on the enzyme, it starts to oligomerize. When one substrate is bound on an enzyme molecule, it can grow to a dimer by adding another enzyme molecule. The binding of a substrate molecule on to the second site of the dimer has a higher affinity than on a monomer. When both sites of the dimer are occupied, the dimer
626
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
can grow to a trimer. The saturated trimer can also grow to a tetramer. The affinities of the third and fourth sites are equal. Do the following: (a) Derive a relationship for the overall enzyme binding saturation as a function of the substrate concentration. (b) If the catalytic reaction can occur only with tetramers, determine the catalytic reaction rate as a function of substrate concentration, assuming the concentration substrate is much higher than the concentration of the total enzyme mixture and the binding on substrate on the enzyme oligomers is much faster than the catalytic reaction itself. Product P does not bind on the enzyme at all. (c) Redo part (b), if the catalytic reaction can only occur with the dimers. (d) Compare your results of (b) and (c). Describe the differences in the catalytic behavior. 10.9. A Multiplying Polymorph. An enzyme E exists in a monomeric form when no ligand is present. When the enzyme is employed to catalyze the reaction: S!P The substrate is first bound to the monomeric enzyme E. There are two reactions can occur to a substrate-bound enzyme: (1) Catalytic reaction occurs and product P is not bound on the enzyme; (2) It can form a dimer by reacting with another substratebound monomer. A substrate-saturated dimer can double to a substrate-saturated tetramer. Do the following: (a) Derive a relationship for the overall enzyme binding saturation as a function of the substrate concentration. (b) If the catalytic reaction can occur only with the monomers, determine the catalytic reaction rate as a function of substrate concentration, assuming the concentration substrate is much higher than the concentration of the total enzyme mixture. (c) Discuss the catalytic rate as obtained in part (b). 10.10. Human glucokinase is known to be a monomeric allosteric protein containing two glucose-binding sites. Table P10.10 shows a set of experimental data on the initial rate of glucose phosphorylation (G6P). TABLE P10.10
The Initial Rate of Glucose Phosphorylation as a Function of Glucose Concentration
[G], mM
0
0.78
1.56
2.92
4.86
7.39
14.98
29.77
49.81
r0, mmol/min/g
0
0.24
0.48
1.28
2.57
4.25
6.74
8.34
8.98
Data from Kamata, K., et al. 2004. Structure, 12(3), 429–438.
Derive an appropriate kinetic model to describe the data. 10.11. Human glucokinase is known to be a monomeric allosteric protein containing two hexose binding sites. Table P10.11 shows a set of experimental data on the initial rate of G6P and fructose phosphorylation (F6P).
627
PROBLEMS
TABLE P10.11
The Initial Rates (mmol/min/g) of Glucose and Fructose Phosphorylation as Functions of Glucose and Fructose Concentrations
[F], mM
0
50
[G], mM
rG6P0
rG6P0
100 rF6P0
rG6P0
0
0
0
4
0
0.25
0.53
0.53
5.54
0.53
0.5
0.96
0.96
6.77
1
1.82
1.82
2.5
6.42 18.50
5
200 rF6P0
rG6P0
9.85
rF6P0
0
25.85
13.85
0.53
32.00
0.96
17.54
0.96
37.54
9.85
1.82
22.15
1.82
44.00
6.42
15.38
5.78
32.62
5.45
55.38
16.68
22.15
14.87
40.92
12.19
65.23
Data from Moukil, M.A., Van Schaftingen, E., 2001. J. Biol. Chem. 276(6), 3872–3878.
Derive an appropriate kinetic model to describe the data. If you were to do experiments again, what would you do it differently to elucidate the kinetic model? 10.12. ATP is a known inhibitor for the synthesis of glycogen by glycogen phosphorylase. Table P10.12 shows a set of experimental data on the initial rate of glycogen synthesis.
TABLE P10.12
[ATP] 5 0 [Pi] (mM)
r0/ rmax
The Initial Rate or Catalytic Activity of Rabbit Muscle Glycogen Phosphorylase as a Function of Phosphate and ATP Concentrations. AMP Concentration is Held the same, and the Phosphorylase Concentration is Fixed at 28 μM 3.2 mM [Pi] (mM)
r0/ rmax
6.4 mM [Pi] (mM)
r0/ rmax
10.2 mM [Pi] (mM)
r0/ rmax
15 mM [Pi] (mM)
r0/ rmax
0.102
0.032
0.784
0.069
0.832
0.026
0.866
0.007
1.350
0.007
0.201
0.065
1.117
0.110
1.157
0.035
1.222
0.013
2.073
0.015
0.301
0.100
1.785
0.208
1.848
0.069
1.904
0.024
3.380
0.034
0.414
0.140
3.120
0.311
3.199
0.142
3.247
0.054
4.987
0.066
0.608
0.209
4.697
0.437
4.744
0.240
4.768
0.105
7.213
0.122
0.731
0.198
6.828
0.474
6.862
0.348
6.965
0.174
12.360
0.261
0.776
0.255
12.237
0.590
12.299
0.536
12.299
0.370
16.261
0.376
1.047
0.284
16.180
0.678
16.261
0.594
16.672
0.464
23.638
0.484
1.106
0.341
23.171
0.734
23.520
0.667
23.756
0.584
33.514
0.514 Continued
628
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
TABLE P10.12
[ATP] 5 0
The Initial Rate or Catalytic Activity of Rabbit Muscle Glycogen Phosphorylase as a Function of Phosphate and ATP Concentrations. AMP Concentration is Held the same, and the Phosphorylase Concentration is Fixed at 28 μM—cont’d 3.2 mM
6.4 mM
10.2 mM
r0/ rmax
[Pi] (mM)
r0/ rmax
[Pi] (mM)
r0/ rmax
[Pi] (mM)
r0/ rmax
1.750
0.402
32.689
0.803
32.852
0.714
33.182
0.660
1.776
0.447
3.089
0.600
3.089
0.545
4.105
0.653
5.455
0.718
6.794
0.708
6.828
0.759
8.089
0.797
33.016
0.848
33.514
0.875
[Pi] (mM)
15 mM [Pi] (mM)
r0/ rmax
Data from Madsen, N.B., Shechosky, S., 1967. J. Biol. Chem. 242(14), 3301-3307.
Rabbit muscle phosphorylase is known to be a dimer, and AMP and ATP are bound on allosteric sites. Derive a suitable kinetic model and evaluate the parameters.
10 Interactive Enzyme and Molecular Regulation O U T L I N E 10.1 Protein Oligomerization and Interactive Enzyme 538 10.1.1 Covalently Bound Oligomers 539 10.1.2 Noncovalent Association Oligomerization 544 10.1.3 Domain Swapping Oligomerization 552 10.1.4 Interactive Enzyme Oligomer Mixture Model 557 10.2 Ligand Binding and Cooperativity 565 10.2.1 Single Ligand Binding on Homosteric Enzymes 569 10.2.2 Sequential Single Ligand Binding on Allosteric Enzymes 575 10.2.3 Single-Ligand Binding on Random-Access Allosteric Enzymes 579 10.3 Competitive Multiligand Binding on an Interactive Enzyme 581 10.3.1 Competitive Ligand Binding on a Homosteric Enzyme 581 10.3.2 Site-Sequential Multiligand Binding on Allosteric Enzymes 590 10.3.3 Multiligand Binding on Random-Access Allosteric Enzymes 593
Bioprocess Engineering http://dx.doi.org/10.1016/B978-0-444-63783-3.00010-1
10.3.4 Enzymes With Homosteric Paired Allosteric Sites 10.4 Catalytic Reaction Rate on Interactive Enzymes 10.4.1 Catalytic Reactions on Homosteric Sites 10.4.2 Catalytic Reactions on Site-Sequential Allosteric Sites 10.4.3 Random-Access Allosteric Enzymes 10.5 Kinetics of Polymorphic Catalysis and Allosteric Modulation 10.5.1 Substrate-Free Polymorph Interconversion 10.5.2 Substrate-Inert Polymorph Interconversion 10.5.3 Oligomers of Paired Allosteric Sites
597 598 598
601 603
604 604 607 612
10.6 Influence of a Competitive Effector on Interactive Enzymes
615
10.7 Summary
618
Bibliography
623
Problems
624
535
# 2017 Elsevier B.V. All rights reserved.
536
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
What is an interactive enzyme? How is an interactive enzyme formed? How does an interactive enzyme regulate biotransformations? All chemical reactions necessary for life are inadequately slow and thermodynamically complex to selectively occur spontaneously. One or more unique enzyme catalysts are required to direct and accelerate the reaction. Almost all enzymes are proteins that fold into domains. The majority of enzymes contain one domain in each molecule. In particular, a lot of enzymes contain one active center only that can interact with a specific type of ligand. This is what we call a simple enzyme in Chapter 7. Still many are composed of two or more domains allowing complex interactions between the enzyme and the molecules with which it can interact (for example, multifunctional proteins). An enzyme molecule contain more than one active centers is called an interactive enzyme, or duoweimei (多位酶) or multiple-sited enzyme. Mutual (two-way) interactions can only occur when multiple active centers are available. Most enzymes are designed to function at a constant rate, but interactive enzymes are sensitive to physiological controls, and thereby they adjust their rate and determine the flux through the metabolic pathway that they control. Most simple enzymes abide by the Michaelis-Menten kinetics quantitatively where the highest rate is the saturation rate that can be achieved (or closely approached) at relatively low substrate concentrations as we have learned in Chapter 7. As a result, a relatively constant rate can be delivered by the enzymes in normal conditions to the living organism they support. The critical concentration at which an enzyme approaches saturation rate (or half the saturation rate) is directly related to the affinity of the substrate to the enzyme, and the saturation rate is proportional to the turnaround frequency of the enzyme. The enzyme is regulated by changing the conformal structures (through interactions). Multiple sites or apparent multiple sites can appear because of the “aggregation” or conformal structure of multiple simple enzymes together to form an enzyme complex that becomes interactive. The conformal structural change of a complex enzyme induces the change of its affinity for a substrate and/or the enzyme turnaround frequency. This constitutes an interactive enzyme. In Chapter 7, we have learned the functions and fundamental approaches to the kinetics of enzyme-catalyzed reactions. All enzymes are remarkable for their ability to bind one or more substrates with appropriate specificity, and then facilitate a particular type of chemical reaction, producing one or more new products that are essential for the function of a living cell. For simple enzymes, Michaelis-Menten model can describe the kinetic behavior quite well even though mechanistically not an ideal model as compared with pseudosteady state approximations to the kinetic steps involved. A brief summary of the Michaelis-Menten kinetics is reviewed below that will set the stage for our discussions in this chapter. Michaelis-Menten model can be mechanistically presented as: Km
kc
E + S ! ES ! E + P
(10.1)
where E stands for the enzyme molecule, S stands for the substrate, and P stands for the product. There are two reaction steps represented by Eq. (10.1), the first step being the fast equilibrium step of enzyme (E) binding with the substrate (S) to produce the enzyme-substrate complex: Km ¼
½E½S ½ES
(10.2)
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
537
In Eq. (10.2), [E] is the concentration of the enzyme that is not bound with the substrate S, and [S] is the concentration of the substrate (that is not bound with the enzyme), [ES] is the concentration of enzyme-substrate complex, and Km is the Michaelis-Menten saturation constant. One can observe from Eqs. (10.1) and (10.2) that Km is the inverse of the equilibrium constant of binding, and thus the inverse of affinity of substrate to enzyme. In an enzyme reactive system, the concentration of the enzyme (or more precisely the enzyme loading) is much lower than the concentration of the substrate. Therefore, the effect of the concentration of the substrate by the enzyme-substrate binding is negligible, while substrate-enzyme complex consumes most of the enzyme. That is: E ¼ ½E + ½ES
(10.3)
where E is the total enzyme in the system, which is independent of enzyme-substrate binding. Solving Eqs. (10.2) and (10.3), one can obtain the ratio of enzyme-substrate complex to the total enzyme, θS ¼
½ES ½S ¼ ½ E Km + ½ S
(10.4)
where θS may also be termed as the substrate saturation ratio on the enzyme. It is the fraction of enzyme that binds with the substrate. The second step represented in Eq. (10.1) is the catalytic conversion of [ES] to E + P, or the turnover of the enzyme from the enzyme-substrate complex to free enzyme (again) and the released product. The overall reaction rate is then given by: rP ¼ kc ½ES
(10.5)
where kc is the enzyme turnaround frequency, or the catalytic rate constant. Combining Eqs. (10.2) and (10.3) into (10.4), one obtains the Michaelis-Menten equation: rP ¼
kc E½S rmax ½S ¼ Km + ½S Km + ½S
(10.6)
Enzymes can be amazingly fast. For normal chemical reactions, we have the example of a turnaround frequency (kc) greater than 106 s1 for catalase and for carbonic anhydrase. Enzymes can significantly increase the reaction rate; for example, orotidine monophosphate (OMP) decarboxylase can increase the rate of the decarboxylation of OMP by a factor of 1017 over the spontaneous rate. Over 5000 different enzymes have been characterized, and almost all of these are proteins. A limited number of catalytic reactions have been demonstrated with certain types of RNA molecules, and such catalytic RNAs are now called ribozymes. These first two types of enzymes are normal biological molecules that have evolved to have the features that make them so essential. In Chapter 7, we have learned some characteristics of allosteric enzymes. An allosteric enzyme, as the name implies, has more than one binding site available. In this chapter, we will learn more about how the enzymes have multiple sites and how they catalyze biotransformations.
538
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME Enzymes, or proteins in general, can consist of multiple subunits or oligomers. The presence of multiple active domains within one enzyme molecule or complex is the prefix for the enzyme to be interactive as the multiple domains are interconnected. Homo- or heterounits of proteins are commonly associated with each other through covalent bonds that are almost always irreversibly stable, through often reversible associations mediated by electrostatic and hydrophobic interactions, or through hydrogen bonds. The term “oligomer” is usually restricted to be less than 25 for the number of subunits. Marianayagam et al. (2004) characterized 450 enzymes, of which only about 140 are monomeric. Among the other 310 oligomeric enzymes, 200 are homo-oligomers; specifically, 125 homodimers, 50 homotetramers, and 25 structures larger than tetramers. The remaining about 110 are hetero-oligomers. Oligomers, particularly molecules of varying number of subunits, are common for enzymes. Oligomerization is the process of association or the addition of subunits to a base molecule to form a new molecule(s); therefore, it can be generalized as polymerization. However, the term “oligomerization” usually refers to polymerization with a low degree. Protein oligomerization has also been called various other terms, including assembly and multimerization. There is no clear distinction amongst the use of the terms; thus, they are interchangeable. Protein oligomerization is often accompanied by the onset of bioactivity and thus is a phenomenon crucial in triggering various physiological pathways. Biotransformations are often enabled by oligomeric enzymes that can respond to different/changing environmental conditions. For example, proteins or enzymes found with thermophiles or microbes living in high-temperature environments are often associated together by charge clusters, networks of hydrogen bonds, optimization of packing, and hydrophobic interactions, which is an indication that protein oligomers are more stable and can sustain a severe environment. Indeed, some hyperthermostable proteins have larger oligomeric size when compared to their mesophilic homologues. As such, one can expect that protein oligomerization is important to bioengineering or biotechnology in general. Proteins can self- or cross-associate either naturally or artificially. Protein oligomerization can occur when environmental conditions are changed, or crosslinking chemical reagent(s) are introduced. If a monomeric protein, ie, lacking quaternary structure, is considered as the starting species, we can obtain the first polymeric seed when a dimer, or different dimers, form. Polymerization can then continue towards trimer(s), tetramer (s), pentamer(s), and so on. The smallest subset of different subunits forming an oligomer is the structural unit of an oligomeric protein and is called a protomer. A protomer can be a protein subunit or several different subunits that assemble(s) in a defined stoichiometry to form an oligomer. When the oligomers are made of identical protomers, they are called homo-oligomer complexes; otherwise, they are hetero-oligomers. The heterostructures refer to chains of different sequences, which undergo association in a manner less statistically favorable and easily controllable, either qualitatively or quantitatively, than protein self-association. Protein hetero-oligomerization represents a very important phenomenon in the formation of molecular machines, like for motor proteins (eg, kinesin, microtubules), or alternatively, can be obtained artificially by the use of asymmetric bifunctional reagents.
10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
539
10.1.1 Covalently Bound Oligomers The type of protein oligomerization that first comes to mind is perhaps covalently linking protein residue units, just like well-known polymers from small monomers, such as cellulose (β-1,4-glucopyranose polymer), starch (mostly α-1,4-glucopyranose polymer), etc. The only difference here is that protein oligomers are formed from large monomers: proteins, which are already polymers of amino acids, or polypeptides having a DP or number of amino acid residues usually greater than 50. Protein crosslinking can occur by naturally forming covalently linked species (usually through a nonpeptide bond, for obvious reasons, as a peptide bond is what forms protein in the first place) that display quaternary structures, or active covalent complexes starting from inactive protomer precursors. Post-transduction modifications, photochemical process(es), or coenzyme binding (ie, going from apo- to halo-forms) can also induce protein self- or crosslinking. Crosslinking can sometimes occur in nature through free cysteines (ie, proteins containing thiol or –SH groups) of two different subunits that can couple to form intermolecular disulfides: R SH + HS R0 ! R S S R0
(10.7)
Disulfide bond is the common way to crosslinking proteins: A well-known example of oligomerization through disulfide bonding is represented by the antibodies Immunoglobulin M (IgM) and Immunoglobulin A (IgA). Antibodies are heavy (150 kDa) globular plasma proteins. Antibodies have sugar chains added to some of their amino acid residues; in other words, antibodies are glycoproteins. The basic functional unit of each antibody is an immunoglobulin (Ig) monomer (containing only one Ig unit); secreted antibodies can also be dimeric with two Ig units such as IgA, tetrameric with four Ig units like teleost fish IgM, or pentameric with five Ig units, like mammalian IgM. IgA is known to be able to form dimer, trimer, and tetramer. Fig. 10.1 shows a schematic of an IgM FIG. Light chains
Ig M monomer unit
Heavy chains J chain
Disulfide bonds
10.1 Immunoglobulin M (IgM) pentameric antibody molecule structure. The region enclosed by the dashed line represents a monomer that resembles that in Fig. 2.16. With permission from Liu, S., 2015. IJPEM, 16(13), 2731-2760.
540
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
pentamer. In fact, an antibody monomer itself may be regarded as a heterotetramer. The Ig monomer is a “Y”-shaped molecule that consists of four polypeptide chains, two identical heavy chains (450–550 amino acid residues each), and two identical light chains (211–217 amino acid residues each) connected by disulfide bonds. Each chain is composed of structural domains called immunoglobulin domains. These domains contain about 70–110 amino acid residues each and are classified into different categories (eg, variable or IgV, and constant or IgC) according to their size and function. They have a characteristic immunoglobulin fold in which two beta sheets create a “sandwich” shape, held together by interactions between conserved cysteines and other charged amino acids. To induce multiple functionalities, heterodimeric antibodies can be artificially made. Heterodimeric antibodies, which are also asymmetrical antibodies, allow for greater flexibility and new formats for attaching a variety of drugs to the antibody arms. One of the general formats for a heterodimeric antibody is the “knobs-into-holes” format. This format is specific to the heavy chain part of the constant region in antibodies. The “knobs” part is engineered by replacing a small amino acid with a larger one. It fits into the “hole,” which is engineered by replacing a large amino acid with a smaller one. What connects the “knobs” to the “holes” are the disulfide bonds between each chain. The “knobs-into-holes” shape facilitates antibodydependent cell-mediated cytotoxicity. Single chain variable fragments are connected to the variable domain of the heavy and light chain via a short linker peptide. The linker is rich in glycine, which gives it more flexibility, and serine/threonine, which gives it specificity. Two different single-chain variable fragments can be connected together via a hinge region to the constant domain of the heavy chain or the constant domain of the light chain. This gives the antibody bispecificity, allowing for the binding specificities of two different antigens. The “knobs-into-holes” format enhances heterodimer formation but doesn’t suppress homodimer formation. To further improve the function of heterodimeric antibodies, artificial antibodies have been constructed with largely diverse protein motifs that use the functional strategy of the antibody molecule, but aren’t limited by the structural constraints of the natural antibody. Being able to control the combinational design of the sequence and three-dimensional space could transcend the natural design and allow for the attachment of different combinations of drugs to the arms. Heterodimeric antibodies have a greater range in shapes they can take, and the drugs that are attached to the arms don’t have to be the same on each arm, allowing for different combinations of drugs to be used in cancer treatment. Pharmaceuticals are able to produce highly functional bispecific and even multispecific antibodies. Bovine seminal ribonuclease (BS-RNase) is another example that natural homooligomerization can occur through disulfide bonding. BS-RNase is the unique member of the large pancreatic-type secretory ribonuclease superfamily, which is dimeric in nature. Other proteins that can form covalent supramolecular structures are structural proteins such as collagen or elastin. The rubber properties of elastin, for example, are caused by desmosine (Fig. 10.2). Desmosine may appear to be a crosslinking agent, even though oligomerization was the cause of desmosine’s appearance, which resulted from the joining of lysine residues of the protomers (proteins). Elastic stress or relaxation can cause desmosine bridge formation or disruption in elastin in tissues or vessels. The J chain in IgA and IgM oligomers facilitates protein oligomerization through covalent binding to protein oligomers. Agents like the J chain that are not part of the original protomers
541
10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
NH2
O
NH2
O
OH
OH
HO
HO
O
NH2 N+
NH2
HO
O
O
NH2
NH2
OH
HO
desmosine
(A)
O
N+
NH2
NH2
O
O
OH
isodesmosine
(B)
FIG. 10.2 Isoforms of desmosine, apparent covalent crosslinker found in elastin.
are crosslinking agents. As we have alluded to, protein oligomerization can be induced artificially. Chemical crosslinking can be and has been extensively employed to make protein oligomers artificially through additional dehydrating molecules, such as 1-ethyl-3-(3-dimethylaminopropyl)carbodiimide (EDC) or carbodiimides in general, or through the use of several bifunctional reagents, such as dialdehydes or diimidoesters, shown in Fig. 10.3. The two chemicals shown in Fig. 10.3, or more precisely crosslinkers, display two terminal reactive
−
O
O
Cl+NH2
(A) Glutaraldehyde
[
−Cl+NH 2
]n
O
O
(B) Diiimidoesters: n = 2, dimethyl-adipimidate; n = 3, -pimelimidate; n = 4, -suberimidate R-NH2 O
[
]4
NH2+Cl−
NH2+Cl−
R-NH
R-NH
[
R-NH2+ O
O
]4
NH2+Cl−
R-NH - 2 CH3OH
O
O
]4
NH2Cl−
NH2+Cl−
NH2+Cl−
[
+ R-NH2
R-NH
[
]4
NH3+Cl−
O NH3+Cl−
(C) Cross-linking reaction of dimethylsuberimidate FIG. 10.3 Glutaraldehyde and diimidoesters, and the crosslinking of proteins (in particular with a lysine residue) by dimethylsuberimidate.
542
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
groups separated by a variable number of unreactive spacers, such as methylenes. They are often symmetric, but asymmetric ones have also been employed for special properties. Some of these crosslinking reagents were extensively used to produce protein oligomers that have been useful, after limited proteolysis, to study protein primary and tertiary structures and, thus, protein conformations. Covalently linked oligomers can have increased activity, higher stability against proteases, and so on. Both disulfide bonding and dehydration crosslinkers react mainly with lysine residues, which act as nucleophiles towards the aldehydic or imidic carbon of the crosslinker under slightly basic conditions (pH ranging from 7.5 to 9). Fig. 10.3 shows two types of crosslinking reagents and an illustration of dehydration crosslinking by imidate, where R-NH2 stands for a protein (N-terminal), especially of that from a lysine residue. The yield of dimers is the highest, which can be more than 20%, while decreasing amounts of higher DP oligomers can be obtained: trimers, tetramers, and traces of even higher DP oligomers. The advantages of protein crosslinker diimidoesters over dialdehydes are that diimidoesters are nontoxic, less reactive and consequently inducing less undesired changes in the desired protein product. For example, dialdehydes can result in side intramolecular reactions potentially inactivating the protein. Diimidoester reactivity can be controlled and lysines may be modified allowing the overall charge of the protein to remain unmodified. Despite a lower oligomerization yield, the products obtained from diimidoesters are more specific and active. The longer the spacer, the higher the intermolecular yield and the less rigid the oligomeric product. Thus, dimethyl-suberimidate is more useful to artificially oligomerize a protein than the shorter diimidoesters. The symmetrical bifunctional N-substituted maleimide derivatives as shown in Fig. 10.4 are useful in protein oligomerization. The reaction occurs between the maleimide and the free R-SH
O
O
O
O
R-S N
N
N
Spacer
O
Spacer
N
O
O
O + R′-SH
Spacer O
O
R-S S N
S
Spacer
N S-R′
O O
O
O
FIG. 10.4 Reaction of a bismaleimide derivative with sulfhydryl groups. The spacer can be a benzene residue or other groups, such as those shown in the insert.
543
10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
sulfhydryl group of a cysteine of the protein as an adduction, which saturates the double link of the maleimide conjugated with two carbonyls. The spacer can be benzene, N,N-1,2- or 1,3- or 1,4 phenyldimaleimide (ortho-, meta-, or para-PDM), or derivatives displaying spacers of varying length between the two reactive maleimides. Protein crosslinking is irreversible unless the spacers contain disulfide bonds (Fig. 10.4 with a disulfide spacer). Disulfide bonds can be broken and as a result, species can be drawn inside cells as heterodimers and then released as dissociated monomers exploiting the reducing environment of the cytosol. Thus, spacers containing disulfides can be applied to all bifunctional reagents, if reversibility is desired. There are spacers other than those shown in Fig. 10.4 that can lead to asymmetric structures. Asymmetric bifunctional crosslinking reagents can provide special properties, especially for covalently linking antibodies (or part of the light/heavy chains) with proteins, protein domains, or toxins. They can be a combination of maleimides or succinimides with or without dithio-derivatives in the spacers, and coupled, in the second terminus, with imidoesters, diones, thiones, 2-iminothiolane, etc. Almost all of these bifunctional crosslinking agents cause a two-step reaction, in which one of the partners is modified and activated (for example, with 2-iminothiolane) before it can react with the appropriate partner and form the new heterodimeric adduct. Some of the most important reactants used for artificial heterodimerization are shown in Fig. 10.5.
O
O
O
O S
O
N
N
O
N
S
O
N O
O
O
(B) SPDP, or N- succinimidyl-3-(2-pyridyldithio)
(A) MBS, or 1-(3-((2,5-dioxopyrrolidinyl)
propionate
oxy-carbonyl)phenyl)-1H-pyrrole-2,5dione O
O
O
N
O
SSO3−
O
N
O
S
S N
O
(C) SMBT, or S-4-succinimidyloxycarbonyla-methyl benzyl thiosulfate
O
(D) SMPT, or 4-succinimidyloxycarbonyl-amethyl-a(2pyridyldithio)toluene
FIG. 10.5 Examples of asymmetric bifunctional protein crosslinking reagents.
544
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
O
NO2
F
F
NO2
S O
(A) Divinyl sulfone (DVS)
FIG. 10.6
(B) Dinitrodifluorobenzene (DFDNB)
Chemical structures of short bifunctional reagents (A) DVS and (B) DFDNB.
Carbodiimides (R1 N ¼ C ¼ C R2), such as EDC, are bifunctional crosslinkers, which forms new isopeptide bonds between the side chains of lysine and glutamic acid or aspartic acid residues of proteins, producing “zero-length” crosslinks. They can be applied to covalently fix existing oligomeric protein aggregates avoiding undesired insertions of group(s) or net charge modifications in the protein complex. Fig. 10.6 shows the chemical structures of two “short” bifunctional reagents. They lack spacers and stabilize preformed structures without affecting oligomers’ conformations. DVS is specific for histidines, while DFDNB reacts with lysines. The limited dimensions of both molecules allow crosslinking only between residues that are very close to each other. The crosslinking can produce intra- or intermolecular adducts, and can thus chromatographically or electrophoretically reveal if the protein was monomeric or already oligomeric before covalent stabilization. Covalent protein oligomerization can also be carried out without the additional crosslinking agent by sealing a lyophilized protein in a vacuum at high temperature (<85°C) for 24–96 h. Dimers, trimers, and traces of tetramers of ribonuclease A (RNase A) and lysozyme can be made without introducing chemical groups that could negatively affect the protein residues. In fact, the heat vacuum treatment of proteins induces the dehydration of some of the lysine and aspartic acid or glutamic acid sidechains, producing newly formed intermolecular isopeptide bonds. This reaction, thought to be specific to a single couple of lysine and glutamic acid residues of RNase A, was found to involve more than one of these acid or basic residues, producing a mixture of heterogeneous products. Crosslinking can also be achieved through UV and/or microwave photochemistry. UV irradiation of a monomer or pre-formed non-covalent oligomers is known to produce covalently stabilized oligomers. However, irradiation treatments can inactivate the native protein or possibly its preformed oligomers.
10.1.2 Noncovalent Association Oligomerization While covalently bound oligomers are intuitive to understand, noncovalent protein oligomers are commonly involved in biotransformations. Protein association can occur often naturally without covalent modification(s), through a homo- or heteroassociation mediated by a weak bond network formed by electrostatic-hydrophobic interactions, and/or specific hydrogen bond(s). If the interface between monomers or protomers is large, almost all types of interactions can occur and are more frequently conserved. These interactions can be crucial for the active forms of several classes of proteins, such as enzymes and transporters. The
10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
545
interaction(s)/association(s) may occur naturally because of the sequence and structural features of the subunits, which build the oligomeric complex(es). The noncovalent associations can also result due to environmental changes, such as pH or ionic strength, or as a consequence of the increase of the monomers’ local concentration. These interactions allow the protein to overpass its dimerization dissociation constant(s) KD: E + E ! E2 KD ¼
(10.8)
½E2 ½E2
(10.9)
A further increase in the concentration can augment the degree of polymerization (DP) with the formation of trimers (E3, DP ¼ 3), tetramers (E4, DP ¼ 4), and larger oligomers (En, DP ¼ n > 4), if the entropy cost is balanced by favorable interface interaction(s): E + E2 ! E3
(10.10)
E + E3 ! E4
(10.11)
E + En1 ! En
(10.12)
or stoichiometrically, the oligomerization in general can be expressed as: nE ! En
(10.13)
The free energy of oligomerization can be represented by: ΔG ¼ ΔG0 + RT ln
½ En ½ E n
(10.14)
in which [E] is the concentration of protein segment(s) exposed, or protein interfaces that are able to interact with other protomers. The simplistic stoichiometry representation in Eqs. (10.8) through (10.14) does not mean that oligomerization can be arbitrary, nor that n can be large. For example, the DNA-binding protein Arc repressor is predominantly monomeric and unfolded at low concentrations, while mostly dimeric at high concentrations in 10 mM KH2PO4/K2HPO4 and 100 mM KCl with pH 7.3. Higher and/or different oligomers are not found unless the structure is altered chemically. There are many factors that can determine the number of oligomers and/or the number of subunits/protomer residues in an oligomer. The protomer concentration, temperature, and solvent are the key parameters governing oligomerization. Protomer chemical properties are key in deciding whether oligomerization could occur, and/or how many protomers could be associated together in a certain fashion. Chemical bonding can only occur between specific groups. The oxygen transporter hemoglobin (Hb) (Fig. 10.7) is an example of a natural oligomeric protein. Structurally, Hb is consisted of mostly α-helices (illustrated as ribbons in Fig. 10.7). Hb is functionally active only as an α2β2 tetramer (or, in the fetus, α2γ2, endowed with a greater affinity to oxygen), with its α and β subunits being associated through salt bridges and other
546
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION hems a1
b2
N
N Fe2+ N
N
a2
b1
O OH
(A) A schematic of human hemoglobin (hHb)
O
OH
(B) The chemical structure of hem
FIG. 10.7 Structure of hemoglobin and its oxygen-binding hem. The functional protein is an oligomer containing two α subunits, two β subunits, and an iron-containing heme group for each protein subunit. Both α and β subunits are consisted of mostly α-helices. With permission from Liu, S., 2015. IJPEM, 16(13), 2731–2760. O OH
OH O
P HO O
N
FIG. 10.8 Pyridoxal 5-phosphate (PLP).
weak interactions. The loss of these interactions leads Hb to switch its conformation from the deoxygenated to the oxygenated form; this modification is due to the cooperative binding interactions triggered by the first oxygen bound. Hb, unless it is tetrameric, cannot be active, while myoglobin, the oxygen collector in tissues, is active as a monomer. Pyridoxal 5-phosphate (PLP) enzymes are also typical non-covalent associated proteins. Fig. 10.8 shows the chemical structure of PLP. For example, the aspartic aminotransferase (EC 2.6.1.1, AST), alanine-glyoxylate-transaminase (EC 2.6.1.44, AGT) (Fig. 10.9A), aromatic L-amino acid decarboxylase (EC 4.1.1.28, AADC) (Fig. 10.9B), and cystalisin (E.C. 4.4.1.1) are inactive as monomers and active only when they are in the dimer forms. The PLP-dependent enzyme family is known to have at least 145 distinct enzymes. It can be classified into 5 enzyme categories: 1-oxido-reductases (EC 1; one enzyme), 2-transferases (EC 2; eighty enzymes), 3-hydrolases (EC 3; two enzymes), 4-lyases (EC 4; Forty nine enzymes), and 5-isomerases (EC 5; thirteen enzymes). Basically, all the PLP-dependent-enzymes are mainly involved in amino acid transformations as decarboxylation, transamination, racemization, β,γ-elimination (elimination or truncation from β or γ location), and β,γ-substitution. The multiple functional catalytic modes of PLP-enzymes, according to the position of net reaction, are summarized in Table 10.1, along with the type of the enzyme oligomers. The PLP coenzyme
547
10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
Cα
Nα
PLP Cα
O
OH
OH O P
O
HO O
N OH
OH O P
HO
N
O
Nβ
α-Subunit Nβ
Cβ
PLP
Cβ
α-Subunit
FIG. 10.9 Structure of holo natively dimeric PLP enzymes: AGT and AADC.
plays a pivotal role in catalysis of various enzymatic reactions. Structurally, aldehyde groups of PLP are bound covalently as internal aldimine (Schiff base)/imine linkage to the ε-amine group of lysine residues close to the N-terminus, as in Fig. 10.9A. Glycogen phosphorylase (EC 2.4.1.1) has inactive monomer and tetramer forms and an active homodimer form. It has a PLP at each active site. Glycogen phosphorylase is the enzyme responsible for catalyzing the conversion of a terminal glucose residue in linear starch or glycogen to glucose-1-phosphate (G1P). Orthophosphate recognition at allosteric binding sites is a key feature for the regulation of enzyme activity in glycogen (or starch) phosphorylases. Fig. 10.10 shows the recovery of catalytic activity of denatured bacterial starch phosphorylase (from Corynebacterium callunae) CcGlgP when PLP is introduced into the medium with different amounts of orthophosphate addition. One can observe that the addition of PLP can restore catalytic activity in partly inactivated enzyme preparations. Orthophosphate has a positive effect on the catalytic activity of CcGlgP. Another interesting example of an oligomeric protein is phenylalanine hydroxylase (PAH, EC 1.14.16.1). The molar mass of a PAH monomer is 51.9 kDa. Prokaryotic PAH is monomeric, whereas eukaryotic PAH exists in an equilibrium between homotetrameric and homodimeric forms. The catalytic active forms are either homodimer or homotetramer, with a homotetramer being more active. Its activity depends on its tetrameric structure and the presence of tetrahydrobiopterin (BH4) and is cooperatively regulated (also known as allosterically regulated by the substrate, in this case Phe, itself). Some mutations become pathogenic because they destabilize and inactivate the tetramer, which consequently results in the organism to become phenylketonuria (PKU). PAH catalyzes the hydroxylation of the aromatic sidechain of phenylalanine to generate tyrosine: +
NH3
O
+
NH3
O
O2 +
PAH
O–
O–
OH
ð10:15Þ NH
NH2
N
OH
NH OH
N
NH
NH
NH OH
O
NH2
N
OH
OH
O
Oligomeric Type and Functional Catalytic Modes of PLP-Enzymes According to the Position of Net Reaction
Reaction Site
Reaction Type
α
Racemization
Mechanism of Reaction
Example of PLP-Enzyme
R
H
C H
R
NH2
NH2 COOH
CH
CH R
NH2
COOH
COOH
NH2
COOH CH
CH H NH2
NH2 Replacement
COOH
COOH R1 β
O
EC 2.1.2.1 Glycine hydroxylmethyltransferase A dimer
H
R
β
R1
R2 O
NH2
CH
+
CH
R2
EC 2.6.1.5 Tyrosine amino-transferase A dimer
COOH
COOH
CH
+
CH R1
Elimination
+ CO2
R NH2
Transamination
EC 4.1.1.17 Ornithine decarboxylase A homodimer
H
CH α NH2
R2 β
CH α NH2
EC 4.2.1.20 Tryptophan synthase A tetramer (αββα)
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Decarboxylation
EC 5.1.1.1 Alanine racemase A dimer
COOH
COOH C
548
TABLE 10.1
TABLE 10.1 Oligomeric Type and Functional Catalytic Modes of PLP-Enzymes According to the Position of Net Reaction—cont’d Reaction Type
Mechanism of Reaction
Elimination
R β γ
Example of PLP-Enzyme
COOH
COOH
CH α
CH
C O
NH2
Replacement
NH2
COOH
R1
γ
β
Elimination
R
γ
β
EC 4.3.1.17 Serine dehydratase A homodimer
COOH
CH α
R2
γ
β
NH2 COOH
COOH
CH α
CH
NH2
EC 2.5.1.48 Cystathionine γ-synthase A tetramer
COOH CH α NH2
EC 4.4.1.11 Methionine γ-lyase A dimer of dimers
COOH C O
NH2
10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
Reaction Site
549
550
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION N
N C
N
(B) Ebola VP 40 antiparallel dimer
(A) Ebola VP40 protein structure. C-domain is colored green, while N-domain is colored brown. All the α-helics and β-sheets are labeled
N N
(C) Ebola VP 40 parallel dimer
° 80A
(D) Ebola VP 40 heximer
° 20A
(E) Ebola VP 40 octamer
FIG. 10.10
The structures of Ebola matrix protein VP40: (A) detailed monomeric unit structure, (B) an antiparallel dimer, (C) a butterfly-shaped (parallel) dimer structure critical for membrane trafficking, (D) a rearranged hexamer structure essential for building and releasing nascent virions, and (E) an RNA-binding octamer ring that controls transcription in infected cells. Reprinted with permission from Liu, S., 2015. IJPEM, 16(13), 2731–2760.
A natively oligomeric protein can also switch towards higher DP oligomers. For example, the natively homotetrameric L-rhamnulose-1-phosphatase aldolase becomes an octamer only upon a single A88F mutation. The introduction of a single residue displaying large nonpolar sidechains (Phe, Trp) can be sufficient to induce the formation of larger oligomeric complex(es). Protein oligomerization can also occur after its production or purified from fermentation broth. Monomeric proteins can naturally and noncovalently undergo oligomerization during post translation, which can become a switch between active and inactive products. For example, several transmembrane receptors are known to oligomerize during applications, which often display kinase activity. Upon ligand binding, the intracellular domain dimerizes, which triggers (auto)phosphorylation of the intracellular domain undergoing conformational changes able to activate a signal transduction cascade that induces or tunes important physiological phenomena. Examples of families of this type of receptors are growth hormone (GH), interferon (IFN), cytokine and Tyr-kinase, and G- protein-coupled receptor families
10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
551
(GPCRs). For example, one of the Tyr-kinase receptors, insulin receptor is, once released into the bloodstream, insulin can bind to receptors on the surface of cells in muscle or other tissues. The insulin receptor is a protein with an (αβ)2 quaternary structure. The two large α-subunits are extracellular, while the smaller β-subunits have a transmembrane domain as well as extraand intracellular domains. In the absence of insulin, the two intracellular domains of the β subunits are separated. Binding with insulin triggers a conformational change in the receptor that brings them closer together (dimerization). Each β subunit intracellular domain is a tyrosine kinase that phosphorylates its partner in the receptor. A family of proteins involved in apoptosis, the Caspase-3, -7, and -9, are interesting oligomeric proteins. Under normal physiological conditions, Caspase-9 exists as an inactive monomer, forming a 1:1 complex with the Apaf-1 cofactor in the presence of cytochrome c and ATP to produce a heteromultimer. This complex colocalizes with a multiple array of Caspase-9 molecules, which consequently increase their concentration above the Caspase-9 homodimer dissociation constant KD. This process allows the homodimer to be formed through an activation loop, and the active dimer provides the catalytic activity necessary to activate Caspase-3 and-7. The membrane channel-forming tetrameric natural oligomer complexes allow specific ions (Na+ or K+) or water to permeate cells, such as aquaporins or aquaglyceroporins. Aquaporins form tetramers in the cell membrane, with each monomer acting as a water channel. The different aquaporins differ in their peptide sequences, which allows for the size of the pore in the protein to differ between aquaporins. The resultant size of the pore directly affects what molecules are able to pass through the pore, with small pore sizes only allowing small molecules like water to pass through the pore. Aquaporins selectively conduct water molecules in and out of the cell, while preventing the passage of ions and other solutes. As such, aquaporins are integral membrane pore proteins and are also known as water channels. Some of them, known as aquaglyceroporins, also transport other small, uncharged solutes, such as glycerol, CO2, ammonia, and urea across the membrane, depending on the size of the pore. For exam˚ and allows the passage of hydrophilic ple, the aquaporin 3 channel has a pore width of 8–10 A molecules between 150 and 200 Da. However, the water pores are completely impermeable to charged species, such as protons, which is a property critical for the conservation of the membrane’s electrochemical potential difference. Proteins can also form large pathogenic oligomers that can evolve towards pathogenic supramolecular structures. Important examples of these malignant processes are the uncontrolled aggregation of the Glu6Val Hb mutant of Hb (E6V-Hb or HbS) in sickle cell anemia and the formation of amyloid or amyloid-like fibrils, as it occurs with several proteins related to severe neurodegenerative diseases. Viruses can be under tremendous evolutionary pressure for the economy of genomic information. The Ebola virus, which causes hemorrhagic fevers with up to 90% lethality, for example, encodes only seven genes transcripting seven proteins, the protein products of which must achieve all of the different steps of the virus life cycle. As a result of this genomic economy, each of the proteins it encodes is essential and commonly multifunctional. One of these proteins is EBOVE VP40 (shown in Fig. 10.10), the viral structural matrix protein that builds the protein shell underneath the viral membrane to assemble and release progeny viruses from the infected cell. VP40 alone is necessary and sufficient for the assembly and release of Ebola virus-looking particles from transfected cells. The general fold of VP40
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10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
has weakly associated N- and C-terminal domains, and a ring-like arrangement made by the N-terminal domains alone. RNA was observed bound to the ring. Purified VP40 protein is shown to be dimeric, not monomeric. This dimeric VP40 grew multiple crystal forms. In fact, two dimeric structures have been confirmed; both are associated by the N-termini, with one being parallel and the other being antiparallel, as shown in Fig. 10.10. No matter what space group the crystals belonged to or what species of Ebolavirus was analyzed, the VP40 dimers assembled end to end in identical linear filaments. The filaments were assembled by protein-protein interfaces that were distinct from those that assembled the RNA-binding ring. The RNA-binding ring assembly of VP40 was not involved in virus assembly. The protein VP40 is called a “transformer” to reflect its ability to refold its structure in order to achieve new functions: a dimer for membrane trafficking, an RNA-binding ring structure for viral transcription, and a filamentous oligomeric assembly for building and budding new virions. One other transformer has been identified, ie, the transcriptional regulatory protein RfaH.
10.1.3 Domain Swapping Oligomerization A peculiar way of forming protein oligomers is through the exchange of a section of peptide(s) or entire domain(s) among the monomeric subunits. Monomers exploit short, flexible hinge regions present in their sequence to address a given domain (or more) in the corresponding partner subunit that can reciprocally swap an identical domain with the former, as illustrated in Fig. 10.11. Three regions of the monomers can be identified in the swapped horse cytochrome c dimer: the main domain (containing α-helix carboxylic terminal CA or CB); the swapped domain (or secondary domain, containing α-helix amine terminal NA or NB); and the hinge region. This mechanism is called DS, which is also known as threedimensional domain swapping or 3D-DS and schematically shown in Fig. 10.12. Beyond dimers, this mechanism, which is known for diphtheria toxin (DT) as well as for other proteins, can also lead to the formation of higher DP oligomers exploiting small, flexible hinge regions that are able to adopt different conformations within various different environmental conditions.
Hinge region NB CB
CB
CA
+ CA NA
NA
NB A
B
Hinge region Swapped domains
FIG. 10.11 An example of domain swapping: two hose cytochrome c monomers A and B to α-helix amine terminal swapped horse cytochrome c dimer. Redraw based on Deshpande, M.S., et al. 2014. Biochemistry, 53, 4696–4703.
10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
Domain swapping
+
553
FIG. 10.12 Schematic view of the mutually reciprocal domain swapping mechanism. The opening of the hinge region in the “starting” monomer (left) allows the formation of the dimer by recreating the intramolecular interdomain interface present in the monomer in an intermolecular-dimeric interface instead. This subsequently forms a new dimeric interface (between the hinge regions), which is absent in the monomer. Through this mechanism, a protein can form an active dimer while still maintaining functional units (F.U.).
The domain-swapped oligomer reconstitutes the native contacts present in the monomer except the hinge region, while a new interface (between the hinge regions) forms in the oligomer only to stabilize it, as one can infer from Figs. 10.11 and 10.12. One can imagine that a stable oligomer can be formed by mutually (or reciprocally) swapping secondary domains from each protomer. However, this would allow a dimer form only for protomers with singular secondary (swappable) domains. Potential stable oligomers of single-swappable domain protomers with DP greater than 2 are cyclic oligomers. Moving beyond dimers without cyclic swapping will require additional oligomerization means for singularswappable domain protomers. As proteins have two end groups, ie, C-terminus and N-terminus, it is not difficult to imagine that a lot of proteins/enzymes can swap two secondary domains. The capability of swapping two subdomains allows for a high degree of oligomerization, as illustrated in Fig. 10.13. The RNase A oligomers are in general mutually and reciprocally domain-swapped oligomers, as shown in Fig. 10.13. In appropriate conditions, such as 40% ethanol/60% water (v/v) at 60°C, two RNase A monomers (E1) can dimerize by DS the C-terminal β-strand to form Cdimer (DC, or CE2), or their N-terminal α-helix to form N-dimer (DN, or NE2). DN can combine with another monomer to produce a cyclic N-swapped trimer (cNP3). Whereas only DC can combine with another monomer to produce a cyclic C-swapped trimer (cCE3), both dimers can combine with another monomer to produce a linear trimer (NCE3). The addition of another monomer via N-terminal swapping to cCE3 produces the tetramer NcCE4. The addition of a monomer to NCE3 can produce the linear tetramers CNCE4 and NCNE4. NCNE4 is also in equilibrium with its bent form bNCNE4. Alternatively, the NCNE4 and CNCE4 tetramers may also be formed via the combination of two dimers. Isomerization of the protein can lead to various forms of the same stoichiometry. For example, cis $ trans isomerization of the Asn 113-Pro 114 peptide bond in one subunit in the central interface is responsible for the two NCNE4 shown with cis in bNCNE4 and trans in linear NCNE4. The formation of cyclic C-swapped tetramers (cCE4) can start with two C-swapped dimers or a cyclic C-swapper trimer and a monomer. However, rearrangement in the swapping of C-termini amounts to multiple swapping steps. RNase A also forms pentamers (E5), hexamers (E6) and higher DP oligomers (En), up to tetradecamers (E14). The linear oligomers are all mutually domain swapped. However, there are cooperative domain swapped oligomers, such as the cyclic oligomers in Fig. 10.13(v), (vi), (vii), and (xi). Cooperative DS means swapping among multiple protomers,
554 10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
FIG. 10.13
Schematic of RNase A domain-swapped oligomers formations. With permission from Liu, S., 2015. IJPEM, 16(13), 2731–2760.
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10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
FIG. 10.14 Schematic view of the cooperative domain swapping. The swapping of secondary domain is not mutually reciprocal. E1
14
DC or CE2 DN or NE2
12
Concentration, mM
E3 10
E4
8 6 4 2 0 0
20
40
60 80 Time, min
100
120
140
FIG. 10.15
Ribonuclease A oligomerization in 40% aqueous ethanol at 60°C with 200 mg/mL (14.6 mM ¼ 0.0146 M) proteins. Five species are shown: E1, monomer; DC, C-swapped dimer; DN, N-swapped dimer; E3, trimers; E4, tetramers. Data from Geiger, R., et al. 2011. J. Biol. Chem., 286, 5813–5822.
or else the swapping is not mutually reciprocal, as illustrated in Fig. 10.14. Cooperative swapping can lead to higher degree of oligomerization than mutual swapping. The oligomerization is fast and can occur in minutes. Fig. 10.15 shows the kinetics of oligomerization for RNase A in 40% aqueous ethanol solution at 60°C. After 60 min, all concentrations of the oligomers plateaued. There are multiple oligomer species present at equilibrium. One can observe that there are more monomers (P1) in the solution than dimers (P2), trimers (P3), tetramers (P4), and oligomers of higher DP at equilibrium. Some of the oligomerization rate constants and equilibrium constants are shown in Table 10.2. Beyond RNases, several other proteins involved in important biologic processes form domain-swapped oligomers. One DS-prone protein which is highly structurally versatile is cyanovirin-N, an 11 kDa protein that inhibits HIV. It can be active either as a monomer or as a metastable domain-swapped dimer. Interestingly, some mutants that become active only as domain-swapped dimers were found to form two different relatively stable DS dimeric conformers, one DS trimer, and two DS tetramers. Cystatins, a class of proteins comprised of stefins that inhibit cysteine proteases, can also dimerize through DS. Cytochrome c, which was known to polymerize, was shown to form
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10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
TABLE 10.2 Parameters Describing the Oligomerization of Ribonuclease A Oligomerization Step
Terminus Swapped
Keq
kf (M21 s21)
kb (s21)
E + E ! CD (or CE2)
C
20
1.0 10
E + E ! ND (or NE2)
N
11
4.0 103
3.63 104
CD + E ! cCE3
C
12
2.0 102
1.67 103
CD + E ! NCE3
N
34
8.0 103
2.35 104
ND + E ! NCE3
C
60
2.0 102
3.33 104
! NcCE4
N
23
8.0 103
3.48 104
NCE3 + E
! NCNE4
N
23
8.0 103
3.48 104
NCE3 + E
! CNCE4
N
40
2.0 102
5.0 104
ND + ND ! NCNE4
C
125
2.0 102
1.60 104
CD + CD ! CNCE4
N
68
8.0 103
1.18 104
0.1 s1
0.1 s1
cCE3 + E
NCNE4
! bNCNE4
1
2
5.00 104
Reprinted with permission from Lo´pez-Alonso, J.P. et al., Arch. Biochem. Biophys. 2009, 489(1–2):41–47.
domain-swapped inactive supramolecular structures via a cooperative DS of its C-termini. In particular, domain-swapped dimers, trimers, tetramers, and polymers up to 40-mers have been characterized. The L68Q human cystatin-C (hcC) mutant, in particular, can initialize the cooperative DS, inducing severe massive amyloidosis in brain arteries and lethal cerebral hemorrhages. There are three more important examples of proteins able to form domain-swapped structures. The first is BCL-XL, an anti-apoptotic protein belonging to the BCL-2 family, which can form active C-terminal-swapped dimers when highly concentrated or alternatively when heated up to 50°C. The second is cadherins, which are cell adhesion proteins that dimerize through β-strand swapping to mediate the adhesion itself. It is worth mentioning that several protein cell receptors are known to dimerize to become active, but less is known about the mechanism responsible for the dimerization. Thus, it is possible that some of them could undergo DS. Finally, histones are also known to fold through DS in their evolutionary pathway. While protein oligomerization is generally beneficial in health, uncontrolled oligomerization or polymerization can lead to severe problems or diseases. Oligomerization could be viewed either as beneficial or detrimental depending on the situation or desired application. Protein oligomers can be the first deleterious seed leading to protein fibrilization (polymerization to large/long molecules), a process implicated in several devastating neurodegenerative diseases. Amyloidosis, for example, is a rare disease that results from accumulation of inappropriately folded proteins, or more appropriately, protein oligomers of a high DP. These misfolded proteins or high DP oligomers are called amyloids. Technically, these high DP macromolecular proteins are not oligomers anymore; they are properly termed polymers. Only brief discussion will be made in this paper on the larger than normal oligomers. While proteins are normally soluble in water, amyloids are insoluble and deposit in organs or tissues, disrupting normal
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557
function. The type of protein that is misfolded into an amyloid and the organ or tissue in which the amyloids are deposited determine the clinical manifestations of amyloidosis. For example, amyloidosis plays a role in the pathologic process of amyloid plaque formation in Alzheimer’s disease and prion infections such as scrapie and Creutzfeld-Jacob disease. While stable mutual domain-swapped oligomers are common and practical applications abound, there are protomers known to follow cooperative DS as well. Some amyloidogenic proteins form fibrils through cooperative DS as illustrated. For example, the structural similarity existing between the Asn-based open interface of RNase A C-swapped dimer, –CE2– or –CD– and the fibrillogenic nature of poly-Q expansions, which are structured as cross-β-spines. Either prion protein (PrP) or cystatin-C, two amyloidogenic cross-βspine-prone proteins, form DS dimers. The PrP, which is associated with the lethal neurodegenerative Creutzfeld-Jacobs Disease and Scrapie, dimerizes through DS. The cooperative domain-swapped oligomers are formed with open-ended edges “neutralized” or stabilized by intermolecular newly formed disulfide bonds, but do not affect the overall tertiary structure of the globular main domain of PrP. The finding that the prevention of DS effectively inhibits cystatin-C dimerization and oligomerization and studies of the hinge region governing the DS process confirm that cooperative DS plays a key role in cystatin-C fibrillogenesis. Another important amyloidogenic protein able to self-associate through cooperative DS is β2-micro globulin (β2-m), the 10.9 kDa light chain of type-I histocompatibility complex, which can seed as amyloid fibrils during long-term hemodyalisis treatments. Like PrP, β2-m dimerizes to form an open-ended dimer and subsequently propagate to open-ended domainswapped amyloid fibrils stabilized by disulfide bonds. Three more amyloidogenic proteins are known to dimerize and massively aggregate through cooperative DS. The first worth mentioning is the immunoglobulin G-binding B1 domain, which forms DS conformational different dimers and tetramers induced by coredomain mutations before forming fibrils. The second one is T7-endonuclease I, which forms cooperative domain-swapped fibrils stabilized by core-domain intermolecular disulfides. The last one is the cell cycle protein Cks1, which fibrilizes through the preliminary formation of a domain-swapped dimer. Conversely, for another important amyloidogenic protein, transthyretin (TTR), cooperative DS is hypothesized, but the direct stacking model interaction through disulfide bonds between subunits is the one still accepted. Domain-swapped oligomers generally have known monomers of identical primary and secondary domains. In rare occasions, natural protein oligomers resembling domainswapped oligomers have no known protomers, and these multidomain proteins are called quasidomain-swapped oligomers. Examples of quasidomain swapping include crystalline pheromone/odorant binding/transport proteins, rice yellow mottle viral capsid protein, human cystatin C (protease inhibitor), Mycobacterium tuberculosis CarD, etc.
10.1.4 Interactive Enzyme Oligomer Mixture Model When an interactive enzyme can oligomerize while binding ligand(s) or catalyzing biotransformations, we have an interactive mixture of enzymes to deal with that was generally termed allostericity. The allostericity was coined to describe the “different” binding sites.
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10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Naturally, we have been introduced to allostericity in Chapter 7. Early models of enzyme mixture are all restricted to two “distinct oligomers” or molecules, known as allosteric modulators. Monod et al. (1963) proposed their model for the kinetics of allostericity, which is later known as the MWC model, based on a tense oligomer and a relaxed oligomer, with both having the same degree of oligomerization. The following year, Koshland et al. (1966) extended this model to add additional modes for conformational change, leading to the KNF model, as known later. The morpheerin model was proposed by Jaffe and coworkers (eg, Jaffe, 2005) to modify MWC model allowing the two oligomers to have different degrees of oligomerization. A summary of the three models is shown in Fig. 10.16. All three models impose different modes of interaction between the oligomers:
The concerted MWC model
0
(A)
ET
ER
ET + ER
ER + S
ER•S
ERSi + S
ET + S
ET•S
ETSi + S
ER•Si+1 ET•Si+1
ER•S
ERSi + S
ER•Si +1
The sequential KNF model
0
(B)
ET
ER
ET + ER
ER + S
The morpheein model
0
(C)
1M
2M
1M + 2M
iM + S
iM•S
Increasing ligand concentration
FIG. 10.16 Schematics of the ligand binding on a pair of enzyme oligomers as described by three classic models: MWC, KNF and morpheein. The MWC (Monod-Wyman-Changeux) model considers the interconversion between two states or forms of an oligomer (ET and ER of a tetramer in the drawing), while ligands can bind on either states. The KNF (Koshland-Ne´methy-Filmer) model restricts the ligands binding to one state, while the interconversion between the two states or forms is subunit-wise (shown for a tetramer). The morpheein model imposes on strict homogeneity among subunits on each enzyme molecule and no conversion directly between enzyme oligomers (shown a trimer and a tetramer); effectively, each subunit can be treated separately rather than the oligomers themselves, ie, morpheein 1M and 2M.
10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
559
FIG. 10.17
T
KT
KT Ke
R
S KR
Reaction scheme of a MonodWyman-Changeux (MWC) model of a protein (or protein complex) made up of two protomers. The protomer can exist under two states (ET and ER), each with a different affinity for the ligand (S): KT and KR. Ke is the ratio of states (concentrations) in the absence of ligand (or equilibrium constant).
S KR
MWC model explored the “allostericity” based on thermodynamics and three-dimensional conformations. Assuming an oligomeric protein with symmetrically arranged identical subunits, each of which has one active center (ligand binding site); two (or more) interconvertible conformational states of a complex protein were postulated to coexist in a thermodynamic equilibrium. The states, often termed tense (T) and relaxed (R), differ in affinity for the ligand molecule. The distribution between the two states (or allosteric sites) is affected by the binding of ligand molecules, as illustrated in Fig. 10.17. All subunits of an enzyme molecule change states at the same time, a phenomenon known as “concerted transition.” The resulting model is better known today as the MWC model for concerted allosteric transitions. When n ligand molecules can bind with one single protein complex (n ¼ 2 as shown in Fig. 10.17), the fractional occupancy based on MWC model is given by: ½S ½S n1 ½S ½S n1 1+ + αKe 1+α K K K K R R (10.16) θ S ¼ R R n ½S ½S n 1+ + Ke 1 + α KR KR where Ke is the isomerization constant or equilibrium constant between the two states when no ligand molecule is bound and α is the ratio of the dissociation constants for the ligand from the R and T states: Ke ¼
½ET0 ½ER0
(10.17)
KR (10.18) KT The corresponding catalytic reaction rate equation is given by: kT ½S ½S n1 ½S ½S n1 1+ + αKe 1+α k K KR K K n R n R rP ¼ rmax R R (10.19) ½S ½S 1+ + Ke 1 + α KR KR where kT is the catalytic reactivity of T state while kR is the catalytic reactivity of R state. α¼
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10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Since its inception, the MWC framework has been extended and generalized. Variations have been proposed; eg, to cater to proteins with more than two states (Edelstein et al., 1996), proteins that bind to several types of ligands (Mello and Tu, 2005) or several types of allosteric modulators (Najdi et al., 2006), and allosteric proteins (with nonidentical subunits or ligand-binding sites) (Stefan et al., 2009). It became known that protein changes structure upon binding a ligand. Koshland et al. (1966) refined the biochemical explanation of the mechanism described by Pauling (1935) to reflect the conformational structure change of protein. The Koshland-Ne´methy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as “induced fit.” For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula: θS ¼
3 4 3 3 6 KAB ðKS Ke ½SÞ + 3KAB KBB ðKS Ke ½SÞ2 + 3KAB KBB ðKS Ke ½SÞ3 + KBB ðKS Ke ½SÞ4 3 4 K ðK K ½SÞ2 + 4K 3 K 3 ðK K ½SÞ3 + K6 ðK K ½SÞ4 1 + 4KAB ðKS Ke ½SÞ + 6KAB BB S e S e S e BB AB BB
(10.20)
where KS is the association constant for S, Ke is the ratio of the proteins in B and A states in the absence of ligand ("transition"), and KAB and KBB are the relative stabilities of pairs of neighboring subunits relative to a pair where both subunits are in the A state. The KNF model is thus a model more suited for homosteric enzyme binding than polymorphs. While MWC and KNF models took the protein/enzyme modeling to a height, it does not explain the arrival of the protein structure, or assuming the two structures are of the same DP. Jaffe (2005) took the protein/enzyme modeling to the next peak by introducing the morpheein model highlighting the importance of conformational flexibility for protein oligomer functionality. Morpheein is a protein that exists simultaneously in two or more different forms or homo-oligomers (morpheein forms), but the subunits must come apart and change shape to convert between forms as illustrated in Fig. 10.16. The shape of the subunit dictates which oligomer is formed. Each oligomer has a finite number of subunits (stoichiometry). Morpheeins can interconvert between forms under physiological conditions and can exist as an equilibrium of different oligomers. The different oligomers have distinct functionalities. Interconversion of morpheein forms can be a structural basis for “allosteric regulation.” A mutation that shifts the normal equilibrium of morpheein forms can serve as the basis for a conformational disease. Features of morpheeins can be exploited for drug discovery. The one protein that is established to function as a morpheein is porphobilinogen synthase (PBGS). The morpheein model postulates that an enzyme mixture of “noninteractive” enzymes (morpheeins). Equal and independent (noninteractive) sites on each morpheein lead to: θS ¼ rP ¼ k 1 E
f1 ½S ð1 f1 Þ½S + K1 + ½S K2 + ½S
(10.21)
f1 ½S ð1 f1 Þ½S + k2 E K1 + ½S K2 + ½S
(10.22)
where f1 is the fraction of enzyme that belongs to morpheein 1, and K1 and K2 are the Michaelis-Menten saturation constants of morpheein 1 and 2, respectively.
10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
561
(A) 1E2, Hugging hetero-dimer (B) 2E2, Detached hetero dimer
(C) E8, Hetero octamer, or homo-tetramer of hugging heterodimers
FIG. 10.18
(D)
E6, Hetero hexamer, or homo-trimer of detached heterodimers
Structural characteristics of wild-type human porphobilinogen synthase (PBGS). Based on Selwood et al.
(2008).
The morpheein model was applied to PBGS by Jaffe et al., with a basis shown in Fig. 10.18. The basic unit is not a monomer, but a dimer of the PBGS: Fig. 10.19 illustrates a more realistic dynamic relationship among protein oligomers. Polymorphs are allowed, as opposed to the morpheein model, where only morpheeins (or paired forms) are allowed for a protein in a given system. A polymorph is an oligomeric protein that can exist as an ensemble of physiologically significant and functionally distinct alternate quaternary assemblies (monomers and/or oligomers), which can be simplified to a morpheein if it is a paired system. Strictly speaking, morpheein can exist only for proteins that only form dimers and maybe trimers of the smallest stable enzyme BCHOU), from which strictly a
562
(v)
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
30E3
(ii)
(vi)
13E4
(i)
10E2
(x)
00E1
(vii)
(viii)
03E3
(iii)
01E2
04E4
(iv)
(ix)
22E5
(xii)
23E6
11E3
21E4
(xi)
12E4
FIG. 10.19 A schematic of interconversions between enzyme polymorphs. Twelve polymorphs are shown: one monomer (E1), two dimers (E2), three trimers (E3), four tetramers (E4), one pentamer (E5), and one hexamer (E6) are shown. For each enzyme oligomer of a given DP (EDP), there can be multiple folding isomers or conformers, especially for high DP oligoenzymes. Some of these isomers or conformers are distinct enough to be classified as polymorphs. With permission from Liu, S., 2015. IJPEM, 16(13), 2731-2760.
single-step transit to an oligomer is required. As we learned in chemical kinetics that trimolecular reaction is already rare, higher oligomers cannot be formed from a single-step “combination” without intermediates formed. Polymorphs exist in nature and use conformational equilibria between different tertiary structures to form distinct oligomers as a means of regulating their function. In some situations, the transition from one polymorph to another can be important, whereas in other situations, the transition is effectively completed before the cooperative regulatory effects of the polymorph take effect. Alternate polymorph forms are not misfolded forms of a protein; rather, they are differently assembled native states that contain alternate subunit conformations. Transitions between alternate polymorph assemblies involve conformational changes, eg, dissociation, further association, or even oligomer dissociation; conformational change in the dissociated state; and reassembly to a different oligomer. These transitions occur in response to the environment changes, eg, effector molecules or allosteric effects, and represent a new model of cooperative/allosteric regulation. The unique features of polymorphs are being revealed through RNase: Table 10.3 shows a collection of some of the polymorphs known today. Some of these polymorphs were identified as morpheeins (Selwood and Jaffe, 2012). The list is expanded based on the latest known data, especially that which are discussed in this paper. As illustrated in Fig. 10.19, the protein oligomers can interchange, with or without the modulation of monomeric forms. Therefore, all the protein forms, either monomers or oligomers, are polymorphs, so long as they exist in the system. Although dynamic interchange relationships exist among the polymorphs, polymorphs may not share the same physiological and/or kinetic properties. As such, each polymorph needs to be treated differently.
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10.1 PROTEIN OLIGOMERIZATION AND INTERACTIVE ENZYME
TABLE 10.3
Polymorphic Enzymes
Protein
Example Species
E.C. Number
CAS Number
Polymorphsa
Acetyl-CoA carboxylase-1
Gallus domesticus
EC 6.4.1.2
9023-93-2
TE2, RE2,
α-Acetylgalactos-aminidase
Bos taurus
EC 4.3.2.2
9027-81-0
TE1, RE4
S-Adenosyl-L-homocysteine hydrolase
Dictyostelium discoideum
EC 3.3.1.1
9025-54-1
E4, En
Adenylosuccinate lyase
Bacillus subtilis
EC 4.3.2.2
9027-81-0
E1, E2, E3, E4
Alanine-glyoxylate transaminase
Mammalian
EC 2.6.1.44
9015-67-2
E1, E2
D-Amino
Homo sapiens
EC 1.4.3.3
9000-88-8
E1, E2, En
Aristolochene synthase
Penicillium roqueforti
EC 4.2.3.9
94185-89-4
E1, En
Aromatic L-aminoacid decarboxylase
Homo sapiens
EC 4.1.1.28
9042-64-2
E1, E2
L-Asparaginase
Leptosphaeria michotii
EC 3.5.1.1
9015-68-3
E2, E4, TE8
Aspartate kinase
Escherichia coli
EC 2.7.2.4 & EC 1.1.1.3
9012-50-4
E1, E2, E4
Aspartic aminotransferase
Homo sapiens
EC 2.6.1.1
9000-97-9
E1, E2
ATPase of the ABCA1 transporter
Homo sapiens
EC 3.6.1.3
9000-83-3
E2, E4
Biodegrative threonine dehydratase/threonine ammonialyase
Escherichia coli
EC 4.3.1.19
774231-81-1
aE1, bE1, aE4,
Biotin—(acetyl-CoA-carboxylase) ligaseholoenzyme synthetase
Escherichia coli
Caspase-9
Homo sapiens
Chorismate mutase
Escherichia coli
EC 5.4.99.5
9068-30-8
E1, E3, E6
Citrate synthase
Escherichia coli
EC 2.3.3.1
9027-96-7
E1, E2, E3, E4, E5, E6, E12
Cyanovirin-N
Nostoc ellipsosporum
918555-82-5
E1, E2
3-Oxoacid CoA-transferase
Sus scrofa domestica
EC 2.8.3.5
9027-43-4
E2, E4
Cystalysin
Treponema denticola
EC 4.4.1.1
Cystathionine beta-synthase
Homo sapiens
EC 4.2.1.22
9023-99-8
E2, …, E16
Dihydrolipoamide dehydrogenase
Sus scrofa domestica
EC 1.8.1.4
9001-18-7
E1, aE2, bE2, E4
Dopamine beta-monooxygenase
Bos taurus
EC 1.14.17.1
9013-38-1
E2, E4
acid oxidase
bE4
EC 6.3.4.15
37340-95-7
E1, E2 E1, E2
E1, E2
Ebolavirus VP 40 Geranylgeranyl pyrophosphate synthase/farnesyltrans-transferase
En
aE2, bE2,
Homo sapiens
EC 2.5.1.29
9032-58-0
E4, E8
E6, E8 Continued
564
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
TABLE 10.3 Polymorphic Enzymes—cont’d Protein
Example Species
E.C. Number
CAS Number
Polymorphs
GDP-mannose 6-dehydrogenase
Pseudomonas aeruginosa
EC 1.1.1.132
37250-63-8
E3, aE4, bE4, E6
Glucosamine-6-phosphate synthase
Escherichia coli
EC 2.6.1.16
9030-45-9
RE2, TE6
Glutamate dehydrogenase
Bos taurus
EC 1.4.1.2
9001-46-1
TE5, RE6,
Glutamate racemase
Mycobacterium tuberculosis, Escherichia coli, Bacillus subtilis, Aquifex pyrophilus
EC 5.1.1.3
9024-08-02
E1, aE2, bE2, E4
Glyceraldehyde-3-phosphate dehydrogenase
Oryctolagus cuniculus, Sus scrofa domestica
EC 1.2.1.12
9001-50-7
E1, E2, E4
Glycerol kinase
Escherichia coli
EC 2.7.1.30
9030-66-4
E1, aE4, bE4
EC 2.4.1.1
9035-74-9
TE1, RE2, TE4
Glycogen phosphorylase
En
Growth hormone receptor
Homo sapiens
E1, E2
GPCRs
Homo sapiens
E1, E2
HIV-integrase
Human immunodeficiency virus-1
EC 2.7.7.-
HPr-kinase/phosphatase
Bacillus subtilis, Lactobacillus casei, Mycoplasma pneumoniae, Staphylococcus xylosus
EC 2.7.1.EC 3.1.3.-
Interferon γ receptor
Homo sapiens
Lactate dehydrogenase
Bacillus stearothermophilus
EC 1.1.1.27
9001-60-9
aE1, bE1,
Lon protease
Escherichia coli, Mycobacterium smegmatis
EC 3.4.21.53
79818-35-2
E1, E2, E3, E4
Mitochondrial NAD(P) + malic enzyme/malate dehydrogenase (oxaloacetate-decarboxylating) (NADP+)
Homo sapiens
EC 1.1.1.40
9028-47-1
E1, aE2, bE2, E4
Peroxiredoxins
Salmonella typhimurium
EC 1.6.4.-& EC 1.11.1.15
207137-51-7
aE2, bE2,
Phenylalanine hydroxylase
Eukaryotes
EC 1.14.16.1
9029-73-6
E2, E4
Phosphoenol-pyruvate carboxylase
Escherichia coli, Zea mays
EC 4.1.1.31
9067-77-0
TE2, RE4
Phosphofructo-kinase
Bacillus stearothermophilus, Thermus thermophilus
EC 2.7.1.11
9001-80-3
TE2, RE4
Polyphenol oxidase
Agaricus bisporus, Malus domestica, Lactuca sativa L.
EC 1.10.3.1
9002-10-2
E1, E3, E4, E6, E12
Porphobilinogen synthase
Drosophila melanogaster, Danio rerio
EC 4.2.1.24
9036-37-7
E2, E6, E8
E1, E2, E4, En 9026-43-1
E1, E2, E3, E6
E1, E2, E4 E2, E4
E10
565
10.2 LIGAND BINDING AND COOPERATIVITY
TABLE 10.3
Polymorphic Enzymes—cont’d
Protein
Example Species
E.C. Number
CAS Number
Pyruvate kinase
Homo sapiens
EC 2.7.1.40
9001-59-6
E1, TE2, RE2, E3, RE4, E5
Ribonuclease A
Bos taurus
EC 3.1.27.5
9901-99-4
E1, E2, E3, E4, E5, E6, En
Ribonucleotide reductase
Mus musculus
EC 1.17.4.1
9047-64-7
E4, E6
Tyr-kinase receptors
Homo sapiens
EC 2.7.10.1
Tyrosine aminotransferase
Homo sapiens
EC 2.6.1.5
9014-55-5
E1, E2
β-Tryptase
Homo sapiens
EC 3.4.21.59
97501-93-4
TE1, RE1, TE4,
Polymorphs
E1, E2
RE4
Tumor necrosis factor-alpha
Homo sapiens
Uracil phosphoribosyl-transferase
Escherichia coli
EC 2.4.2.9
94948-61-5
E1, E2, E3
9030-24-4
E3, E5
a
Subscripts (right) indicate the number of subunits (DP), where n stands for other or higher oligomers. Left subscripts T and R indicate inactive (or tense) and active (or relaxed); left subscripts a and b indicate different forms. Augmented from Liu, S., 2015. “A review on protein oligomerization process,” IJPEM, 16(13), 2731–2760.
10.2 LIGAND BINDING AND COOPERATIVITY The specificity of enzymes is due to the binding site-specific nature of enzyme. Reviewing the many results with various enzymes, Koshland (1958) proposed the concept of induced fit to explain how enzymes can discriminate for the correct substrate, and explained how this would allow them to ignore smaller, but incorrect analogs. One enzyme unit contains one binding site, which leads to the specific nature of the enzyme kinetics, the Michaelis-Menten equation. However, in many occasions, the Michaelis-Menten equation fails to describe the reaction rate changes with substrate concentration. The single binding site for enzymes is thus too restrictive, leading to the cooperativity of enzymes. For simple enzymes, Eq. (10.4) can be rearranged to give: θS ¼ Km ½S 1 θS
(10.23)
One can notice this is a linearized form of the Michaelis-Menten binding. The linearity of Eq. (10.23) serves as the comparison for complex enzymes. To follow the development of interactive enzyme binding, we start with hemoglobin, which is an enzyme. When plotting the equilibrium hemoglobin fractional occupancy of oxygen (ie, θS) as a function of the partial pressure of oxygen, Christian Bohr in 1904 obtained a sigmoidal (or “S-shaped”) curve as shown in Fig. 10.20. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind, until all binding sites are saturated. In addition, Bohr noticed that increasing CO2 pressure shifted this curve to the right, as shown in Fig. 10.20. One can infer that higher concentrations of CO2 make it more difficult
FIG. 10.20
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Oxygen binding by dog blood as a function of oxygen partial pressure at different partial pressures of carbon dioxide at 37.8°C as shown by Bohr (1904).
100
80 O2 Saturation, %
566
60 pCO2 = 5 Torr
40
pCO2 = 10 Torr pCO2 = 20 Torr
20
pCO2 = 40 Torr pCO2 = 80 Torr
0 0
25
50
75 pO2, Torr
100
125
150
for hemoglobin to bind oxygen (Bohr et al., 1904). This latter phenomenon, together with the observation that hemoglobin’s affinity for oxygen increases with increasing pH, is known as the Bohr effect. The cause of the oxygen binding on hemoglobin differing from the Michaelis-Menten binding behavior lies on the multiple domains of a hemoglobin molecule. However, due to the lack of proper analytical model, earlier studies have focused on the phenomenological observations alone. For example, cooperative binding or cooperativity was coined for this phenomenon. A receptor molecule is said to exhibit cooperative binding if it is binding to ligand scales nonlinearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor’s apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The empirical concept of cooperative binding only applies to molecules or complexes with more than one ligand-binding site. If several ligand-binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be noncooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO2 reduces hemoglobin’s facility to bind oxygen.) Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill (1910) suggested a phenomenological equation that has since been named after him: θS ¼
½Sn n + ½Sn KH
(10.24)
where n is the “Hill coefficient” or cooperativity index, and KH is a microscopic dissociation constant. If n < 1, the system exhibits negative cooperativity, whereas cooperativity is positive
10.2 LIGAND BINDING AND COOPERATIVITY
567
if n > 1. The total number of ligand binding sites is an upper bound for n. Hill equation can be linearized to give: log
θS ¼ n log ½S n log Kd 1 θS
(10.25)
θS versus log[S]. In the case of the Hill equa1 θS tion, it is a line with slope n and intercept log Kd. The Hill equation is an improvement over the Michaelis-Menten expression as Eq. (10.24) can correlate the bulk of the data in Fig. 10.20, ie, a wider range of data than Eq. (10.4) does. Hill equation is extended to catalytic reactions: The “Hill plot” is obtained by plotting log
rP ¼
rmax ½Sn n + ½Sn KH
(10.26)
A combination of experimental observations and theoretical treatments reviewed so far leads from a simple enzyme that has a single binding site to an aggregate or conformal complex structure of enzyme molecules that has multiple binding sites. It is the multiple binding sites that give rise to numerous features of enzymes that enable regulations of reactions in living cells. The Hill empirical equation can correlate some of the experimental data to some degree, but it fails to give an account of the structure of the enzyme, and the parameters have no physical meaning other than the perceived empirical concept of cooperativity. We will examine the treaties for the complex enzymes. KNF model was the first to treat an interactive enzyme from a mechanistic point of view. An enzyme is said to be interactive if the binding of one ligand molecule will affect the further binding of more ligand molecule(s), ie, resulting in the “cooperation” among the bound ligands and enzyme. The binding of ligand on an interactive enzyme progresses one at a time. One sometimes calls this one at a time event as sequential, although sequential could have a more restrictive meaning, as the receptor sites themselves can be selective leading to preferential binding. A more precise term should be “facilitate” or cooperative. When one substrate molecule is bound to one site on an interactive protein/enzyme (of multiple subunits), it facilitates the binding of other substrate molecules on the other n 1 sites of the same protein/ enzyme molecule/complex: K1
En + S ! En S Ki
En Si1 + S ! En Si 81 i n
(10.27) (10.28)
where n is the total number of binding sites available on the protein/enzyme complex/molecule, En denotes for a protein/enzyme complex having n-subunits, S denotes for the binding substrate, and EnSi denotes for one enzyme complex/molecule bound with i substrate molecules. The binding on the first site by the substrate can facilitate further binding on the other sites. In enzyme analysis, we often use Michaelis-Menten kinetic Eq. (10.6). The MichaelisMenten saturation coefficient corresponds to the ratio of the dissociation rate constant over the binding rate constant. The analogy to kinetics on a simple enzyme is applied here for the binding coefficient K’s. The first binding coefficient is denoted as K1 (instead of Km for a
568 Homosteric
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Allosteric sequential
(A)
Allosteric random
(B)
(C) FIG. 10.21 Schematics of ligand binding on an interactive enzyme with (A) homosteric sites, (B) allosteric sites with preference in binding, and (C) allosteric sites without preference in binding. Each enzyme molecule is illustrated with three interactive binding sites. An allosteric enzyme has different sites as illustrated with squares, ovals, and hexagons in (B and C). The binding on an allosteric enzyme can be sequential if there is a preference of the ligand on particular sites. However, a homosteric enzyme contains identical sites as illustrated by the identical shapes. The binding occurs one at a time, but no order on the sites is available. In all the cases, when the concentration of ligand molecules increases (from left to right), increasing number of ligand molecules (shown as yellow circles) bound on the sites of the enzyme molecules. Binding of one ligand molecule will result in the change of conformal structure of the enzyme, and thus the bindings are also color coded to illustrate the interactions or the interactive nature of the enzyme.
simple enzyme) and for the subsequent binding sites, Eq. (10.28) shows that the binding rate constant is Ki for binding the ith substrate on the enzyme En. The interaction of the ligandenzyme-ligand is captured through the variation of binding affinity (or constant Ki’s). Fig. 10.21 shows the cooperative binding schematics of three different types of interactive enzymes: (a) homosteric (or homotropic) enzymes, (b) sequential allosteric (or heterotropic), and (c) random allosteric (or heterotropic). Homosteric enzymes contain multiple identical binding sites (homotropic sites). Each site on a homosteric enzyme molecule interacts with the ligand molecule in the same fashion; thus, the homosteric sites “compete” equally for binding with the ligand molecule. In other words, the complexing of one ligand molecule with any one of the available sites on the enzyme molecule is random. However, when a ligand molecule is bound on the homosteric enzyme molecule, the conformal structure change of the enzyme molecule can be induced; thus, the interaction of the vacant sites with the ligand molecule can be altered, or different from those already bound. As the ligand molecules are bound on the homosteric enzyme, and they possess identical interactions with the enzyme. Therefore, depending on how the ligand can bind on the multiple sites available on an enzyme molecule, the kinetics and equilibrium of the binding itself will be different. There is a marked interaction of the enzyme (with ligands). This is what we call cooperative binding.
569
10.2 LIGAND BINDING AND COOPERATIVITY
Allosteric enzymes, on the other hand, have different types of binding sites (hetero binding sites). The difference in chemical and form/steric properties can present a scaled preference (or sequence), for which the ligand molecule can bind. The preference or sequence in the approach of the ligand can thus be expected, but in some cases, preference may not exist. Binding and reactivity on each site of an allosteric enzyme molecule are also expected to be unique. The binding on an allosteric enzyme can thus be either sequential (or preferential) or random. In either case, binding of one ligand molecule will cause the change of conformal structure of the enzyme, and thus modify the affinity and reactivity afterwards. This cooperativity is clear at least for sites that are randomly accessible. However, for preferential binding, the cooperativity is obscured due to the sequence imposed. We shall discuss the three extreme situations (of interactions) in the following sections.
10.2.1 Single Ligand Binding on Homosteric Enzymes For enzyme homo-oligomers, all the binding sites are equal. Monod et al. (1963) pioneered the analysis on cooperative homosteric binding, while the KNF model was more accurate by inclusion of ligand-enzyme molecular interactions. Fig. 10.22 shows an illustration of the ligand binding on an interactive homosteric enzyme. Since all the binding sites are identical, one can interpret the rates in reactions (10.27) and (10.28) as: En + S ! En S, En S!En + S, En S + S!En S2 , En S2 !En S + S, En Si1 + S!En Si , En Si ! En Si1 + S,
KS
FIG. 10.22
a1KS
r + n1 ¼ nkn1 + ½En ½S
(10.29a)
rn1 ¼ kn1 ½En S
(10.29b)
r + n2 ¼ ðn 1Þkn2 + ½En S½S rn2 ¼ 2kn2 ½En S2
(10.30b)
r + ni ¼ ðn i + 1Þkni + ½En Si1 ½S rn2 ¼ ikni ½En Si
a2KS
(10.30a)
(10.31a) (10.31b)
a3KS
Schematics of ligand binding on a homosteric interactive enzyme with four subunits. The ligand binding occurs one at a time. When the concentration of the ligand molecules increases, an increasing number of ligand molecules bind on the sites of the enzyme molecule. Successive binding of ligand molecules results in the changes in the conformal structure (shape) of the enzyme, and thus the bindings affinity changes. The affinity change factor is denoted as α’s.
570
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
where n is the total number of binding sites available on the protein/enzyme complex/molecule, En denotes for a protein/enzyme complex having n-subunits, S denotes for the binding substrate, EnSi denotes for one complex enzyme complex/molecule bound with i substrate molecules, kni+ is the binding rate constant for substrate molecule on each of the (n i + 1) vacant sites, and kni is the dissociation of a substrate molecule from each of the i bound substrate molecules. The binding of the first substrate molecule can facilitate further binding on the homosteric enzyme molecule. The subsequent binding can have a different affinity as the conformational change could occur because of the binding of the first site. When binding on the first site, there are n equal sites available to bind. and thus the rate of binding on the enzyme is n-times that of each site. When binding on the ith site, there are a total of (n i + 1) equal vacant sites; thus, the binding rate is (n i + 1)-times that of each site. To determine the isotherm of substrate binding on a protein/enzyme, we assume that thermodynamic equilibrium applies to the binding of substrates on all the sites, that is: nkn1 + ½En ½S n½En ½S ¼ kn1 KS Y i1 ðn i + 1Þkni + ½En Si1 ½S ½En ½Si n ½En Si ¼ ¼ αj i ikni KSi ½En S ¼
(10.32)
(10.33)
j¼1
where n ðn 1Þ ðn i + 1Þ n ¼ i 12 i KS ¼
kn1 kn1 +
kni + kn1 + 1 αi1 ¼ kni kn1
(10.34a)
(10.34b)
(10.34c)
where αj is the affinity change factor (or cooperativity factor) due to the binding of the jth ligand molecule. When αi > 1, the affinity is increased due to the binding of the previous i substrates. Otherwise when αi < 1, the affinity is decreased due to the binding of the previous i substrates. The overall enzyme balance: n n Y i1 X ½En ½S X ½Si1 n E n ¼ ½ En + ½En Si ¼ ½En + αj i1 (10.35) KS i¼1 i j¼1 KS i¼1 Leads to: ½En ¼
En i1 i1 n Y ½S αj i1 i KS j¼1
n X
½S 1+ KS i¼1
(10.36)
10.2 LIGAND BINDING AND COOPERATIVITY
Y i1 ½Si n αj i1 i KS j¼1 ½En Si ¼ k1 n k1 X n Y ½S KS + ½S αj k1 k KS j¼1 k¼1
571
En
(10.37)
The total bound substrates on the interactive enzyme of n-binding sites, BnS, can be computed by: n Y i1 X i½Si n En αj i1 i n KS X i¼1 j¼1 i½En Si ¼ (10.38) Bn S ¼ i n i1 X n Y ½S i¼1 KS + αj i1 i KS i¼1 j¼1 which retains the details for thermodynamic properties of each successive binding. In addition, Eq. (10.38) reflects the kinetics of ligand binding where the affinity can vary with the number of sites already bound. When [S] ! 1, Eq. (10.38) is reduced to: B n S 1 ¼ En n
(10.39)
ie, the amount of saturation-bound substrate is identical to the number of binding sites on an enzyme molecule times the amount of enzyme (as each homosteric enzyme molecule/ complex has n binding sites available), which is intuitive. Therefore, Eq. (10.38) can also be written as: n Y i1 1X i½Si n αj i1 n i¼1 i j¼1 KS Bn S ¼ (10.40) θnS ¼ n Y k1 Bn S1 X ½Sk1 n KS + ½S αj k1 k KS j¼1 k¼1 where θnS is the ratio of substrate saturation on the homosteric enzyme of n-binding sites. Before we start to examine the features of the binding characteristics of homosteric enzymes, let us examine the extremes or asymptotic behaviors of Eq. (10.40). When n ¼ 1, Eq. (10.40) is reduced to: θ1S ¼
½S KS + ½ S
(10.41)
which is the identical to Eq. (10.4). When the ligand concentration is low, ie, [S] < KS, Eq. (10.40) is reduced to: 0 1 n Y i½S1 αj 11 n 1 j¼1 KS ½S θnS (10.42) ¼ Y 0 0 + n½S K S ½S n KS + ½S αj 11 1 KS j¼1
572
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Not surprisingly, it is very similar to Eq. (10.4). However, Eq. (10.42) renders a lower saturation than that of Eq. (10.4). When [S] is very high, that is, [S]> > KS, Eq. (10.40) is reduced to: 9 8 nY n1 n1 = 2 1< n Y n½Sn ð n 1 Þ ½ S n n1 αj + αj KS + ½S n1 ; n : n j¼1 KSn1 KSn2 j¼1 α 9 ¼ n1 8 (10.43) θnS n nY < nY 1 2 n½Sn ðn 1Þ½Sn1 = α KS + ½S n n KS + ½ S αj n1 + αj n1 n1 ; : n KS KSn2 j¼1
j¼1
which is, again, strikingly similar to Eq. (10.40). However, the intercept or when [S] is extended to 0, θnS ¼ (n 1)/n 6¼ 0, unless n ¼ 1. We next examine the extreme case of stiff enzyme oligomers. If there is no cooperativity or the further enzyme binding is not affected by the bound substrate, αj 1. In this case, Eq. (10.40) is reduced to: i n 1X n i½S n i¼1 i KSi1 ½S (10.44) ¼ θnS ¼ k1 n X n ½S KS + ½S KS + ½S k KSk1 k¼1 which is identical to the Michaelis-Menten saturation, Eq. (10.4). Therefore, if the binding of the ligand does not affect the enzyme-folding or binding affinity, a homo-oligomer can be treated as multiple simple enzymes. More frequently, the binding of a ligand on an interactive enzyme changes the conformal structure of the enzyme, thus leading to a change in the binding affinity of the remaining binding sites. Without loss of generality, we let: Y i1 n αj (10.45) ani ¼ i j¼1
be the (cumulative) effective affinity modification factor. Then, Eqs. (10.37) and (10.40) are reduced to: ½Si KSi ½En Si ¼ En n X ½Sk 1+ ank k KS k¼1 ani
(10.46)
n X iani ½Si
θnS ¼
1 n
i¼1
1+
KSi
n X ani ½Si i¼1
KSi
(10.47)
10.2 LIGAND BINDING AND COOPERATIVITY
573
While it is plausible to have the affinities vary extensively with the successive binding of each ligand, one may find some situations where the variation of affinity becomes negligible after the binding of a few ligands. If we let: α2 ¼ α3 ¼ … ¼ αi ¼ … ¼ αn
82 i n
(10.48)
That is, the binding of third and subsequent ligands will not further affect the structure of the enzyme, thus binding affinity. In this case: i1 Y αj ¼ α1 α2i2
8i > 1
(10.49a)
when i ¼ 1
(10.49a)
j¼1 i1 Y
αj ¼ 1,
j¼1
n Y i1 X ½Si α 1 ½S n α1 ½S α1 n αj i ¼ 2 1 + α2 2 +n 1 i α2 KS KS α2 KS α 2 i¼1 j¼1
(10.50)
n Y i1 X i½Si α1 ½S ½S n1 ½S α1 n αj i ¼ n 1 + α2 +n 1 i α2 α2 KS KS KS KS i¼1 j¼1
(10.51)
Thus: α1 ½S n1 α1 1 + α2 +1 n X nEn ½S α2 α2 KS n Bn S ¼ i½En Si ¼ KS α1 ½S α1 ½S α1 i¼1 1 + α2 + 1 2 +n 1 α2 KS KS α22 α2
(10.52)
α1 ½S n1 α 1 + α2 + 1 n2 B n S ½S KS α2 αn1 n θnS ¼ ¼ Bn S1 KS α1 ½S α1 ½S αn2 + n 1 + α + 1 1 2 αn1 KS KS α22 α22
(10.53)
Now Eqs. (10.52) and (10.53) are closed form expressions for any n 1. The overwhelmingly large number of studies with hemoglobin has provided ample opportunity to use this molecule to show the cooperative binding behavior. Fig. 10.23 shows the oxygen saturation with horse hemoglobin as a function of the oxygen partial pressure. One can observe that the curve fits with the experimental data (symbols) quite well. The fitting parameters, based on Eq. (10.53), is given by KS ¼ 8.715 Torr-O2, α1 ¼ 0.04349, α2 ¼ 64.56, n ¼ 4. Fig. 10.24 shows the oxygen saturation isotherm of human hemoglobin at 25°C and pH 7.3. The curve shown in Fig. 10.24 has parameters given by KS ¼ 61.634 Torr-O2, α1 ¼ 0.03199, α2 ¼ 88.26, n ¼ 4. One can observe that the model, Eq. (10.53), can describe the human hemoglobin binding on oxygen quite well. There are 4 oxygen-binding sites available on each hemoglobin molecule, as indicated from the model fit.
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
FIG. 10.23 Oxygen saturation isotherm for horse hemoglobin. The symbols are data as calibrated by Pauling (1935) for pH 8.3. The curve is data fitting to Eq. (10.53) with KS ¼ 8.715 Torr-O2, α1 ¼ 0.04349, α2 ¼ 64.56, n ¼ 4.
1.0
0.8 O2 saturation
574
0.6
0.4
0.2
0.0 0
2
4
6
8
10
pO2, Torr
1.0
0.8 O2 saturation
FIG. 10.24 Oxygen saturation isotherm for human hemoglobin at pH 7.4 and 25°C. The symbols are experimental data taken from Imai, K., 1973. Biochemistry 12, 798–808) and the curve is based Eq. (10.53).
0.6
0.4
0.2
0.0 1
10
pO2, Torr
100
The human hemoglobin is known to have four subunits or to be tetramers. Based on the isotherm fitting, there are 4 oxygen-binding sites, which give one oxygen-binding site per subunit. Also, the binding affinity of the first oxygen-binding site on a hemoglobin molecule corresponds to KS ¼ 61.634 Torr, which is lower than for the second binding site as α1 ¼ 0.03199. In effect, the binding coefficient for the first site is equivalent to 61.634/4 Torr-O2 ¼ 15.4085 Torr-O2 because all four sites are equally accessible. The subsequent sites have an oxygen affinity corresponding to KS/α2 ¼ 0.6986 Torr. Therefore, hemoglobin is a remarkable cooperative transporter that has an appropriately low affinity (82.634 Torr) in the lungs, allowing more oxygen to be bound when oxygen is abundant and an appropriately high affinity (0.6986 Torr) in the deoxygenated capillaries, allowing the oxygen to be carried farther.
575
10.2 LIGAND BINDING AND COOPERATIVITY
10.2.2 Sequential Single Ligand Binding on Allosteric Enzymes Fig. 10.25 shows an illustration of the ligand binding on an allosteric enzyme. Adair (1925) and Klotz et al. (1946) pioneered the analysis on allosteric binding. Klotz (2003) likened the ligand binding on allosteric enzymes to the dissociation of multivalent acid or base. The binding and dissociation are strictly sequential and all the binding sites are distinctively different. Following the framework of Klotz et al. (1946), we generalize the sequential binding of substrate for reactions (10.27) and (10.28) by: En + S !En S, En S !En + S, En S + S !En S2 , En S2 !En S + S, En Si1 + S !En Si ,
r + n1 ¼ kn1 + ½En ½S
(10.54a)
rn1 ¼ kn1 ½En S
(10.54b)
r + n2 ¼ kn2 + ½En S½S
(10.54c)
rn2 ¼ kn2 ½En S2
(10.54d)
r + ni ¼ kni + ½En Si1 ½S
(10.54e)
rn2 ¼ kni ½En Si
(10.54f)
En Si !En Si1 + S,
Only when binding occurred on the first site can the substrate further bind on the second site. The bindings on the allosteric sites are expected to have different affinities. To determine the isotherm of substrate binding on protein/enzyme, we assume that thermodynamic equilibrium applies to the binding of substrates on all the sites, ie, the overall rate for each site is zero. One can show that: ½En ½S K1S
(10.55)
½En Si1 ½S ½En ½Si ¼ i Y KiS KjS
(10.56)
½En S ¼ ½En Si ¼
j¼1
where KiS ¼
K1S
K2S
kni kni +
(10.57)
K3S
K4S
FIG. 10.25 Schematics of ligand binding on an enzyme with four allosteric sites. The ligand binding occurs one at a time in an orderly manner. When the concentration of ligand molecules increases, increasing number of ligand molecules bound on the sites of the enzyme molecule. Successive binding of ligand molecules can result in the changes in the conformal structure (as indicated by the green arrow) of the enzyme, and the binding affinity is different for the different (allosteric) sites.
576
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
is the substrate binding constant on the ith allosteric site. With Eq. (10.57), we have included the affinity change factor into the affinity coefficient itself. KiS is a combination of the affinity coefficient and the cooperativity factor due to the binding of the prior substrate molecules. The overall enzyme balance leads to: En ¼ ½ E n +
n X
n X ½Si ½En Si ¼ ½En + ½En i Y i¼1 i¼1 KjS
(10.58)
j¼1
Leads to: ½En ¼
En n X ½Si 1+ i Y i¼1 KjS
(10.59)
j¼1
En ½En Si ¼
½Si i Y KjS j¼1
n X
½Sk 1+ k Y k¼1 KjS
(10.60)
j¼1
The total bound substrates on the allosteric enzyme of n-binding sites, BnS, can be computed by: En Bn S ¼
n X
i½En Si ¼
i¼1
n X i½Si i Y i¼1 KjS j¼1
n X ½Si 1+ i Y i¼1 KjS
(10.61)
j¼1
When n ¼ 1, the substrate binding isotherm Eq. (10.61) is reduced to: B1 S ¼
E1 ½ S K1 + ½S
(10.62)
which is similar to the Langmuir adsorption isotherm of solid adsorbates. When [S] ! 1, Eq. (10.4) is reduced to: Bn S1 ¼ En n
(10.63)
577
10.2 LIGAND BINDING AND COOPERATIVITY
ie, the amount of saturation bound substrate is identical to the number of binding sites on an enzyme molecule times the amount of enzyme (as each allosteric enzyme molecule/complex has n binding sites available), which is intuitive. Therefore, Eq. (10.61) can also be written as: n 1X i½Si i n i¼1 Y KjS
θnS ¼
Bn S ¼ B n S1
j¼1
(10.64)
n X ½Sk 1+ k Y k¼1 KjS j¼1
where θnS is the ratio of substrate saturation on the allosteric enzyme of n-binding sites. The two asymptotes for Eq. (10.64) are similar to those for Eq. (10.40). When [S] is small, ie, [S] < KjS, Eq. (10.64) reduces to its limiting behavior: θnS
½S nðK1S + ½SÞ
(10.65)
and when [S] is high, ie, [S] > > KjS, then: n1 KnS + ½S θnS n KnS + ½S
(10.66)
Both Eqs. (10.65) and (10.66) can be easily confused with the Michaelis-Menten equation (10.6). However, Eq. (10.65) has a lower saturation value, while (10.66) has a nonzero intercept when extended to a zero substrate concentration. Eq. (10.64) can be reduced to Eq. (10.47), if we let: ani ¼
Kni i Y KjS
(10.67a)
j¼1
Or αi ¼
ði + 1ÞnK1S ðn iÞKi + 1S
(10.67b)
TABLE 10.4
Stoichiometric Binding Constants for Binding of Azosulfathiazole by Serum Albumin
I
1
2
3
4
5
6
7
8
9
10
125
45.0
21.6
11.7
67.0
3.99
2.43
1.51
0.95
0.60
1
0.8
0.3830
0.3403
0.3222
0.3420
0.4801
1
1/KiS, mM αi1
Data from Klotz (2003).
0.6481
0.5348
0.4466
578
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
1.0
VC I–II, K1S = 0.0166 mM, K2S = 0.0992 mM VC II, KS = 0.0166 mM
Glycine Saturation
0.8
VC I, KS = 0.0992 mM
0.6 0.4 0.2 0.0 10−8
10−7
10−6
10−5 10−4 10−3 Glycine Concentration (M)
10−2
10−1
FIG. 10.26 Allosteric interaction demonstrated on ligand binding isotherm. A comparison of binding curves for glycine with the normal VC I-II containing both allosteric sites and with the altered VC II containing only one site. VC is an RNA riboswitch. The symbols are data taken from Mandal et al., 2004. The curves are fittings to Eq. (10.64), with parameters shown on the upper left corner.
Experimentally, Eqs. (10.40) and (10.64) cannot be distinguished. The coefficients can be related through ani or redefining αi, (Eqs. 10.45, 10.67a and 10.67b). Therefore, the allosteric enzymes and homosteric enzymes have similar binding behaviors experimentally. Table 10.4 shows the binding coefficients of azosulfathiazole by serum albumin. The total number of binding sites is n ¼ 10, and K10 ¼ 0.08 mM. Both parameters for Eqs. (10.40) and (10.64) are listed. Fig. 10.26 shows the allosteric interaction based on substrate binding isotherms between an allosteric enzyme and a modified enzyme that contains only one of its two allosteric sites. When only a single binding site is available on a protein/enzyme/RNA molecule, the isotherm follows Eq. (10.64) with n ¼ 1. This is clearly shown in Fig. 10.26, where one of the site in the two-sited RNA molecule has a binding affinity equal to that of the single-site binding affinity. One can observe that K2S ¼ 0.0166 mM for VC I-II is the same as the K1S for VC II alone. The binding curve for VC I alone is also plotted in Fig. 10.26. One can observe that the sequential binding of the two allosteric sites falls right between the two binding behaviors of the constitutive sites. The identical affinity of the allosteric binding site on the allosteric enzyme as that when present alone indicates that the binding is strictly sequential on allosteric enzymes and there are no conformational structural changes induced by the binding of the VCI site. This is intuitive, as the sites are not identical and they do not compete in the same way for the ligand molecule. While enzymes are known mostly as proteins, we are aware of a few essential RNA ribozyme activities. Bacteria use many RNA molecules as riboswitches, which upon binding some specific metabolic ligand alter their structure and are then able to bind to a DNA regulatory site to influence the expression of genes coding for enzymes that specifically metabolize the ligand that activates that particular riboswitch. Most such riboswitches appear to have a single-ligand binding site (Mandal et al., 2004). This example demonstrating the effect of cooperativity is taken from Mandal et al. (2004) who demonstrated that a specific RNA
579
10.2 LIGAND BINDING AND COOPERATIVITY
riboswitch from Vibrio cholerae has two aptamers with binding sites for glycine, VC I and VC II. The genes controlled by this riboswitch express enzymes for the catabolism of glycine as an energy source, and ideally should only be turned on when glycine is abundant in order to permit its normal use for protein synthesis. Protein enzymes show cooperativity due to the fact that they have more than one binding site within one oligomer, even though this is, in most cases, proportional to a stoichiometry of one per subunit. While RNAs do not normally form oligomers, a cooperative response is possible with this riboswitch by virtue of the fact that it contains two binding sites on a single RNA chain. Binding of a glycine at the first of these sites then induces a conformational change in the RNA, leading to formation of the adjacent site to enable its binding of glycine, which explains the classic cooperative binding curve shown in Fig. 10.26. The comparison to a single-site binding isotherm is made possible by the authors by making a modified version of the riboswitch, in which site I has been deleted. As shown in Fig. 10.26, the modified RNA (VC II) now has a simple saturation-binding curve, consistent with a single site at a constant affinity. The cooperative response is ideal for a switch, since it greatly narrows the concentration range over which the riboswitch becomes fully active. While it is highly improbable for allosteric enzymes, if the binding affinities are identical for all the allosteric sites, then: KS ¼ K1S ¼ K2S ¼ … ¼ KiS ¼ … ¼ KnS
(10.68)
n + 1 n ½S ½S 1+n ðn + 1 Þ ½S KS KS θnS ¼ KS ½S 2 ½S n n 1 1+ KS KS
(10.69)
Eq. (10.64) is reduced to:
Eq. (10.69) represents the binding isotherm for site-sequential or preferential binding of a single ligand on an allosteric enzyme containing sites of identical affinities.
10.2.3 Single-Ligand Binding on Random-Access Allosteric Enzymes We discuss another case where all the binding sites are equally accessible, but they are all different. This is similar to homosteric enzyme; however, each site is different. Fig. 10.27 shows a schematic of random access of a single ligand on an allosteric enzyme. When first substrate molecule approaches the enzyme molecule: Ki1
En + S ! En Si1
(10.70)
where the subscript i1 stands for the location of the binding site among (1,2, …, n). The subsequent binding can be written as: Ki j
En Si1 …Sij1 + S ! En Si1 …Sij1 Sij
(10.71)
As expected, the affinity will change factor after each successive binding of ligands:
580
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
FIG. 10.27
A schematic of random access of a single ligand on an interactive enzyme with three allosteric sites. The ligand can bind on any one of the three sites at any time, and each successive binding results in a change in the enzyme conformal structure, leading to a change in the affinity of binding on the next available allosteric site.
a1−1K2
K1
−1 K3 a12
a1−1K3 a2−1K1
K2
−1 K2 a13
a2−1K3 a3−1K1 −1 a23 K1
a3−1K2
K3
h i ½En ½Sj En Si1 …Sij1 Sij ¼ α i1 , …, ij1 j Y Kim
(10.72)
m¼1
where α(i1, …, ij1) is the affinity change factor after j 1 sites (i1, …, ij1) are bound by ligand S. In terms of the power law of the substrate, one can obtain: n X n X ⋯ E n Sj ¼ i1 ¼1 i2 ¼1 i2 6¼i1
¼
n X n X i1 ¼1 i2 ¼1 i2 6¼i1
h
n X
En Si1 …Sij1 Sij
i
ij ¼1 ij 6¼i1 , …, ij1
⋯
n X
½En ½Sj α i1 , …, ij1 j Y ij ¼1 ij 6¼i1 , …, ij1 Kim
(10.73)
m¼1
Overall enzyme balance leads to: ½En ¼ 1+
n X
½S
n X n X j
En ⋯
i1 ¼1 i2 ¼1 i2 6¼i1
j¼1
n X
α i1 , …, ij1
ij ¼1 ij 6¼i1 , …, ij1
j Y
(10.74)
Kim
m¼1
½Sk
n X n X i1 ¼1 i2 ¼1 i2 6¼i1
½En Sk ¼ En 1+
n X j¼1
½S
⋯
n X
αði1 , …, ik1 Þ i Y ij ¼1 Kim ik 6¼i1 , …, ik1
n X n X j i1 ¼1 i2 ¼1 i2 6¼i1
⋯
n X ij ¼1 ij 6¼i1 , …, ij1
α i1 , …, ij1
m¼1
j Y m¼1
K im
(10.75)
10.3 COMPETITIVE MULTILIGAND BINDING ON AN INTERACTIVE ENZYME
581
When the affinity change is negligible, ie, α ¼ 1, we can obtain: n n X n n X X 1X 1 j½Sj ⋯ n j n j¼1 Y d Y ½S i1 ¼1 i2 ¼1 ij ¼1 1 + Kim i2 6¼i1 ij 6¼i1 , …, ij1 Kj ½S d½S j¼1 m¼1 ¼ θnS ¼ n n Y Y n ½S ½S 1+ 1+ Kj Kj j¼1 j¼1
¼
(10.76)
n 1 ½S X Kj + ½S n j¼1
Now Eq. (10.76) is binding saturation for a nonsite-sequential allosteric enzyme with negligible affinity change. One can show that Eq. (10.76) is a special case of Eq. (10.47).
10.3 COMPETITIVE MULTILIGAND BINDING ON AN INTERACTIVE ENZYME Although enzymes are very specific in the ligand in that they can interact, still there are ligands that are similar in stereostructural and/or chemical forms. These similar ligands can act on the same enzyme binding sites. When multiple ligands act on an interactive enzyme competitively, the situation is more complicated. We will now have a brief discussion on the competitive ligand binding on interactive enzymes.
10.3.1 Competitive Ligand Binding on a Homosteric Enzyme Fig. 10.28 shows a schematic of two ligands A and B competitively binding on an interactive enzyme of four subunits. One can observe that there is an increasing number of states for the binding over the single ligands. In this section, we shall examine the competitive binding over multiple sites for two-sited, three-sited and four-sited systems. This will allow one to appreciate how the system can be modeled and understand the procedure better. Instead of looking at one species a time (as that shown in Fig. 10.28 in rows), we will examine the competitive binding one site a time sequentially (column-wise from left to right in Fig. 10.28). 10.3.1.1 Two-Site Homosteric Enzyme There are three situations for the binding when focusing on one particular species: (1) one molecule of species j is bound only: [E2(Sj)]; (2) Species j bound on one of the two sites: [E2(Si) (Sj)] + [E2(Sj)(Si)]; and (3) Species j bound on both sites: [E2(Sj)(Sj)]: 2½ E2 S j E2 S j ¼ (10.77) Kj
582
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
E4
Ki1
a −1 i1Ki2 E4Si1
a −1 i1i2Ki3 E4Si1Si2
E4Si1Si2Si3
−1 a i1i2i3 Ki4
E4Si1Si2Si3Si4
A A A
A
A
a−1AAAKA
a−11AAKA
A
A A
KA
a−1AKA
A
A
a−1AAKB
A B
B
a−1AABKA
A
A
A A A
A
a−1AKB
A
a−1AABKB
A A
B A
A
a−1ABKA
a−1AABKB
B
A
A B
A
B A
B
a−1ABKB
B A
a−1ABBKA
A
a−1BKA
B B B B
B
B
a−1ABBKB
A B B
A
KB
B
a−1BKB
B
a−1BBKA
a−1BBBKA
B B
B
a−1BBKB
a−1BBBKB
B B
B
B
B B
B
B
B FIG. 10.28
A schematic of two ligand species A and B competitively binding on an interactive enzyme of four
subunits.
which describes binding on the first site. The binding on the second site is given by: 2 E2 Sj Sj ½ E2 S j ¼ αjð jÞ (10.78) E2 Sj Sj ¼ αjð jÞ 2Kj Kj2 and,
E2 Sj ½Si ½E2 Sj ½Si 0 0 ½E2 Si Sj E2 Sj Si ¼ αið jÞ + αjðiÞ ¼ ðαið jÞ + αjðiÞ Þ Ki Ki K j Kj
(10.79)
10.3 COMPETITIVE MULTILIGAND BINDING ON AN INTERACTIVE ENZYME
583
The concentration or saturation of species j (bound species j) can be computed by
NS X E2 S j + E2 Si Sj + 2 E2 Sj Sj
B2 S j ¼ ½ E2
(10.80)
i¼1 i6¼j
½ E2
NS 2 Sj Sj X Sj ½Si ½Si + 2αjð jÞ + αjðiÞ + αið jÞ ¼2 Kj Kj Ki Ki Kj Kj i¼1 i6¼j
where αi(j) denotes for the affinity change factor for species i after species j is bound on the enzyme. This equation can be reduced to NS B2 S j Sj Sj X ½Si ¼2 + ðαið jÞ + αjðiÞ Þ (10.81) ½ E2 Ki Kj Kj i¼1 The site balance is given by NS X
E2 ¼1+ ½ E2
j¼1
NS X NS NS X X E2 Sj + E2 Si Sj + E2 S j S j j¼1 i¼1 i6¼j
j¼1
½ E2
(10.82)
) ( 2 NS NS NS X X X Sj Sj ½Si 2+ ðαið jÞ + αjðiÞ Þ αjð jÞ 2 ¼1+ Ki Kj Kj j¼1 i¼1 i¼1 The saturation of species j is given by NS Sj Sj X ½Si 2 + ðαið jÞ + αjðiÞ Þ Ki K K j j i¼1 1 B2 Sj 1 ( ) θ2j ¼ ¼ NS NS NS 2 E2 2 X X X ½Sk ½Si ½Si 2 1+ 2+ ðαiðkÞ + αkðiÞ Þ αiðiÞ 2 Kk Ki Ki i¼1 i¼1 k¼1
(10.83)
For noninteracting homosteric enzyme, all the affinity change factors are unity. Eqs. (10.81) and (10.83) are then reduced to NS B2 S j Sj Sj X ½Si ¼2 +2 (10.84) ½E2 Kj Kj i¼1 Ki
θ2j ¼ 1+
Sj Kj NS X ½Si i¼1
Ki
(10.85)
584
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
10.3.1.2 Three-Site Homosteric Enzyme For a three-site homosteric enzyme, there are four situations in binding: (1) Species j bound only: [E3Sj]; (2) Species j bound on one of the three sites: [E3SiSj] + [E3SiSjSk]; (3) Species j bound on two of the three sites: [E3SjSj] + [E3SjSjSk]; and (4) Species j bound on all three sites: [E3SjSjSj]. Binding on the first site, 3½E3 Sj E3 Sj ¼ (10.86) Kj on the second site,
2 2 E3 Sj Sj ½ E3 S j ¼ 3αjð jÞ E3 Sj Sj ¼ αjð jÞ 2Kj Kj2
½E3 Sj ½Si 0 2½E3 Si Sj 0 2 E3 Sj ½Si + αið jÞ ¼ 3ðαið jÞ + αjðiÞ Þ E3 Sj Si ¼ αjðiÞ Kj Ki Ki K j
(10.87)
(10.88)
and on the third site, 3 E 3 Sj Sj Sj ½ E3 S j ¼ αjð j, jÞ αjð jÞ E3 Sj Sj Sj ¼ αjð j, jÞ 3Kj Kj3
E 3 Si Sj Sj E3 Sj Sj ½Si 0 + αið j, jÞ Ki 2Kj 2 2 ½E3 ½Si Sj ½E3 ½Si Sj 3 0 0 ¼ αjði, jÞ ðαið jÞ + αjðiÞ Þ + αið j, jÞ 3αjð jÞ 2 Ki Kj2 Ki Kj2 i ½E ½S S 2 3h 0 3 i j 0 ¼ αjði, jÞ ðαið jÞ + αjðiÞ Þ + 2αið j, jÞ αjð jÞ 2 Ki Kj2 i ½E ½S S 2 3h 3 i j ¼ αjði, jÞ ðαið jÞ + αjðiÞ Þ + 2αið j, jÞ αjð jÞ 4 Ki Kj2 ½E3 Sj ½Si ½Sk E3 Sj Si Sk ¼ αði, j, kÞ Ki Kj Kk
(10.89a)
E3 Sj Sj Si ¼ α0jði, jÞ
(10.89b)
(10.89c)
where αj,i denotes for the affinity change factor after species j and i bound on the enzyme, and αða, b, cÞ ¼ αaðb, cÞ ðαbðcÞ + αcðbÞ Þ + αbðc, aÞ ðαcðaÞ + αaðcÞ Þ + αcða, bÞ ðαaðbÞ + αbðaÞ Þ
(10.90)
The bound species j can be computed by NS NS X NS X X B3 Sj ¼ E3 Sj + E3 Sj Si + 2 E3 Sj Sj + E3 Sj Si Sk i¼1 i6¼j
+2
NS X i¼1 i6¼j
E 3 Sj Si Sj + 3 E 3 Sj Sj Sj
i¼1 k¼1 i6¼j k6¼j
(10.91)
10.3 COMPETITIVE MULTILIGAND BINDING ON AN INTERACTIVE ENZYME
585
Which gives NS NS X NS X 1 Kj B3 Sj ½Si 1 X ½Si ½Sk f1 ¼ ¼1+ ðαið jÞ + αjðiÞ Þ + αði, j, kÞ 3 Sj ½E3 K 3 Ki Kk i i¼1 i¼1 k¼1 i6¼j
(10.92)
i6¼j k6¼j
NS h 2 i ½S Sj X Sj Sj i + αjði, jÞ ðαið jÞ + αjðiÞ Þ + 2αið j, jÞ αjð jÞ + 2αjð jÞ + αjð j, jÞ αjð jÞ 2 Ki 2Kj i¼1 Kj Kj i6¼j
and NS NS NS NS NS X Sj X S j XX E3 ½Si X ½Si ½Sk f0 ¼ ¼1+3 ðαið jÞ + αjðiÞ Þ + αði, j, kÞ ½E3 Ki j¼1 Kj i¼1 k¼1 Ki Kk Kj i¼1 j¼1 i6¼j
i6¼j k6¼j
2 N S h 2 NS NS i ½S X X Sj X Sj i + αjði, jÞ ðαið jÞ + αjðiÞ Þ + 2αið j, jÞ αjð jÞ + αjð jÞ 2 2 K 2Kj i¼1 Kj i j¼1 j¼1 i6¼j
+
NS X
αjð j, jÞ αjð jÞ
j¼1
(10.93)
3 Sj Kj3
The enzyme saturation is then given by
1 B 3 Sj f1 Sj θ3j ¼ ¼ f0 Kj 3 E3
(10.94)
For noninteracting homosteric enzyme, all the α’s are unity. Eqs. (10.91) and (10.94) are then reduced to NS NS X NS X Sj Sj ½Si X Sj ½Si ½Sk B3 S j ¼3 +6 +3 (10.95) ½E3 K Kj K Ki Kj Kk i j i¼1 i¼1 k¼1 NS NS X NS X Sj Sj ½Si X Sj ½Si ½Sk +2 + K Kj K Ki Kj Kk i j i¼1 i¼1 k¼1
θ3j ¼ 1+3
NS X NS NS X NS X NS X ½Sk ½Si X ½Sm ½Si ½Sk +3 + K Ki K Ki Km Kk i k m¼1 i¼1 k¼1 k¼1 i¼1
NS X ½Si i¼1
¼ 1+
Sj Kj NS X ½Si i¼1
(10.96)
Ki
10.3.1.3 Four-Site Homosteric Enzyme For a four-site homosteric enzyme, there are five situations in binding: (1) Species j bound only: [E4Sj]; (2) Species j bound on one of the four sites: [E4Si1Sj] + [E4Si1SjSi2] + [E4Si1SjSi2Si3]; (3) Species j bound on two of the four sites: [E4SjSj] + [E4SjSjSi] + [E4SjSjSi1Si2]; (4) Species j
586
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
bound on three of the four sites: [E4SjSjSj] + [E4SjSjSjSi] + [E4SjSjSjSi1]; and (5) Species j bound on all four sites: [E4SjSjSjSj]. Binding on the first site, 4½E4 Sj (10.97a) E4 Sj ¼ Kj on the second site, 2 2 ½E4 Sj 0 3 E4 Sj Sj 0 ½E4 Sj ¼ 6αjð jÞ ¼ 6αjð jÞ E4 Sj Sj ¼ αjð jÞ 2Kj Kj2 Kj2
3½E4 Si Sj 3 E4 Sj ½Si ½E4 Sj ½Si E4 Sj Si ¼ α0ið jÞ + α0jðiÞ ¼ 6ðαið jÞ + αjðiÞ Þ Kj Ki Ki Kj
(10.98a)
(10.98b)
on the third site, 3 3 2 E4 Sj Sj Sj ½E4 Sj ½E4 Sj 0 0 ¼ 4αjð j, jÞ αjð jÞ ¼ 4αjð j, jÞ αjð jÞ E4 Sj Sj Sj ¼ αjð j, jÞ 3Kj Kj3 Kj3 2 E4 S i S j S j 2 E4 Sj Sj ½Si 0 0 E4 Sj Sj Si ¼ αjði, jÞ + αið j, jÞ Ki 2Kj 2 2 ½E4 ½Si Sj ½E4 ½Si Sj 0 0 ¼ αjði, jÞ 6ðαið jÞ + αjðiÞ Þ + 2αið j, jÞ 6αjð jÞ Ki Kj2 Ki Kj2 h i ½E ½S S 2 4 i j 0 0 ¼ 6 αjði, jÞ ðαið jÞ + αjðiÞ Þ + 2αið j, jÞ αjð jÞ 2 Ki Kj h i ½E ½S S 2 4 i j ¼ 3 αjði, jÞ ðαið jÞ + αjðiÞ Þ + 2αið j, jÞ αjð jÞ Ki Kj2 2½E4 Si1 Si2 Sj 3½E4 Si1 ½Si2 Sj 0 0 0 + ⋯ ¼ 2αjði1, i2Þ αi1ði2Þ +⋯ E4 Sj Si1 Si2 ¼ αjði1,i2Þ Kj Ki2 Kj ½E4 ½Si1 ½Si2 Sj ½E4 ½Si1 ½Si2 Sj + ⋯ ¼ 4αði1,i2, jÞ ¼ 4!α0jði1, i2Þ α0i1ði2Þ Ki1 Ki2 Kj Ki1 Ki2 Kj
(10.99a)
(10.99b)
(10.99c)
and on the fourth site,
E4 Si1 Si2 Sj Sj E4 Si2 Sj Sj ½Si1 0 + αi1ðj, i2, jÞ +⋯ E4 Sj Si1 Si2 Sj ¼ 2Kj Ki1 h i ¼ α0jði1,i2, jÞ 2αði1,i2, jÞ + α0i2ðj, i1, jÞ 3 αjði1, jÞ ðαi1ð jÞ + αjði1Þ Þ + 2αi1ð j, jÞ αjð jÞ
α0jði1,i2, jÞ
+ α0i1ðj,i2, jÞ 3
h i ½E ½S ½S S S 4 i1 i2 j j αjði2, jÞ ðαi2ð jÞ + αjði2Þ Þ + 2αi2ð j, jÞ αjð jÞ Ki1 Ki2 Kj Kj
587
10.3 COMPETITIVE MULTILIGAND BINDING ON AN INTERACTIVE ENZYME
h i 2 αjði1, i2, jÞ αði1,i2, jÞ + αi2ðj,i1, jÞ αjði1, jÞ ðαi1ð jÞ + αjði1Þ Þ + 2αi1ð j, jÞ αjð jÞ 3 h i ½E ½S ½S S S 4 i1 i2 j j (10.100a) + αi1ðj, i2, jÞ αjði2, jÞ ðαi2ð jÞ + αjði2Þ Þ + 2αi2ð j, jÞ αjð jÞ Ki1 Ki2 Kj Kj E4 Si Sj Sj Sj E4 Sj Sj Sj ½Si 0 0 + αið j, j, jÞ +⋯ E4 Sj Si Sj Sj ¼ αjði, j, jÞ 3Kj Ki ¼
n h io ½E ½S S S S 4 i j j j 0 0 ¼ αið j, j, jÞ 4αjð j, jÞ αjð jÞ + αjði, j, jÞ 3 αjði, jÞ ðαið jÞ + αjðiÞ Þ + 2αið j, jÞ αjð jÞ Ki Kj Kj Kj h i ½E ½S S S S 3 4 i j j j ¼ αið j, j, jÞ αjð j, jÞ αjð jÞ + α0jði, j, jÞ αjði, jÞ ðαið jÞ + αjðiÞ Þ + 2αið j, jÞ αjð jÞ KKKK 4 i
½E4 Si1 Si2 Si3 E4 Sj Si1 Si2 Si3 ¼ α0i1, i2,i3 Kj
Sj
j
j
j
(10.100b) +⋯
½E4 ½Si1 ½Si2 ½Si3 ¼ α0jði1,i2,i3Þ 4αði1, i2, i3Þ Ki1 Ki2 Ki3 Kj
Sj
+⋯
(10.100c)
½E4 ½Si1 ½Si2 ½Si3 Sj ¼ αði1,i2, i3, jÞ Ki1 Ki2 Ki3 Kj where αða, b, c, dÞ ¼ αaðb, c, dÞ αðb, c, dÞ + αbða, c, dÞ αða, c, dÞ + αcða, b, dÞ αða, b, dÞ + αdða, b, cÞ αða, b, cÞ
(10.101)
When four different ligands bind on all four homosteric sites, there are four interaction factors assuming there is no difference in the order (or sequence) of binding. The bound species j can be computed by NS NS X X B4 Sj ¼ E4 Sj + 2 E4 Sj Sj + E4 Sj Si + 3 E4 Sj Sj Sj + 2 E4 Sj Si Sj i¼1 ij
+
NS X NS X
i¼1 ij
NS X E4 Sj Si1 Si2 + 4 E4 Sj Sj Sj Sj + 3 E4 Sj Si Sj Sj
i1¼1 i2¼1 i16¼j i26¼j
+2
i¼1 ij
NS X NS NS X NS X NS X X E4 Sj Si1 Si2 Sj + E4 Sj Si1 Si2 Si3 i1¼1 i2¼1 i16¼j i26¼j
i1¼1 i2¼1 i3¼1 i16¼j i26¼j i36¼j
(10.102)
588
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
f1 ¼
B4 Sj Kj ½ E4 4 S j
(10.103)
And overall site balance leads to 8 > NS > NS NS < X X X E4 Sj + E4 Sj Sj + E 4 Sj Si + E 4 Sj Sj Sj + E4 Sj Sj Si2 E 4 ¼ ½ E4 + > j¼1 > i¼1 i2¼1 : i6¼j i26¼j +
NS X NS NS X X E4 Sj Si1 Si2 + E4 Sj Sj Sj Sj + E4 Sj Sj Sj Si3 i1¼1 i2¼1 i16¼j i26¼j
+
NS X NS X i2¼1 i3¼1 i26¼j i36¼j
i3¼1 i36¼j
E4 Sj Sj Si2 Si3 +
NS X NS X NS X
E4 Sj Si1 Si2 Si3
i1¼1 i2¼1 i3¼1 i16¼j i26¼j i36¼j
(10.104)
9 > > =
E4 f0 ¼ > ½ E4 > ;
The enzyme saturation is then given by θ4j ¼
B4 Sj f1 Sj ¼ 4E4 f0 Kj
(10.105)
For noninteracting homosteric enzyme, Eqs. (10.103) and (10.105) are reduced to NS NS X NS NS X NS X NS X X X B4 Sj Sj Sj ½Si ½Si1 ½Si2 Sj ½Si1 ½Si2 ½Si3 Sj ¼4 + 12 + 12 +4 (10.106) ½ E4 Ki Kj Kj Ki1 Ki2 Kj Ki1 Ki2 Ki3 Kj i¼1 i1¼1 i2¼1 i1¼1 i2¼1 i3¼1
θ4j ¼ 1+
Sj Kj NS X ½Si i¼1
(10.107)
Ki
10.3.1.4 n-Site Homosteric Enzyme For the competitive binding of two species (S and P) on an n-site homosteric enzyme, the concentration of a given binding is given by
10.3 COMPETITIVE MULTILIGAND BINDING ON AN INTERACTIVE ENZYME
En Si Pj ¼ αði, jÞ
n! ½Si ½Pj ½E n i j i! j! ðn i jÞ! KP K
589 (10.108)
P
where αði, jÞ ¼ αSði1, jÞ αði1, jÞ + αPði, j1Þ αði, j1Þ
(10.109)
αS ði1, jÞ ¼ αPS 8 i, j > 0
(10.110a)
αPði, j1Þ ¼ αSP 8 i, j > 0
(10.110b)
αð j, 0Þ ¼ αðj1, 0Þ αP0
(10.110c)
αð0, iÞ ¼ αð0, i1Þ αS0
(10.110d)
αð1, 0Þ ¼ αð0, 1Þ ¼ 1
(10.110e)
letting
Site balance leads to n X ni En X n! ½Si ½Pj ¼ αði, jÞ ½En i j ½En i¼0 j¼0 i! j! ðn i jÞ! KS K P
½S ½P n ½S ½P ¼ 1 + αPS + αSP + nð1 αS0 Þ + nð1 αP0 Þ KS KP KS KP ½S n ½S n ½P n ½P n + 1 + αS0 1 + αPS + 1 + αP0 1 + αSP KS KS KP KP
(10.111)
And the S-binding is given by n X ni Bn S 1X n! ½Si ½Pj ¼ i αði, jÞ ½ En i j n½En n i¼1 j¼0 i! j! ðn i jÞ! KS K P
½S ½S ½P n1 ½S ¼ αPS 1 + αPS + αSP + ð1 αS0 Þ KS KS KP KS
(10.112)
½S ½S n1 ½S ½S n1 + αS0 1 + αS0 αPS 1 + αPS KS KS KS KS θnj ¼
Bn Sj nEn
(10.113)
590
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
And when all of the interactive coefficients are unity or are for a noninteracting homosteric enzyme, the binding saturation is reduced to Bn Sj ½S ½S ½P n1 ¼ 1+ + (10.114) θnj ¼ nEn KS KS KP
10.3.2 Site-Sequential Multiligand Binding on Allosteric Enzymes 10.3.2.1 Two-Site Sequential Allosteric Multiple-Ligand Interactive Binding When multiple ligands are acting on the same two-sited allosteric enzyme, there is a combination of randomness (no preference on the ligands) and order (site 1 and site 2). The reaction scheme can be represented by: E2 + Sj ! E2 Sj
(10.115a)
E2 Si + Sj ! E2 Si Sj
(10.115b)
E2 Sj + Si ! E2 Sj Si
(10.115c)
E2 Sj + Sj ! E2 Sj Sj
(10.115d)
where E2Sj stands for the two-site allosteric enzyme, with ligand species j bound on the first site, and j i denotes for the two-site allosteric enzyme, with ligand species j bound on the first site and species i bound on the second (and last) site. Chemical equilibria give rise to: ½E2 Sj E2 Sj ¼ (10.116a) K1j E2 Sj ½Si ½E2 Sj ½Si ¼ αj E2 Sj Si ¼ αj K2i K1j K2i ½E2 Si Sj ½E2 ½Si Sj ¼ αi E2 Si Sj ¼ αi K2j K1i K2j
(10.116b)
(10.116c)
Enzyme balance gives rise to: E 2 ¼ ½ E2 +
NS X j¼1
NS X NS X E2 S j + E2 Sj Si
(10.117)
i¼1 j¼1
Substituting Eqs. (10.116a) and (10.116b) into (10.117), we obtain: NS NS NS X Sj X Sj X ½Si E2 ¼1+ + αj K1j j¼1 K1j i¼1 K2i ½ E2 j¼1
(10.118)
591
10.3 COMPETITIVE MULTILIGAND BINDING ON AN INTERACTIVE ENZYME
The ligand species j binding: NS NS X X E 2 Sj Si + E 2 Si Sj + 2 E 2 Sj Sj B2 S j ¼ E 2 S j + i¼1 i6¼j
(10.119)
i¼1 i6¼j
Substituting Eq. (116) into (10.119), we obtain: NS NS B 2 Sj Sj Sj X ½Si Sj X Sj Sj ½Si ¼ + αj + αi + 2αj ½E2 K1j K1j i¼1 K2j K2j i¼1 K1i K1j K2j i6¼j
i6¼j
(10.120)
NS NS Sj Sj X ½Si Sj X ½Si ¼ + αj + αi K1j K1j i¼1 K2i K2j i¼1 K1i
The saturation of species j on the two-site sequential allosteric enzyme is given by: NS NS Sj Sj X ½Si Sj X ½Si + αj + αi K K1i K K K 1j i¼1 2j 2j i¼1 1 B2 Sj 1 1j θ2j ¼ ¼ (10.121) NS NS NS 2 E2 2 X Sj X Sj X ½Si 1+ + αj K1j j¼1 K1j i¼1 K2i j¼1
10.3.2.2 Three-Site Sequential Allosteric Multiple Ligand Interactive Binding When multiple ligands are acting on the same three-sited allosteric enzyme, the reaction scheme can be represented by: ⎯ ⎯→ E3 + Sj ← ⎯⎯ ⎯ ⎯→ E3Sj + Si1 ← ⎯⎯ ⎯ ⎯→ E3SjSi1 + Si2 ← ⎯⎯ ⎯ ⎯→ E3Si1 + Sj ← ⎯⎯
E3Sj, E3SjSi1, E3SjSi1Si2,
E3Si1Sj, ⎯ ⎯→ E3Si1Sj + Si2 ← E ⎯⎯ 3Si1SjSi2, ⎯ ⎯→ E3Si1Si2Sj, E3Si1Si2 + Sj ← ⎯⎯
ð10:122aÞ ð10:122bÞ ð10:122cÞ ð10:122dÞ ð10:122eÞ ð10:122fÞ
where E2Si1SjSi2 denotes for the three-site allosteric enzyme with ligand species i1 bound on the first site, species j bound on the second site, and species i2 bound on the third site. Chemical equilibria give rise to: ½ E3 S j E3 S j ¼ (10.123a) K1j E3 Sj ½Si1 ½E3 Sj ½Si1 ¼ αj (10.123b) E3 Sj Si1 ¼ αj K2i K1j K2i1
592
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
E3 Si1 Sj ½Si2 ½E3 ½Si1 Sj ½Si2 0 ¼ αi1, j αj K3i2 K1i1 K2j K3i2 ½E3 ½Si1 Sj ½Si2 ¼ αi1, j K1i1 K2j K3i2
E3 Si1 Sj Si2 ¼ α0i1, j
(10.123c)
where αi1,j is the overall affinity change of the enzyme owing to the binding of the first site by species i1 and the second site by species j. Enzyme balance gives rise to: E3 ¼ ½E3 +
NS X
NS X NS NS X NS X NS X X E3 S j + E3 Sj Si1 + E3 Sj Si1 Si2
j¼1
i1¼1 j¼1
(10.124)
i2¼1 i1¼1 j¼1
Substituting Eqs. (10.123) into (10.124), we obtain: ! ! NS NS NS NS NS NS X X E3 ½Si ½Si1 X ½Si2 X ½Si2 X ½Si1 X ½Si3 ¼1+ + αi1 + αi1, i2 ½ E3 K K K K K 1i1 i2¼1 2i2 i2¼1 2i2 i1¼1 1i1 i3¼1 K3i3 i¼1 1i i1¼1
(10.125)
The ligand species j binding: NS X B3 Sj ¼ E3 Sj + E3 Sj Si + E3 Si Sj + 2 E3 Sj Sj i¼1 i6¼j
+
NS X NS X E3 Sj Si1 Si2 + E3 Si1 Sj Si2 + E3 Sj Si1 Si2 i2¼1 i1¼1 i26¼j i16¼j
+2
NS X
(10.126)
E3 Si Sj Sj + ½E3 Sj Si Sj + E3 Sj Sj Si Þ + 3 E3 Sj Sj Sj
i¼1 i6¼j
Substituting Eqs. (10.123) into (10.126), we obtain: ! NS NS NS X B3 S j S j ½Si X ½Si2 X ½Si1 ¼ 1 + αj + αi1 ½E3 K1j K K K1i1 i¼1 2i i2¼1 2i2 i1¼1 ! ! NS NS NS NS NS Sj X ½Si X Sj X ½Si1 X ½Si2 ½S2i2 X ½Si1 + αi + α α + K2j i¼1 K1i i1¼1 i1j K1i1 i2¼1 K3i2 K3j i2¼1 K2i2 i1¼1 i1i2 K1i1
(10.127)
The saturation of species j on the three-site sequential allosteric enzyme is given by: θ3j ¼
1 B3 Sj 3 E3
(10.128)
593
10.3 COMPETITIVE MULTILIGAND BINDING ON AN INTERACTIVE ENZYME
10.3.2.3 n-Site Sequential Allosteric Multiple Ligand Interactive Binding In general, for an n-site allosteric enzyme, competitive ligand binding is given by: 8 NS NS NS n X Sji2 X Sj
i2
i1
NS NS X X S j1 Sji1 + ⋯ αj1 ⋯ji1 j K K j ¼1 1j1 j ¼1 i1ji1 1
i1
NS NS 9 NS NS NS n X X X X X Sj1 Sji1 S ji + 1 Sjm1 X Sjm = + ⋯ ⋯ αj1 ⋯ji1 jji + 1 ⋯jm1 K K K Km1jm1 j ¼1 Kmjm ; m¼i + 2 j ¼1 1j1 j ¼1 i1ji1 j ¼1 i + 1ji + 1 j ¼1 1
i1
m1
i+1
m
(10.129) 0 1 NS NS NS NS NS NS n X X X Sj X Sj X ½Sm X Sji @X Sj1 Sji1 A En ¼1+ (10.130) + αm + ⋯ α ½En K1j j¼1 K2j m¼1 K1m i¼3 j ¼1 Kiji j ¼1 K1j1 j ¼1 j1 ⋯ji1 Ki1ji11 j¼1 1
i
i1
The saturation of species j on the n-site allosteric enzyme is given by: θnj ¼
1 Bn S j n En
(10.131)
One can appreciate that conformal structural change due to different ligand species and the competitive bindings on an allosteric enzyme. For a site-sequential noninteractive allosteric enzyme binding of multiligands, the above equations give: k NS n X k X Sj Y X ½Sl K K k¼1 m¼1 mj i¼1 l¼1 il i6¼m
θnj ¼ 1+
NS n Y k X X k¼1 i¼1
½Sl K l¼1 il
(10.132)
10.3.3 Multiligand Binding on Random-Access Allosteric Enzymes 10.3.3.1 Two-Site Random-Access Allosteric Enzyme Multiple species binding on a two-site random-access allosteric enzyme is the simplest case of multiligand interactive binding on a random access allosteric enzyme. In this, the maximum number of any species that can be bound on a given enzyme molecule is 2. There are three situations for the binding when focusing on one particular species: (1) One molecule of species j is bound only: [E2(Sj)1] + [E2(Sj)2]; (2) Species j bound on one of the two sites: [E2(Si)1(Sj)2] + [E2(Sj)1(Si)2]; and (3) Species j bound on both sites: [E2(Sj)1(Sj)2].
594
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
The concentration or saturation of species j (bound species j) can be computed by: NS X ½E2 Sj 1 + E2 Sj 2 + E2 ðSi Þ1 Sj 2 + ½E2 Sj 1 ðSi Þ2 + 2½E2 Sj 1 Sj 2 i¼1 i6¼j
B2 S j ¼ ½ E2
½ E2 2 h
NS i X αj=1 + αi=2 αj=2 + αi=1 Sj Sj Sj +2 + + Sj ½Si + αj=1 + αj=2 ¼ K1j K2i K2j K1i K1j K2j K1j K2j i¼1 i6¼j
where αj/n denotes for the affinity change factor after species j bound on the site n. This equation can be reduced to:
NS αj=1 + αi=2 αj=2 + αi=1 Sj Sj X B 2 Sj ¼ + + Sj ½Si + ½E2 K1j K1j K2i K2j K1i K2j i¼1
(10.133)
The site balance is given by: E2 ¼½E2 +
NS NS X NS X 1X E2 Sj 1 + ½E2 Sj 2 + E2 ðSi Þ1 Sj 2 + ½E2 Sj 1 ðSi Þ2 2 j¼1 j¼1 i¼1 i6¼j
+
NS X
E 2 Sj 1 Sj 2
j¼1
¼½E2 +
NS NS X NS X 1X E2 Sj 1 + ½E2 Sj 2 + E2 ðSi Þ1 Sj 2 + ½E2 Sj 1 ðSi Þ2 2 j¼1 i¼1 j¼1
which can be yielded to: ) ) ( ( NS NS NS NS X X X X Sj Sj E2 ½Si ½Si ¼1+ 1+ αi=2 1+ αi=1 + ½ E2 K2i K1i K1j K2j j¼1 i¼1 j¼1 i¼1
(10.134)
The saturation of species j is given by:
NS αj=1 + αi=2 αj=2 + αi=1 Sj Sj X + + Sj ½Si + K1j K2i K2j K1i K1j K2j 1 i¼1 ( ) ( ) θ2j ¼ NS NS NS NS 2 X X X X Sj Sj ½Si ½Si + 1+ αi=2 1+ αi=1 1+ K2i K1i K1j K2j j¼1 i¼1 j¼1 i¼1
(10.135)
595
10.3 COMPETITIVE MULTILIGAND BINDING ON AN INTERACTIVE ENZYME
For non-interacting allosteric enzyme, Eqs. (10.133) and (10.135) are reduced to: NS NS B2 S j Sj Sj Sj X ½Si Sj X ½Si ¼ + + + ½E2 K1j K2j K1j i¼1 K2i K2j i¼1 K1i 0 NS NS 1 Sj Sj Sj X ½Si Sj X ½Si Sj Sj + + + C B K2j K1j i¼1 K2i K2j i¼1 K1i 1 B C K K 1 K1j 1j 2j C B ! ! θ2j ¼ + ¼ C B N N N N S S S S 2 2@ X X X X ½Si ½Si ½Si ½Si A 1+ 1+ 1+ 1+ K K K K i¼1 1i i¼1 2i i¼1 1i i¼1 2i
(10.136)
(10.137)
10.3.3.2 Three-Site Allosteric Enzyme For a three-site allosteric enzyme, there are four situations in binding: (1) Species j bound only: [E3(Sj)1] + [E3(Sj)2] + [E3(Sj)3]; (2) Species j bound on one of the three sites: [E3(Si)1(Sj)2] + [E3(Sj)1(Si)2] + [E3(Si)1(Sj)3] + [E3(Sj)1(Si)3] + [E3(Si)3(Sj)2] + [E3(Sj)3(Si)2] + [E3(Si1)1(Sj)2(Si2)3] + [E3(Sj)1(Si1)2(Si2)3] + [E3(Si)1(Sj)3(Sk)2] + [E3(Sj)1(Si1)3(Si2)2] + [E3(Si)3(Sj)2(Sk)1] + [E3(Sj)3(Si)2(Sk)1]; (3) Species j bound on two of the three sites: [E3(Sj)1(Sj)2] + [E3(Sj)1(Sj)3] + [E3(Sj)2(Sj)3] + [E3(Sj)1(Sj)2(Si)3] + [E3(Sj)1(Sj)3(Si)2] + [E3(Sj)2(Sj)3(Si)1] and (4) Species j bound on all three sites: [E3(Sj)1(Sj)2(Sj)3]. The bound species j can be computed by: B3 Sj ¼
3 X
NS X NS X NS X 3 X 3 3 X X E3 S j n + E3 Sj n ðSi Þn1 + E3 Sj n ðSi1 Þn0 ðSi2 Þ6nn0
n¼1
+2
i¼1 n¼1 n1¼1 n16¼n i6¼j 3 h X
i1¼1 i2¼1 n¼1 i16¼j i26¼j
NS X 3 h i h i i X E3 Sj n Sj n0 + 2 E3 ðSi Þn Sj n0 Sj 6nn0 + 3 E3 Sj 1 Sj 2 Sj 3
n¼1
i¼1 n¼1 i6¼j
(10.138) where n0 ¼ n + 1 if n < 3, or n0 ¼ 1. ( n0 ¼
n + 1, 8n < 3 1,
if n ¼ 3
(10.139)
596
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
NS 3 3 3 B3 S j X Sj XX X αj=n + αi=n1 ¼ + Sj ½Si ½E3 n¼1 Knj i¼1 n¼1 n1¼1 Knj Kn1i i6¼j
n16¼n
NS X NS X 3 α 3 α X X ðj=n, i=n0, k=0nÞ::2 j=n + αj=n0 2 + Sj ½Si1 ½Si2 + 2 Sj Knj Kn0i K0nk Knj Kn0j n¼1 i1¼1 i2¼1 n¼1
(10.140)
i16¼j i26¼j
+2
NS X 3 α X αðj=1, j2, j=3Þ::2 3 ðj=n, j=n0,i=0nÞ::2 2 Sj ½Si + 3 Sj Knj Kn0j K0ni K1j K2j K3j i¼1 n¼1 i6¼j
where 0n ¼ 6 n n0, αj/n,i/m denotes for the affinity change factor after species j first bound on the site n and then species i bound on site m, and: αða, b, cÞ::2 ¼ αa,b + αa,c + αb,a + αb,c + αc,a + αc,b
(10.141)
That is there are 6 ¼ 3! binding interactions when all three allosteric sites are bound with three different ligands. Eq. (10.140) can be rearranged to give: 0 1 NS α NS X NS α 3 3 X X X C B3 S j X Sj B j=n + αi=n1 ðj=n, i1=n0, i2=0nÞ::2 B1 + ¼ ½ S + ½Si1 ½Si2 C i @ A ½E3 n¼1 Knj Kn1i Kn0i1 K0ni2 n1¼1 i¼1 i2¼1 i1¼1
(10.142)
n16¼n
And overall site balance leads to: ! NS X NS NS NS NS 3 3 X X Sj E3 1X ½Si1 X ½Si2 X ½Si1 X ½Si2 ¼1 + + α + α ½ E3 2! n¼1 i1¼1 Kn0i1 i2¼1 i2=n Kni2 i1¼1 Kni1 i2¼1 i2=n0 Kn0i2 K j¼1 n¼1 nj NS NS NS 1X ½Si1 X ½Si2 X ½Si3 + α 3! i1¼1 K1i1 i2¼1 K2i2 i3¼1 ði3=1, i2=2,i1=3Þ::2 K3i3
(10.143)
For random binding on a noninteracting three-site allosteric enzyme, Eqs. (10.142) and (10.143) are reduced to: 3 NS 3 NS NS 3 3 Sj X Sj X X ½Si X Sj X ½Si2 X B3 S j X ½Si1 ¼ + + ½E3 n¼1 Knj n¼1 Knj i¼1 n1¼1 Kn1i n¼1 Knj i2¼1 K0ni2 i1¼1 Kn0i1 n16¼n
(10.144)
10.3 COMPETITIVE MULTILIGAND BINDING ON AN INTERACTIVE ENZYME
And overall site balance leads to: 3 NS NS NS NS NS NS 3 X X Sj XX ½Si1 X E3 ½Si2 X ½Si1 X ½Si2 X ½Si3 ¼1 + + + ½ E3 Knj n¼1 i1¼1 Kn0i1 i2¼1 Kni2 i1¼1 K1i1 i2¼1 K2i2 i3¼1 K3i3 n¼1 j¼1 0 1 NS 3 Y X Sj A ¼ @1 + K nj n¼1 j¼1
597
(10.145)
The saturation of species j on a noninteracting random allosteric enzyme is given by: !1 3 NS X Sj X 1 1 B3 S j ½Si ¼ K 1+ (10.146) θ3j ¼ 3 E3 K 3 n¼1 nj i¼1 ni One can appreciate that bindings of multiple ligands are more complicated than that of single ligand. For non-site sequential noninteractive allosteric enzyme binding of multi ligands, we can show the binding saturation as given by: ! NS n X d Y ½Sm !1 1+ n NS K X Sj d Sj i¼1 Sj X 1 ½Sm m¼1 im ! ¼ K 1+ (10.147) θnj ¼ NS n K n n i¼1 ij X Y ½Sm m¼1 im 1+ K m¼1 im i¼1 which turned out to be rather “simple.”
10.3.4 Enzymes With Homosteric Paired Allosteric Sites Fig. 10.29 shows examples of a special class of enzyme oligomers, where each subunit contains a pair of active centers: site type 1 and site type 2. The two site types are distinct and represented by diamonds and triangles. Assuming substrates Si 8i ¼ 1, 2, …, NS can bind on site type 1, while ligands Aj 8j ¼ 1, 2, …, NP can bind on the site type 2. Since the bindings of Si and Aj on the allosteric sites, the bindings can be computed almost separately. Site type 1
(i) Monomer
FIG. 10.29
Site type 2
(ii) Dimer
(iii) Trimer
(iv) Tetramer
Oligomeric enzymes of homosteric paired allosteric sites. Homosteric subunits of the enzyme with each subunit containing a pair of allosteric sites: site type 1 and site type 2. The distinctly different sites allow different ligands to be bound.
598
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Following Eq. (10.108), the binding on homosteric pairs of allosteric sites is as given by: ! ! m l Y Y Ajk ½Sik n n Aj1 …Ajl En Si1 …Sim ¼ (10.148) αði1 , …, im , j1 , …, jl Þ ½En l m KSik KAjk k¼1
"
αði1 , i2 , …, im , j1 , …, jl Þ ¼
k¼1
m 1 X αði , i , …, ik1 , ik + 1 , …, im , j1 , …, jl Þ αSði1 , i2 , …, ik1 , ik + 1 , …, im , j1 , …, jl Þ! + ik m + l k¼1 1 2 # l X αði1 , i2 , …, im , j1 , …, jk1 , jk + 1 , …, jl Þ αAði1 , i2 , …, im , j1 , …, jk1 , jk + 1 , …, jl Þ! + jk + k¼1
(10.149) Enzyme balance: En ¼
n X n X Aj1 …Ajl En Si1 …Sim
(10.150)
m¼0 l¼0
The binding concentration or saturation can be computed for each species. We can see the additional complexity over the homosteric enzyme. However, the equations are quite similar:
10.4 CATALYTIC REACTION RATE ON INTERACTIVE ENZYMES As noted in the previous section, binding on interactive enzymes can be allosteric or homosteric. The enzymatic catalysis of cooperative binding can be summarized following Eqs. (10.27) and (10.28). Again, there are two distinctly different situations: a homosteric enzyme, a site-sequential allosteric enzyme, and a random site allosteric enzyme, as illustrated in Fig. 10.21. To simplify the discussions, we shall restrict our discussions on the initial rates, ie, the effect of product(s) and their bindings are negligible. The binding and catalytic reaction on the first site of an interactive enzyme can be stoichiometrically represented by: Kn
kn
En + S ! En S ! En + P
(10.151)
where Kn is the ratio of binding and dissociation rate constants for the first site of the n-site enzyme En, and kn is the turnaround frequency of the catalytic ability of the first binding site. As we have learned from the above: discussions, the interpretation of the binding equilibrium can be different for different types of binding sites.
10.4.1 Catalytic Reactions on Homosteric Sites For a homosteric enzyme, all the binding sites are equal. The binding and catalytic reaction can be represented by: α1 KS i
ði + 1Þβi kn
En Si + S ! En Si + 1 ! En Si + P
81 i n
(10.152)
10.4 CATALYTIC REACTION RATE ON INTERACTIVE ENZYMES
599
The binding of the first substrate molecule can facilitate the reaction to occur, as well as further binding of additional substrate molecules. When an interactive ligand/substrate is bound on a homosteric enzyme, it can induce a change in the conformal structure of the enzyme as whole and thus immediately change the interactive capability of the vacant binding sites. The subsequent binding of substrate molecules can have different affinities, as the conformational change could occur because of the binding of a substrate molecule. The reactivity can also be changed. The Michaelis-Menten saturation coefficient for the first binding is denoted as KS and the catalytic reaction rate constant is denoted as kn, in Eq. (10.151). For the subsequent binding, Eq. (10.152) shows that the “Michaelis-Menten saturation coefficient” is KS/αi and the catalytic reaction rate constant is βikn. Eq. (10.152) shows that the reactivity for each bound site on a homosteric enzyme is the same when multiple sites are bound. The overall catalytic reaction rate for an enzyme with n-equally reactive binding sites, rPn, as illustrated by Eq. (10.152) is given by: rPn ¼ kn ½En S + kn
n X
iβi1 ½En Si
(10.153)
i¼2
Substituting Eq. (10.39) in (10.160), we obtain the general catalytic rate expression for enzymes with n-equally reactive binding sites: n Y i1 X iβ ½Si1 n n+ αj i1 1 i KS i¼2 j¼1 (10.154) rPn ¼ kn En ½S k1 n k1 X n Y ½S KS + ½S αj k1 k KS j¼1 k¼1 Letting: rmax n ¼ nkn En
(10.155)
Eq. (10.154) is reduced to: n Y i1 1X iβ ½Si1 n αj i1i1 n i¼2 i j¼1 KS n ½S k1 n k1 X n Y ½S KS + ½S αj k1 k KS j¼1 k¼1
1+
rPn ¼ rmax
(10.156)
The two asymptotes are worth mentioning. The first one is at low substrate concentration. When [S] < KS, Eq. (10.156) is reduced to: rPn ¼ rmax n ½S
1 + ðn 1Þα1 β1 KS + n½S
½S KS
(10.157)
which can be confused with the Michaelis-Menten equation (10.6), if we had neglected the second-order term in the numerator. Not to mention that when n ¼ 1, Eq. (10.156) is reduced
600
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
to Eq. (10.6). Another case is at high substrate concentrations, [S] ≫ KS; therefore, Eq. (10.156) is reduced to:
rPn ¼ rmax
βn2 n
n1 KS + βn1 ½S αn1 n KS + ½S αn1
(10.158)
which could be easily confused with the Michaelis-Menten equation. However, the intercept at zero substrate concentration is not zero. One can rewrite Eq. (10.156) without loss of generality to: n 1X iβ ani ½Si1 n i¼2 i1 n ½S n X KS + ½S ani ½Si1
1+
rPn ¼ rmax
(10.159)
k¼1
While Eqs. (10.156) and (10.159) are valid in general, it is inconvenient to use them, as there are multiple terms in the equation, and the number of terms increase with increasing number of binding sites. If the binding affinity and catalytic activity are not altered by the binding of substrate(s), both Eqs. (10.156) and (10.159) are reduced to rPn ¼
rmax n ½S KS + ½S
(10.160)
which is identical to the Michaelis-Menten equation. When the variations of affinity and reactivity become negligible from the third site on: α2 ¼ α3 ¼ ….. ¼ αi ¼ … ¼ αn 82 i n
(10.161)
β2 ¼ β3 ¼ … ¼ βi ¼ … ¼ βn1 82 i n
(10.162)
and:
Eq. (10.158) is reduced to:
rPn ¼ rmax n
½S KS
1
α1 β β2 α1 ½S n1 + ðn 1Þ½Sα1 1 + β2 1 + α2 α2 KS α2 KS α1 α1 ½S n ½S α1 1 2 + 2 1 + α2 +n 1 α2 KS KS α2 α2
(10.163)
Chen et al. (2014) studied the structure and catalytic behaviors of a recombinant Clonorchis sinensis hexokinase (rCsHK). In this case, the enzyme is known to be homo-trimeric. Fig. 10.30 shows the initial catalytic rate variation with substrate concentration for three different substrates: glucose, mannose and fructose. Fitting of the data to Eq. (10.163) gives the kinetic parameters for the different substrate as shown in Table 10.5. One can observe from Fig. 10.30 that Eq. (10.163) can describe the kinetic behavior quite well.
601
10.4 CATALYTIC REACTION RATE ON INTERACTIVE ENZYMES
FIG. 10.30
The catalytic activity of a homotrimeric recombinant Clonorchis sinensis hexokinase (rCsHK) on hexose conversion to hexose 6-phosphate. The symbols are data taken from Chen et al., 2014. The system consists of 3 mM ATP, 100 mM KCl, 15 mM MgCl2, 0.4 mM NADH, 5 mM phosphoenolpyruvate, 5 U/mL of rabbit muscle pyruvate kinase, 10 U/mL rabbit muscle lactate dehydrogenase, and 100 mM Tris-Cl (pH 8.5). The curve is fitted with Eq. (10.163).
5
rP/E, mmol/min/g
4
3 Mannose
Glucose
Fructose
2
1
0 0.01
0.1
1
10
[S], mM
TABLE 10.5
Kinetic Parameters of Recombinant Clonorchis sinensis Hexokinase (rCsHK) Responding to Different Substrates. The Number of Binding Sites per Enzyme Molecule is n ¼ 3
Substrate
rmax/E (mmol/(min g))
Fructose Mannose Glucose
KS (mM)
19.43
α1
6.207
373.9
19.10
1857
99.89
α2
β1
β2
17.09
2.254
0.20212
154.7
4.318
8.21 10
2.080
0.01191
1.48 10
1.816
0.002512
33.13
4 6
10.4.2 Catalytic Reactions on Site-Sequential Allosteric Sites For a site-sequential allosteric enzyme, the ligand has preference on the binding sites. The difference of the binding sites and sequence of the binding are only part of the characteristics of the allosteric enzyme. It is intuitive that the reactivity of the sites are different as well: KiS
kci
En Si1 + S ! En Si ! En Si1 + P
81 i n
(10.164)
The binding on the first site by the substrate can facilitate the reaction to occur, and further binding on the other sites and reactions. The catalytic reaction rate constant is different for different sites. For the ith site, the catalytic rate constant is denoted as kci in Eq. (10.164). The overall catalytic reaction rate for an enzyme with n-allosteric reactive binding sites, rnP, as illustrated by (10.164) is given by: rnP ¼ kc1
n X i¼1
½En Si + kc2
n X i¼2
½En Si + … + kci
n X
½En Si + … + kcn ½En Sn
(10.165)
i¼m
if each site on the allosteric enzyme can catalyze the reaction independently and the reactivity is not altered by the binding of substrate.
602
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
By substituting Eq. (10.62) into (10.165), we obtain the catalytic rate expression for sitesequential allosteric enzymes with n-reactive allosteric sites: i X n X i¼1
kcj ½Si
j¼1 i Y
KjS
j¼1
rPn ¼ En 1+
n X ½Si i Y i¼1 KjS
(10.166)
j¼1
which is a general site-sequential allosteric enzyme catalytic rate expression. If the variations of affinity and reactivity becomes negligible from the third site on, or: K3S ¼ K4S ¼ … ¼ KiS 83 i n
(10.167)
kc3 ¼ kc4 ¼ … ¼ kci ¼ … ¼ βcn 83 i n
(10.168)
and:
Eq. (10.166) is reduced to:
kc1 + rnP ¼ En ½S
½S K2S
n2 ½S 1 kc3 ½S2 K3S + ½S K2S K3S ½S 2 1 1 K3S K3S 8 (10.169) n1 9 > > ½S > > > > 1 = < ½S K3S K1S + ½S 1 + ½S > > K2S > > > > 1 : K3S ;
kc1 + kc2 ½kc1 + kc2 + ðn 2Þkc3
½S K3S
n1
Eq. (10.169) is the catalytic rate expression for random-access allosteric enzymes with variable binding and reactivity on the first three sites. To show how the kinetic model fits with actual experimental data, Fig. 10.31 shows the relative catalytic activity of allosteric enzyme, ie, glucokinase (or hexokinase IV). The fitting of the data showed that glucokinase has 2 allosteric binding sites for glucose (n ¼ 2), and the parameters are shown in Table 10.6. The curves in Fig. 10.31 are based on Eq. (10.169), it fitted the data quite well. When an activator is added, the kinetic parameters changed (for both glucose binding sites). One can observe from Fig. 10.31 that hexokinase IV can be modeled with a 2-site allosteric enzyme.
603
10.4 CATALYTIC REACTION RATE ON INTERACTIVE ENZYMES
FIG. 10.31 Allosteric behaviors
1.6
Relative glucokinase activity
1.4 1.2
No effector addition BAD SAHBA GKA (RO0281675) BAD SAHBA + GKA
of glucokinase (hexokinase IV). The symbols are experimental data take from Szlyk, B., et al. 2014. Nat. Struct. Mol. Biol. 21(17), 36–42. The curves are fittings with Eq. (10.169) with parameters shown in Table 10.6.
1.0 0.8 0.6 0.4 0.2 0.0 10−2
10−1
100
101
102
[Glucose], mM
TABLE 10.6
Kinetic Parameters of Glucokinase With Eq. (10.169) kc1En
Effector addition 1. No effector addition
K1S (mM)
0.02121
2. With BAD SAHBA (S118D)
8.640
34.52
10.09
K2S (mM)
kc2/kc1
N
3.690
4903
2
3.391
3.655
2
3. With GKA (RO0281675)
144.8
5.083
1.299
0
2
4. With both BAD SAHBA and GKA
105.5
4.738
0.6764
0.2069
2
10.4.3 Random-Access Allosteric Enzymes For multiligand binding on a random-access allosteric enzymes, the binding saturation of species j is given by n n X n n X X X α i1 , …, ij1 j ½S ⋯ kcj i Y j¼1 i1 ¼1 i2 ¼1 ij ¼1 Kim i2 6¼i1 ij 6¼i1 , …, ij1 m¼1 (10.170) rPn ¼ En n n X n n X X X α i1 , …, ij1 j 1+ ½S ⋯ j Y j¼1 i1 ¼1 i2 ¼1 ij ¼1 i2 6¼i1 ij 6¼i1 , …, ij1 Kim m¼1
Letting:
α i1 , …, ij1 anj ¼ ⋯ i Y i1 ¼1 i2 ¼1 ij ¼1 Kim i2 6¼i1 ij 6¼i1 , …, ij1 n X n X
n X
m¼1
(10.171)
604
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Eq. (10.170) is reduced to: n X
rPn ¼ En
kci ani ½Si
i¼1
1+
n X
(10.172) ani ½S
i
i¼1
where ani is the cumulative affinity to the order of i, and kci is the effective reactivity of the ith order. Eq. (10.172) is conveniently valid for single species binding on all the interactive enzymes.
10.5 KINETICS OF POLYMORPHIC CATALYSIS AND ALLOSTERIC MODULATION In the above section, we have learned that enzyme catalytic rate deviates from the Michaelis-Menten rate if enzymes aggregate to form complex of multiple binding sites. Still, we have dealt only with enzymes that exist in a single form. When polymorphs all show catalytic activity, the situation is more complex. Both binding isotherm and catalytic rate are affected by polymorphs. For a polymorphic system, the total number of enzyme subunits is distributed among the polymorphs. Again, we shall restrict our discussions to the initial rates. Polymorphs can interconvert. The interconversion of polymorphs can affect the enzymatic catalysis. We must then consider how polymorphs interconvert. For this reason, if equilibrium can be assumed, let: nE1 >j En At thermodynamic equilibrium:
h KjEn ¼
(10.173) i
j En
(10.174)
½E1 n
And the overall enzyme unit balance leads to:
! jn jn N N n h i h i X X X X X i n ¼ n E¼ j E n S + j En j En S i n¼1
j¼1
n¼1
j¼1
(10.175)
i¼1
where jn is the total number of polymorph members that each has n subunits and N is number of subunits of the polymorph that has maximum number of subunits.
10.5.1 Substrate-Free Polymorph Interconversion When enzymes have substrate bound, a polymorph may be stabilized against conversion to different polymorphs. Fig. 10.32 illustrates a substrate-free polymorph. In this case,
605
10.5 KINETICS OF POLYMORPHIC CATALYSIS AND ALLOSTERIC MODULATION
Ligand-Free Polymorph E1
E2
E3
En Ei + Ek
jEi+k
j Ei
+S
jEi
S
jEi
Sm + S
jEi
Sm+1
Increasing ligand concentration
FIG. 10.32 A schematic of ligand-free polymorph system. A polymorph is a mixture of oligomers of enzyme protomer E1. The oligomers or polymorph members can interconvert only when no ligands are bound.
polymorphs can only interconvert when no substrates were bound. Eq. (10.174) can be expressed for ligand-free enzyme polymorphs: h i n (10.176) j En ¼ KjEn pe where jEn is the jth polymorph member with n substrate subunits, KjEn is the equilibrium constant between the singular polymorph member and the jth polymorph member with n subunits, and pe is polymorph enzyme parameter, an analog to the concentration of protomer. Since the interconversion can occur among the polymorphs, the overall enzyme mixture composition is dynamic. Assuming that the interconversion of enzyme polymorphs is fast, we consider the enzyme polymorphs in thermodynamic equilibrium only here. Substituting Eqs. (10.33) and (10.176) into (10.175), one can obtain after rearrangements: ( ) jn N X n Y k1 X X ½Sk n n nKjEn pe 1 + αjnl k (10.177) E¼ k Kjn n¼1 j¼1 k¼1 l¼1 which can be solved for pe, irrespectively of whether any form of the protomer (monomeric enzyme) is present in the system. Here, pe is simply a parameter for the polymorph, having the unit of enzyme concentration.
606
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Substituting Eq. (10.176) into (10.33), one can obtain: h ih ii Y i1 i1 h i Y j En S pne KjEn ½Si n n αjnl ¼ α jnl j En Si ¼ i i i i Kjn Kjn l¼1 l¼1
(10.178)
With Eq. (10.178), one can derive the enzyme saturation and the catalytic reaction rate. The total bound enzyme is given by: jn X jn N X n h N X n Y i1 i X X pne KjEn X i½Si n i j En S i ¼ αjnl i1 (10.179) BS ¼ Kjn i¼1 i l¼1 Kjn n¼1 j¼1 i¼1 n¼1 j¼1 The enzyme saturation is given by: θS ¼
jn N n Y i1 KjEn X BS X pne X i½Si n αjnl i1 ¼ E n¼1 E j¼1 Kjn i¼1 i l¼1 Kjn
(10.180)
The overall catalytic reaction rate for an enzyme with n-equally reactive binding sites of invariable reactivity, rPn, as illustrated by Eq. (10.153) is given by: rPn ¼kn ½En S + kn
n X
iβni1 ½En Si
i¼2
Y n X n i1 pne KEn ½S pn KEn ½Si + kn iβni1 αjnl e i ¼kn n Kn Kn i l¼1 i¼2 Summing up all the polymorphs, one obtains the overall rate: ( ) Y jn N n i1 X X KjEn kjn 1X ½Si1 n n npe 1+ iβ α rP ¼ ½S Kjn n i¼2 jni1 i l¼1 jnl Kni1 n¼1 j¼1
(10.181)
(10.182)
which is the general overall catalytic rate expression for a substrate-free interconvertible polymorph. 10.5.1.1 Substrate-Free Interconvertible Polymorph Of Limiting Structural Changes If Eq. (10.48) holds, or: αjn2 ¼ αjn3 ¼ … ¼ αjni ¼ … ¼ αjnn 82 i n Then the polymorph ratio is given by: ( !) jn N X X αjn1 αjn1 αjn1 ½S n ½S n npe KjEn 1 2 + 2 1 + αjn2 +n 1 E¼ αjn2 Kjn Kjn αjn2 αjn2 n¼1 j¼1
(10.183)
(10.184)
When both Eqs. (10.161) and (10.162) are applicable, Eq. (10.182) is reduced to: ( ) jn N X X αjn1 αjn1 βjn1 βjn2 KjEn kjn ½S n1 n npe 1 βjn2 + ðn 1Þ½Sαjn1 + βjn2 1 + αjn2 rP ¼ ½ S Kjn αjn2 Kjn αjn2 Kjn n¼1 j¼1 (10.185)
607
10.5 KINETICS OF POLYMORPHIC CATALYSIS AND ALLOSTERIC MODULATION
which is the catalytic rate form for a substrate-free interconvertible polymorph of homosteric enzymes, when the substrate-binding affecting affinity and reactivity are only in the first two successive bindings. 10.5.1.2 Substrate-Free Interconvertible Polymorph of No Structural Changes In the extreme case where substrate binding is not affecting the enzyme structure, ie, αjnl 1, Eq. (10.177) is reduced to: E¼
N X
npne
½S n KjEn 1 + Kjn j¼1
jn X
n¼1
(10.186)
Furthermore, when βjnl 1, Eqs. (10.185) or (10.182) is reduced to: rP ¼ ½S
N X
n½E1 n
n¼1
jn X KjEn kjn ½S n1 1+ Kjn Kjn j¼1
For a bimorph having the same number of subunits, Eq. (10.186) is reduced:
½S n ½S n E ¼ npne K1En 1 + + K2En 1 + K1n K2n
(10.187)
(10.188)
which can be solved for the parameter pe: npne ¼
E n ½S ½S n K1En 1 + + K2En 1 + K1n K2n
(10.189)
Substituting Eq. (10.189) into (10.187), we obtain: K1En k1n ½S n1 K2En k2n ½S n1 1+ + 1+ K1n K2n K K2n 1n n rP ¼ E½S ½S ½S n K1En 1 + + K2En 1 + K1n K2n
(10.190)
which is readily reduced to Eq. (10.16). Therefore, the MWC model is applicable only in the case where the interaction of substrate and enzyme is restricted to the form change only. The binding itself does not change the affinity (nor reactivity) of the enzyme molecule. As one substrate molecule is bound on the enzyme, it does not induce any conformal changes that would alter the affinity or the reactivity of the rest of the binding sites.
10.5.2 Substrate-Inert Polymorph Interconversion Fig. 10.33 shows a schematic of a substrate-inert polymorph system. When there are no effects of substrate binding on the polymorph interconversion: j En
¼ KjEn pne
(10.191)
608
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Ligand-Inert Polymorph
• • •
E1
• • •
• • •
• • •
E2
E3 • • •
• • •
En
•••
Ei + Ek
•••
jEi+k
j Ei
+S
Ei S + Ek
•••
j Ei
S
jEi+k
jEi
S
Sm + S
Ei Sm1 + Ek Sm2
j Ei
Sm+1
jEi+k Sm+1
Increasing ligand concentration
FIG. 10.33
A schematic of the ligand-inert polymorph system. A polymorph is a mixture of oligomers of enzyme protomer E1. The oligomers or polymorph members can interconvert without any effects from the ligand binding.
and the enzyme unit balance Eq. (10.175) leads to: jn jn N N X X X X n n npe KjEn E¼ j En ¼ n¼1
j¼1
n¼1
(10.192)
j¼1
The equations with regards to individual polymorphs are still valid. Using Eqs. (10.33) and (10.191), one can derive at the enzyme saturation ratio as: n Y i1 X i½Si n n K p α jEn jnl i1 e jn N X i Kjn BS 1 X i¼1 l¼1 (10.193) θS ¼ ¼ n i 1 X n Y E E n¼1 j¼1 ½Si Kjn + αjnl i1 i Kjn i¼1 l¼1 The overall catalytic reaction rate when all subunits in an enzyme complex are equally reactive can be obtained by summing up all the polymorphs with each giving a rate described by Eq. (10.153). That is: n Y i1 X iβjni1 ½Si1 n n + α jnl jn i1 N i X X Kjn i¼2 l¼1 pne KjEn kjn (10.194) rP ¼ ½S n Y i1 X ½Si1 n n¼1 j¼1 Kjn + ½S αjnl i1 i Kjn k¼1 l¼1
10.5 KINETICS OF POLYMORPHIC CATALYSIS AND ALLOSTERIC MODULATION
609
with the parameter pe determined by Eq. (10.192). When both Eqs. (10.161) and (10.162) are applicable, Eq. (10.194) is reduced to: ½S n1 1 βjn2 + ðn 1Þ½Sαjn1 + βjn2 1 + αjn2 αjn2 Kjn αjn2 Kjn ! n αjn1 αjn1 αjn1 ½S ½S 1 2 + 2 1 + αjn2 +n 1 αjn2 Kjn Kjn αjn2 αjn2 αjn1
rP ¼ ½S
N X n¼1
npne
jn X KjEn kjn j¼1
Kjn
βjn1 βjn2
αjn1
(10.195) When substrate binding does not alter the binding affinity nor the reactivity, ie, αjnl 1 and βjnl 1, Eqs. (10.193) and (10.195) are reduced to: rP ¼ ½S
N X n¼1
npne
jn X KjEn kjn ½S 1 1+ Kjn Kjn j¼1
jn N X KjEn ½S X ½S 1 n np 1+ θS ¼ E n¼1 e j¼1 Kjn Kjn
(10.196)
(10.197)
For a bimorph, Eqs. (10.192) and (10.196) give: n2 E ¼ n1 En1 1 KEn1 + n2 E1 KEn2
1 1 ) K k ½ S K k ½ S En1 n1 En2 n2 1+ + n2 pn2 1+ rP ¼ ½S n1 pn1 e e Kn1 Kn2 Kn1 Kn2
(10.198)
(
(10.199)
letting: f1 ¼
n1 pn1 e KEn1 E
(10.200)
Eq. (10.198) gives: 1 f1 ¼
n2 pn2 e KEn2 E
(10.201)
Substituting Eqs. (10.200) and (10.201) into (10.199), we obtain: (
) f1 kn1 ½S 1 ð1 f1 Þkn2 ½S 1 rP ¼ E½S 1+ + 1+ Kn1 Kn2 Kn1 Kn2
(10.202)
which is readily reduced to Eq. (10.21). Therefore, we have recovered the Morpheein model.
610
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Let us examine the kinetics of human PBGS catalyzed 5-aminolevulinic acid (ALA) to asymmetrically condense to porphibilinogen (PBG): OH
O O
OH
O
ð10:203Þ 2 NH
OH
2
+ 2H 2O
NH 2
O
NH
which was used as a model reaction system for the morpheein model by Selwood et al. (2008). Human PBGS exist as a bimorph, a hexamer, and an octamer, as shown in Fig. 10.15. At neutral pH, human PBGS exist predominantly as the high-activity octamer, while increasing pH will have the octamer converted to the low-activity hexamer. For each polymorph (or bimorph, in this case), the binding and catalytic reaction can be represented by: Kn
En + S ! En S α1 K ni n
iβni kn
! En Si + S En Si + 1 ! En Si1 + PBG + 2H2 O
(10.27) 81 i n
(10.204)
where Kn is the binding saturation coefficient (Michaelis-Menten constant) for the first site of the n-site enzyme En, and 2βn1kn/2 is the turnaround frequency of the catalytic ability of the first pair of the binding sites. Following the derivations above, one can obtain: n Y i1 X iβ ½Si n αnl ni1i Kn kn En i¼2 i l¼1 (10.205) rPn ¼ i n i1 X 2 n Y ½S 1+ αnl i i Kn k¼1 l¼1 and the enzyme unit balance, Eq. (10.192), for the bimorph of 6 and 8 subunits leads to: E ¼ 6p6e KE6 + 8p8e KE8
(10.206)
letting: 6p6e KE6 E
(10.207a)
8p8e KE8 ¼ 1 f6 E
(10.207b)
f6 ¼ Similarly: f8 ¼
The overall initial PBG production rate is given by: rPBG ¼ rP6 + rP8
(10.208)
10.5 KINETICS OF POLYMORPHIC CATALYSIS AND ALLOSTERIC MODULATION
When both Eqs. (10.161) and (10.162) are applicable, Eq. (10.205) gives: ½S n1 β =β 1 1 + α 1 + ðn 1Þ½Sαn2 n1 n2 n2 n nα β KEn pe kn Kn Kn ½S rPn ¼ n1 n2 2αn2 Kn αn1 αn1 ½S n ½S αn1 1 2 + 2 1 + αn2 +n 1 αn2 Kn Kn αn2 αn2 Substituting Eqs. (207) and (10.209) into (10.208), we obtain: ½S 5 β =β 1 1 + α62 1 + 5½Sα62 61 62 rPBG α61 β62 k6 K6 K6 ¼ f6 ½S 6 E 2α62 K6 α61 α61 ½S ½S α61 1 2 + 2 1 + α62 +6 1 K6 K6 α62 α62 α62 7 ½S β =β 1 1 + α82 1 + 7½Sα82 81 82 α81 β82 k8 K8 K8 ½S + f8 2α82 K8 α81 α81 ½S 8 ½S α81 1 2 + 2 1 + α82 +8 1 α82 K8 K8 α82 α82
611
(10.209)
(10.210)
Unlike the rate expressions presented so far, Eq. (10.210) shows the catalytic rate dependence on the substrate concentration to the second order at low substrate concentrations. The reason is that the reaction consumes two moles of the substrate ALA to make one mole of product PBG, as indicated by the reaction (10.203). Fig. 10.34 shows the catalytic behavior of PBGS as affected by the pH, where the lines are based on Eq. (10.210). Assuming that pH only affects the form, we effectively have a polymorph (or bimorph of a hexamer and octamer) that the interconversion is controlled by pH (not the substrate ALA). The binding and reactivity are not affected by pH. The assumptions here may be too strong. In this case, Eq. (10.210) can be applied to fit the experimental data. The kinetic parameters are listed in Table 10.7. FIG. 10.34 Specific production rate of porphobilinogen (PBG) from 5-aminolevulinic acid (ALA), as catalyzed by wild type human PBG synthase. The symbols represent the data are obtained from Selwood et al., 2008. The lines are based on Eq. (10.210) with parameters given in Table 10.7.
40
rPBG/E, mmol/h/g
E = 5 mg/L, pH = 7 30
E = 10 mg/L, pH = 8
20
10 E = 20 mg/L, pH = 8.8 0 0.001
0.01
0.1 1 [ALA], mM
10
100
612
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
TABLE 10.7 Kinetic Parameters Used in Fig 10.30. The Dimers are Assumed to be Inactive in Catalyzing the Reaction, k2 ¼ 0, and the Amount of Hexamer at pH 7 is Negligible n/2fnβn2kn (h21) Polymorph
n
pH 7
pH 8
pH 8.8
Kn (mM)
αn1
αn2
βn1/βn2
Octamer
8
40.01
10.12
2.741
0.3785
294.6
3.130
0.9911
Hexamer
6
14.96
4.416
0.1025
596.3
9.560
0.9856
0
One can observe from Fig. 10.34 that the catalytic rate increases with increasing substrate concentration, decrease with increasing pH except at very low substrate concentrations. The polymorph model can describe the experimental well:
10.5.3 Oligomers of Paired Allosteric Sites Fig. 10.29 shows schematics of some oligomeric enzymes with paired allosteric sites. The allosteric binding of a substrate S and an effector I on an oligomeric enzyme with paired allosteric sites is illustrated in Fig. 10.35. One can derive the kinetic expression straightforwardly by noting that the substrate can bind on one type of sites while the effector always go to the other type of sites. Since the bindings of I and S on the allosteric sites, the bindings can be computed almost separately following Section 10.3.4 as given by: Y i1 ½En ½Si ½En ½Si n n ½En Si ¼ αj ¼ α (10.211a) ð0, iÞ i i KSi KSi j¼1
KS E+S + I
E•S +
+ S
a−1ISKI a−1ISKI
+ I
a −1I0KI I2•E + S
I•E•S2 + S −1 a SIKS I +
+ S I•E•S a−1SIKS + a −1ISKI I
I•E + S
FIG. 10.35
E•S2 + S + a−1S0KS I
I KI
•••
a−1S0KS
a −1ISKI a −1SIKS
I2•E•S +
a−1SIKS
S
a −1SIKS
a−1S0KS
I
a−1ISKI a−1ISKI + S
I•E•S3 + I
a−1SIKS
I2•E•S4 + S
a −1SIKS
I•E•S4 + I
a−1ISKI
a−1ISKI
I2•E•S2 + S
•••
E•S4 +
E•S3 + S + I
I2•E•S3
a−1SIKS •••
An illustration of the kinetics of the effector acting on an oligomeric enzyme of paired homosteric sites. The effector is bound on a different site than where catalytic reaction takes place.
613
10.5 KINETICS OF POLYMORPHIC CATALYSIS AND ALLOSTERIC MODULATION
½En ½Ii n ½Ii En ¼ αði, 0Þ i KIi
(10.211b)
½En ½Si ½Ij n n αð j, iÞ Ij En Si ¼ j j i Ki K
(10.211c)
1 jαðj1, iÞ αIS + iαðj, i1Þ αSI i+j
(10.212a)
S
αð j, iÞ ¼
I
αð j, 0Þ ¼ αðj1, 0Þ αI0
(10.212b)
αð0, iÞ ¼ αð0, i1Þ αS0
(10.212c)
αð1, 0Þ ¼ αð0, 1Þ ¼ 1
(10.212d)
where α(i,j) denotes for the affinity change due to the binding of i-effector molecules (I) and j-substrate molecules (S), αIS stands for the affinity change factor of effector binding when multiple S already bound, and αSI stands for the affinity change factor of substrate (S) binding when multiple effector I molecules already bound. In writing Eqs. (10.211a) through (10.212d), we have assumed that the conformal structural change only occurs once when each effector and substrate molecules are bound, as shown in Fig. 10.35. These equations can be reduced directly from (10.148) and (10.149). The catalytic rate is determined by the rate-limiting steps: iβð j, iÞ kc Ij En Si ! Ij En Si1 + P
rP ¼ kc ½ES + kc
n X i¼2
iβð0, iÞ ½ESi + kc
81 i n n X n X
(10.213)
iβð j, iÞ Ij ESi
(10.214)
j¼1 i¼1
Enzyme balance: En ¼ ½ E n +
n X j¼1
n X n X ½Ii En + En Sj + I i En S j
(10.215)
i¼1 j¼1
Substituting Eqs. (10.211a) through (10.211c) into (10.215), one can obtain: ( ) n n X n X X ½Si ½Ii ½Si ½Ij n n n αð0,i1Þ i + αði1, 0Þ i + ½En (10.216) αð j, iÞ En ¼ ½En + ½En j j i i KS KI Ki K i¼1
i¼1 j¼1
S
Eq. (10.214) can be reduced to: n n X n X i½Si X i½Si ½Ij n n n αð0, iÞ βð0, iÞ i + αð j, iÞ βð j, iÞ j i i KS i¼1 j¼1 j KSi KI i¼1 ( ) rP ¼ k c E n n n X n X X ½Si ½Ii ½Si ½Ij n n n αð0, iÞ i + αði, 0Þ i + 1+ αð j, iÞ j j i i KS KI K i KI i¼1 i¼1 j¼1 S
I
(10.217)
614
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
or: n n X n X i½Si X i½Si ½Ij n n n αð0, iÞ βð0, iÞ i + αð j, iÞ βð j, iÞ j i i KS i¼1 j¼1 j KSi KI i¼1 rmax ( ) rP ¼ n n X n n X X ½Si ½Ii ½Si ½Ij n n n αð0, iÞ i + αði, 0Þ i + 1+ αð j, iÞ j j i i KS KI Ki KI i¼1 i¼1 j¼1
(10.218)
S
where: rmax ¼ nkc En
(10.219)
Solutions of Eqs. (10.212a) through (10.212c) yield αð j, 0Þ ¼ αI0
j1
(10.220a)
i1 αð0, iÞ ¼ αS0
(10.220b)
j1
j1
j1
i1 i1 i1 αð j, iÞ ¼ AαSI αIS + BαS0 αIS + CαSI αI0
8i, j > 0
(10.220c)
With B¼
αSI ðαIS + αSI Þ 2αS0 αIS 6ðαSI αS0 Þ
(10.221a)
C¼
αIS ðαIS + αSI Þ 2αI0 αSI 6ðαIS αI0 Þ
(10.221b)
αIS + αSI BC 2
(10.221c)
βð0, iÞ ¼ β0i1
(10.222a)
A¼ we further let
j
βð j, iÞ ¼ βI β1 β2i1 8i, j > 0
(10.222b)
Substituting Eqs. (10.220a) through (10.222b) into (10.218), one can obtain, rP ¼ rmax where
½Sf1 KS f0
½S n ½I n En 1 1 1 1 ¼ 1 αS0 αI0 + αS0 1 + αS0 + αI0 1 + αI0 f0 ¼ KS ½ En I n K
½S ½I n A + 1 + αSI 1 1 + αIS 1 KS KI αSI αIS
n n B ½S ½I + 1 + αS0 1 1 + αIS 1 αS0 αIS KS KI
C ½S n ½I n + 1 + αSI 1 1 + αI0 1 αSI αI0 KS KI
(10.223)
(10.224a)
615
10.6 INFLUENCE OF A COMPETITIVE EFFECTOR ON INTERACTIVE ENZYMES
0.025 2.5 × 10−4 M 0.020
r, OD291/min
1.0 × 10−4 M 0.015 5.0 × 10−5 M 0.010
2.5 × 10−5 M
FIG. 10.36 Catalytic activity of CTP synthetase as affected by an ignitor: GTP, for the conversion of glutamine. The symbols are data taken from Levitzki, A., Koshland, Jr., D.E., 1969. Negative cooperativity in regulatory enzymes. Proc. Natl. Acad. Sci. U.S.A. 62 (20), 1121–1128. The lines are based on Eq. (10.223) with parameters given below
[GTP] 0.005
0.000 10−2
10−1
100
101
102
[Glutamine], mM
rmax, OD291/ min
KS, mM
KI, mM
αS0
αI0
αSI
αIS
β0
βI
β1
β2
n
14.19
0.0888
29.59
0.1143
141.68
0.22159
163.73
0
0.3605
3.0228
1.048
4
½S n1 AβI ½S n1 ½I n + 1 + αSI β2 1 + αIS β1 1 f1 ¼ 1 + αS0 β0 αIS KS KS KI n1 n
Bβ ½S ½I 1 + αIS β1 1 + I 1 + αS0 β2 αIS KS KI n1 n
Cβ ½S ½I + I 1 + αSI β2 1 + αI0 β1 1 αIS KS KI
(10.224b)
For catalytic enzyme applications, allosteric effector binding can greatly enhance the catalytic activity as the number of catalytic sites is unaffected. Fig. 10.36 shows such a case, where the catalytic rate is increased by the addition of GTP for the conversion of glutamine to CTP by CTP synthetase. The CTP synthetase here is of a tetramer with each monomer unit containing both GTP and Glutamine (different) sites. It is understandable that the great increase of catalytic activity is found for an enzyme with allosteric sites for effector(s). The binding of effector on different (other than substratebinding) sites also made the kinetic equation easier to apply.
10.6 INFLUENCE OF A COMPETITIVE EFFECTOR ON INTERACTIVE ENZYMES We have discussed the competitive binding of ligands on interactive enzymes in Section 10.3. The theory on competitive binding can be applied to examine the influence of a competitive effector on interactive enzymes.
616
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
When competitive binding is examined for interactive enzymes, one may think of homosteric enzymes. As discussed in Section 10.3, competitive binding can occur in both homosteric and allosteric enzymes. In this section, we shall use the example of glucokinase, which is an allosteric enzyme. Glucokinase has two allosteric sites on each molecule and they both can bind glucose and fructose. This is a simple situation of the class of problems we have discussed in Section 10.5.3. Fig. 10.37 shows a schematic of the competitive binding of substrate S and effector I on a two-sited allosteric enzyme. The sequential or preferential competitive effector binding on a site-sequential allosteric enzyme for a two allosteric site enzyme can represented stoichiometrically by: K1S
kc1
K2S
kc2
E + S ! ES ! E + P
(10.225a)
ES + S ! ES2 ! ES + P
(10.225a)
K1I
E + I ! EI
(10.225c)
K2I
EI + I ! EI2 α1 K2I S
kI1
α1 K2S I
kI2
(10.225d)
ES + I ! ESI ! EI + P
(10.225e)
EI + S ! EIS ! EI + P
(10.225f)
The fast equilibrium steps of binding, as shown in reactions (10.225a) through (10.225f), can be written as ½E½S K1S
(10.226a)
½ES½S ½E½S2 ¼ K1S K2S K2S
(10.226b)
½ES ¼ ½ES2 ¼
aSK2S K1S
S
S
aSK2I
S
I
aIK2S
I
S
I
I
S
K1I I
aIK2I
FIG. 10.37
A schematic of competitive sequential binding of S and I on a two-sited allosteric enzyme.
10.6 INFLUENCE OF A COMPETITIVE EFFECTOR ON INTERACTIVE ENZYMES
617
½E½I K1I
(10.226c)
½EI½I ½E½I2 ¼ K1I K2I K2I
(10.226d)
½EIS ¼ αI
½EI½S ½E½I½S ¼ αI K2S K1I K2S
(10.226e)
½ESI ¼ αS
½ES½I ½E½S½I ¼ αS K2I K2I K1S
(10.226f)
½EI ¼ ½EI2 ¼
The enzyme balance leads to: E ¼ ½E + ½ES + ½ES2 + ½EI + ½EI2 + ½EIS + ½ESI
(10.227)
Substituting Eqs. (10.226a) through (10.226f) into (10.227), we obtain: ½ E ¼
E
½S ½S ½I ½I2 αS αI + + + + ½I½S + 1+ K1S K1S K2S K1I K1I K2I K1S K2I K2S K1I 2
(10.228)
The catalytic rate is determined by the rate-limiting steps: rP ¼ kc1 ð½ES + ½ES2 Þ + kc2 ½ES2 + kI1 ½ESI + kI2 ½EIS
(10.229)
which can be reduced to:
kc1 kc1 + kc2 αS kI1 αI kI2 + ½S + ½I + K1S K1S K2S K1S K2I K2S K1I rP ¼ E½S ½S ½S2 ½I ½I2 αS αI + + + + ½I½S + 1+ K1S K2I K2S K1I K1S K1S K2S K1I K1I K2I
(10.230)
When the substrate concentration is fixed while measuring the catalytic rate variation with the competitive effector concentration, one can reduce Eq. (10.230) to: rP ¼ a 0
1 + a1 ½I 1 + b1 ½I + b2 ½I2
(10.231)
with the four constants a0, a1, b1, and b2 defined by: kc1 kc1 + kc2 + ½S K K1S K2S a0 ¼ E½S 1S ½S ½S2 + 1+ K1S K1S K2S
(10.232a)
αS kI1 αI kI2 + K K K2S K1I a1 ¼ 1S 2I kc1 kc1 + kc2 + ½S K1S K1S K2S
(10.232b)
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
FIG. 10.38 Variation of Fructose phosphorylation rate on human hexokinase IV with glucose concentration. While hexokinase IV phosphorylates glucose as well, the data shown is for fructose phosphorylation. The symbols represent data taken from Moukil, M.A., van Schaftingen, E., 2001. J. Biol. Chem. 276(6), 3872– 3878, while the line is drawn based on Eq. (10.231).
50
40 rF6P, mmol/min/g
618
30
20
10
0 0
20
40
60
80
100
[Glucose], mM
1 αS αI + ½S + K1S K2I K2S K1I K b1 ¼ 1I ½S ½S2 + 1+ K1S K1S K2S 1
b2 ¼ K1I K2I
½S ½S2 1+ + K1S K1S K2S
!
(10.232c)
(10.232d)
The kinetic nature of the competitive activation can be demonstrated with human hexokinase IV. Fructose is a poor substrate due to having much lower affinity. Human hexokinase IV is known to be monomeric, but has two active heterosteric sites (allosteric) that are accessible to both fructose and glucose. The addition of low concentrations of glucose produces an almost five-fold increase in the enzyme’s activity with fructose (Fig. 10.38). The line drawn in Fig. 10.38 is based on Eq. (10.231) with a0 ¼ 10.27 mmol-F6P/min/g, a1 ¼ 1.587 mM1, b1 ¼ 0.1172 mM1, b2 ¼ 0.01466 mM2. There is a maximum fructose conversion rate, while the activation of the enzyme catalyzed fructose reaction is less at low and high glucose concentrations.
10.7 SUMMARY An enzyme having more than one active center is called an interactive enzyme, or dueoweimei (多位酶), a multisited enzyme. Enzymes or proteins are commonly present in oligomeric forms for active biological functions. The formation of protein oligomers, either in nature or by artificial means, can take place via covalent bonding or through weak bond network associations. Disulfide bonds are more common in forming covalent protein
619
10.7 SUMMARY
oligomers. Covalent bound protein oligomers usually have additional elements introduced, while proteins associated through weak bond networks usually do not involve any additional bound chemical groups. Proteins are commonly present in stable forms or folds. Oligomerization can occur when stable proteins unfold, creating exposed interfaces for association interactions. One well-known process of weak bond network oligomerization is DS. Separating the two touching/interacting domains creates opportunities for similar interactions with different protein molecules, leading to oligomerization. Both disulfide bonding and DS oligomerization can be dynamic (or reversible), at least at low DPs. The occurrence of oligomerization is usually in response to environmental changes. The dynamic nature of the protein oligomerization is important for bioactivity control. The morpheein model and the polymorph model are thus important in quantifying enzyme-catalyzed biotransformations. Disulfide bonding and DS can act together to form supramolecules. The formation of supramolecules, such as amyloids, in nature can be benign or harmful. The oligomerization of proteins or enzymes allow for multiple active centers or binding sites to be available on one enzyme/protein complex molecule. The multiple active centers provide what is necessary for the regulation of molecule flow or interactions with ligands. Oligomeric enzymes are common interactive enzymes, although interactive enzymes can be monomeric. Oligomeric enzymes provide homosteric active centers (or binding sites). Interactive enzymes can also be allosteric, in that multiple active centers have distinctly different ligand banding affinities and/or catalytic activities. In general, ligand binding on an interactive enzyme results in the interactive enzyme to undergo conformal structural changes. Therefore, even a homosteric interactive enzyme can exhibit variable affinity and/or catalytic activity among different active centers. Treaties have been developed to quantitatively describe the behaviors of interactive enzymes with single and multiple ligands. The complex structure of an interactive enzyme provides opportunity for an effector (or cofactor) or an inhibitor to regulate the biotransformations through either competitive or allosteric regulations. Mechanistic kinetic model expressions have been developed for the complex interactions. The general binding single ligand saturation is given by: ½Si KSi ½ En S i ¼ E n n X ½Sk 1+ ank k KS k¼1 ani
(10.46)
n X iani ½Si
θnS ¼
1 n
i¼1
1+
KSi
n X ani ½Si i¼1
(10.47)
KSi
where En is the total concentration of an interactive enzyme having n-active centers, [EnSi] is the concentration of the interactive enzyme bound with i-substrate S molecules, ani is the effective affinity ratio (or cooperativity) of the ith order, and KS is the characteristic affinity of the substrate on the interactive enzyme.
620
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
When a simple reaction: En
S ! P is catalyzed by an interactive enzyme, the catalytic reaction rate is given by: n X
rPn ¼ En
kci ani ½Si
i¼1
1+
n X
(10.172) ani ½Si
i¼1
For multiple ligands acting on an interactive enzyme, the binding behavior is more complex due to the interactions. For the competitive binding of two species S and P on an n-site homosteric enzyme, the competitive binding is governed by E n S i P j ¼ αð i , j Þ
n! ½Si ½Pj ½En i j i! j! ðn i jÞ! KS K
(10.108)
P
in the case of constant shape change factor, n X ni En X n! ½Si ½Pj ¼ αði, jÞ ½ En i j ½En i¼0 j¼0 i! j! ðn i jÞ! KS KP n ½S ½P ½S ½P ¼ 1 + αPS + αSP + nð1 αS0 Þ + nð1 αP0 Þ KS KP KS KP n n n ½S ½S ½P ½P n + 1 + αS0 1 + αPS + 1 + αP0 1 + αSP KS KS KP KP n X ni Bn S 1X n! ½Si ½Pj ¼ i αði, jÞ ½ En i j n½En n i¼1 j¼0 i! j! ðn i jÞ! KS KP n1 ½S ½S ½P ½S ¼ αPS 1 + αPS + αSP + ð1 αS0 Þ KS KS KP KS ½S ½S n1 ½S ½S n1 + αS0 1 + αS0 αPS 1 + αPS KS KS KS KS
(10.111)
(10.112)
And when all the interactive coefficients are unity or for noninteracting homosteric enzyme, the binding saturation is reduced to Bn Sj ½S ½S ½P n1 ¼ 1+ + (10.114) θnj ¼ nEn KS KS KP For a three-sited homosteric enzyme, the binding of species j in a total of NS ligand species is governed by NS NS NS NS NS X Sj X S j XX E3 ½Si X ½Si ½Sk f0 ¼ ¼1+3 ðαið jÞ + αjðiÞ Þ + αði, j, kÞ ½ E3 Ki j¼1 Kj i¼1 k¼1 Ki Kk Kj i¼1 j¼1 i6¼j
i6¼j k6¼j
621
10.7 SUMMARY
2 N S h 2 NS NS i ½S X X Sj X Sj i + αjði, jÞ ðαið jÞ + αjðiÞ Þ + 2αið j, jÞ αjð jÞ + αjð jÞ 2 2 K 2K Kj i j i¼1 j¼1 j¼1 i6 ¼ j 3 N S X Sj + αjð j, jÞ αjð jÞ 3 K
(10.93)
j
j¼1
NS NS X NS X 1 Kj B3 Sj ½Si 1 X ½Si ½Sk ¼1+ ðαið jÞ + αjðiÞ Þ + αði, j, kÞ f1 ¼ 3 Sj ½E3 K 3 Ki Kk i i¼1 i¼1 k¼1 i6¼j
i6¼j k6¼j
NS h 2 i ½S Sj X Sj Sj i + α ðα + αjðiÞ Þ + 2αið j, jÞ αjð jÞ + 2αjð jÞ + αjð j, jÞ αjð jÞ 2 (10.92) Ki 2Kj i¼1 jði, jÞ ið jÞ Kj Kj i6¼j
1 B 3 Sj f1 Sj θ3j ¼ ¼ f0 Kj 3 E3 For a three-sited sequential allosteric enzyme: ! ! NS NS NS NS NS NS X X E3 ½Si ½Si1 X ½Si2 X ½Si2 X ½Si1 X ½Si3 ¼1+ + αi1 + αi1, i2 ½ E3 K K1i1 i2¼1 K2i2 i2¼1 K2i2 i1¼1 K1i1 i3¼1 K3i3 i¼1 1i i1¼1
! NS NS NS X B3 Sj Sj ½Si X ½Si2 X ½Si1 ¼ 1 + αj + αi1 ½E3 K1j K K K1i1 i¼1 2i i2¼1 2i2 i1¼1 ! ! NS NS NS NS NS Sj X ½Si X Sj X ½Si1 X ½Si2 ½S2i2 X ½Si1 + αi + α α + K2j i¼1 K1i i1¼1 i1j K1i1 i2¼1 K3i2 K3j i2¼1 K2i2 i1¼1 i1i2 K1i1
(10.94)
(10.125)
(10.127)
and for a three-site random accessible allosteric enzyme: ! NS X NS NS NS NS 3 3 X X Sj E3 1X ½Si1 X ½Si2 X ½Si1 X ½Si2 ¼1 + + α + α ½ E3 2! n¼1 i1¼1 Kn0i1 i2¼1 i2=n Kni2 i1¼1 Kni1 i2¼1 i2=n0 Kn0i2 K j¼1 n¼1 nj NS NS NS 1X ½Si1 X ½Si2 X ½Si3 α + 3! i1¼1 K1i1 i2¼1 K2i2 i3¼1 ði3=1, i2=2, i1=3Þ::2 K3i3
0 1 N N N 3 S S S X X αj=n + αi=n1 X X αðj=n, i1=n0, i2=0nÞ::2 C B3 S j Sj B B1 + ¼ ½Si + ½Si1 ½Si2 C @ A ½E3 n¼1 Knj Kn1i Kn0i1 K0ni2 n1¼1 i¼1 i2¼1 i1¼1 3 X
(10.143)
(10.142)
n16¼n
αða, b, cÞ::2 ¼ αa, b + αa, c + αb,a + αb,c + αc,a + αc,b
(10.141)
622
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
Interactive enzymes can exist as a mixture of multiple oligomers that can interconvert. This type of interactive enzymes is called a polymorph. Each oligomer is a member of polymorph (or a polymorph). When there are only two members in a polymorph, the polymorph is called a bimorph. For example, morpheein model deals with a bimorph of constant (invariable) affinity and catalytic activity homosteric enzymes that can interconvert without any effect from the substrate binding. MWC model assumes a bimorph of equal number of homosteric units with invariable affinity and catalytic activity so that interbimorph conversion can occur only when substrate is not bound. When the substrate binding stabilizes the polymorph members, the interconversion between polymorphs can only occur if substrate(s) is no bound. For polymorph of substrate-free interconversion, the substrate-free polymorph concentration is given by: h i n (10.176) j En ¼ KjEn pe where jEn is the jth polymorph member with n substrate subunits, KjEn is the equilibrium constant between the singular polymorph member and the jth polymorph member with n subunits, and pe is polymorph enzyme fraction parameter analogic to the concentration of protomer. The enzyme saturation is given by: jn N n Y i1 KjEn X BS X pne X i½Si n αjnl i1 (10.180) θS ¼ ¼ E n¼1 E j¼1 Kjn i¼1 i l¼1 Kjn and the overall catalytic rate is given by: ( ) Y jn i1 N n i1 X X X K k 1 ½ S n jEn jn npne 1+ iβ α rP ¼ ½S Kjn n i¼2 jni1 i l¼1 jnl Kni1 n¼1 j¼1
(10.182)
which is the general overall catalytic rate expression for a substrate-free interconvertible polymorph. When the interconversion amongst the polymorph members is not affected by the substrate, the total concentration of a polymorph member is given by: j En
¼ KjEn pne
(10.191)
The enzyme saturation ratio is given by:
θS ¼
jn N X X
BS 1 ¼ E E n¼1
j¼1
KjEn pne Kjn +
n Y i1 X n
i¼1 n X
i
l¼1
i1 n Y i
i¼1
αjnl
i½Si i1 Kjn
½Si αjnl i1 Kjn l¼1
(10.193)
And the overall catalytic reaction rate is:
rP ¼ ½S
N X n¼1
pne
jn X j¼1
n+ KjEn kjn
n Y i1 X n
i
αjnl
iβjni1 ½Si1
i1 Kjn n i1 X ½Si1 n Y Kjn + ½S αjnl i1 i Kjn i¼2
k¼1
l¼1
l¼1
(10.194)
10.7 SUMMARY
623
The effect of an effector (or cofactor), either competitive or allosteric, can be derived based on the same principles. For an interactive enzyme with n-pair of active sites (n-homosteric sites for substrate S and another n-homosteric sites for effector I), the rate expression is given by: n n X n X i½Si X i½Si ½Ij n n n αð0, iÞ βð0, iÞ i + αð j, iÞ βð j, iÞ j i i KS i¼1 j¼1 j KSi KI i¼1 rmax ( ) rP ¼ (10.223) n n X n X X n ½Si ½Ii ½Si ½Ij n n n αð0, iÞ i + αði, 0Þ i + 1+ αð j, iÞ j j i i KS KI KSi KI i¼1 i¼1 j¼1
Bibliography Adair, G.S., 1925. The hemoglobin system. VI. The oxygen dissociation curve of hemoglobin. J. Biol. Chem. 63, 529–545. Bohr, C., 1904. Die Sauerstoffaufnahme des genuinen Blutfarbstoffes und des aus dem Blute dargestellten Ha¨moglobins. Zentralbl. Physiol. 23, 688–690. Bohr, C., Hasselbalch, K., Krogh, A., 1904. Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensa¨urespannung des Blutes auf dessen Sauerstoffbindung u¨bt 1. Skandinavisches Archiv fu¨r Physiologie 16, 402–412. Chen, T., et al., 2014. Sequence analysis and molecular characterization of clonorchis sinensis hexokinase, an unusual trimeric 50-kDa glucose-6-phosphate-sensitive allosteric enzyme. PLoS One 9 (9), e107940. Edelstein, S.J., Schaad, O., Henry, E., Bertrand, D., et al., 1996. A kinetic mechanism for nicotinic acetylcholine receptors based on multiple allosteric transitions. Biol. Cybern. 75, 361–379. Fersht, A., 1999. Structure and Mechanism in Protein Science: A Guide to Enzyme Catalysis and Protein Folding. W.H. Freeman, San Francisco. Hill, A.V., 1910. The possible effects of the aggregation of the molecules of hæmoglobin on its dissociation curve. J. Physiol. 40, iv–vii. Jaffe, E.K., 2005. Morpheeins—a new structural paradigm for allosteric regulation. Trends Biochem. Sci. 30, 490–497. Klotz, I.M., 2003. Ligand-receptor complexes: origin and development of the concept. J. Biol. Chem. 279 (1), 1–12. Klotz, I.M., Walker, F.M., Pivan, R.B., 1946. The binding of organic ions by proteins. J. Am. Chem. Soc. 68, 1486–1490. Koshland Jr., D.E., 1958. Applications of a theory of enzyme specificity to protein synthesis. Proc. Natl. Acad. Sci. U. S. A. 44, 98–104. Koshland Jr., D.E., Ne´methy, G., Filmer, D., 1966. Comparison of experimental binding data and theoretical models in proteins containing subunits. Biochemistry 5, 365–385. Lawrence, S.H., Jaffe, E.K., 2008. Expanding the concepts in protein structure-function relationships and enzyme kinetics: teaching using morpheein. Biochem. Mol. Biol. Educ. 36 (20), 274–283. Mandal, M., Lee, M., Barrick, J.E., Weinberg, Z., Emilsson, G.M., Ruzzo, W.L., Breaker, R.R., 2004. A glycinedependent riboswitch that uses cooperative binding to control gene expression. Science 306, 275–279. Marianayagam, N.J., Sunde, M., Matthews, J.M., 2004. The power of two: protein dimerization in biology. Trends Biochem. Sci. 29, 618–625. Mello, B.A., Tu, Y., 2005. An allosteric model for heterogeneous receptor complexes: understanding bacterial chemotaxis responses to multiple stimuli. Proc. Natl. Acad. Sci. U. S. A. 102, 17354–17359. Monod, J., Changeux, J.P., Jacob, F., 1963. Allosteric proteins and cellular control systems. J. Mol. Biol. 6, 306–329. Monod, J., Wyman, J., Changeux, J.P., 1965. On the nature of allosteric transitions: a plausible model. J. Mol. Biol. 12, 88–118. Najdi, T.S., Yang, C.R., Shapiro, B.E., Hatfield, G.W., Mjolsness, E.D., 2006. Application of a generalized MWC model for the mathematical simulation of metabolic pathways regulated by allosteric enzymes. J. Bioinf. Comput. Biol. 4 (2), 335–355.
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Pauling, L., 1935. The oxygen equilibrium of hemoglobin and its structural interpretation. Proc. Natl. Acad. Sci. U. S. A. 21, 186–191. Selwood, T., Jaffe, E.K., 2012. Dynamic dissociating homo-oligomers and the control of protein function. Arch. Biochem. Biophys. 519, 131–143. Selwood, T., Tang, L., Lawrence, S.H., Anokhina, Y., Jaffe, E.K., 2008. Kinetics and thermodynamics of the interchange of the morpheein forms of human porphobilinogen synthase. Biochemistry 47, 3245–3257. Stefan, M.I., Edelstein, S.J., Nove`re, N.L., 2009. Computing phenomenologic Adair-Klotz constants from microscopic MWC parameters. BMC Syst. Biol. 3, 68.
PROBLEMS 10.1. What is a covalent protein oligomer? List the examples of covalent bonding for protein oligomers. 10.2. Why do one study protein oligomerization? How could protein or enzyme oligomerize if no covalent bonds form? 10.3. What is a PLP enzyme? 10.4. What is domain-swapping oligomerization? 10.5. What is a polymorphic enzyme? 10.6. Experimental data of Bohr (1904) for oxygen binding on normal dog blood at 37.8°C is given in Table P10.6. TABLE P10.6
Oxygen Binding of Normal Dog Hemoglobin at 37.8°C as a Function of Partial Pressures of CO2 and O2 Observed
No
pCO 2 (Torr)
pO 2 (Torr)
O2 Uptake
VI.1
2.3
7.2
35.5
VI.2
8.2
6.7
13.1
48.8
6.8
3.4
IV.1
5.7
12.2
37.2
V.3
10.6
12.2
25.9
V.4
28.3
12
16.6
V.2
54.3
12.1
11.8
IV.2
4.6
25.9
85.8
V.6
8.1
26.7
77.2
I.2
8.6
25.4
68.4
II.2
12.9
25.7
63.2
III.2
12.9
26
57.4
V.7
17
31.7
74.6
VIII.1
625
PROBLEMS
TABLE P10.6
Oxygen Binding of Normal Dog Hemoglobin at 37.8°C as a Function of Partial Pressures of CO2 and O2—cont’d Observed
No
pCO 2 (Torr)
pO 2 (Torr)
O2 Uptake
IV.3
89.2
33.2
35
VI.3
3
55.6
94.9
VI.4
25
55.1
87.1
VI.5
88.4
63.9
72.9
IV.4
0–6
150
VII.1
75.2
152.9
99.6
151.3
155.8
96.2
100
Determine the binary oxygen-carbon dioxide-binding isotherm. 10.7. A “Noninteracting” Complex Enzyme. An enzyme E contains two allosteric nonintereacting sites. The two sites behave independently except when a competing allosteric ligand is bound on the other site. When a competitive allosteric ligand is bound on any of the two sites, it causes the enzyme to undergo conformal structural changes and thus change the binding affinity, as well as the catalytic reactivity towards the competitive substrate. Let the reaction: S!P be catalyzed by such an enzyme. (a) Derive a relationship for the overall enzyme-binding saturation as a function of the substrate concentration, if the substrate is the only molecule that can interact with the enzyme. Assume that the two sites have different affinities. (b) Derive a relationship for the overall enzyme binding saturation by the substrate, if an allosteric ligand I can interact with the enzyme and causing conformal changes to the enzyme. Assume that the two sites have different affinities. (c) If the catalytic reaction can occur, following part b), determine the catalytic reaction rate as a function of substrate and ligand concentrations. 10.8. A Growing Polymorph. An enzyme E is employed to convert substrate S to P. The overall stoichiometry is represented by S!P The enzyme E exists in a monomeric form when S is absent. However, when substrate S is bound on the enzyme, it starts to oligomerize. When one substrate is bound on an enzyme molecule, it can grow to a dimer by adding another enzyme molecule. The binding of a substrate molecule on to the second site of the dimer has a higher affinity than on a monomer. When both sites of the dimer are occupied, the dimer
626
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
can grow to a trimer. The saturated trimer can also grow to a tetramer. The affinities of the third and fourth sites are equal. Do the following: (a) Derive a relationship for the overall enzyme binding saturation as a function of the substrate concentration. (b) If the catalytic reaction can occur only with tetramers, determine the catalytic reaction rate as a function of substrate concentration, assuming the concentration substrate is much higher than the concentration of the total enzyme mixture and the binding on substrate on the enzyme oligomers is much faster than the catalytic reaction itself. Product P does not bind on the enzyme at all. (c) Redo part (b), if the catalytic reaction can only occur with the dimers. (d) Compare your results of (b) and (c). Describe the differences in the catalytic behavior. 10.9. A Multiplying Polymorph. An enzyme E exists in a monomeric form when no ligand is present. When the enzyme is employed to catalyze the reaction: S!P The substrate is first bound to the monomeric enzyme E. There are two reactions can occur to a substrate-bound enzyme: (1) Catalytic reaction occurs and product P is not bound on the enzyme; (2) It can form a dimer by reacting with another substratebound monomer. A substrate-saturated dimer can double to a substrate-saturated tetramer. Do the following: (a) Derive a relationship for the overall enzyme binding saturation as a function of the substrate concentration. (b) If the catalytic reaction can occur only with the monomers, determine the catalytic reaction rate as a function of substrate concentration, assuming the concentration substrate is much higher than the concentration of the total enzyme mixture. (c) Discuss the catalytic rate as obtained in part (b). 10.10. Human glucokinase is known to be a monomeric allosteric protein containing two glucose-binding sites. Table P10.10 shows a set of experimental data on the initial rate of glucose phosphorylation (G6P). TABLE P10.10
The Initial Rate of Glucose Phosphorylation as a Function of Glucose Concentration
[G], mM
0
0.78
1.56
2.92
4.86
7.39
14.98
29.77
49.81
r0, mmol/min/g
0
0.24
0.48
1.28
2.57
4.25
6.74
8.34
8.98
Data from Kamata, K., et al. 2004. Structure, 12(3), 429–438.
Derive an appropriate kinetic model to describe the data. 10.11. Human glucokinase is known to be a monomeric allosteric protein containing two hexose binding sites. Table P10.11 shows a set of experimental data on the initial rate of G6P and fructose phosphorylation (F6P).
627
PROBLEMS
TABLE P10.11
The Initial Rates (mmol/min/g) of Glucose and Fructose Phosphorylation as Functions of Glucose and Fructose Concentrations
[F], mM
0
50
[G], mM
rG6P0
rG6P0
100 rF6P0
rG6P0
0
0
0
4
0
0.25
0.53
0.53
5.54
0.53
0.5
0.96
0.96
6.77
1
1.82
1.82
2.5
6.42 18.50
5
200 rF6P0
rG6P0
9.85
rF6P0
0
25.85
13.85
0.53
32.00
0.96
17.54
0.96
37.54
9.85
1.82
22.15
1.82
44.00
6.42
15.38
5.78
32.62
5.45
55.38
16.68
22.15
14.87
40.92
12.19
65.23
Data from Moukil, M.A., Van Schaftingen, E., 2001. J. Biol. Chem. 276(6), 3872–3878.
Derive an appropriate kinetic model to describe the data. If you were to do experiments again, what would you do it differently to elucidate the kinetic model? 10.12. ATP is a known inhibitor for the synthesis of glycogen by glycogen phosphorylase. Table P10.12 shows a set of experimental data on the initial rate of glycogen synthesis.
TABLE P10.12
[ATP] 5 0 [Pi] (mM)
r0/ rmax
The Initial Rate or Catalytic Activity of Rabbit Muscle Glycogen Phosphorylase as a Function of Phosphate and ATP Concentrations. AMP Concentration is Held the same, and the Phosphorylase Concentration is Fixed at 28 μM 3.2 mM [Pi] (mM)
r0/ rmax
6.4 mM [Pi] (mM)
r0/ rmax
10.2 mM [Pi] (mM)
r0/ rmax
15 mM [Pi] (mM)
r0/ rmax
0.102
0.032
0.784
0.069
0.832
0.026
0.866
0.007
1.350
0.007
0.201
0.065
1.117
0.110
1.157
0.035
1.222
0.013
2.073
0.015
0.301
0.100
1.785
0.208
1.848
0.069
1.904
0.024
3.380
0.034
0.414
0.140
3.120
0.311
3.199
0.142
3.247
0.054
4.987
0.066
0.608
0.209
4.697
0.437
4.744
0.240
4.768
0.105
7.213
0.122
0.731
0.198
6.828
0.474
6.862
0.348
6.965
0.174
12.360
0.261
0.776
0.255
12.237
0.590
12.299
0.536
12.299
0.370
16.261
0.376
1.047
0.284
16.180
0.678
16.261
0.594
16.672
0.464
23.638
0.484
1.106
0.341
23.171
0.734
23.520
0.667
23.756
0.584
33.514
0.514 Continued
628
10. INTERACTIVE ENZYME AND MOLECULAR REGULATION
TABLE P10.12
[ATP] 5 0
The Initial Rate or Catalytic Activity of Rabbit Muscle Glycogen Phosphorylase as a Function of Phosphate and ATP Concentrations. AMP Concentration is Held the same, and the Phosphorylase Concentration is Fixed at 28 μM—cont’d 3.2 mM
6.4 mM
10.2 mM
r0/ rmax
[Pi] (mM)
r0/ rmax
[Pi] (mM)
r0/ rmax
[Pi] (mM)
r0/ rmax
1.750
0.402
32.689
0.803
32.852
0.714
33.182
0.660
1.776
0.447
3.089
0.600
3.089
0.545
4.105
0.653
5.455
0.718
6.794
0.708
6.828
0.759
8.089
0.797
33.016
0.848
33.514
0.875
[Pi] (mM)
15 mM [Pi] (mM)
r0/ rmax
Data from Madsen, N.B., Shechosky, S., 1967. J. Biol. Chem. 242(14), 3301-3307.
Rabbit muscle phosphorylase is known to be a dimer, and AMP and ATP are bound on allosteric sites. Derive a suitable kinetic model and evaluate the parameters.