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Significance of reduced dimensionality in reaction kinetics: impact of multi-site particles
Chemistry and Physics of Lipids 101 (1999) 37 – 44 www.elsevier.com/locate/chemphyslip
Significance of reduced dimensionality in reaction kinetics: impact of multi-site particles Gary L. Nelsestuen * Department of Biochemistry, Uni6ersity of Minnesota, 1479 Gortner A6e., St. Paul, MN 55108, USA
Abstract This review examines novel kinetic properties of enzymes on membrane surfaces or states of restricted diffusion. A leading feature is the presence of multiple enzymes and/or substrates per particle. In these states, enzymes can be influenced by parameters such as the number of substrates or enzymes per particle, particle size, the rates of exchange of substrate or enzyme from the particle, or substrate diffusion to the particle. These steps are independent of the enzyme site parameters which are described by classical enzymology. The results make it clear that non-classical behaviors are important to biological systems, are the basis for some enzyme expression levels and are determinants of cellular design. To identify more unique functions of these states, descriptions of catalysis in the non-solution state should become a part of kinetic education in biology. © 1999 Elsevier Science Ireland Ltd. All rights reserved. Keywords: Enzyme kinetics; Periplasm; Alkaline phosphatase; Beta-lactamase; Surface catalysis
1. Introduction The biochemical approach of cell dissection and study of isolated components presents difficulty for detection of enzyme kinetic properties in reduced dimension. An intact cell has expansive surfaces, high packing of macromolecules and other organization properties that decline or disappear when a cell is ruptured. These properties are difficult to mimic in purified and/or reconstituted systems. It follows that forms of regulation, created by biophysical properties or structural organization in the cell, are difficult to detect. In * Tel.: +1-612-624-3622; fax: + 1-612-625-5780. E-mail address: [email protected] (G.L. Nelsestuen)
addition, the popular description of kinetics in biochemical education is dominated by relationships appropriate for soluble enzymes. These systems permit shortcuts in theory which do not apply to other situations. The goal of classical enzyme kinetics is the characterization of enzyme reaction mechanism. However, biological reaction kinetics may have different goals. This review utilizes the principal unit of soluble enzyme kinetics, the rate constant, to describe systems in reduced dimension. The intended audience are those, like myself, whose educational background included classical enzyme kinetics. Given that soluble enzyme kinetic behavior can expand to a nearly 1000-page description (Segel, 1975), it follows that this review can cover only a few points about reduced dimension. The primary
0009-3084/99/$ - see front matter © 1999 Elsevier Science Ireland Ltd. All rights reserved. PII: S 0 0 0 9 - 3 0 8 4 ( 9 9 ) 0 0 0 5 3 - 5
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G.L. Nelsestuen / Chemistry and Physics of Lipids 101 (1999) 37–44
subjects of this review are kinetic systems for which data are available and which show behaviors that are outside of standard steady state enzymology. The principle topic is the impact of multiple sites per particle.
possible. Furthermore, the Adair equation does not imply a mechanism for change in the rate constant as sites are filled. Consequently, most reactions are analyzed by other relationships.
2.2. Alternati6es to the Adair equation 2. Multi-enzyme particles
2.1. The Adair equation: the proper solution for multisite kinetic situations While enzymes in reduced dimension can display a number of unique properties, one which stands out is the presence of multiple enzyme sites on surfaces or in a restricted space. At a recent symposium on the immune response, sponsored by the University of Minnesota Department of Applied Mathematics, a speaker outlined a mathematical model for ligand binding to dimerized and/or multiple cell surface receptors. An audience member asked whether all this had been described at about the turn of the century by Adair (Adair, 1925). In some ways, the questioner was correct. The law of the rate constant applies correctly to the Adair equation. Attempts to analyze multisite particles by other expressions may create oversight. The Adair equation indicates that each addition of a substrate (S) to a multisite particle (P) has its own unique rate and equilibrium constant (Eqn (1)).
Binding of S by sites on P must involve initial collision of two particles. If the sites are attached to a cell, initial collision is between the cell and the substrate molecule. If that cell contains 100 000 enzymes or binding sites, the first collision involves a cell with 100 000 empty binding sites, the second involves a cell with 99 999 empty binding sites, etc. In other words, the Adair equation stipulates 100 000 different steps for ligand binding to this cell surface receptor. Of course, experimental solution of 100 000 rate expressions is not
2.2.1. Independent sites A first approach for analysis of multi-site particle behavior assumes that the sites have no impact on each other. The equilibria in Eqn (1) can be described by a single equilibrium. This allows analysis by the classic equations for first order dissociation rate and second order or pseudo-first order association rates. These methods and derivations are widely available in textbooks (Voet and Voet, 1995) and will not be discussed here. Shortcomings of these assumptions are made apparent by known behaviors (below). 2.2.2. Cooperati6ity Cooperativity is a common behavior that deviates from the independent site model. Each binding event has a different affinity due to site–site interaction. The symmetry model of Monod, Wyman and Changeaux provided an important mechanistic concept for cooperativity (Monod et al., 1965). Since cooperativity is also well-described in most textbooks, it will not be discussed in this review. 2.2.3. Collision-limited beha6ior Collision-limited behavior is an important consideration for membrane-associated enzymes that is not considered in standard textbooks. In response to kinetics of phage binding to a bacterial receptor, Berg and Purcell (1977) presented a model for the association rate constant which consisted of two steps, collision and site binding (Eq. (2)). kobs = (4pNAvDa/1000)(Ns/(Ns + pa))
(2)
The rate constant for collision of two spherical particles is provided by Smoluchowski theory (kcoll = 4pNAvDa/1000) and the probability of binding by the term, Ns/(Ns + pa). In these relationships, D is the sum of diffusion constants of the colliding particles, NAv is Avogadro’s Num-
G.L. Nelsestuen / Chemistry and Physics of Lipids 101 (1999) 37–44
ber, N is the number of empty binding sites on a particle, a is the sum of colliding particle radii and s is the effective capture radius of the binding site. This relationship shows that the rate constant (kobs) will vary with each binding event, since each involves a different value of N. This provides 100 000 different rate constants for a cell with 100 000 binding sites. Like the symmetry model for cooperativity, Eq. (2) provides a mechanism for rate constant change as sites are filled. It also helps identify the changes that are significant. Two extremes can be considered. At one extreme (pa Ns), the relationship simplifies to Eq. (3). kobs =4NAvDs/1000
(3)
This extreme applies to all soluble enzymes. The rate constant, kobs, is a function of the site properties of the enzyme or binding site, s (units=cm/site). The rate constants of classical enzyme kinetics are more appropriately described as ‘site rate constants’. The reaction solution is homogeneous, since the enzyme does not deplete substrate in its immediate vicinity. This behavior also applies to so-called ‘diffusion-limited enzymes’ which are limited by diffusion of substrate to the enzyme site. These enzymes capture a small percentage of substrate-enzyme collision events and do not deplete substrate in their vicinity. The size of the enzyme-bearing particle (a) has no impact on the reaction, other than its contribution to D. If the substrate is much smaller than the enzyme-bearing particle, D of the large particle is small so its size has no impact on the reaction rate. Thus, simple attachment of an enzyme to a membrane will not necessarily change kobs. The other extreme (Ns pa in Eq. (2)) generates the relationship in Eq. (4). kobs =4pNAvDa/1000
(4)
This condition removes site properties, s, from kobs. The result is a ‘particle rate constant’ (units of a= cm/particle). At this extreme, the particle becomes a perfectly reactive surface that captures every substrate with which it collides. The sites are now interactive since substrate binding by one site lowers exposure of other sites to substrate molecules. Diffusional equilibrium is not main-
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tained and the substrate concentration in the vicinity of the surface is depleted. This type of kinetics is also called ‘heterogeneous catalysis’ to signify that the substrate concentration in solution is not uniform. Generally, a very high value of N is needed to reach this level.
2.3. Enzyme kinetics in an intact bacterium: an example of multisite enzyme beha6ior with heterogeneous catalysis Heterogeneous catalysis is well known for neurotransmitters such as acetylcholine (Land et al., 1981). The concentration of acetylcholine esterase in the synaptic cleft is adequate to hydrolyze all the neurotransmitter before it diffuses from the cleft. Unfortunately, this system is not very accessible to steady state kinetic analysis. A more accessible example of flux-limited kinetics was provided by enzymes of the Escherichia coli periplasm (Martinez et al., 1996). This review will discuss qualitative behaviors and the original literature should be consulted for more complete description. In this intact organism, the reaction can be limited by diffusion of substrate into the periplasm. The presence of a partially permeable outer membrane requires a minor modification of Eq. (2), in which f(a) is a constant that lowers the effective collisional rate constant of substrate and enzymes in the periplasm. This provides Eq. (5), where f(a) is a number less than 1.0 and is a function of the porins expressed in the outer membrane (Martinez et al., 1996). kobs = (4pNAvDaf(a)/1000)(Ns/(Ns + paf(a))) (5) Further modification of s (by f(s)) is needed to correct for catalysis rather than simple binding. Further derivation is not needed to describe the general impact on reaction kinetics. Fig. 1 shows the behavior of alkaline phosphatase in the E. coli periplasm. At low substrate, the rate constant is provided by Eq. (4) (modified by f(a)) and the reaction is independent of site properties. The Eadie–Hofstee plot is horizontal, suggesting a Vmax of infinity. This results from rate limitation by diffusion, a non-saturable process. However, as more substrate is added, N
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declines and the reaction switches to be limited by site behavior (Eq. (3)). The outcome is a non-linear kinetic plot that involves a switch from one type of kinetics to another as substrate is added. The reaction can no longer be described by simple constants such as (apparent) KM and Vmax. An alternative model has been used to analyze another periplasmic enzyme, beta-lactamase (Nikaido et al., 1982). This model assumed diffusional equilibrium on each side of the partially permeable outer membrane. This approach should be valid, since f(a) was about 0.001 (Nelsestuen and Martinez, 1997), indicating that only one of 1000 substrate-cell collisions resulted in substrate entry into the periplasm. This might
Fig. 1. Kinetic behavior of alkaline phosphatase in the Escherichia coli periplasm. The plots are Eadie–Hofstee type. When plots are compared vertically, those with higher values represent more efficient catalysis. The inset shows the arrangement of enzyme (E) in the periplasm and substrate (S) that enters by diffusion through porins of the outer membrane. Enzyme activity was monitored with para-nitrophenylphosphate. The curve shows the best theoretical fit to steady state kinetic data for bacteria containing about 150 000 enzymes per cell (from Martinez et al., 1996). The Eadie–Hofstee plot for the same amount of enzyme in solution (right, linear plot) gives a slope equal to −1/KM (KM = 10 mM). The intercept on the horizontal axis is Vmax. The cellular enzyme gave a nonlinear relationship. At low velocity, the horizontal portion of the graph corresponds to full capture of every substrate that enters the periplasm (Eq. (4), see text). The plot curves to intercept the horizontal axis at Vmax for the enzyme site. Plot curvature occurs as N (Eq. (2), text) declines and the rate constant becomes limited by enzyme site behavior (Eq. (3), text).
be described as a two-chamber model. It allows use of diffusional equilibrium and enzyme site constants (KM and Vmax) to estimate substrate concentrations in the periplasm. However, full steady state kinetic analysis was not developed and subsequent study found some non-ideal behavior for in vivo substrate titrations of this enzyme (Martinez, 1995). The two-chamber model cannot be used for substrates with high permeability where diffusional equilibrium on both sides of the membrane is not valid. It will not extrapolate to systems that do not have a membrane barrier. The Berg and Purcell model is more robust since it describes a situation without a membrane barrier and is easily modified to one that contains a partially permeable membrane.
2.3.1. The impact of flux-limited kinetics on the interpretation of an enzyme’s role and beha6ior A first concept is that expression levels of alkaline phosphatase and beta-lactamase are designed to capture a high proportion of substrate that enters the periplasm (Nelsestuen and Martinez, 1997). The very different levels of these enzymes in the cell represent the amounts needed to achieve efficient capture of the respective substrates. These levels may seem excessive for other goals such as modulation of substrate flow through a metabolic pathway. High alkaline phosphatase should allow capture of substrates that enter the periplasm from either the cytosol or from the environment. This may increase phosphate scavenging actions by the cell. High betalactamase will prevent inhibition of periplasmic enzymes by penicillin antibiotics. The substrate concentration at the midpoint of in vivo titrations of alkaline phosphatase is 1 mM or higher, far above the KM of the free enzyme (about 10 mM; Martinez et al., 1996). Inorganic phosphate, a product of the alkaline phosphatase reaction and a competitive inhibitor of the enzyme (KI of about 40 mM; Martinez et al., 1996), is not an effective inhibitor in vivo, where millimolar concentrations are needed to reduce reaction velocity by 50%. Poor inhibition arises from the need to reduce N to a level where site kinetics become dominant (Eq. (3)). These high substrate and/or inhibitor concentrations are
G.L. Nelsestuen / Chemistry and Physics of Lipids 101 (1999) 37–44
meaningless to the biological system; alkaline phosphatase is expressed when free phosphate is less than about 1 mM. Since product inhibition is not important for enzyme regulation, it may serve other purposes. For example, phosphate binding to alkaline phosphatase may perform a role similar to that of the amino acid binding proteins in the periplasm. That is, several proteins in the periplasm bind sugars or amino acids and appear to assist the transport of these substrates into the cytosol (Ames, 1986). A similar function for alkaline phosphatase and inorganic phosphate may provide a dual role for this protein. These points are presented to emphasize that knowledge of kinetics in reduced dimension is needed to understand behavior of intact organisms. Other possible functions of this kinetic behavior have been suggested for other periplasmic enzymes (Martinez et al., 1996). 3. Multi-substrate particles
3.1. Phospholipase enzymes The simplest example of a multisubstrate particle may be the phospholipid membrane itself. Phospholipase A2 catalyzes hydrolysis of phospholipids to lysophospholipids. Multi-substrate particles allow processive enzyme action with many catalytic events before dissociation of the enzyme from the substrate particle. The presence of multiple substrates enhances the reaction, but can become detrimental. For example, upon hydrolysis of all substrates on one particle, which may require only a few seconds, the enzyme must dissociate and bind to another vesicle in order to continue reaction. This exchange is extremely slow and becomes rate limiting, in vitro. This behavior has been described in recent reviews (Gelb et al., 1995; Jain et al., 1995) and will not be discussed in detail here. The problem of slow enzyme exchange may be limited to the reconstituted system and the use of small phospholipid vesicles. The biological situation may involve large cell membranes with much larger numbers of substrates per particle. Thus, enzyme exchange in the biological system may not be rate limiting.
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3.2. Blood clotting enzymes Blood clotting enzymes provide more complex examples where protein substrates (S, Fig. 2, top) are located in solution and on the membrane particle. A general picture of an enzyme complex is shown in Fig. 2. Blood clotting provides several steps of this type. The enzyme has very slow exchange and is considered to be fixed to the membrane. Substrate can arrive at the enzyme from solution or from a membrane-bound substrate pool. The product must dissociate from the membrane to allow more substrate to bind. Examples of experimental outcomes are shown in the graphs in Fig. 2. The most important substrate pool is determined by the affinity of substrate for the membrane (Lu and Nelsestuen, 1996). Low affinity presents rapid dissociation rates and little lateral diffusion of substrate before equilibration with solution. This behavior provides a simple kinetic situation (Condition D, Fig. 2) where the membrane enhances capture of substrate from solution, without creation of an intermediate pool of membrane-bound molecules. Classical steady state kinetic plots such as Eadie–Hofstee should be linear (Condition D, Fig. 2). KM(apparent) and Vmax values should be applicable. Substrate concentrations at half-maximum velocity are typically about 0.1 mM, within the biological concentrations of these proteins. Membrane particle size has little impact on this reaction mechanism (Lu and Nelsestuen, 1996). Membranes with high affinity for substrate provide slow dissociation rates, enabling lateral diffusion of substrate to fill the enzyme site. The impact of this situation on reaction velocity will vary with reaction parameters. If product dissociation rate is very slow, this step can become rate limiting, similar to rate limitation by phospholipase exchange from the membrane (above). An example of this is shown in Condition C (Fig. 2), where reaction properties correspond to substrate/ product association and dissociation throughout the substrate titration. Maximum velocity is limited by the number of substrate/product sites on the membrane particle and the dissociation rate. At no time is this reaction limited by the enzyme
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G.L. Nelsestuen / Chemistry and Physics of Lipids 101 (1999) 37–44
Fig. 2. Kinetic behaviors of blood clotting enzymes (results plotted by Eadie – Hofstee are from Lu and Nelsestuen, 1996). The drawing (top) depicts some of the events that may occur: (1) Substrate (S) binds to the membrane surface; (2) Substrate diffuses on the membrane to the enzyme; (3) Substrate is bound to the enzyme (E) from solution; and (4) Product (P) dissociates from the membrane. Plot A–B shows the result for enzymes assembled on large vesicles (100 nm diameter, 1400 substrate binding sites per vesicle) that have high affinity for substrate. In this case, the enzyme captures all substrates that bind to the membrane (condition A) until the enzyme becomes saturated (Condition B). Small vesicles (30 nm diameter, 100 substrate binding sites per vesicle, Condition C) of the same composition and affinity for substrate provide complete capture of all substrates that bind to the membrane over the entire range of substrate concentrations. The plot follows substratemembrane binding (KD =5 nM) where the enzyme site is never limiting. Condition D shows a result for a membrane with low substrate affinity. Direct capture of substrate from solution provides a linear plot that is limited by the enzyme and is described by KM.
active site. One enzyme site per particle is sufficient to cleave all of the substrates that bind to a small vesicle. Change of particle size, without change of binding affinity, alters the behavior described in the preceding paragraph. Once again, substrate affinity for the membrane is high, allowing lateral diffusion. The enzyme captures virtually every substrate that binds to the particle (Condition A, Fig. 2). However, the larger particle provides more substrates and a single enzyme site per particle becomes saturated before the substrate sites on the membrane are filled. Consequently, the plot must curve toward maximum velocity of the enzyme active site (kcat, Condition B, Fig. 2). For this condition, large particles offer great advantage over small particles (Lu and Nelsestuen, 1996). Another important parameter is the number of enzymes. The results in Fig. 2 are limited to behaviors with one enzyme per particle. Multiple enzymes will reduce the number of substrates available to each enzyme and can make large membrane particles (Condition A, B, Fig. 2) behave in the manner of small membrane particles (Condition C, Fig. 2). If enzyme density on the membrane surface is high, blood clotting enzymes may also act in a flux-limited mechanism, similar to that described for enzymes in the E. coli periplasm (Fig. 1). This behavior has been observed for enzymes assembled on whole cells (McGee et al., 1992). Flow of solvent over the surface can also deliver substrate to the surface and two different enzyme systems have been shown to be flow-limited in tubes that are lined with phospholipids and blood clotting enzymes (Andree et al., 1994; Billy et al., 1995). All situations with diffusion or flow-limited reactions will be site-independent and have general properties similar to those shown in Fig. 1. However, at high substrate, the reaction will become limited by site catalysis. The frequent use of apparent KM to describe these situations is not appropriate, since they must produce non-linear Eadie–Hofstee plots. Thus, in vivo systems have abundant and variable behaviors that might be used to regulate enzyme reaction rates. The periplasmic enzymes in
G.L. Nelsestuen / Chemistry and Physics of Lipids 101 (1999) 37–44
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E. coli represent an intact organism and it is likely that the kinetic properties shown here are important to biological function. In contrast, the blood clotting reactions represent in vitro assays. The biological membrane used for coagulation is poorly characterized. In fact, to support the lateral diffusion mechanism in vitro, non-physiological levels of phosphatidylserine were required. Substrate concentrations appear adequate to support the solution substrate model (Condition D, Fig. 2). Thus, the lowered substrate requirements of lateral diffusion mechanisms (A, B, C Fig. 2) may not be relevant. On the other hand, limited diffusion of substrates within a blood clot may create flux-limited enzyme kinetics due to diffusion barriers. Delisi (1983) has estimated that membrane association by a substrate/ligand will only enhance reaction for systems with widely separated receptors/enzymes (5 100 per cell). In other cases, the much more rapid diffusion of substrate through solution will fill the active site more rapidly than diffusion on the membrane. This general prediction fits the outcomes described in Fig. 2 for blood clotting enzymes. Thus, it is possible to over-state the contribution of a pool of membrane-associated substrates to reaction rate. Benefits are restricted to certain conditions such as low enzyme levels and high affinity substrate binding.
acted by random diffusion in the plane of the membrane, diffusion was not rate-limiting. This appeared similar to the cases shown in Fig. 2. In fact, it may be common for membrane-bound intermediates to eliminate one diffusion step from the pathway, enhancing reaction rates but simplifying some aspects of reaction analysis. Cases where intermediate substrate pools are important are those with very low enzyme and substrate levels (e.g. B 100 enzymes per cell; Delisi, 1983). Overall, it is clear that kinetic features of surface-bound and/or multisite enzyme particles, that are distinct from classical enzyme kinetic behaviors, are important to biology. These unique behaviors provide the basis for expression levels of some enzymes and for some features of cell design. Kinetic properties of soluble enzymes, referred to as classical steady state enzyme kinetics, are the only exposure for most biological scientists in training. This background can provide difficulty in detection of novel behaviors of interfacial systems. Thus, to advance our understanding of these systems on biology, we suggest modification of current practices in kinetic education. While complete descriptions are impossible, novel kinetic behaviors of intact organisms or organelles can be outlined. This change would greatly assist in future advances in our knowledge of interfacial kinetic impacts on biology.
3.3. Other aspects of multienzyme/substrate particles
Acknowledgements
Many complex reaction situations may arise, some of which will present important new reaction features. Others may simplify mathematical descriptions. One example may be sequential enzymes of a reaction pathway with membranebound intermediates. The membrane-bound substrate pool in Fig. 2 fulfils this description in some ways. Membrane association by substrate may be compared with a first catalytic step in a pathway. Thus, sequential enzymes may show behaviors similar to those in Fig. 2. Reaction of cytochrome b5 with NADH-cytochrome b5 reductase has been examined (Rogers and Strittmatter, 1974, 1975). While substrate and enzyme inter-
This work is supported in part by Grant HL60859 from the National Institutes of Health.
References Adair, G.S., 1925. The hemoglobin system. VI. The oxygen dissociation curve of hemoglobin. J. Biol. Chem. 63, 529. Ames, G.F.-L., 1986. Bacterial periplasmic transport systems: structure, mechanism and evolution. Ann. Rev. Biochem. 55, 397 – 425. Andree, H.A.M., Contino, P.B., Repke, D., Gentry, R., Nemerson, Y., 1994. Transport rate limited catalysis on macroscopic surfaces. The activation of factor X in a continuous flow enzyme reactor. Biochemistry 33, 4368 – 4374.
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Berg, H.C., Purcell, E.M., 1977. Physics of chemoreception. Biophys. J. 20, 193 – 219. Billy, D., Speijer, H., Willems, G., Hemker, H.C., Lindhout, T., 1995. Prothrombin activation by prothrombinase in a tubular flow reactor. J. Biol. Chem. 270, 1029–1034. Delisi, C., 1983. Role of diffusion regulation in receptor–ligand interactions. Methods Enzymol. 93, 95–109. Gelb, M.H., Jain, M.K., Hanel, A.M., Berg, O.G., 1995. Interfacial. enzymology. of. glycerolipid. hydrolases: lessons. from. secreted. phospholipases A2. Ann. Rev. Biochem. 64, 653 – 688. Jain, M.K., Gelb, M.H., Rogers, J., Berg, O.G., 1995. Kinetic basis for interfacial catalysis by phospholipase A2. Methods Enzymol. 249, 567–614. Land, B.R., Salpeter, E.E., Salpeter, M.M., 1981. Kinetic parameters for acetylcholine interaction in intact junction. Proc. Natl. Acad. Sci. USA 78, 7200–7204. Lu, Y., Nelsestuen, G.L., 1996. The prothrombinase reaction: mechanism switching between Michaelis–Menten and nonMichaelis – Menten behaviors. Biochemistry 35, 8201– 8209. Martinez, M.B., 1995. An Investigation of Enzyme Kinetics at the Diffusional Limit. Enzymes in the Periplasm of Escherichia coli. PhD Thesis, University of Minnesota. Martinez, M.B., Flickinger, M., Nelsestuen, G.L., 1996. Accurate kinetic modeling of alkaline phosphatase in the Es-
.
chericia coli periplasm: implications for enzyme properties and substrate diffusion. Biochemistry 35, 1179 – 1186. McGee, M.P., Li, L.C., Xiong, H., 1992. Diffusion control in blood coagulation. Activation of factor X by factors IXa/ VIIIa assembled on human monocyte membranes. J. Biol. Chem. 267, 24333 – 24339. Monod, J., Wyman, J., Changeux, J.-P., 1965. J. Mol. Biol. 12, 88. Nelsestuen, G.L., Martinez, M.B., 1997. Steady state enzyme velocities that are independent of [Enzyme}: an important behavior in many membrane and particle bound states. Biochemistry 36, 9081 – 9086. Nikaido, H., Rosenberg, E.Y., Foulds, J., 1982. Porin channels in Escherichia coli: studies with b-lactams in intact cells. J. Bacteriol. 153, 232 – 240. Rogers, M.J., Strittmatter, P., 1974. Evidence for random distribution and translational movement of cytochrome b5 in endoplasmic reticulum. J. Biol. Chem. 250, 5713 – 5718. Rogers, M.J., Strittmatter, P., 1975. The interaction of NADH-cytochrome b5 reductase and cytochrome b5 bound to egg lecithin liposomes. J. Biol. Chem. 250, 5713 – 5718. Segel, I.H., 1975. Enzyme Kinetics: Behavior and Analysis of Rapid Equilibrium and Steady State Enzyme Systems. Wiley, New York. Voet, D., Voet, J.G., 1995. Biochemistry, 2nd edition. Wiley, New York.
Significance of reduced dimensionality in reaction kinetics: impact of multi-site particles Gary L. Nelsestuen * Department of Biochemistry, Uni6ersity of Minnesota, 1479 Gortner A6e., St. Paul, MN 55108, USA
Abstract This review examines novel kinetic properties of enzymes on membrane surfaces or states of restricted diffusion. A leading feature is the presence of multiple enzymes and/or substrates per particle. In these states, enzymes can be influenced by parameters such as the number of substrates or enzymes per particle, particle size, the rates of exchange of substrate or enzyme from the particle, or substrate diffusion to the particle. These steps are independent of the enzyme site parameters which are described by classical enzymology. The results make it clear that non-classical behaviors are important to biological systems, are the basis for some enzyme expression levels and are determinants of cellular design. To identify more unique functions of these states, descriptions of catalysis in the non-solution state should become a part of kinetic education in biology. © 1999 Elsevier Science Ireland Ltd. All rights reserved. Keywords: Enzyme kinetics; Periplasm; Alkaline phosphatase; Beta-lactamase; Surface catalysis
1. Introduction The biochemical approach of cell dissection and study of isolated components presents difficulty for detection of enzyme kinetic properties in reduced dimension. An intact cell has expansive surfaces, high packing of macromolecules and other organization properties that decline or disappear when a cell is ruptured. These properties are difficult to mimic in purified and/or reconstituted systems. It follows that forms of regulation, created by biophysical properties or structural organization in the cell, are difficult to detect. In * Tel.: +1-612-624-3622; fax: + 1-612-625-5780. E-mail address: [email protected] (G.L. Nelsestuen)
addition, the popular description of kinetics in biochemical education is dominated by relationships appropriate for soluble enzymes. These systems permit shortcuts in theory which do not apply to other situations. The goal of classical enzyme kinetics is the characterization of enzyme reaction mechanism. However, biological reaction kinetics may have different goals. This review utilizes the principal unit of soluble enzyme kinetics, the rate constant, to describe systems in reduced dimension. The intended audience are those, like myself, whose educational background included classical enzyme kinetics. Given that soluble enzyme kinetic behavior can expand to a nearly 1000-page description (Segel, 1975), it follows that this review can cover only a few points about reduced dimension. The primary
0009-3084/99/$ - see front matter © 1999 Elsevier Science Ireland Ltd. All rights reserved. PII: S 0 0 0 9 - 3 0 8 4 ( 9 9 ) 0 0 0 5 3 - 5
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G.L. Nelsestuen / Chemistry and Physics of Lipids 101 (1999) 37–44
subjects of this review are kinetic systems for which data are available and which show behaviors that are outside of standard steady state enzymology. The principle topic is the impact of multiple sites per particle.
possible. Furthermore, the Adair equation does not imply a mechanism for change in the rate constant as sites are filled. Consequently, most reactions are analyzed by other relationships.
2.2. Alternati6es to the Adair equation 2. Multi-enzyme particles
2.1. The Adair equation: the proper solution for multisite kinetic situations While enzymes in reduced dimension can display a number of unique properties, one which stands out is the presence of multiple enzyme sites on surfaces or in a restricted space. At a recent symposium on the immune response, sponsored by the University of Minnesota Department of Applied Mathematics, a speaker outlined a mathematical model for ligand binding to dimerized and/or multiple cell surface receptors. An audience member asked whether all this had been described at about the turn of the century by Adair (Adair, 1925). In some ways, the questioner was correct. The law of the rate constant applies correctly to the Adair equation. Attempts to analyze multisite particles by other expressions may create oversight. The Adair equation indicates that each addition of a substrate (S) to a multisite particle (P) has its own unique rate and equilibrium constant (Eqn (1)).
Binding of S by sites on P must involve initial collision of two particles. If the sites are attached to a cell, initial collision is between the cell and the substrate molecule. If that cell contains 100 000 enzymes or binding sites, the first collision involves a cell with 100 000 empty binding sites, the second involves a cell with 99 999 empty binding sites, etc. In other words, the Adair equation stipulates 100 000 different steps for ligand binding to this cell surface receptor. Of course, experimental solution of 100 000 rate expressions is not
2.2.1. Independent sites A first approach for analysis of multi-site particle behavior assumes that the sites have no impact on each other. The equilibria in Eqn (1) can be described by a single equilibrium. This allows analysis by the classic equations for first order dissociation rate and second order or pseudo-first order association rates. These methods and derivations are widely available in textbooks (Voet and Voet, 1995) and will not be discussed here. Shortcomings of these assumptions are made apparent by known behaviors (below). 2.2.2. Cooperati6ity Cooperativity is a common behavior that deviates from the independent site model. Each binding event has a different affinity due to site–site interaction. The symmetry model of Monod, Wyman and Changeaux provided an important mechanistic concept for cooperativity (Monod et al., 1965). Since cooperativity is also well-described in most textbooks, it will not be discussed in this review. 2.2.3. Collision-limited beha6ior Collision-limited behavior is an important consideration for membrane-associated enzymes that is not considered in standard textbooks. In response to kinetics of phage binding to a bacterial receptor, Berg and Purcell (1977) presented a model for the association rate constant which consisted of two steps, collision and site binding (Eq. (2)). kobs = (4pNAvDa/1000)(Ns/(Ns + pa))
(2)
The rate constant for collision of two spherical particles is provided by Smoluchowski theory (kcoll = 4pNAvDa/1000) and the probability of binding by the term, Ns/(Ns + pa). In these relationships, D is the sum of diffusion constants of the colliding particles, NAv is Avogadro’s Num-
G.L. Nelsestuen / Chemistry and Physics of Lipids 101 (1999) 37–44
ber, N is the number of empty binding sites on a particle, a is the sum of colliding particle radii and s is the effective capture radius of the binding site. This relationship shows that the rate constant (kobs) will vary with each binding event, since each involves a different value of N. This provides 100 000 different rate constants for a cell with 100 000 binding sites. Like the symmetry model for cooperativity, Eq. (2) provides a mechanism for rate constant change as sites are filled. It also helps identify the changes that are significant. Two extremes can be considered. At one extreme (pa Ns), the relationship simplifies to Eq. (3). kobs =4NAvDs/1000
(3)
This extreme applies to all soluble enzymes. The rate constant, kobs, is a function of the site properties of the enzyme or binding site, s (units=cm/site). The rate constants of classical enzyme kinetics are more appropriately described as ‘site rate constants’. The reaction solution is homogeneous, since the enzyme does not deplete substrate in its immediate vicinity. This behavior also applies to so-called ‘diffusion-limited enzymes’ which are limited by diffusion of substrate to the enzyme site. These enzymes capture a small percentage of substrate-enzyme collision events and do not deplete substrate in their vicinity. The size of the enzyme-bearing particle (a) has no impact on the reaction, other than its contribution to D. If the substrate is much smaller than the enzyme-bearing particle, D of the large particle is small so its size has no impact on the reaction rate. Thus, simple attachment of an enzyme to a membrane will not necessarily change kobs. The other extreme (Ns pa in Eq. (2)) generates the relationship in Eq. (4). kobs =4pNAvDa/1000
(4)
This condition removes site properties, s, from kobs. The result is a ‘particle rate constant’ (units of a= cm/particle). At this extreme, the particle becomes a perfectly reactive surface that captures every substrate with which it collides. The sites are now interactive since substrate binding by one site lowers exposure of other sites to substrate molecules. Diffusional equilibrium is not main-
39
tained and the substrate concentration in the vicinity of the surface is depleted. This type of kinetics is also called ‘heterogeneous catalysis’ to signify that the substrate concentration in solution is not uniform. Generally, a very high value of N is needed to reach this level.
2.3. Enzyme kinetics in an intact bacterium: an example of multisite enzyme beha6ior with heterogeneous catalysis Heterogeneous catalysis is well known for neurotransmitters such as acetylcholine (Land et al., 1981). The concentration of acetylcholine esterase in the synaptic cleft is adequate to hydrolyze all the neurotransmitter before it diffuses from the cleft. Unfortunately, this system is not very accessible to steady state kinetic analysis. A more accessible example of flux-limited kinetics was provided by enzymes of the Escherichia coli periplasm (Martinez et al., 1996). This review will discuss qualitative behaviors and the original literature should be consulted for more complete description. In this intact organism, the reaction can be limited by diffusion of substrate into the periplasm. The presence of a partially permeable outer membrane requires a minor modification of Eq. (2), in which f(a) is a constant that lowers the effective collisional rate constant of substrate and enzymes in the periplasm. This provides Eq. (5), where f(a) is a number less than 1.0 and is a function of the porins expressed in the outer membrane (Martinez et al., 1996). kobs = (4pNAvDaf(a)/1000)(Ns/(Ns + paf(a))) (5) Further modification of s (by f(s)) is needed to correct for catalysis rather than simple binding. Further derivation is not needed to describe the general impact on reaction kinetics. Fig. 1 shows the behavior of alkaline phosphatase in the E. coli periplasm. At low substrate, the rate constant is provided by Eq. (4) (modified by f(a)) and the reaction is independent of site properties. The Eadie–Hofstee plot is horizontal, suggesting a Vmax of infinity. This results from rate limitation by diffusion, a non-saturable process. However, as more substrate is added, N
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declines and the reaction switches to be limited by site behavior (Eq. (3)). The outcome is a non-linear kinetic plot that involves a switch from one type of kinetics to another as substrate is added. The reaction can no longer be described by simple constants such as (apparent) KM and Vmax. An alternative model has been used to analyze another periplasmic enzyme, beta-lactamase (Nikaido et al., 1982). This model assumed diffusional equilibrium on each side of the partially permeable outer membrane. This approach should be valid, since f(a) was about 0.001 (Nelsestuen and Martinez, 1997), indicating that only one of 1000 substrate-cell collisions resulted in substrate entry into the periplasm. This might
Fig. 1. Kinetic behavior of alkaline phosphatase in the Escherichia coli periplasm. The plots are Eadie–Hofstee type. When plots are compared vertically, those with higher values represent more efficient catalysis. The inset shows the arrangement of enzyme (E) in the periplasm and substrate (S) that enters by diffusion through porins of the outer membrane. Enzyme activity was monitored with para-nitrophenylphosphate. The curve shows the best theoretical fit to steady state kinetic data for bacteria containing about 150 000 enzymes per cell (from Martinez et al., 1996). The Eadie–Hofstee plot for the same amount of enzyme in solution (right, linear plot) gives a slope equal to −1/KM (KM = 10 mM). The intercept on the horizontal axis is Vmax. The cellular enzyme gave a nonlinear relationship. At low velocity, the horizontal portion of the graph corresponds to full capture of every substrate that enters the periplasm (Eq. (4), see text). The plot curves to intercept the horizontal axis at Vmax for the enzyme site. Plot curvature occurs as N (Eq. (2), text) declines and the rate constant becomes limited by enzyme site behavior (Eq. (3), text).
be described as a two-chamber model. It allows use of diffusional equilibrium and enzyme site constants (KM and Vmax) to estimate substrate concentrations in the periplasm. However, full steady state kinetic analysis was not developed and subsequent study found some non-ideal behavior for in vivo substrate titrations of this enzyme (Martinez, 1995). The two-chamber model cannot be used for substrates with high permeability where diffusional equilibrium on both sides of the membrane is not valid. It will not extrapolate to systems that do not have a membrane barrier. The Berg and Purcell model is more robust since it describes a situation without a membrane barrier and is easily modified to one that contains a partially permeable membrane.
2.3.1. The impact of flux-limited kinetics on the interpretation of an enzyme’s role and beha6ior A first concept is that expression levels of alkaline phosphatase and beta-lactamase are designed to capture a high proportion of substrate that enters the periplasm (Nelsestuen and Martinez, 1997). The very different levels of these enzymes in the cell represent the amounts needed to achieve efficient capture of the respective substrates. These levels may seem excessive for other goals such as modulation of substrate flow through a metabolic pathway. High alkaline phosphatase should allow capture of substrates that enter the periplasm from either the cytosol or from the environment. This may increase phosphate scavenging actions by the cell. High betalactamase will prevent inhibition of periplasmic enzymes by penicillin antibiotics. The substrate concentration at the midpoint of in vivo titrations of alkaline phosphatase is 1 mM or higher, far above the KM of the free enzyme (about 10 mM; Martinez et al., 1996). Inorganic phosphate, a product of the alkaline phosphatase reaction and a competitive inhibitor of the enzyme (KI of about 40 mM; Martinez et al., 1996), is not an effective inhibitor in vivo, where millimolar concentrations are needed to reduce reaction velocity by 50%. Poor inhibition arises from the need to reduce N to a level where site kinetics become dominant (Eq. (3)). These high substrate and/or inhibitor concentrations are
G.L. Nelsestuen / Chemistry and Physics of Lipids 101 (1999) 37–44
meaningless to the biological system; alkaline phosphatase is expressed when free phosphate is less than about 1 mM. Since product inhibition is not important for enzyme regulation, it may serve other purposes. For example, phosphate binding to alkaline phosphatase may perform a role similar to that of the amino acid binding proteins in the periplasm. That is, several proteins in the periplasm bind sugars or amino acids and appear to assist the transport of these substrates into the cytosol (Ames, 1986). A similar function for alkaline phosphatase and inorganic phosphate may provide a dual role for this protein. These points are presented to emphasize that knowledge of kinetics in reduced dimension is needed to understand behavior of intact organisms. Other possible functions of this kinetic behavior have been suggested for other periplasmic enzymes (Martinez et al., 1996). 3. Multi-substrate particles
3.1. Phospholipase enzymes The simplest example of a multisubstrate particle may be the phospholipid membrane itself. Phospholipase A2 catalyzes hydrolysis of phospholipids to lysophospholipids. Multi-substrate particles allow processive enzyme action with many catalytic events before dissociation of the enzyme from the substrate particle. The presence of multiple substrates enhances the reaction, but can become detrimental. For example, upon hydrolysis of all substrates on one particle, which may require only a few seconds, the enzyme must dissociate and bind to another vesicle in order to continue reaction. This exchange is extremely slow and becomes rate limiting, in vitro. This behavior has been described in recent reviews (Gelb et al., 1995; Jain et al., 1995) and will not be discussed in detail here. The problem of slow enzyme exchange may be limited to the reconstituted system and the use of small phospholipid vesicles. The biological situation may involve large cell membranes with much larger numbers of substrates per particle. Thus, enzyme exchange in the biological system may not be rate limiting.
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3.2. Blood clotting enzymes Blood clotting enzymes provide more complex examples where protein substrates (S, Fig. 2, top) are located in solution and on the membrane particle. A general picture of an enzyme complex is shown in Fig. 2. Blood clotting provides several steps of this type. The enzyme has very slow exchange and is considered to be fixed to the membrane. Substrate can arrive at the enzyme from solution or from a membrane-bound substrate pool. The product must dissociate from the membrane to allow more substrate to bind. Examples of experimental outcomes are shown in the graphs in Fig. 2. The most important substrate pool is determined by the affinity of substrate for the membrane (Lu and Nelsestuen, 1996). Low affinity presents rapid dissociation rates and little lateral diffusion of substrate before equilibration with solution. This behavior provides a simple kinetic situation (Condition D, Fig. 2) where the membrane enhances capture of substrate from solution, without creation of an intermediate pool of membrane-bound molecules. Classical steady state kinetic plots such as Eadie–Hofstee should be linear (Condition D, Fig. 2). KM(apparent) and Vmax values should be applicable. Substrate concentrations at half-maximum velocity are typically about 0.1 mM, within the biological concentrations of these proteins. Membrane particle size has little impact on this reaction mechanism (Lu and Nelsestuen, 1996). Membranes with high affinity for substrate provide slow dissociation rates, enabling lateral diffusion of substrate to fill the enzyme site. The impact of this situation on reaction velocity will vary with reaction parameters. If product dissociation rate is very slow, this step can become rate limiting, similar to rate limitation by phospholipase exchange from the membrane (above). An example of this is shown in Condition C (Fig. 2), where reaction properties correspond to substrate/ product association and dissociation throughout the substrate titration. Maximum velocity is limited by the number of substrate/product sites on the membrane particle and the dissociation rate. At no time is this reaction limited by the enzyme
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Fig. 2. Kinetic behaviors of blood clotting enzymes (results plotted by Eadie – Hofstee are from Lu and Nelsestuen, 1996). The drawing (top) depicts some of the events that may occur: (1) Substrate (S) binds to the membrane surface; (2) Substrate diffuses on the membrane to the enzyme; (3) Substrate is bound to the enzyme (E) from solution; and (4) Product (P) dissociates from the membrane. Plot A–B shows the result for enzymes assembled on large vesicles (100 nm diameter, 1400 substrate binding sites per vesicle) that have high affinity for substrate. In this case, the enzyme captures all substrates that bind to the membrane (condition A) until the enzyme becomes saturated (Condition B). Small vesicles (30 nm diameter, 100 substrate binding sites per vesicle, Condition C) of the same composition and affinity for substrate provide complete capture of all substrates that bind to the membrane over the entire range of substrate concentrations. The plot follows substratemembrane binding (KD =5 nM) where the enzyme site is never limiting. Condition D shows a result for a membrane with low substrate affinity. Direct capture of substrate from solution provides a linear plot that is limited by the enzyme and is described by KM.
active site. One enzyme site per particle is sufficient to cleave all of the substrates that bind to a small vesicle. Change of particle size, without change of binding affinity, alters the behavior described in the preceding paragraph. Once again, substrate affinity for the membrane is high, allowing lateral diffusion. The enzyme captures virtually every substrate that binds to the particle (Condition A, Fig. 2). However, the larger particle provides more substrates and a single enzyme site per particle becomes saturated before the substrate sites on the membrane are filled. Consequently, the plot must curve toward maximum velocity of the enzyme active site (kcat, Condition B, Fig. 2). For this condition, large particles offer great advantage over small particles (Lu and Nelsestuen, 1996). Another important parameter is the number of enzymes. The results in Fig. 2 are limited to behaviors with one enzyme per particle. Multiple enzymes will reduce the number of substrates available to each enzyme and can make large membrane particles (Condition A, B, Fig. 2) behave in the manner of small membrane particles (Condition C, Fig. 2). If enzyme density on the membrane surface is high, blood clotting enzymes may also act in a flux-limited mechanism, similar to that described for enzymes in the E. coli periplasm (Fig. 1). This behavior has been observed for enzymes assembled on whole cells (McGee et al., 1992). Flow of solvent over the surface can also deliver substrate to the surface and two different enzyme systems have been shown to be flow-limited in tubes that are lined with phospholipids and blood clotting enzymes (Andree et al., 1994; Billy et al., 1995). All situations with diffusion or flow-limited reactions will be site-independent and have general properties similar to those shown in Fig. 1. However, at high substrate, the reaction will become limited by site catalysis. The frequent use of apparent KM to describe these situations is not appropriate, since they must produce non-linear Eadie–Hofstee plots. Thus, in vivo systems have abundant and variable behaviors that might be used to regulate enzyme reaction rates. The periplasmic enzymes in
G.L. Nelsestuen / Chemistry and Physics of Lipids 101 (1999) 37–44
43
E. coli represent an intact organism and it is likely that the kinetic properties shown here are important to biological function. In contrast, the blood clotting reactions represent in vitro assays. The biological membrane used for coagulation is poorly characterized. In fact, to support the lateral diffusion mechanism in vitro, non-physiological levels of phosphatidylserine were required. Substrate concentrations appear adequate to support the solution substrate model (Condition D, Fig. 2). Thus, the lowered substrate requirements of lateral diffusion mechanisms (A, B, C Fig. 2) may not be relevant. On the other hand, limited diffusion of substrates within a blood clot may create flux-limited enzyme kinetics due to diffusion barriers. Delisi (1983) has estimated that membrane association by a substrate/ligand will only enhance reaction for systems with widely separated receptors/enzymes (5 100 per cell). In other cases, the much more rapid diffusion of substrate through solution will fill the active site more rapidly than diffusion on the membrane. This general prediction fits the outcomes described in Fig. 2 for blood clotting enzymes. Thus, it is possible to over-state the contribution of a pool of membrane-associated substrates to reaction rate. Benefits are restricted to certain conditions such as low enzyme levels and high affinity substrate binding.
acted by random diffusion in the plane of the membrane, diffusion was not rate-limiting. This appeared similar to the cases shown in Fig. 2. In fact, it may be common for membrane-bound intermediates to eliminate one diffusion step from the pathway, enhancing reaction rates but simplifying some aspects of reaction analysis. Cases where intermediate substrate pools are important are those with very low enzyme and substrate levels (e.g. B 100 enzymes per cell; Delisi, 1983). Overall, it is clear that kinetic features of surface-bound and/or multisite enzyme particles, that are distinct from classical enzyme kinetic behaviors, are important to biology. These unique behaviors provide the basis for expression levels of some enzymes and for some features of cell design. Kinetic properties of soluble enzymes, referred to as classical steady state enzyme kinetics, are the only exposure for most biological scientists in training. This background can provide difficulty in detection of novel behaviors of interfacial systems. Thus, to advance our understanding of these systems on biology, we suggest modification of current practices in kinetic education. While complete descriptions are impossible, novel kinetic behaviors of intact organisms or organelles can be outlined. This change would greatly assist in future advances in our knowledge of interfacial kinetic impacts on biology.
3.3. Other aspects of multienzyme/substrate particles
Acknowledgements
Many complex reaction situations may arise, some of which will present important new reaction features. Others may simplify mathematical descriptions. One example may be sequential enzymes of a reaction pathway with membranebound intermediates. The membrane-bound substrate pool in Fig. 2 fulfils this description in some ways. Membrane association by substrate may be compared with a first catalytic step in a pathway. Thus, sequential enzymes may show behaviors similar to those in Fig. 2. Reaction of cytochrome b5 with NADH-cytochrome b5 reductase has been examined (Rogers and Strittmatter, 1974, 1975). While substrate and enzyme inter-
This work is supported in part by Grant HL60859 from the National Institutes of Health.
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