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A method for writing enzyme rate equations: application to the estimation of the number and size of key proton families
A Method for Writing Enzyme Rate Equations: Application to the Estimation of the Number and Size of Key Proton Families MONIQUE VASSEUR, GUY VAN MELLE,* AND FRANCISCO ALVARADO Centre de Recherches sur la Nut&ion, Centre National de la Recherche Scientifique, Meudon,
France
Received
25 April 1988; revised I3 September
92190
1988
ABSTRACT We develop a method to derive the pa-dependent activation. Our presentation for sucrase, the three-key-proton model existence, in the acid ionization reaction, respectively responsible for either V-type
rate equation for enzyme models that include is based on a kinetic model recently described of Vasseur and coworkers, which considers the of two functionally distinct prototropic groups, or K-type kinetic effects. In contrast, as concerns
the basic ionization reaction, the model conforms to classical concepts of pH-dependent activation, whereby a single proton participates in either V-type or K-type effects but not in both at the same time. Enzymes
with more than three key protons
have been described,
indicating
that, rather
than isolated protons, groups of protons should be considered, and therefore the model can be better described as a three-proton-family model, where a proton family is defined as one or several protons that are gained or lost as a block and perform the same kinetic function. The resulting model is treated here as a useful framework upon which other models can be built. To facilitate the writing of the rate equations, we define two new entities: (1) intralevel coefficients, which describe the various combinations of the enzyme with either the substrate(s), the allosteric effector( or both at a given protonation level, and (2) interlevel coefficients, which describe the interplay between the various protonation levels, The resulting rate equation can be used in a global fit procedure permitting in a single computer run the estimation of (1) the entire set of dissociation and microscopic ionization constants of the model, (2) the number and kinetic function of proton families characterizing the enzyme under consideration, and (3) the number of key protons constituting each family, which is derived from the derivatives of the kinetic parameters, V,/K,,,, V,, and K,,,.
INTRODUCTION To explain the alkali metal ion and H+ dependent activation and/or inhibition of rabbit intestinal sucrase (sucrose a-D-glucohydrolase,
*Present address: Institut Lausanne, Switzerland.
MATHEMATICAL
Universitaire
BIOSCIENCES
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14
M. VASSEUR,
G.
VAN
MELLE,
AND F. ALVARADO
EC.3.2.1.48), we have proposed a kinetic model involving three key protons [l, 21. In contrast with classical models of pH-dependent enzyme activation [3-61, the three-proton model considers the existence, in a single acid ionization reaction, of two functionally distinct prototropic groups, each constituting a different proton family. One proton (family) participates in V-type activation, the other in K-type kinetic effects.’ This fact is reflected experimentally by the slope of + 1 exhibited by each of the log V, versus pH and the pK, versus pH plots; the logVm/Km versus pH plot exhibits a slope of + 2. Sucrase, however, represents a relatively simple case because each of the observed proton families contains a single member. In the present paper we develop a more general procedure for interpreting enzyme kinetic models that takes into account the fact that enzymes with larger numbers of key protons exist, yeast hexokinase being an example [9]. If a proton family is defined as one or several key protons that are gained or lost as a block and perform the same kinetic function [2], the three-proton model can be given the general form:
x
z
Protonation levels: L(3)
h
h
L(o)
where the enzyme is subdivided into four different protonation levels, one of which, L,, contains two functionally distinct enzyme forms, YyEZ, and X,EZz. The lowercase subscripts x, y, and z represent the number of protons constituting, respectively, the three proton families X, Y, and Z. The acid ionization constant Kl involves n, = x + y protons, and the basic ionization constant K,, no = z protons. The illustrated microscopic ionization constants concern the one-step displacement of a whole family and appear raised to the power of the number of protons involved. X and Y are the
‘Our nomenclature has been defined in detail previously [2, 71. We use the term “allosteric” in its most ample sense: indirect interactions between distinct sites that do not necessarily occur in separate monomers. K-kinetics means allosteric affinity-type kinetics; V-kinetics, allosteric capacity-type kinetics [8].
KEY PROTON
FAMILIES
15
OF ENZYMES
proton families respectively responsible for the K-type and V-type effects observed at the acid end of the spectrum. At the basic end, and depending on whether or not the fully deprotonated enzyme form, E, is capable of substrate binding, the Z family would be responsible for either K-type or V-type effects. Equation (1) constitutes a framework within which the behavior of a given enzyme can be explored by considering the possible specific interactions between each of the indicated free enzyme forms, the substrate(s), and any suitable modifier (allosteric activator, inhibitor, and so forth). To illustrate our procedure, we will use the simple but useful case of sucrase, illustrated in Figure 1. Previously, we have shown that this monomeric enzyme has only one substrate-binding site but possesses several modifier sites, each capable of specifically interacting with an alkali metal ion [2]. Whereas there is only one activator or A site that accepts either Na+ or Li+, there are several inhibitory or I sites: two that can bind Na+, seven that can bind Li+. The three key proton families each have one member, i.e., x = y = z = 1 [2]. Utilizing this model as the basis, in this paper we describe an original procedure that permits us to easily write rate equations readily adaptable for global, one-step computer analysis of enzyme kinetic data. We furthermore show how to identify the number and size of the proton families characterizing an enzyme, which can be done by computing the derivatives of the double-logarithmic functions of the fundamental kinetic parameters describing it. THE
RATE
EQUATION
FOR pH-DEPENDENT
ENZYME
ACTIVATION
To illustrate our procedure, we shall first establish the state [ET] and velocity u equations for the three-proton model, which are necessary to define the u/[ET] ratio that yields the overall rate equation. We shall use the simplifying assumption that the various enzyme species are all in rapid equilibrium and that the product-forming steps are rate-limiting, as is the case with sucrase. The State Equation. If [ET] is the sum of all possible enzyme forms constituting the model, and [ET], is the sum of all the forms present at each given protonation level L,, the state equation is
LETI=~[ETl,.
(2)
P
At levels L2,,, L,, and L,, (see Figure alone, bound to S, bound to A, or bound
l), the enzyme can exist either to each S and A. By using the
16
M. VASSEUR,
G.
VAN
MELLE,
AND F. ALVARADO
FIG. 1. Three-dimensional representation of the three-proton-family model for intestinal sucrase, depicting the pH-dependent interaction between a monomeric enzyme, the substrate S, and an allosteric modifier A (or I). For sucrase, A and I represent the same species, an alkali metal ion that can bind either to octiuafory or to inhibitory sites. In accord with Equation (l), the various enzyme forms have been grouped into four different protondon levels (taken from Fig. 2 of Ref. 2). One key feature of the model is the splitting of the acid ionization constant K, into two distinct pathways, which explains the coexistence, in this single ionization step, of two distinct proton families responsible for changes in either enzyme affinity (K-type kinetics) or catalytic activity (V-type kinetics). On the basic side, the situation conforms to classical concepts of pH-dependent activation, whereby a given proton family participates in either V-type or K-type effects, but not in both at the same time. Further details in [2] and/or in the text.
17
KEY PROTON FAMILIES OF ENZYMES
appropriate dissociation constants of the free enzyme, we obtain
to express each enzyme complex in terms
[ET1=~[%I,,
(3)
P
where I, represents
the intralevel coefficient characterizing
level L,, e.g.,
For later use, and in order to render our treatment more general, we find it useful to split I, into two intralevel subcoefficients, such that
,,=1;+g4;, =P
where 1; concerns those enzyme forms not involving to all enzyme complexes that include S (SE, SEA). From Equations (3) and (5), we obtain
S (E, EA) and 1; refers
Special situations (some are illustrated in the model in Figure 1) are easily taken care of. Thus, for instance, at levels where neither the substrate nor the activator bind (e.g., at L2J, Equation (5) simplifies to I, = 1; =l. At levels where there is no substrate binding but the modifier binds to several I sites (e.g., at L, where [A] is [I]), the corresponding 1; coefficients are raised to the power of nip, the number of existing inhibitory sites:
,’
P
=
1
i
\
[Al
Kp
i
“ip .
Because the various levels are interconnected by protonation-deprotonation reactions, the amount of free enzyme at a given level can be expressed by linking it to the corresponding amount of free enzyme at the reference
18
M. VASSEUR,
G.
VAN
MELLE,
AND F. ALVARADO
TABLE 1 The Three-Proton-Family Protonation level L3
(I+
Sucrase Model”
I’P
I”P
PI )“”
1
K,,
nP
%
PI+ lx+.”
x+Y
(4x)x(&“)y L 2x L 2”
1 1+
1 1+
LWG
I+ [AI/K,, LW4z,
1+
X
([H+ I/K,,)”
LWC,
(W+
1+ W/G I+ LWG,
I/K,,)’
Y
1
(4,/W+
0
I)=
“Definition of the intralevel coefficients I; and I;, the interlevel and the number of protons np in excess by reference to 15,.
- z
coefficients
hp.
level L, (where, for sucrase, L, = L,), so that
PI, = [Elrhpt where h, is the interhei coefficient defining the interplay between levels.’ Because the interlevel coefficients deal with either the gaining or the losing of np key protons, they can take one of two forms, (K,/[H+ 1)“~ or ([H’ ]/K,)“p, respectively. By combining Equations (6) and (S), it is now possible to express the total amount of enzyme in the system as
(9)
The complete set of interlevel and intralevel three-proton model are listed in Table 1.
coefficients
constituting
the
The Velocity Equation. To render the procedure general, we shah assume that catalysis can occur at any level containing enzyme forms capable of binding S-for instance, at Lty, L,, and L, (Figure 1). This assumption
‘A symbol such as hp., would signify the interlevel coefficient linking level Lp to some other level L,. However, it is more practical to define all coefficients in terms of a given reference level L,, preferably which we simplify to hp.
the catalytically
active
one. This yields
the notation
h,, r,
19
KEY PROTON FAMILIES OF ENZYMES
generates
the expression U=C(kp[SEl~
(10)
P where kp and k; are rate constants that concern the enzyme complexes at the corresponding levels. By applying the appropriate dissociation constants to express each enzyme species in terms of a chosen level L,, Equation (10) acquires the more general form
(11) This equation shows clearly that u is pH-dependent through the interlevel coefficients. However, if catalysis occurs exclusively at one level, L,, as is the case with sucrase, Equation (11) simplifies to
where u is pH-independent. The practical significance of the two quite different situations exemplified by Equations (11) and (12) is discussed in detail below. The Ouerull Rate Equation. ratio can now be derived:
&=
From Equations
([Sl/K,,)(~,h,(K,,/K,)[k,
(9) and (11) the u/[ET]
+ wwK:P)ll
(13)
~php[ll, +([Sl/Kp&‘]
Following Alvarado et al. [lo], by letting 5 = k,[ET] and &’ = k;[ET], and rearranging, Equation (13) simplifies to give an overall rate equation that has the form of the classical Michaelis-Menten equation:
where the fundamental
kinetic parameters
are
=PGhP Km = Ksr&(
K,,/K,,)$‘h,
(15)
M. VASSEUR, G.
20
When the catalytically equation simplifies to
VAN
MELLE, AND F. ALVARADO
active enzyme forms occur only at level L,, this
K + K’([Al/K,‘r)
v,=C,( K,,/K&‘h,
.
(17)
From the above equations, two facts emerge that, we shall see, can be exploited to estimate the size np of each of the proton families constituting a given model. First, Km involves both of the intralevel subcoefficients, 1; and lp, whereas V, has to do only with lg. As a result of this, the Q/K, ratios concern only lb. This result, which reflects the fact that only those enzyme forms that bind S can participate in catalysis, has general validity and is not restricted to the specific example used here to illustrate the sucrase threeproton-family model. Second, the fact that V, is defined quite differently depending on whether catalysis occurs at several levels [Eq. (16)] or at only one [Eq. (17)] is reflected by the absence of the interlevel coefficient h, in the numerator of Equation (17). THE NUMBER
AND
SIZE
OF KEY
PROTON
FAMILIES
Equation (14) is readily adaptable for the nonlinear regression analysis of enzyme kinetic data; details concerning our numerical technique and statistical analyses as applied to intestinal brush-border sucrase have been fully described elsewhere [2, 111. If the global fit can yield the best values for all the intrinsic parameters characterizing a model and, in particular, all the microscopic ionization constants, application of the appropriate equation should permit estimating in a single run both the number and the size of the proton families involved. In a second step, the adequacy of the model can be assessed by imposing appropriate restrictions on some of the estimated parameters, followed by an F test [2]. The size of the key proton families fully characterizing a given enzyme is given by the derivatives of the double-logarithmic functions of K,,,, V,, and V,/K, with respect to pH. In practice, to transform [H+] [Eqs. (15) and (16)] into pH, the known principle can be applied that, for any continuous function U([H+ I),
4logU) = -d(W)
[H+] U
dU d[H+]’
21
KEY PROTON FAMILIES OF ENZYMES
By solving Equation (18), the limiting slopes of the various 1ogU versus pH functions can be obtained. These limiting slopes furnish directly np, the number of protons constituting each given family. Application to the Three-Proton-Family Sucrase Model. To illustrate our procedure, the sucrase three-proton-family model will be used as the base, although we shall consider certain variations that illustrate the applicability of the procedure to other, more complicated situations. The various solutions envisaged are listed in Table 2, where we distinguish two fundamental cases. The simplest, case 1, fully corresponds to sucrase, where catalysis occurs only at one level (L, = L,). For this simple case, the results in Table 2 illustrate one interesting fact, namely, that the splitting of Kl yields the simplest possible solution, one that involves mixed-type kinetics representing the sum of V-type and K-type kinetic effects, which depend, respectively, on the participation of one or the other of two functionally distinct proton families [2]. In contrast, in the absence of splitting, as is the case for the basic ionization constant, four different solutions can be envisaged, V-type and K-type effects being mutually exclusive. The solutions preconized by us are compared with those defined by Tipton and Dixon [12], who did not consider the splitting of K, and therefore offer an identical series of solutions for the acid and basic branches of pH profiles. (a) me PK, versus pH function. Our procedure will be illustrated first by solving the pK,,, versus pH function, which is the simplest. The negative logarithm of K,,, [Eq. (15)] can be written as
CP’PP xp( K,,/K,,)l;h,
1 .
(19)
The first term on the right-hand side does not depend on [H+ 1, and therefore its derivative with respect to [H+ ] is zero. By applying Equation (18), we obtain
w+1
- ~,(WK,,)~;:h,
(20)
The solution for the derivatives in brackets in this equation will always exhibit the same form, namely, d h, /d [H+ ] = np h, /[H+ 1. We shall use the
M. VASSEUR,
22
G.
VAN
MELLE,
AND F. ALVARADO
TABLE 2 The Three-Proton-Family
Sucrase Modela
Limiting values of slopes derived from log (I&)
log (K/K,)
Inhibitory
log (l/K,)
effect
of protons
Kinetic effect
Case I (Catalysis takes place only at level 1) Acid branch 1A: Both Y,,EZz and EZ, bind S (x+y)
x
Y
Ictv
Mixed type
Basic branch lB,: Only EZz binds S --z lB,: lB,,:
lB,,:
Fully competitiveb
K
Both EZz and E bind S All ionization constants exhibit finite values and K,,= K,,
* lB,,:
--z
0
0
--z
Fully noncompetitiveb
V
exhibit finite values but K,,+ K,,
All ionization
constants
--z
--z
0
--z
+Z
V
Mixed type’
K,-rO 0
Case 2 (Catalysis
can take place at two levels)
Acid branch 2A: Both Y,EZz x
Uncompetitiveb
and EZ, bind S x
0
Basic branch 2B: Both EZ, and E bind S 0
0
0
“The number and function of key prototropic groups in each family, as deduced from the derivatives of the fundamental kinetic parameters defining the model. bThe nomenclature of Tipton and Dixon [12] is used here.
convention that np represents the number of protons at each level, in excess by comparison with L,, the reference level. Accordingly (see Table l), np is positive for levels on the left of L, [Eq. (l)], negative for those on the right. From Equation (20) we obtain
C,(Kr/K,)n,l~hp
VP’PP S1ope(pKm) =
Cpl;hp
-
~,(K,,/K
)l”h ’ SF P P
(21)
23
KEY PROTON FAMILIES OF ENZYMES
Because only the limiting values for the slope of pK,,, at either extreme of the pH profiles are of interest for our purposes, only the highest and the lowest protonation levels in the model under consideration need to be solved to establish the size of the proton families involved in K-type effects. Depending on the enzyme species capable of binding S, several different solutions can be conceived; these are listed in Table 2. At the acid side of the pH spectrum, for sucrase, the highest level involving S-binding is &,,. At low pH (where, mathematically speaking, [H+ ] can be said to tend to infinity), by applying the nP coefficients relevant to either 1; or 1; (respectively, 1; and l;‘y, Table l), the limiting values for the slopes of the pK,,, versus pH plot are (cases 1A and 2A): Limiy+ope(pK,)
=(x+y)-y=x.
At the basic side, as mentioned, the situation conforms to classical models and several distinct solutions are of pH-dependent enzyme activation, possible. First, at high pH where [H+ ] tends to zero, we have the following cases. Case lB,.
When only EZ, binds S, meaning Limiting slope(pK,)
that K,, + 0,
= - z.
[H+]+O
Cases lB,,, lB,, 2B. When both the EZ, and E enzyme forms bind S and all the basic ionization constants exhibit finite values, Limiting slope(pK,)
= 0.
[H+]-+O
Case 1B,, . In contrast, if both the EZ, and E enzyme forms bind S but K,, + 0, we obtain Limiting slope(pK,)
= + z.
[H+]-+O
It should be noted that a limiting slope of zero does not necessarily imply that pK,,, is pH-independent throughout the entire basic pH range. This particular situation would occur only when substrate binding does not modify enzyme ionization, i.e., when K0 = K,, (see Figure 1). If substrate binding affects enzyme ionization such that K0 # Kos, then the pK,,, versus pH plot will exhibit a “wave” [13]. (b) The logVm versus pH function. Depending on whether catalysis occurs at either one or several protonation levels, the V, parameter will be
M. VASSEUR,
24
G.
VAN
MELLE,
AND F. ALVARADO
defined quite differently, as already mentioned. Again, we will discuss each situation in turn. First, from Equation (16), and using the same procedure as for pK,, we obtain
Slope( log V,) = -
~,n,(Kr/K,) [ Vp+ V,‘([Al/K:,)]h, &(KJK,,)[
Vp+ V,‘([Al/K:,)]h,
+ ~pnp(KJKrp)l;hp ~,(K,JK,&‘h,
.
(22)
Because, in this situation, both S-binding and catalysis occur at either side of the pH profile (levels L,, and La; see, respectively, cases 2A and 2B, Table 2), the limiting values for the slope of log V, are the same at the acidic and basic ends of the spectrum: Limiting slope (log V, ) = 0. [H+ ] -+ m (or 0)
Acid branch-case 1A. When the catalytically active enzyme forms occur at only one level, Equation (16) simplifies to the second term on the right-hand side [see Eq. (17)]. The limiting value for the slope of log V, at low pH will then be Limiting slope (log V, ) = y [H+]-+co
Basic branch-case lB,. In contrast, at high pH, two different situations may again occur. First, if E does not bind substrate, we will have Limiting slope (log V, ) = 0, [H+]+O
but, (case lB,) if E binds substrate yielding the same solution:
there are then three possible variants,
all
Limiting slope (log V,) = - 2. [H+]-*O
(c)
The
log V, + pK,,,, function can log V, versus results, listed
versus pH function. Because log(I/‘,/K,,,) = the limiting values for the slope of the log( VW/Km) versus pH be obtained simply by adding the results obtained from the pH and pK,,, versus pH functions, as already explained. The in Table 2, are self-explanatory.
log(I/,/K,,,)
KEY PROTON FAMILIES OF ENZYMES
CONCLUDING
25
REMARKS
The three classical double-logarithmic functions are redundant, meaning that two of them are sufficient to obtain all the necessary information. For reasons explained in detail by Cleland [3], it has become common practice to analyze u versus pH kinetic data exclusively in terms of the log V, and log( V, /K,,,) versus pH plots, whereas pK,,, is ignored. While fully recognizing the reasons underlying this approach, the analysis marshalled in the present article furnishes a useful, complementary manner of interpreting pH effects on enzymes. According to our procedure, it appears that focusing on the V, and K,,, parameters is appropriate, particularly because the logT/, versus pH and pK, versus pH transformations furnish directly the value of n,,, the number of protons pertaining to each of the proton families respectively involved in V-type and K-type kinetic effects. The log(V,/K,,,) versus pH plot merely represents the sum of the other two. As illustrated in Table 2 (case 2), when more than one catalytic level is present, the total information is difficult to obtain, particularly with respect to the number of protons involved in V-type effects. As for those enzymes in which only one catalytic level is present (case l), the kinetic behavior of the relevant protons differs for each of the two ionization reactions. At the basic side, one of the three logarithmic functions yields limiting slope values of zero (cases lB, and lB,), a situation often encountered. The relevant double-logarithmic representations will therefore give either a wave or a straight line indicating true pH independence at either end of the pH range under study. Under these conditions, a single proton family is involved in the (basic) ionization reaction and can exhibit either V-type or K-type kinetics, but not both simultaneously. In contrast, at the acid side, all three logarithmic functions exhibit slopes with finite limiting values, revealing the coexistence of two distinct, K and V, proton families (case 1A). In conclusion, the three-proton-family model represents a useful approach for characterizing enzyme mechanisms. Nonetheless, the practical problem remains that of being able to work at pH levels sufficiently extreme, yet not too extreme, that the functional integrity of the enzyme is preserved, It is only when working as far as possible at either side of the bell-shaped velocity versus pH curves that the experimentally observed limiting slopes can be regarded as the true limiting slopes, capable of supplying the correct information. Even if the data are suitable, however, it must be borne in mind that, in practice, either one of the two proton families existing at acid pH might be nondemonstrable experimentally, in which case the three-protonfamily model will simplify to originate the linear Michaelis-Davidsohn model, characterized by having only two key proton families [14]. As an
26
M. VASSEUR,
G.
VAN
MELLE,
AND F. ALVARADO
example, if the X family is not readily apparent, the model in Equation (1) will involve only the two protonation levels hy and L,, and the corresponding limiting slopes for log( V, /K,,,), log V,, and pK, would be identical with those of cases lB, and 2B (Table 2). Conversely, if the Y family is absent, meaning that the model would include levels &, and L,, the corresponding slopes for log( V, /K,,,), log V,, and pK, would be those of case lB,. To be complete, we should mention in concluding that enzymes having two distinguishable proton families at the basic end of the pH spectrum can be envisaged [l] even though, to the best of our knowledge, such cases have not yet been described. This work was supported in part by contract UB.80.7.1094 from the Delkgation G&ale de la Recherche Scientifique et Technique (DGRST) and contract 87.7001 from the Institut National de la SantC et Recherche Mgdicale (INSERM). The work of Guy van Melle at Meudon was supported in part by the Fondation pour la Recherche Medicale, Paris. REFERENCES 1 2
M. Vasseur, M. Vasseur,
3
(1988). W. W. Cleland,
4
These d’Etat, Universite de Paris-Sud, Orsay, G. van Melle, R. Frangne, and F. Alvarado, Ado. Enzymol. 45:273-387
France, 1985. Biochem. J. 251:667-675
(1977).
5
A. Cornish-Bowden, Fundamentals of Enzyme Kinetics, 3rd ed., Butterworths, London, 1979, pp. 130-141. M. Dixon and E. C. Webb, Enzymes, 3rd ed., Longmans, London, 1979, pp. 138-164.
6 7
I. H. Segel, Enzyme Kinetics, Wiley, New York, 1975, pp. 884-914. M. Vasseur, C. Tellier, and F. Alvarado, Arch. Biochem. Biophys. 218:263-274
8 9
J. Monod, J.-P. Changeux, and F. Jacob, J. Mol. Biol. 6:306-329 (1963). R. E. Viola and W. W. Cleland, Biochemistry 17:4111-4117 (1978).
10 11
F. Alvarado, A. Mahmood, C. Tellier, and M. Vasseur, Biochim. 613:140-152 (1980). J. W. L. Robinson, G. van Melle, and S. Johansen, in Intestinal
12 13 14
Gilles-Baillien and R. Gilles, Eds., Springer-Verlag, Berlin, 1983, pp. 64-75. K. F. Tipton and H. B. F. Dixon, Methods Enzymol. 63:183-234 (1979). U. Mura and C. Bauer, J. Theoret. Biof. 75:181-188 (1978). L. Michaelis and H. Davidsohn, Biochem. 2. 35:386-412 (1911).
(1982).
Biophys. Transport,
Acta M.
France
Received
25 April 1988; revised I3 September
92190
1988
ABSTRACT We develop a method to derive the pa-dependent activation. Our presentation for sucrase, the three-key-proton model existence, in the acid ionization reaction, respectively responsible for either V-type
rate equation for enzyme models that include is based on a kinetic model recently described of Vasseur and coworkers, which considers the of two functionally distinct prototropic groups, or K-type kinetic effects. In contrast, as concerns
the basic ionization reaction, the model conforms to classical concepts of pH-dependent activation, whereby a single proton participates in either V-type or K-type effects but not in both at the same time. Enzymes
with more than three key protons
have been described,
indicating
that, rather
than isolated protons, groups of protons should be considered, and therefore the model can be better described as a three-proton-family model, where a proton family is defined as one or several protons that are gained or lost as a block and perform the same kinetic function. The resulting model is treated here as a useful framework upon which other models can be built. To facilitate the writing of the rate equations, we define two new entities: (1) intralevel coefficients, which describe the various combinations of the enzyme with either the substrate(s), the allosteric effector( or both at a given protonation level, and (2) interlevel coefficients, which describe the interplay between the various protonation levels, The resulting rate equation can be used in a global fit procedure permitting in a single computer run the estimation of (1) the entire set of dissociation and microscopic ionization constants of the model, (2) the number and kinetic function of proton families characterizing the enzyme under consideration, and (3) the number of key protons constituting each family, which is derived from the derivatives of the kinetic parameters, V,/K,,,, V,, and K,,,.
INTRODUCTION To explain the alkali metal ion and H+ dependent activation and/or inhibition of rabbit intestinal sucrase (sucrose a-D-glucohydrolase,
*Present address: Institut Lausanne, Switzerland.
MATHEMATICAL
Universitaire
BIOSCIENCES
@Elsevier Science Publishing 655 Avenue of the Americas,
de Medecine
95:13-26
Sociale
et Preventive,
Bugnon,
13
(1989)
Co., Inc., 1989 New York, NY 10010
0025-5564/89/$03.50
14
M. VASSEUR,
G.
VAN
MELLE,
AND F. ALVARADO
EC.3.2.1.48), we have proposed a kinetic model involving three key protons [l, 21. In contrast with classical models of pH-dependent enzyme activation [3-61, the three-proton model considers the existence, in a single acid ionization reaction, of two functionally distinct prototropic groups, each constituting a different proton family. One proton (family) participates in V-type activation, the other in K-type kinetic effects.’ This fact is reflected experimentally by the slope of + 1 exhibited by each of the log V, versus pH and the pK, versus pH plots; the logVm/Km versus pH plot exhibits a slope of + 2. Sucrase, however, represents a relatively simple case because each of the observed proton families contains a single member. In the present paper we develop a more general procedure for interpreting enzyme kinetic models that takes into account the fact that enzymes with larger numbers of key protons exist, yeast hexokinase being an example [9]. If a proton family is defined as one or several key protons that are gained or lost as a block and perform the same kinetic function [2], the three-proton model can be given the general form:
x
z
Protonation levels: L(3)
h
h
L(o)
where the enzyme is subdivided into four different protonation levels, one of which, L,, contains two functionally distinct enzyme forms, YyEZ, and X,EZz. The lowercase subscripts x, y, and z represent the number of protons constituting, respectively, the three proton families X, Y, and Z. The acid ionization constant Kl involves n, = x + y protons, and the basic ionization constant K,, no = z protons. The illustrated microscopic ionization constants concern the one-step displacement of a whole family and appear raised to the power of the number of protons involved. X and Y are the
‘Our nomenclature has been defined in detail previously [2, 71. We use the term “allosteric” in its most ample sense: indirect interactions between distinct sites that do not necessarily occur in separate monomers. K-kinetics means allosteric affinity-type kinetics; V-kinetics, allosteric capacity-type kinetics [8].
KEY PROTON
FAMILIES
15
OF ENZYMES
proton families respectively responsible for the K-type and V-type effects observed at the acid end of the spectrum. At the basic end, and depending on whether or not the fully deprotonated enzyme form, E, is capable of substrate binding, the Z family would be responsible for either K-type or V-type effects. Equation (1) constitutes a framework within which the behavior of a given enzyme can be explored by considering the possible specific interactions between each of the indicated free enzyme forms, the substrate(s), and any suitable modifier (allosteric activator, inhibitor, and so forth). To illustrate our procedure, we will use the simple but useful case of sucrase, illustrated in Figure 1. Previously, we have shown that this monomeric enzyme has only one substrate-binding site but possesses several modifier sites, each capable of specifically interacting with an alkali metal ion [2]. Whereas there is only one activator or A site that accepts either Na+ or Li+, there are several inhibitory or I sites: two that can bind Na+, seven that can bind Li+. The three key proton families each have one member, i.e., x = y = z = 1 [2]. Utilizing this model as the basis, in this paper we describe an original procedure that permits us to easily write rate equations readily adaptable for global, one-step computer analysis of enzyme kinetic data. We furthermore show how to identify the number and size of the proton families characterizing an enzyme, which can be done by computing the derivatives of the double-logarithmic functions of the fundamental kinetic parameters describing it. THE
RATE
EQUATION
FOR pH-DEPENDENT
ENZYME
ACTIVATION
To illustrate our procedure, we shall first establish the state [ET] and velocity u equations for the three-proton model, which are necessary to define the u/[ET] ratio that yields the overall rate equation. We shall use the simplifying assumption that the various enzyme species are all in rapid equilibrium and that the product-forming steps are rate-limiting, as is the case with sucrase. The State Equation. If [ET] is the sum of all possible enzyme forms constituting the model, and [ET], is the sum of all the forms present at each given protonation level L,, the state equation is
LETI=~[ETl,.
(2)
P
At levels L2,,, L,, and L,, (see Figure alone, bound to S, bound to A, or bound
l), the enzyme can exist either to each S and A. By using the
16
M. VASSEUR,
G.
VAN
MELLE,
AND F. ALVARADO
FIG. 1. Three-dimensional representation of the three-proton-family model for intestinal sucrase, depicting the pH-dependent interaction between a monomeric enzyme, the substrate S, and an allosteric modifier A (or I). For sucrase, A and I represent the same species, an alkali metal ion that can bind either to octiuafory or to inhibitory sites. In accord with Equation (l), the various enzyme forms have been grouped into four different protondon levels (taken from Fig. 2 of Ref. 2). One key feature of the model is the splitting of the acid ionization constant K, into two distinct pathways, which explains the coexistence, in this single ionization step, of two distinct proton families responsible for changes in either enzyme affinity (K-type kinetics) or catalytic activity (V-type kinetics). On the basic side, the situation conforms to classical concepts of pH-dependent activation, whereby a given proton family participates in either V-type or K-type effects, but not in both at the same time. Further details in [2] and/or in the text.
17
KEY PROTON FAMILIES OF ENZYMES
appropriate dissociation constants of the free enzyme, we obtain
to express each enzyme complex in terms
[ET1=~[%I,,
(3)
P
where I, represents
the intralevel coefficient characterizing
level L,, e.g.,
For later use, and in order to render our treatment more general, we find it useful to split I, into two intralevel subcoefficients, such that
,,=1;+g4;, =P
where 1; concerns those enzyme forms not involving to all enzyme complexes that include S (SE, SEA). From Equations (3) and (5), we obtain
S (E, EA) and 1; refers
Special situations (some are illustrated in the model in Figure 1) are easily taken care of. Thus, for instance, at levels where neither the substrate nor the activator bind (e.g., at L2J, Equation (5) simplifies to I, = 1; =l. At levels where there is no substrate binding but the modifier binds to several I sites (e.g., at L, where [A] is [I]), the corresponding 1; coefficients are raised to the power of nip, the number of existing inhibitory sites:
,’
P
=
1
i
\
[Al
Kp
i
“ip .
Because the various levels are interconnected by protonation-deprotonation reactions, the amount of free enzyme at a given level can be expressed by linking it to the corresponding amount of free enzyme at the reference
18
M. VASSEUR,
G.
VAN
MELLE,
AND F. ALVARADO
TABLE 1 The Three-Proton-Family Protonation level L3
(I+
Sucrase Model”
I’P
I”P
PI )“”
1
K,,
nP
%
PI+ lx+.”
x+Y
(4x)x(&“)y L 2x L 2”
1 1+
1 1+
LWG
I+ [AI/K,, LW4z,
1+
X
([H+ I/K,,)”
LWC,
(W+
1+ W/G I+ LWG,
I/K,,)’
Y
1
(4,/W+
0
I)=
“Definition of the intralevel coefficients I; and I;, the interlevel and the number of protons np in excess by reference to 15,.
- z
coefficients
hp.
level L, (where, for sucrase, L, = L,), so that
PI, = [Elrhpt where h, is the interhei coefficient defining the interplay between levels.’ Because the interlevel coefficients deal with either the gaining or the losing of np key protons, they can take one of two forms, (K,/[H+ 1)“~ or ([H’ ]/K,)“p, respectively. By combining Equations (6) and (S), it is now possible to express the total amount of enzyme in the system as
(9)
The complete set of interlevel and intralevel three-proton model are listed in Table 1.
coefficients
constituting
the
The Velocity Equation. To render the procedure general, we shah assume that catalysis can occur at any level containing enzyme forms capable of binding S-for instance, at Lty, L,, and L, (Figure 1). This assumption
‘A symbol such as hp., would signify the interlevel coefficient linking level Lp to some other level L,. However, it is more practical to define all coefficients in terms of a given reference level L,, preferably which we simplify to hp.
the catalytically
active
one. This yields
the notation
h,, r,
19
KEY PROTON FAMILIES OF ENZYMES
generates
the expression U=C(kp[SEl~
(10)
P where kp and k; are rate constants that concern the enzyme complexes at the corresponding levels. By applying the appropriate dissociation constants to express each enzyme species in terms of a chosen level L,, Equation (10) acquires the more general form
(11) This equation shows clearly that u is pH-dependent through the interlevel coefficients. However, if catalysis occurs exclusively at one level, L,, as is the case with sucrase, Equation (11) simplifies to
where u is pH-independent. The practical significance of the two quite different situations exemplified by Equations (11) and (12) is discussed in detail below. The Ouerull Rate Equation. ratio can now be derived:
&=
From Equations
([Sl/K,,)(~,h,(K,,/K,)[k,
(9) and (11) the u/[ET]
+ wwK:P)ll
(13)
~php[ll, +([Sl/Kp&‘]
Following Alvarado et al. [lo], by letting 5 = k,[ET] and &’ = k;[ET], and rearranging, Equation (13) simplifies to give an overall rate equation that has the form of the classical Michaelis-Menten equation:
where the fundamental
kinetic parameters
are
=PGhP Km = Ksr&(
K,,/K,,)$‘h,
(15)
M. VASSEUR, G.
20
When the catalytically equation simplifies to
VAN
MELLE, AND F. ALVARADO
active enzyme forms occur only at level L,, this
K + K’([Al/K,‘r)
v,=C,( K,,/K&‘h,
.
(17)
From the above equations, two facts emerge that, we shall see, can be exploited to estimate the size np of each of the proton families constituting a given model. First, Km involves both of the intralevel subcoefficients, 1; and lp, whereas V, has to do only with lg. As a result of this, the Q/K, ratios concern only lb. This result, which reflects the fact that only those enzyme forms that bind S can participate in catalysis, has general validity and is not restricted to the specific example used here to illustrate the sucrase threeproton-family model. Second, the fact that V, is defined quite differently depending on whether catalysis occurs at several levels [Eq. (16)] or at only one [Eq. (17)] is reflected by the absence of the interlevel coefficient h, in the numerator of Equation (17). THE NUMBER
AND
SIZE
OF KEY
PROTON
FAMILIES
Equation (14) is readily adaptable for the nonlinear regression analysis of enzyme kinetic data; details concerning our numerical technique and statistical analyses as applied to intestinal brush-border sucrase have been fully described elsewhere [2, 111. If the global fit can yield the best values for all the intrinsic parameters characterizing a model and, in particular, all the microscopic ionization constants, application of the appropriate equation should permit estimating in a single run both the number and the size of the proton families involved. In a second step, the adequacy of the model can be assessed by imposing appropriate restrictions on some of the estimated parameters, followed by an F test [2]. The size of the key proton families fully characterizing a given enzyme is given by the derivatives of the double-logarithmic functions of K,,,, V,, and V,/K, with respect to pH. In practice, to transform [H+] [Eqs. (15) and (16)] into pH, the known principle can be applied that, for any continuous function U([H+ I),
4logU) = -d(W)
[H+] U
dU d[H+]’
21
KEY PROTON FAMILIES OF ENZYMES
By solving Equation (18), the limiting slopes of the various 1ogU versus pH functions can be obtained. These limiting slopes furnish directly np, the number of protons constituting each given family. Application to the Three-Proton-Family Sucrase Model. To illustrate our procedure, the sucrase three-proton-family model will be used as the base, although we shall consider certain variations that illustrate the applicability of the procedure to other, more complicated situations. The various solutions envisaged are listed in Table 2, where we distinguish two fundamental cases. The simplest, case 1, fully corresponds to sucrase, where catalysis occurs only at one level (L, = L,). For this simple case, the results in Table 2 illustrate one interesting fact, namely, that the splitting of Kl yields the simplest possible solution, one that involves mixed-type kinetics representing the sum of V-type and K-type kinetic effects, which depend, respectively, on the participation of one or the other of two functionally distinct proton families [2]. In contrast, in the absence of splitting, as is the case for the basic ionization constant, four different solutions can be envisaged, V-type and K-type effects being mutually exclusive. The solutions preconized by us are compared with those defined by Tipton and Dixon [12], who did not consider the splitting of K, and therefore offer an identical series of solutions for the acid and basic branches of pH profiles. (a) me PK, versus pH function. Our procedure will be illustrated first by solving the pK,,, versus pH function, which is the simplest. The negative logarithm of K,,, [Eq. (15)] can be written as
CP’PP xp( K,,/K,,)l;h,
1 .
(19)
The first term on the right-hand side does not depend on [H+ 1, and therefore its derivative with respect to [H+ ] is zero. By applying Equation (18), we obtain
w+1
- ~,(WK,,)~;:h,
(20)
The solution for the derivatives in brackets in this equation will always exhibit the same form, namely, d h, /d [H+ ] = np h, /[H+ 1. We shall use the
M. VASSEUR,
22
G.
VAN
MELLE,
AND F. ALVARADO
TABLE 2 The Three-Proton-Family
Sucrase Modela
Limiting values of slopes derived from log (I&)
log (K/K,)
Inhibitory
log (l/K,)
effect
of protons
Kinetic effect
Case I (Catalysis takes place only at level 1) Acid branch 1A: Both Y,,EZz and EZ, bind S (x+y)
x
Y
Ictv
Mixed type
Basic branch lB,: Only EZz binds S --z lB,: lB,,:
lB,,:
Fully competitiveb
K
Both EZz and E bind S All ionization constants exhibit finite values and K,,= K,,
* lB,,:
--z
0
0
--z
Fully noncompetitiveb
V
exhibit finite values but K,,+ K,,
All ionization
constants
--z
--z
0
--z
+Z
V
Mixed type’
K,-rO 0
Case 2 (Catalysis
can take place at two levels)
Acid branch 2A: Both Y,EZz x
Uncompetitiveb
and EZ, bind S x
0
Basic branch 2B: Both EZ, and E bind S 0
0
0
“The number and function of key prototropic groups in each family, as deduced from the derivatives of the fundamental kinetic parameters defining the model. bThe nomenclature of Tipton and Dixon [12] is used here.
convention that np represents the number of protons at each level, in excess by comparison with L,, the reference level. Accordingly (see Table l), np is positive for levels on the left of L, [Eq. (l)], negative for those on the right. From Equation (20) we obtain
C,(Kr/K,)n,l~hp
VP’PP S1ope(pKm) =
Cpl;hp
-
~,(K,,/K
)l”h ’ SF P P
(21)
23
KEY PROTON FAMILIES OF ENZYMES
Because only the limiting values for the slope of pK,,, at either extreme of the pH profiles are of interest for our purposes, only the highest and the lowest protonation levels in the model under consideration need to be solved to establish the size of the proton families involved in K-type effects. Depending on the enzyme species capable of binding S, several different solutions can be conceived; these are listed in Table 2. At the acid side of the pH spectrum, for sucrase, the highest level involving S-binding is &,,. At low pH (where, mathematically speaking, [H+ ] can be said to tend to infinity), by applying the nP coefficients relevant to either 1; or 1; (respectively, 1; and l;‘y, Table l), the limiting values for the slopes of the pK,,, versus pH plot are (cases 1A and 2A): Limiy+ope(pK,)
=(x+y)-y=x.
At the basic side, as mentioned, the situation conforms to classical models and several distinct solutions are of pH-dependent enzyme activation, possible. First, at high pH where [H+ ] tends to zero, we have the following cases. Case lB,.
When only EZ, binds S, meaning Limiting slope(pK,)
that K,, + 0,
= - z.
[H+]+O
Cases lB,,, lB,, 2B. When both the EZ, and E enzyme forms bind S and all the basic ionization constants exhibit finite values, Limiting slope(pK,)
= 0.
[H+]-+O
Case 1B,, . In contrast, if both the EZ, and E enzyme forms bind S but K,, + 0, we obtain Limiting slope(pK,)
= + z.
[H+]-+O
It should be noted that a limiting slope of zero does not necessarily imply that pK,,, is pH-independent throughout the entire basic pH range. This particular situation would occur only when substrate binding does not modify enzyme ionization, i.e., when K0 = K,, (see Figure 1). If substrate binding affects enzyme ionization such that K0 # Kos, then the pK,,, versus pH plot will exhibit a “wave” [13]. (b) The logVm versus pH function. Depending on whether catalysis occurs at either one or several protonation levels, the V, parameter will be
M. VASSEUR,
24
G.
VAN
MELLE,
AND F. ALVARADO
defined quite differently, as already mentioned. Again, we will discuss each situation in turn. First, from Equation (16), and using the same procedure as for pK,, we obtain
Slope( log V,) = -
~,n,(Kr/K,) [ Vp+ V,‘([Al/K:,)]h, &(KJK,,)[
Vp+ V,‘([Al/K:,)]h,
+ ~pnp(KJKrp)l;hp ~,(K,JK,&‘h,
.
(22)
Because, in this situation, both S-binding and catalysis occur at either side of the pH profile (levels L,, and La; see, respectively, cases 2A and 2B, Table 2), the limiting values for the slope of log V, are the same at the acidic and basic ends of the spectrum: Limiting slope (log V, ) = 0. [H+ ] -+ m (or 0)
Acid branch-case 1A. When the catalytically active enzyme forms occur at only one level, Equation (16) simplifies to the second term on the right-hand side [see Eq. (17)]. The limiting value for the slope of log V, at low pH will then be Limiting slope (log V, ) = y [H+]-+co
Basic branch-case lB,. In contrast, at high pH, two different situations may again occur. First, if E does not bind substrate, we will have Limiting slope (log V, ) = 0, [H+]+O
but, (case lB,) if E binds substrate yielding the same solution:
there are then three possible variants,
all
Limiting slope (log V,) = - 2. [H+]-*O
(c)
The
log V, + pK,,,, function can log V, versus results, listed
versus pH function. Because log(I/‘,/K,,,) = the limiting values for the slope of the log( VW/Km) versus pH be obtained simply by adding the results obtained from the pH and pK,,, versus pH functions, as already explained. The in Table 2, are self-explanatory.
log(I/,/K,,,)
KEY PROTON FAMILIES OF ENZYMES
CONCLUDING
25
REMARKS
The three classical double-logarithmic functions are redundant, meaning that two of them are sufficient to obtain all the necessary information. For reasons explained in detail by Cleland [3], it has become common practice to analyze u versus pH kinetic data exclusively in terms of the log V, and log( V, /K,,,) versus pH plots, whereas pK,,, is ignored. While fully recognizing the reasons underlying this approach, the analysis marshalled in the present article furnishes a useful, complementary manner of interpreting pH effects on enzymes. According to our procedure, it appears that focusing on the V, and K,,, parameters is appropriate, particularly because the logT/, versus pH and pK, versus pH transformations furnish directly the value of n,,, the number of protons pertaining to each of the proton families respectively involved in V-type and K-type kinetic effects. The log(V,/K,,,) versus pH plot merely represents the sum of the other two. As illustrated in Table 2 (case 2), when more than one catalytic level is present, the total information is difficult to obtain, particularly with respect to the number of protons involved in V-type effects. As for those enzymes in which only one catalytic level is present (case l), the kinetic behavior of the relevant protons differs for each of the two ionization reactions. At the basic side, one of the three logarithmic functions yields limiting slope values of zero (cases lB, and lB,), a situation often encountered. The relevant double-logarithmic representations will therefore give either a wave or a straight line indicating true pH independence at either end of the pH range under study. Under these conditions, a single proton family is involved in the (basic) ionization reaction and can exhibit either V-type or K-type kinetics, but not both simultaneously. In contrast, at the acid side, all three logarithmic functions exhibit slopes with finite limiting values, revealing the coexistence of two distinct, K and V, proton families (case 1A). In conclusion, the three-proton-family model represents a useful approach for characterizing enzyme mechanisms. Nonetheless, the practical problem remains that of being able to work at pH levels sufficiently extreme, yet not too extreme, that the functional integrity of the enzyme is preserved, It is only when working as far as possible at either side of the bell-shaped velocity versus pH curves that the experimentally observed limiting slopes can be regarded as the true limiting slopes, capable of supplying the correct information. Even if the data are suitable, however, it must be borne in mind that, in practice, either one of the two proton families existing at acid pH might be nondemonstrable experimentally, in which case the three-protonfamily model will simplify to originate the linear Michaelis-Davidsohn model, characterized by having only two key proton families [14]. As an
26
M. VASSEUR,
G.
VAN
MELLE,
AND F. ALVARADO
example, if the X family is not readily apparent, the model in Equation (1) will involve only the two protonation levels hy and L,, and the corresponding limiting slopes for log( V, /K,,,), log V,, and pK, would be identical with those of cases lB, and 2B (Table 2). Conversely, if the Y family is absent, meaning that the model would include levels &, and L,, the corresponding slopes for log( V, /K,,,), log V,, and pK, would be those of case lB,. To be complete, we should mention in concluding that enzymes having two distinguishable proton families at the basic end of the pH spectrum can be envisaged [l] even though, to the best of our knowledge, such cases have not yet been described. This work was supported in part by contract UB.80.7.1094 from the Delkgation G&ale de la Recherche Scientifique et Technique (DGRST) and contract 87.7001 from the Institut National de la SantC et Recherche Mgdicale (INSERM). The work of Guy van Melle at Meudon was supported in part by the Fondation pour la Recherche Medicale, Paris. REFERENCES 1 2
M. Vasseur, M. Vasseur,
3
(1988). W. W. Cleland,
4
These d’Etat, Universite de Paris-Sud, Orsay, G. van Melle, R. Frangne, and F. Alvarado, Ado. Enzymol. 45:273-387
France, 1985. Biochem. J. 251:667-675
(1977).
5
A. Cornish-Bowden, Fundamentals of Enzyme Kinetics, 3rd ed., Butterworths, London, 1979, pp. 130-141. M. Dixon and E. C. Webb, Enzymes, 3rd ed., Longmans, London, 1979, pp. 138-164.
6 7
I. H. Segel, Enzyme Kinetics, Wiley, New York, 1975, pp. 884-914. M. Vasseur, C. Tellier, and F. Alvarado, Arch. Biochem. Biophys. 218:263-274
8 9
J. Monod, J.-P. Changeux, and F. Jacob, J. Mol. Biol. 6:306-329 (1963). R. E. Viola and W. W. Cleland, Biochemistry 17:4111-4117 (1978).
10 11
F. Alvarado, A. Mahmood, C. Tellier, and M. Vasseur, Biochim. 613:140-152 (1980). J. W. L. Robinson, G. van Melle, and S. Johansen, in Intestinal
12 13 14
Gilles-Baillien and R. Gilles, Eds., Springer-Verlag, Berlin, 1983, pp. 64-75. K. F. Tipton and H. B. F. Dixon, Methods Enzymol. 63:183-234 (1979). U. Mura and C. Bauer, J. Theoret. Biof. 75:181-188 (1978). L. Michaelis and H. Davidsohn, Biochem. 2. 35:386-412 (1911).
(1982).
Biophys. Transport,
Acta M.