I,. theor. Biol. (1977) 69, 275-285

Active Site-directed and Allosteric Effecters of Regulatory Enzymes: The Activation of Aspartate Transcarbamylase by Substrate and Transition State Analogues GEOFFREY D. SMITH

Department of Biochemistry, Faculty of Science, Australian National University, Canberra, A.C.T., Australia (Received 13 December 1976, and in revisedform 6 June 1977)

An extension of the two-state theory of Monod, Wyman & Changeux (1965) has been used to explain quantitatively the effects of substrate and transition state analogues on the kinetic and physico-chemicalproperties of aspartate transcarbamylase. The derived equations accurately predict the observed activation at low concentrations followed by inhibition at higher concentrations of such analogues. They also predict the binding, kinetic and physical behaviour of the enzyme in the presenceof both active site-directedeffecters, such as succinate, and allosteric effecters, such as cytidine triphosphate, both of which the enzyme is subjected to physiologically. The equations are applicable to other enzymespro-

vided that the system behaves as though there is only one substrate and that binding equations can be converted to kinetic equations by replacing a dissociationconstant by a microscopicMichaelis constant.

1. Introduction Several regulatory enzymes are influenced by effector molecules which compete with substrate for binding at the active sites of the enzyme. Equations have been derived (Smith, Kuchel & Roberts, 1975) which describe the binding and kinetics in such situations when the enzyme exists as an equilibrium mixture of two conformational states as proposed by Monod, Wyman & Changeux (1965). Aspartate transcarbamlyase from E. coli K-12 [Icarbamyl phosphate : L- aspartate carbamyl transferase (EC 2.1.3.2)] is welf established as such a mixture in which a “relaxed”, active state of the enzyme is in equilibrium with a “taut”, inactive or less active form (Gerhart, 1970). The binding and certain physical properties of the enzyme in the presence of the competitive inhibitor, succinate, or the allosteric inhibitor, cytidine triphosphate (CTP), have been quantitatively explained in terms of such a model (Changeux & Rubin, 1968). The treatment of Changeux & Rubin 275

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(1968), however, did not encompass the kinetic behaviour of the enzyme nor did it describe the properties of the enzyme in the presence of active-site directed and allosteric effecters, a situation which the enzyme is subjected to in uiuo. In addition, it has not been shown whether certain kinetic properties of the enzyme, such as its activation by substrate and transition state analogues at low aspartate concentrations, are compatible with, or explicable in terms of the two-state model. It is the purpose of this communication to present a generalized theoretical treatment to allow quantitative description of the binding, kinetic and certain physical properties of the enzyme under a variety of conditions and in terms of meaningful physical parameters. 2. Binding Equations

Consider an enzyme that exists in two isomeric (m = 1) or oligomeric (m > 1) conformations, E, and E,, with a molar equilibrium constant, X. mE 1 =J%,

(1)

x = cw/cm”*

(2)

If E, and E2 have n,, and nZs equivalent and independent sites at which ligands S and A can bind competitively, and also nls and nZ8 alternative equivalent and independent sites at which ligand B can bind allosterically, then it can be shown using methods based on those described previously (Smith, Kuchel & Roberts, 1975; Nichol, Smith & Ogston, 1969; Nichol, O’Dea & Baghurst, 1972) that the number of moles of S bound per base mole of enzyme is given by:

U-JGlCW~,s(l

+ CSll~,, + C~/~LJ”~-

‘(I+ CBl/fh,)“‘“+

r, =

where

I%1 = b%I(1+ Csll~,, + C4/kJ”S(1 + CBIIhJ”lB + mCJ%lU+ CSl/&s + C4I&JY1+ CWW’*“~

(4) and [ ] denotes molar concentration of a given species, K,,, KIA and K,, represent the intrinsic dissociation constants of S, A and B binding to E,, and Kzs, K2,, and K,, are the corresponding intrinsic dissociation constants for E2. It can be shown by induction that, in general, for i forms of enzyme, j types of ligand binding competitively with S and k types of ligand binding allosterically, each at a different class of sites,

T dIEJCSl/&.dl r, =

+ CSl/~i~ + 7 CAl~i~)nis-l v Cl+ C~lIKd”‘”

T mCEJ(l+ iISl/Ks + F LLl/~i~~‘” J-JCl+ CkllKJ’“‘--

’ (5)

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ENZYMES

where m now represents the number of monomeric sub-units in the oligomer EYZand the dissociation constants K and number of binding sites n are subscripted by numbers representing the particular enzymic species i and clompetitive j or allosteric k ligand. 3. Kinetic Equations The usual approaches for describing enzyme kinetics become too complex for the types of enzyme under discussion but it is possible to use an alternative approach which involves transcribing thermodynamically derived binding equations into kinetic equations (Dalziel, 1968; Smith et al., 1975; F’rieden, 1967). The assumption implicit in this approach is that the rate of breakdown to product of any complex between enzyme and substrate is a rate-limiting step and hence the dissociation constants for S can be regarded as microscopic Michaelis constants (Dalziel, 1968). Applying this approach to the enzyme described by equations (l)-(3), where S is now regarded as the single substrate for the enzyme and A a competitive ligand, allows an initial velocity for the enzyme to be written as:

VI CSl/& SC1+ Csl&

+ C4/L)n’S-

‘Cl+ cw/~,B)“‘B +

V2-J WW”-‘CSl/~zsU + CSl/L + c4/&,4)“zs- ‘(1 + wZB)n2B, lJ = (i-~s]/K,,+[A]/K1”)“‘S(l mX[EJ”-‘(1

+[Bl&j)nIB+ +[S]/K,, +[A]/K2J”ZS(1

(6)

+[B]/K,J2”

where V, and V, represent the maximum velocities of E, and E,, respectively. Equation (6) can be simplified to be more applicable to aspartate transcarbamylase since it may be assumed (Changeux & Rubin, 1968; Gerhart, 1970) that E, (the taut form of the enzyme) is inactive and effectively does n!ot bind competitive ligands such as succinate. Hence with m = 1,

~IL-sl/J%~ + CSl/~,, +C4/~,,4)“‘“’ = (1 +[S]/K,s+[A]/K,,)“‘S(l

+[B]/KIB)“‘B+X(l

‘Cl+ CmkI)“‘B +[B]/K,,)“**’

(7’

4. State Functions

In relating binding or kinetic properties of an enzyme undergoing conformational changes to the degree of conformational change it is necessary to have a state function which relates the physical parameter being measured to the amounts of the different enzyme species present (Monod et al., 1965). For the present discussion the state functions will be written in terms of the weight-average sedimentation coefficient, S, but the same arguments

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apply to any weight-average property, including weight-average molecular weight. The Sis defined (Schachman, 1959) for the example under discussion by: j=

sE~ cE,

+ sEz cEz I CO

CO =

(9)

CE, +CE2,

where sE1and sE2are the sedimentation coefficients of E, and E2, respectively, and cEI and cE2 are the respective weight concentrations (g/l) of E, and E, and their complexes; it is assumed that bound ligands do not contribute to the sedimentation coefficient by virtue of their own weight or density. and

c.szz = M,L-Ed1 +CSliG, + ~4/GJ”=t~

+ fW~z,P”,

(11)

where ME, and ME* are the molecular weights of E, and E,. Combining equations (2), (9), (10) and (11) allows c& and cEI to be rewritten as functions not containing [El] and [E,] and hence from equation (8) the S can be written in terms of the known or assumed constants or concentrations m, MEI, MEP [sl, [Al, [BIT nts nm nlBy “2m Km K2,, K,,, K2.4, Km K~B, x> Co, sE1 and SE2; as is usual in enzyme kinetics it is assumed that the free concentrations of substrate and effecters are equal to their total, initial concentrations which are known. In presenting these equations the concentration and binding constants for substrate have been included to make them potentially applicable to active enzyme centrifugation (Claverie, Dreux & Cohen, 1975). This approach can readily be applied to other particular enzyme situations than isomerizations as indicated by the generalized binding equation (5), including dimerizing enzymes; however, for higher order associations solving the equations for the respective enzyme concentrations becomes difficult due to the need to solve higher order polynomial equations than quadratics. 5. Application

to Aspartate

Transcarbamylase

Kinetics

Aspartate transcarbamylase is a two-substrate enzyme whose kinetics have not been completely resolved, even for the catalytic subunit alone. Some workers believe that the enzyme has a rapid equilibrium random mechanism (Heyde, Nagabhushanam & Morrison, 1973) whereas others believe it to be ordered (Gerhart, 1970; Collins 8~ Stark, 1971). For the present work this disagreement is unimportant since it is known that carbamyl

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phosphate binds more rapidly and tightly to the enzyme than does aspartate and much of the work with the enzyme, both kinetic and structural, has been done under conditions where carbamyl phosphate is saturating. Many of the literature values for binding, kinetic and structural properties refer to such a situation. The mechanism is therefore effectively ordered but can be approximated by a one-substrate mechanism. If the interconversion of the central complexes in the mechanism is slow compared with all other steps in the reaction sequence, which is possible for a random (Heyde et al., 1973) or an ordered mechanism, the assumption that the Michaelis constant for aspartate is effectively a dissociation constant is fulfilled and equation (7) is applicable. In this case S represents aspartate, A represents succinate or some other competitive (active site-directed) ligand such as substrate or transition state analogue and B represents an effector such as CTP or ATP which binds at the regulatory subunit. The forms E, and E2 represent the relaxed R and taut T states of the enzyme, respectively. Equation (6) can be simplified to equation (7) in its application to aspartate transcarbamylase by noting that succinate and presumably other active site-directed ligands effectively only bind to one form (E,) of the enzyme (Changeux & Rubin, 1968) and hence that only this form of the enzyme is catalytically active. Thus V, = 0 and V, represents the overall maximum velocity in the absence of an allosteric ligand B. Similar simplification can be made for the binding and state function equations. That the kinetic approach used herein is valid for this enzyme is shown in Fig. 1, in which equation (7) is used to curve fit published kinetic data (Gerhart, 1970) for the enzyme both in the absence of effecters and, in turn, in the presence of CTP and ATP. In fitting these data K,, has been taken to be close to the K,,, of 8.1 x 10e3 M for the free catalytic subunit (Heyde et al., 1973) since it is known that the conformational and binding properties of this subunit are very similar in the free state and in the “relaxed” form of the native enzyme (Changeux & Rubin, 1968). The number of binding sites for all ligands has been taken to be six, in accord with the known hexameric structure of both catalytic and regulatory subunits in the native enzyme (Gerhart, 1970). The values of the dissociation constants of CTP to the E, and E, forms of the enzyme are close to those used by Changeux & :Rubin (1968) for BrCTP. The curve in the presence of ATP was calculated assuming negligible binding of ATP to the E2 form and using a dissociation constant for E, which gave the best fit. The value of 8 for the equilibrium constant describing the equilibrium between the two forms of enzyme is iin a range known to be reasonable for the enzyme in saturating carbamyl Iphosphate (Gerhart, 1970); its value would be expected to vary over a small range depending on the precise experimental conditions.

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In Fig. 2 the equations presented above have been used to curve-fit the interesting activation effect on the enzyme by maleate at low (1 x 10e3 M) aspartate concentration (Gerhart & Pardee, 1963). This effect is clearly consistent with and indeed follows directly from the fact that the enzyme is a two-state system. The relative velocity has been calculated by dividing

the initial velocity calculated from equation (7) at each value of the concentration of the competitive ligand, A, by the initial velocity calculated with

J-----I

rn~ Asporiole

FIG. 1. Plots of initial velocity of aspartate transcarbamylase, in the absence (curve B) and presence (curve C) of the inhibitor CTP or activator ATP (curve A). The experimental points are taken from Fig. 3 of Gerhart (1970). The theoretical curves were calculated using equation (7) with the following parameter values: VI = 7.6 mmol h-l mg,-l KI, = 5.5 x 10-a M, nla = 6, nls = 6, nza = 6, X = 8, [A] = 0 (curves A, B, C); [B] = 0 (curve B); [B] = 2.0 x 10m3 M, KIB = 1 x 1O-5 M, &, = 100 M (curve A); [B] = 0.5 x 10m3M, Kls = 1.95 X 10e5 M, KzB = 1.35 x 10e5 M (curve C).

[A] = 0. Heating the enzyme has the same effect on the enzyme as adding mercurials (Gerhart & Pardee, 1963), viz. to remove the co-operativity of the enzyme. Line B in Fig. 2 was obtained with exactly the same parameters in equation (7) as line A except that X = 0; i.e. the enzyme is all converted into the El form. This is kinetically equivalent to converting the enzyme totally into catalytic subunit because of the kinetic equivalence of E, and the catalytic subunit (Changeux & Rubin, 1968; Gerhart, 1970). In Figure 3 similar data by Collins & Stark (1971) with succinate (or maleate) and the transition state analogue N-(phosphonacetyl)~L-aspartate (PALA) is curvefitted. Clearly the transition state analogue is effective at much lower concentrations, reflecting its much higher affinity for the enzyme than those of the substrate analogues. For this calculation A now represents PALA and because the parameters [A] and KIA only appear as a ratio in equation (7) the data obtained with different competitive ligands can all be

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FIG. 2. Activation of aspartate transcarbamylase by maleate. The experimental data are those of Gerhart & Pardee (1963). Curve A refers to native enzyme and curve B to enzyme whose regulatory properties have been destroyed by heat treatment. The theoretical curves were calculated using equation (7) with the following parameter values: [:?I= 1 X lo-3 M, Kls = 8.1 X 1O-3 M, Kla = 7.5 x 10e4 M, nIs = 6, [B] = 0 and 1’ = 6 (curve A) or X = 0 (curve B).

FIG. 3. Activation of aspartate transcarbamylase by maleate, succinate or PALA. The experimental data are those of Collins & Stark (1971). The theoretical curve was calculated using equation (7) with the following parameter values: [S] = 1 ?: 10e3 M, h;s = 8.1 x 1O-3 M, nls = 6, [B] = 0 and X= 12 (all effecters), Kla = 7.5 x low4 M (succinate and maleate) or KIA = 2.3 x 1O-B M (PALA).

282

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plotted on the same graph by altering the abscissa scale. Having chosen KrA = 7.5 x 10d4 M for succinate, the corresponding value for PALA is 2.3

x 1o-6

M.

In Fig. 4 is shown a theoretical prediction, using the same parameters as in Fig. 2, of the effect of increasing the substrate concentration on such a plot. At higher substrate concentrations, e.g. 20 mM, the activation effect

.

0

?

3

4 itffeclor)

5

6

7

8

9

II

x IO

FIG. 4. Theoretical curves showing the dependence on substrate concentration of the activation by an active site-directed ligand such as succinate in the case of aspartate transcarbamylase. The curves were calculated from equation (7) using the following parameter values: KIs = 8.1 X 10m3 M, nlS = 6, KIA = 7.5 x lo-* M, [B] = 0, X= 6 and [S] = 1 x low3 M (curve A), [s] = 2 x 10m3 M (curve B), [S] = 20 x 10m3 M (curve C).

of the analogues is not seen. The activation effect clearly stems from the crossover which is obtained in the v versus [S] plots in the presence and absence of the competitive ligand (Smith et al., 1975). Such a crossover, relevant to the data of Fig. 2, is illustrated in Fig. 5. In this figure it is also shown that an allosteric inhibitor, such as CTP in the case of aspartate transcarbamylase, lowers the substrate concentration at which the crossover occurs. Since in vivo the enzyme is presumably influenced simultaneously by succinate and CTP, the effect illustrated in Figs 2 and 3 is not likely to be so dramatic in the bacterial cell. This has been suggested by Collins & Stark (1971). The equations presented above allow the kinetics in the presence of both effecters simultaneously to be quantitatively predicted. 6. Quantification of Sedimentation Results

Many of the conclusions concerning the two-state structure of aspartate transcarbamylase were based on the physical studies, including difference sedimentation studies, by Gerhart & Schachman (1968). In Fig. 6 the

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283

FIN. 5. Theoretical curves showing the effect of an allosteric effector, such as GTP, on the crossover in a velocity versus substrate plot caused by an active site-directed effector such as succinate in the case of aspartate transcarbamylase. The curves were calculated from equation (7) using the following parameter values: V, = 7.6, K,, = 8.1 x lO-3 M, &,, = 7.5 x 1O-4 M, nls = 6, nlB = 6, n 2B := 6, X = 6 (curves A, B, C); [A] = [B] = 0 (curve B); [A] = 0.4 x 10e3 M, [B] = 0 (curve A); [A] -~ 0.4 x 10e3 M, [B] = 2 x 1O-5 M, K,,

= 2 X 1o-5

M, K,,

= 1.3 X 1o-5

M.

FIG. 6. The effect of succinate on the sedimentation coefficient of asp&ate tramcarbamylase. The abscissa represents the change in sedimentation coefficient observed upon adding a given concentration of succinate plus saturating carbamyl phosphate to the enzyme. The experimental points are those of Gerhart & Schachman (1968). The theoretical curves were calculated as explained in the text using equations (2) and (8)-(11) %with the following parameter values: m = 1, sEl = 11.2S, sEa = 11.6.5 c0 = 4.2 g I- 1 (equivalent to 1.4 x 1O-6 M), n 1s = 6, Mm = iv&z = 310,000, [S] = 0, [B] = 0, x =: 4 and KIa = 4.75 x lo-* M (curve A) or K,, = 7.5 X IO-” M (curve B). The two points lshown as squares are calculated points using the above parameters and with nIs = nas = 6, &, = 2 X 10m6 M, Kzs = 1.3 x lo-” M and [B] = 1 x lo-* M or [B] = 1 v IO-” M. ‘The abscissa would be represented by [A].

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difference sedimentation data in Fig. 11 of Gerhart & Schachman (1968) have been fitted in terms of the equations and approach presented above (Section 4). In fact the equations presented by Changeux & Rubin (1968) are sufficient for this purpose, representing a specific case of those given above. However, the equations presented herein would allow quantification of sedimentation data obtained in the presence of more than one active site-directed ligand. In Fig. 6 theoretical curves have been calculated for two different dissociation constants of succinate, one (4.75 x 10m4 M) being that assumed by Changeux & Rubin (1968) and the other (7.5x IO-’ M) being that used in curve-fitting Figs 2-5 of this paper. 7. IXscussion The two-state theory of Monod, Wyman & Changeux (1965) and variations of it based on interconversions between different oligomeric forms of an enzyme (Frieden, 1967; Nichol, Jackson & Winzor, 1967) describe sigmoidal binding and physical behaviour of a number of proteins and enzymes. Extensions of the theory have allowed description of such systems in which an allosteric effector (a ligand that binds at separate binding sites from those at which the ligand whose binding is being measured binds) is also present (Frieden, 1967; Rubin & Changeux, 1966; Changeux & Rubin, 1968; Nichol et al., 1972) and recently an extension has been made to describe an alternative situation in which active site-directed effecters are present (Smith et al., 1975). Certain enzymes, such as aspartate transcarbamylase, are subjected in vivo to the influence of both types of effector simultaneously and the present communication extends the previous treatments to incorporate such a situation. Furthermore, the approach of Dalziel (1968) has been used to transform the binding equations into kinetic equations to allow description of the kinetics of such enzymes. This approach is perhaps the only one that makes tractable the kinetics of such complex enzyme systems, although certain assumptions are involved (Dalziel, 1968). The additional assumption that aspartate transcarbamylase can be approximated as a one-substrate enzyme at saturating concentrations of carbamyl phosphate seems valid in view of the tight and rapid binding of this substrate; the reaction would effectively become pseudo-first order with respect to the substrate aspartate. The fact that the procedure works so well in the case of aspartate transcarbamylase gives one confidence that the assumptions are valid in this case. The values of the parameters used in curve-fitting the published data have been taken, as quoted, from the scientific literature. Of course, such parameters as X, the equilibrium constant between the two forms of the enzyme,

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will be expected to vary to an extent depending on the precise experimental conditions used, especially when one considers that the free energy difference between the two conformational states is very small (Gerhart, 1970), and of course there are slight discrepancies in similar published experimental results for the enzyme. It is for this reason that limited licence has been taken in varying the value of Xto give the best possible fit to the data. The variations in parameters used are likely to be well within the experimental error in their determination. The concept of an active site-directed or competitive effector which could activate an enzyme existing in two conformational states in equilibrium was noted by Smith et al. (1975). As shown by them the kinetics shown by an e’nzyme in the presence of such a competitive effector may often be indistinguishable from those shown in the presence of an allosteric effector. However, the observation, at low substrate concentration, of an activation at low c’oncentrations of effector followed by inhibition at higher concentrations (1Figs 2 and 3) would be diagnostic for the competitive rather than the allosteric situation. It would presumably not, however, be definitive for a preexisting two-state conformational equilibrium as opposed to a ligandinduced conformational change. It is of interest that glyceraldehydephosphate dehydrogenase (De Vijlder, Hilvers, van Lis & Slater, 1969) and phosphofructokinase (Mansour, 1972) also exhibit activation followed by inhibition in the presence of NAD and ATP, respectively. REFERENCES CHANGEUX, J.-P. & RUBIN, M. M. (1968). Biochemistry 7, 553. CLAVERIE, J.-M., DREUX, H. & COHEN, R. (1975). Biopolymers 14, 1685, 1701. COLLINS, K. D. & STARK, G. R. (1971). J. biol. Chem. 246, 6599. DALZIEL, K. (1968). FEBS Lett. 1, 346. DE VIJLDER, J. J. M., HILVERS, A. G., VAN Lrs, J. M. J. & SLATER, E. C. (1969).

Biochim. biophys. Acru 191, 221. F:RIEDEN, C. (1967). J. biol. Chem. 242, 4045. GERHART, J. C. (1970). Curr. Top. Cell. Reg. 2, 275. GERHART, J. C. & PARDEE, A. B. (1963). Cold Spring Harb. Symp. quant. Biol. 28, 491. GERHART, J. C. & SCHACHMAN, H. K. (1968). Biochemistry 7, 538. HEYDE, E., NAGABHUSHANAM, A. & MORRISON, J. F. (1973). Biochemisfry 12, 4718. MANSOUR, T. E. (1972). Curr. Top. Cell. Reg. 5, 1. MONOD, J., WYMAN, J. 8c CHANGEUX, J.-P. (1965). J. molec. Biol. 12, 88. NICHOL, L. W., JACKSON, W. J. H. & WINZOR, D. J. (1967). Biochemistry 6, 2449. NICHOL, L. W., O’DEA, K. & BAGHURST, P. A. (1972). J. theor. Biol. 34, 255. NICHOL, L. W., SMITH, G. D. & OGSTON, A. G. (1969). Biochim. biophys. Acta 184, 1. RUBIN, M. M. & CHANGEUX, J.-P. (1966). J. molec. Biol. 21, 265. SCHACHMAN, H. K. (1959). Ultracentrifugation in Biochemistry. London: Academic Press. SMITH, G. D., KUCHEL, P. K. & ROBERTS, D. V. (1975). Biochim. biophys. Acta 377, 197.