I. rheor. Biol. (1977) 68, 391413

A Simple Model for a Regulatory Enzyme STANLEY AINSWORTH

Department of Biochemistr)*, Universit!. of‘ Shefield, Shefield SlQ 2TN, U.K. (Received 2 A4arch 1977) A simple model for a regulatory enzyme is describedwhich leads to relationshipsbetweenthe initial velocity of the catalysedreaction and the varied concentration of a substratethat are of the non-inflected or sigmoidal varietieswithout a maximum. The model assumes that the most relevant measureof protein configuration (itself determining the kinetic behaviour of the enzyme) is the apparent association constant, ai, measuredfor the given fractional saturation of the ligand under investigation. It is further assumedthat the original state of the protein in solution, ao, is destabilizedby an increment of energy, AC:, that is proportional to the fractional saturation of the enzyme by ligand so that the formation of a new configurational state, a,, can be representedby -AGO, = RTln at/c+,. The rate or fractional saturation equation that can be derived from this model predicts both positive and negative cooperativity. Either equation can be transformedfor linear representation, provided the maximum velocity or its equivalent maximum saturation is known, and estimatesof a0and aI (the apparent associationconstantsat zero and complete saturation) can be obtained thereby. A procedure is also describedby which an initial estimate of the maximum velocity or saturation can be improved. The modelis testedby application to a range of data in the literature and it is shown to give fits to the data comparable in quality to thoseprovided by the model of Monod, Wyman & Changeux (1965).

1. Introduction The non-hyperbolic rate equations given by many oligomeric enzymes have attracted considerable attention becauseit is thought that they provide the basis for the short-term control of metabolic reactions. In consequence,the mathematical analysis of models for these enzymes has been developed to a very sophisticated level. Unfortunately, the complexity of the rate equations that are predicted (in particular their powered dependenceon ligand concentrations) has inhibited the type of detailed experimental investigation that is

392

S. AINSWORTH

commonplace with enzymes whose rate equations conform to the Michaelis-Menten expression (Cleland, 1970). The purpose of this paper, therefore, is to describe a simple model for a regulatory enzyme which predicts a rate equation that can be linearly transformed so as to provide estimates of its two disposable constants. It is hoped that this simplicity might facilitate the description of allosteric effects and open the way to the systematic cxperimental investigation of those enzymes whose behaviour can be so described.

2. The Model (A)

DEFINITION

The hyperbolic Michaelis-Menten

OF THE PROBLEM

expression can be written as: __VAu ~.~

v = (1 +AtX)

(1)

where v represents the initial velocity of the catalysed reaction corrected to unit concentration of enzyme. V is an apparent maximum velocity and c1the reciprocal of the apparent Michaelis constant for the substrate whose identity or concentration is represented by A. Both apparent constants are functions of the concentrations of other substrates and effecters (if added to the assay solutions) so that, when their values are determined, they can be subjected to further analysis as functions of these non-varied concentrations (Ainsworth, 1975), a procedure that is facilitated by linear transformation of the rate equation. By contrast, a general expression for the rate of a reaction catalysed by an oligomeric enzyme can be represented as : v _ (n,A=+n,A’-’ _~~~~. ~_ . ~_..~ . n,A) (d,A”+dzAz-’ . . . &+,)

(2)

where the coefficients of both the numerator and denominator are themselves polynomials in the concentrations of other substrates and effecters kept constant throughout the experiment at non-saturating values. Equation (2) can give rise to curves, v =f’(A), with inflections and with maxima (Botts, 1958) and correspondingly, general models for oligomeric enzymes have been devised (Whitehead, 1970; Wyman, 1972; Kurganov, Kagan, Dorozhko & Yakovlev, 1974) which allows this range of behaviour. It is not proposed to seek this generality: instead, the model that is to be examined is intended to lead only to relationships, ~1 = .flA), of the non-inflected and sigmoidal varieties which do not possess a maximum. An extension of the model that would explain maxima is introduced but not considered in detail.

hfODEL

FOR (B)

A BASIC

REGULATORY

ENZYME

?93

ASSUMPTIONS

It is assumed that the enzyme retains a constant molecular weight and reaches a steady state with respect to its substrates and other ligands before measurements of the initial rate of the catalysed reaction are made. The individual protomcric unit of the oligomer are assumed to be identical, each possessing one set of binding sites for the substrates and products of the reaction and the other ligands which affect the rate of catalysis. It is furthcl assumed that the catalytic mechanisms of the individual protomcrs give rise to rate equations that can be represented by a hyperbolic function 01 ~1::: substrate concentration that is varied. (C)

PHYSICAL

BASIS

OF THE

MODEL

The slope of the hyperbolic relationship, u = f(A), progressively dccreascb because it becomes more and more difficult for an A molecule to find an empty binding site as the fractional saturation of the enzyme increases. in the sigmoid curve, by contrast, it becomes easier, over part of the concentration range, for a molecule of A to bind. The bound molecules of A must, therefore, be interacting in some way with the empty sites to increase their affinity for A. With the assumption that each protomer of the enzyme carries one such A binding site, then it must appear that the interaction hasto OCCLI~ between the protomers. It is also possible to imagine that binding becomes progressively more difficult than that required by the Michaclis-Menten caiuation and to distinguish “positive” from “negative” interaction. The physical basis for the interaction between protomers is generally assumedto be in a changed free energy-structure-function relationship of the oligomer caused by the binding of ligand molecules to individual protomet-s.This may involve an observable change in the tertiary or quaternary structure of the protein, what is usually called its conformation. Such changes depend on the well-establishedexistence of protein flexibility and have already been invoked by Koshland (1958) to form the basis of his “induced fit“ theory of enzyme action. In addition, however. there may occur lessobvious 4iifts in electrical forces which are not the lessimportant if they influence the constants for ligand binding or substrate transformation. Such operationally significant changes are called “configurational” by Whitehcud ( 1970) and can be taken to include conformational changes where appropriate. Within ihis contest, it is possible to devise a model for a regulatc>ry enzyme which employs constants for the configurational changesthat occur in the oligotnet a:, binding progresses, together with further constants to describe ths cvnsequential changes in ligand affinity, but such a model, with an unr~stGctrd nllmber of constants. would be of no use in the practical :mnlysis of kinetic

394

S. AINSWORTH

data. The most important models of a regulatory enzyme have, therefore, had to define simplified relationships between structure and function (Monad, Wyman & Changeux, 1965; Koshland, NCmethy & Elmer, 1966). The corresponding simplifications that are now proposed are given in the following statements with their associated special assumptions. (1) It is assumed that the initial velocity of the catalysed reaction is represented by a modified Michaelis-Menten equation : VAEi u = mplete saturation represented by 107.4 in relation to Kirschner’s scale of 0 100. Lines B, C and D show alternative plots of equation (9) where complete saturation was supposedto occur at 100.0, 104.5 and 109.1, respectively, in relation to Kirschner’s scale. The four lines A, B, C and D validate the conclusions of equation (10) and represent a test application of equation ( ! -7)t Kirschner (1967) according to the model of Monad, Wyman & Changcu.x (1965). (c) PYRUVATE KINASE FROM Succhurott~~w.s

r~urlsbergmsis

Figure 5 representsrate data for the forward reaction catalysed by pyru\atc kinase in the presenceand absenceof the activator, fructose- I, h-diphosphatc, and inhibitor, ATP (Fig. I of Johannes & Hess, 1973). The maximum velocities estimated by Johannes & Hess were employed in plotting

0

0.5

I.0

I.5 A=phosphoenolpyruvote

2.0

2.5

no

FIG. 6. Initial rate of reaction catalysed by pyruvate kinase (Rozengurt er N/., 1969). Curve A, 0.1 nw-fructose-1,6-diphosphate and 2 mwATP; curve B, control: curve C‘. 2 mwATP. For symbols see Fig. 4.

MODEL

FOR

A

REGIJLATORY

403

ENZYMl

equation (9) but it is probable that these values arc underestimated for lines A and B but overestimated in respect of line C. The calculated curves fit the data points satisfactorily except at the extremes of curve C: in this connection, it should be noted that Johannes & Hess (1973) also failed to lil these points by the Monod, Wyman & Changeux (1965) model. (D)

PYRUVATE

KINASE

FROM

Figure 6 gives rate data for the forward kinase from rat liver (Fig. 4 of Rozengurt, 1969). Examination of the figure shows that range but fails to represent the low velocity (E)

PHOSPHORYLASE

B FROM

RAT

LIVER

reaction catalysed by pyruvate JimCnez de Astia & Carminatti. equation (9) holds in the middle points satisfactorily. RABBIT

MUSCLE

Figure 7 illustrates rate data for phosphorylase B (Fig. 2 of Madsen 8r SIhechosky, 1967). As before, the curve is calculated by equation (I 3) from the constants of the line given by equation (9). The fit obtained is as good as that provided by Madsen & Shechosky (1967) using the Monod, Wyman CkChangeux (1965) model.

I

0

2

I

4 A : [Glucose-l

I

I

6 -phosphate]

a /K,,,

FIG. 7. Initial rate of reaction catalysed by phosphorylase R in the presence of 10 rnM. ATP (Madsen & Shechosky, 1967). For symbols see Fig. 4.

406

S. AINSWORTEI

(F) PHOSPHORYLASE B FROM RABBIT MUSCLE

Figure 8 also illustrates rate data for phosphorylase B (Fig. 1 of But, 1967). The original data were presented with an arbitrary velocity scale. The curves and lines shown in Fig. 8 were therefore calculated by assuming that the highest velocity recorded on each curve, together with the corresponding maximum velocities, had the following relative values on a uniform scale: A, 6*92,7.125; B, 6~75,7WO; C, 6.26,6.720; D, 5.25, 7.125. Thesevalues secure excellent fits by equations (9) and (13) except for the lowest velocity points. Note that CI~can be plotted as a linear function of the concentration of the activator, AMP. (G)PHOSPHOFRUCKTOKINASE FROM Escherichiu coli Figure 9 shows the initial velocity of the reaction catalysed by phosphofructokinase as a function of the concentration of the substrate, fructose-6phosphate (Fig. 11 of Blangy, But & Monod, 1968). Equations (9) and (13) were employed with a value of V set at 93.63 on the original velocity scale. Table 1 records the constants that were determined as a function of the concentration of the inhibitor, phosphoenolpyruvate. It is evident from these

TABLE 1

The line constants of equation (9) jbr the phosphofructokinase function of phosphoenolpyruvate concentration

reaction as a

Curve

___- -~~-- ____....~~ Phosphoenolpyruvate (mM) k al X lo-' aox10-2M-1

M-’

A

B

C

1.6 3.06 16.4 0.771

4.0 2.39 6.49 0.594

8.0 2.55 4.95 0.387

11 24.0 2.38 3.13 0.289

E 35.0 0.98 0.923 0.346

figures that phosphoenolpyruvate reduces the affinity of the enzyme for its substrate, but with a much more marked effect on ~1, than on c(~: correspondingly, the value of k diminishes as the phosphoenolpyruvate concentration increases. It also appears that the value of cl0 determined for curve E is anomalous, a conclusion consistent with the distinctly different character of the original curve. (H)THREONINE

DEAMINASE FROM Bacillus subtilis

Figure 10 illustrates the effect of substrate concentration on the initial rate of threonine deamination in the presence and absence of the inhibitor, isoleucine (Fig. 4 of Hatfield & Umbarger, 1970). The fit to the original data

408

S. AINSWORTH

I

0

2 isoleucine

3

4

5

x 10’~

FIG. 11. Values of cq,- ’ (G) and CY,- 1 (A) calculated from the lines illustrated in Fig. 10, represented as a function of the concentration of inhibitor, isoleucine.

In[v/(V-“)A]

I -----.I

5

IO

-_

-.--I

~

IS L :Aspar

20

;5

tote rn~

FIG. 12. Initial rate of reaction catalysed by aspartate transcarbamylase (Gerhart, 1970). Curve A, 2 mwATP; curve B, control; curve C, 0.5 mwcytidine triphosphate. For symbols see Fig. 4.

MODEL

FOR

A

REGliLATORY

409

ENZYME

points procided by equations (9) and (13) is good, only one point sho\ving a marked departure from the calculated curve. Figure 11 shows a plot of the apparent Michaelis constants, a,’ and 2; I, as a function of the inhibitor concentration. The relation with cc;’ is linear, but that with a;’ may hc curved. It is evident from Fig. I I that the final form of the enzyme has ;I higher affinity for the substrate than has the initial form and that thi\ differcncc becomesmore marked as the inhibitor concentration increases.In consequence,the values of k provided by the four curves are 0.2 I, I +I. I .Si and 2.35 in the order of increasing isoleucine concentration. It will bc SCCII tllat this order is the opposite of that observed in the inhibition of phospho1‘ructokinase by phosphoenolpyruvate (Table 1). Finally. note that extrapolation of the lines drawn in Fig. 1I to the concentration axis indicates rhat the final form of threonine deaminasehas a much lower aRinity for isolcucino than has the initial form. (I)

ASPARTATE

TRANSCARBAMYLASL

FROM

ESCHERIC’HI

1 C‘OL.1

Figure 12 shows the effect of aspartate on the initial rate of t-caction catalysed by aspartate transcarbamylase both alone and in the presenceot inhibitor, cytidine triphosphate and activator, ATP (unpublished observations of Pigict, Yang & Schachman given in Fig. 3 of Gerhart. 1970). The data point:, were fitted by equation (9) with a value of Vxet at 7.10 O!I the c)riginal velocity scale. It will be observed that the three lines are drawn to pass through a single value of In t~[ which provides xl = 0.423 mM -I. (J)

ASPARTATE

TRANSCARBAMYLASE

FROM

ESCHERICHl.4

C0l.l

Figure I3 illustrates data for the binding of cytidine tiphosphate by alspartate transcarbamylase (Fig. 1 of Winlund & Chamberlin, 1970). The data points are represented on a Scatchard plot and 5.8 binding sites arc assumed.The curve drawn through the points is based on constants derived from the fit to equation (9) also shown in figure. It will be noted that these data arc tested for negative interaction. The value of Mu),O-786x IO” M- ‘, and x,, 3.47 X lo4 M-‘, can be compared, respectively, with association constants for 2.9 high and 2.9 low affinity sites of I .I x IO6 M- ’ and 2.5 X lo4 M-‘, calculated by Winlund & Chamberlin (1970) by fitting the Scatchard plot. (K;)

ALKALINE

PMOSPIIATASE

FROM

CALI-

IN?TSTlh:I‘

A second example with a negative slope is provided by the binding ot‘ inorganic phosphate to alkaline phosphatase (Fig. 7 of Chappelet-Tordo, Fosset, lwatsubo, Gache & Lazdunski, 1974). The data points and fitted

410

S. AINSWORTH

FIG. 13. Scatchard plot of the binding of cytidine triphosphate to aspartate transcarbamylase (Winlund & Chamberlin, 1970). V is the number of molecules of CTP bound per molecule of enzyme and P the maximum such number is assumed to be 5.8. The symbols A represent data points on the Scatchard plot with the vertical axis expanded Y 10. For symbols see Fig. 4.

curves are illustrated in Fig. 14, two binding sites being assumed.Again, the values of q, and CI~,2-23 x lo5 M-’ and 7-14 x IO3 M-‘, respectively, can be compared with the corresponding high and low affinity constants of 2.5 x lo5 M-’ and 7.5 x IO3 M-‘, calculated by fitting the Scatchard plot. 4. Discussion The experimental data examined in the previous section show that the model represents a wide range of behaviour and provides constants that can be further examined as functions of secondary variables. The model is

MODEL

FOR

A

REGULATORY

41 I

ENZSME 1

0

0.2

0.4 L Y

Fla. 14. Scatchard plot of the binding of phosphate to alkaline phosphatase (ChappeletT’ordo et al., 1974). ZTand Pare defined as in Fig. 13, with P = 2. For symbols see Fig. 4.

generatfy most successful in fitting experimentai data points in the middle range of saturation (which, it will be argued, is the most significant part of the range) but shows departures from predicted behaviour at the extremes 01 saturation. The deviations at high saturation probably arise from error in the estimation of complete saturation or V and, assuming this to be the cause, it has been shown that these deviations can be used to provide an improved estimate of either parameter. The deviations from predicted behaviour at low saturation invariably appear as estimates of saturation or velocity that are lower than expected. The error may represent a systematic failure of the model or a systematic failure in experimental technique. In view of the variety of systems examined it might seem unreasonable to suppose that the error could be experimental in origin were it not for the fact that the difficulties of measurement at the extremes of saturation are well-recognised. Indeed, both Walter & Barrett ( 1970) and Philo & Selwyn (1973), emphasize that visual estimates of the initial rates of enzyme catalysed reactions are consistently lower than the true rates and that this error becomes both more likely and significant when the initial rate is small. Walter & Barrett (1970), also point out that this l’ystematic error, if unappreciated, could lead to incorrect conclusions about

412

S . .A I N S W 0 K ‘I I1

the nature of the cnzymc mechanism. The origin of the deviations which occur at low saturations must therefore remain in doubt. The generally good fits achieved by equation (9) in the middle range of saturation provides constants which [when applied to equation (13)] provide calculated curves, A = I, that give an acceptable Iit to experimental data points over the whole range. This effect arises because even a large proportional error at low saturations represents only a small absolute error in (‘ while, at high saturations, a large change in A is associated with only a small change in 2). It can therefore be concluded (quite apart from error) that the middle range of saturation is the most important for establishing the nature and significance of the relationship L’ = f(A). The same conclusion has been reached in a more analytical way by Weber (1965) who has shown that the information content of a value of p (determined for binding at :I single site) is defined by the relationship: -4P) = P l%z p+(l -PI b, (1 -P), which can be approximated with negligible error by the function :

(20)

J(p) = 4p(l -PI. (21) Examination of equation (21) confirms the conclusion advanced above and defines the useful range of p as approximately 0.1-0.9. At the present time, it is quite impossible to test any hypothesis which purports to establish fundamental relationships between the structure and function of a protein. There is, therefore, a need for an interim model that implies no more constants than can be readily determined but which accords to those constants a significance that can be further examined. It is in this context that the present model should be considered.

REFERENCES AINSWORTH, S. (1975). J. theor. Biol. 50, 129. BLANGY, D., But, H. & MONOD, J. (1968). J. molec. Biol. 31, 13. BOWS, J. (1958). Tram Furuu’c~y Sot. 54, 593. BUC, H. (1967). Biochem. biophys. Res. Conmum. 28, 59. CHAPPELET-TORDO, D., FOSSET, M., IWATSUBO, M., GACHE, C. & LAZDUXSKI. M. (1974). Biochemistry 13, 1788. CLELAND, W. W. (1970). In The Ez~/nes, 3rd ed. (P. D. Boyer, ed.), Vol 2, p. 1. New York and London: Academic Press. GERHART, J. C. (1970). Curt-. Topics cell. Regulatiotz 2, 275. HATFIELD, G. W. & UMBARGER, H. E. (1970). J. biol. Chem. 245, 1749. JOHANNES, K.-J. & HESS, B. (1973). J. tnolec. Biol. 76, 181. KIRSCHNER, K. (1967). In Proceedings of the 4th Meetittg of the Federarion of European Biochemical Societies (E. Kvamme & A. Pihl, eds) p. 39. New York and London: Academic Press. KOSHLAND, D. E. (1958). Proc. mtn. Acad. Sri., U.S.A. 44, 98. KOSHLAND, D. E., NBMETHY, G. & FILMER, D. (1966). Biochemistry 5, 365.

MODEL KURGANOV,

Biol.

B. 1.. KAGAN,

FOR 2.

A

REGULATOR\’

S., DOROZIIKO,

A.

I‘NZYMI. I. &

YAKOVLE~‘,

413 V. A. (1974).

J. //Ic~u.

47, 1.

MADSEN, N. B. & SHECHOSKY, S. (1967). J. biol. Chem. 242, 3301. MONOD, J., WYMAN, J. & CHANGUEX, J.-P. (1965). J. molec. Biol. 12, 88. PHILO, R. D. & SELWYN, M. J. (1973). Biochem. J. 135, 525. ROUGHTON, F. J. W. & LYSTER, R. L. J. (1965). Hrulrodels Skrifier 48, 185. RCIZENGURT, E., JIM~NEZ DE Asin\, L. & CARMINATTJ, H. (1969). .I. hiol. C’hetn. 244, 1131. SZABO, A. & KARPLUS, M. (1972). J. tolec. Biol. 72, 163. WALTER, C. & BARRETT, M. J. (1970). Enzymologia 38, 147. WEBER, G. (1965). In Molenrlor Biophysics (B. Pullman & M. Weisshluth, eds) 1’. 369. New York : Academic Press. W:EBER, G. & ANDERSON, S. R. (1965). Biochemistry 4, 1942. WHITEHEAU, E. (1970). Progr. Biophys. tnolec. Biol. 21, 323. WINLUND, C. C. & CHAMBERLIN, M. J. (1970). Biochem. biophys. Res. Commw~. 40, 41 WYMAN, J. (1977). Clrrr. Topics ceil. Regulation 6, 209.

A simple model for a regulatory enzyme.

I. rheor. Biol. (1977) 68, 391413 A Simple Model for a Regulatory Enzyme STANLEY AINSWORTH Department of Biochemistr)*, Universit!. of‘ Shefield, Sh...
996KB Sizes 0 Downloads 0 Views