an evolutionary approach to enzyme kinetics: optimization of ordered mechanisms

Bulletin of Mathematical Biology Vol. 56, No. 1, pp. 65 106, 1994 Printed in Great Britain

0092~8240/9456.00 + 0.00 Pergamon Press Ltd

© 1993 Society for Mathematical Biology

AN E V O L U T I O N A R Y A P P R O A C H T O E N Z Y M E K I N E T I C S : O P T I M I Z A T I O N O F O R D E R E D M E C H A N I S M S

THOMAS WILHELM, EDDA HOFFMANN-KLIPP and REINHART HEINRICH~ Institut ffir Biophysik, Fachbereich Biologie, Humboldt-Universit/it zu Berlin, Berlin, Germany

A theoretical investigation is presented which allows the calculation of rate constants and phenomenological parameters in states of maximal reaction rates for unbranched enzymic reactions_ The analysis is based on the assumption that an increase in reaction rates was an important characteristic of the evolution of the kinetic properties of enzymes_ The corresponding nonlinear optimization problem is solved taking into account the constraint that the rate constants of the elementary processes do not exceed certain upper limits. One-substrate-one-product reactions with two, three and four steps are treated in detail, Generalizations concern ordered uni- uni-reactions involving an arbitrary number of elementary steps_ It could be shown that depending on the substrate and product concentrations different types of solutions can be found which are classified according to the number of rate constants assuming in the optimal state submaximal values. A general rule is derived concerning the number of possible solutions of the given optimization problem. For high values of the equilibrium constant one solution always applies to a very large range of the concentrations of the reactants. This solution is characterized by maximal values of the rate constants of all forward reactions and by non-maximal values of the rate constants of all backward reactions. Optimal kinetic parameters of ordered enzymic mechanisms with two substrates and one product (bi-uni-mechanisms) are calculated for the first time. Depending on the substrate and product concentrations a complete set of solutions is found. In all cases studied the model predicts a matching of the concentrations of the reactants and the corresponding Michaelis constants, which is in good accordance with the experimental data. It is discussed how the model can be applied to the calculation of the optimal kinetic design of real enzymes.

1. Introduction. In contrast to chemical reactions of an inanimate nature the enzymic reactions of living cells are the result of natural selection during evolution. It can be assumed, therefore, that the kinetic parameters of enzymes can be characterized by certain optimum principles. This view is supported by a number of observations, for example by the fact that mutations or other changes in the structure of contemporary enzymes lead in most cases to a worse functioning of cellular metabolism (Belfiore, 1980). Much theoretical work has been devoted to inquiring whether the structural design and functioning of

I Author to whom correspondence should be addressed.

65

66 T. WILHELM et al.

present-day biochemical systems can be rationalized on the basis of optimization principles. In quantitative investigations the crucial point is the formulation of appropriate performance functions whose maximum (or minimum) value might correspond to the outcome of the evolution of cellular metabolism. In the literature the following optimization principles are mostly considered: (a) maximization of reaction rates of individual enzymes or of steady state fluxes in enzymic systems; (b) minimization of concentrations of metabolic intermediates; (c) minimization of transient times; (d) maximization of sensitivity to external signals, and (e) optimization of thermodynamic efficiencies (for recent papers cf. Burbaum et al., 1989; Pettersson, 1989, 1991, 1992; Heinrich et al., 1991; Schuster and Heinrich, 1991; Mel6ndez-Hevia and Torres, 1988; Mel6ndez- Hevia et al., 1993; Heinrich, 1993; Stucki, 1988). In the present theoretical analysis it is assumed that the reaction rates, v, of enzymes were important targets of evolutionary pressure. In particular, the kinetic parameters of enzymes resulting from the optimization principle v = Vma X are considered. Generalizing previous investigations (Fersht, 1974; Cornish-Bowden, 1976; Albery and Knowles, 1976; Chin, 1983; Ellington and Benner, 1987; Burbaum et al., 1989; Pettersson, 1989, 1992) not only one-substrate-one-product reactions but also ordered enzymic reactions with two substrates are considered. Furthermore, previous results concerning an enzymic three-step mechanism (Heinrich and Hoffmann, 1991) are generalized by taking into account an arbitrary number of elementary steps. A special characteristic of the present investigation is that according to experimental observations the determination of optimal states is carried out by taking into consideration upper limits of the microscopic rate constants. This holds not only for the second order rate constants of the binding of substrates and products to the enzyme but also for the first order rate constants describing the release of substrates and products or the interconversion of enzyme-intermediate complexes. Another constraint follows from the thermodynamic condition that none of the evolutionary changes of enzymes resulting in new constellations of the rate constants affect the overall equilibrium constants of the catalyzed reactions. Accordingly, a nonlinear optimization problem with one equality constraint and several inequality constraints has to be solved. It is shown that for all reactions considered a unique solution can be found at given reactant concentrations. However, different combinations of the concentrations of the reactants may result in different types of solutions. A formula is derived yielding the number Z, of solutions which may be found at variations of substrate and product concentrations. For ordered enzyme mechanisms Z,, strongly increases with the number of elementary steps, n. The following types of solutions are of particular interest for all reactions considered: (1) a solution characterized by maximal values of all forward rate constants and submaximal values of all backward rate constants which is valid for a central region within the space of the reactant concentrations ("central solution"); (2)

AN E V O L U T I O N A R Y A P P R O A C H TO ENZYME KINETICS 67

solutions characterized by very high affinities for substrates and products which are found for very low concentrations of all reactants ("high-affinity solutions"); and (3) solutions with low affinities of substrates or products which are obtained if the concentration of one of the reactants becomes very high ("low affinity solutions"). It is shown that in contrast to previous assumptions (e.g. Albery and Knowles, 1976; Brocklehurst, 1977; Burbaum et al., 1989; Pettersson, 1989) an optimal enzyme activity is not compulsorily achieved by maximal values of the second order rate constants.

The analysis reveals the interesting fact that at variations of the concentrations of the outer reactants all possible solutions for the optimal values of the microscopic rate constants can only be observed if the dimension of the concentration space of the reactants is sufficiently high.

The results of the present model are also discussed in terms of the optimal phenomenological parameters, i.e. the Michaelis constants and the maximal activities. It is shown that in accordance with experimental observations optimal states are generally characterized by a matching of Michaelis constants and the concentrations of the corresponding reactants. In particular one also gets vanishing Michaelis constants for uni-uni-reactions in the limit ofvanishing concentrations of the substrate and product. Another result of the model refers to the internal equilibrium constant Ki, t which turns out to be equal to unity for many types of the solutions, independently of the overall equilibrium constant.

2. General Considerations. The reaction mechanism of an unbranched enzyme reaction is depicted in Scheme 1.

k. ~ Xo ~ k~

Xn ~ kl~ 'Xl knllLk,nl, k21Lk

•

X3 S c h e m e 1.

X~ denote the various enzymic species and k i, k_ i are true or apparent first order rate constants. This reaction scheme is valid for ordered reactions with one or more substrates and products. The binding of a substrate S or a product P at step i is described by replacing the apparent rate constant k i or k i with true rate constants multiplied by the concentrations of S or P (Sk i, Pk~) .

68 T. W I L H E L M et al.

With

v~=k~X~_ t -k_~X ~ ( / = 1 , . . . , n), (1)

one obtains under steady state conditions (v = vi) for the concentrations X d

j= lk -~ -Vr= lk -~ - , j= l k - j ( i = l , . . . , n ) . (2)

Taking into account concentration:

the conservation relationship of the total enzyme

n - 1

E,=Xo+ x,, (3) i = 1

and the cyclic property X o = X, of Scheme 1 one gets for the reaction rate

(q- 1)Et v = ~ 1 ~ " - ' 1 i+ j - , k~ ' (4)

i=1~-~_~ + E k ~ [ I k , i = 1 d = l ( t - I ) r = i

and k _ o = k _ .. where by definition k_+(.+0=k_+ ~ equilibrium constant:

q denotes the apparent

Formula (5) also applies to the equilibrium constant q by inserting true rate constants. The following analysis is restricted to the case where the equilibrium constant q ~> 1.

We are interested in those values of the elementary rate constants k~, k_i maximizing the absolute amount Iv[ of the reaction rate under the constraints of fixed values of the concentrations of the reactants and of a fixed value of the equilibrium constant q. According to equation (4) the reaction rate, v, is a homogeneous function of first degree of the elementary rate constants k+i, i.e.

~v(ki, k_ i) = v(c&i, ~k_ i), (6)

with an arbitrary value of a > 0. For that reason, without constraints for the rate constants of the elementary reactions the reaction rate could be increased in an unlimited way. According to the physico-chemical principles of enzyme reactions it is reasonable to take into account upper limits of the individual rate constants at the optimization of the reaction rate. For given values of the concentrations of the reactants the maximum of Iv[ must be located within the subspace of parameter constellations:

q ~ _ f i ki i=1 k - i" (5)

AN EVOLUTIONARY APPROACH TO ENZYME KINETICS 69

O<~k+_i <~ k+_i,ma,. (7)

Due to equation (6) and condition (7) states of maximal enzyme activity have the property that one or more kinetic constants assume their maximal values.

Since the numerator in equation (4) is independent of the rate constants optimal states are characterized by those values of the rate constants minimizing the denominator N(ki, k_ i).

Uniqueness of Solution. Eliminating one of the rate constants using the equilibrium constant the remaining rate constants entering the denominator of equation (2) become independent variables of the optimization problem. Each of these quantities appears in the indiviual addends of N, to the power of one or minus one only. Therefore, the second derivative of N with respect to all these independent variables is always non-negative and neither within nor on the boundary of the parameter subspace maxima exist which could separate different minima of N.

Number of Possible Solutions. Because of the existence of upper limits for the rate constants a possible solution of the optimization problem l v I= Vma x may be characterized by a and fl denoting the number of rate constants k, and k_ i, respectively, assuming non-maximal values (solutions of type T~,~). All types of the elementary kinetic designs are obtained by consideration of all possible combinations of non-maximal rate constants. Since q/> 1 all solutions with non-maximal values of only forward rate constants (T~, o, e ¢ 0 ) are impossible so that the total number K, of allowed combinations reads:

K.= (8)

However, the number of possible combinations which correspond to optimal solutions is drastically reduced, since all combinations can be excluded, where simultaneously k i and k ~ or k~ and k~i_ 1) assume non-maximal values (note the cyclic notation for the indices). This can be proved as follows: Each term in the second addend of the denominator of equation (4) contains k~ only together with k_ i and k_ (i- 1) in the form of kJ(k ~k_ (i- 1)) while k_ ~ also appears in the first addend in the form of 1/k_ i. Therefore, a state with non-maximal values of k i and k _ i or of k i and k_ (i- 1) can be further optimized by increasing k i and k or k~ and k_(~_l) by the same factor, which affects neither the equilibrium constant nor the second addend, but decreases the first addend of the denominator. The number Z, of possible optimal solutions for an ordered n-step mechanism with q > 1 can be expressed in the following way:


z.= Z z;(~)+ ~ z.~s(i,j), (9) j=l i,j=l

where Z~(j) denotes the number of optimal solutions with j submaximal backward rate constants and maximal values of all forward rate constants and Z[~(i,j) that with i submaximal forward rate constants and j submaximal backward rate constants. By combinatorial reasons one gets:

and

with Zig(i, j) = 0 for i + j > n - 1. For example the matrices Z~ r, Z~ ~, Z~ ~ have the following form:

(i 4°°) 11 !155°°t ( oot ooo 1 ooo z ~ r = 0 0 , z { r = , z { r = 0 0 0 0 .

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

(11)

It follows that in states of maximal activity not more than n rate constants can assume submaximal values. Solutions of type To," for which Z~,(n)= 1 have the highest number of submaximal rate constants. Table 1 gives a comparison of the numbers of possible combinations K, and possible optimal solutions Z, .

3. Uni-uni-reactions. The kinetic equation for an ordered enzymic reaction unireactant in both directions with the substrate S and the product P involving n elementary steps can be derived from equation (4) by replacing k 1 and k_ , by Skt and Pk_, , respectively. One gets:

Table 1. Number of possible combinations K, of submaximal kinetic constants in comparison with the number of possible

optimal solutions Z, growing with the number n of elementary steps of the enzymic reaction

n 1 2 3 4 5 6 7 8

K. 2 12 56 240 992 4032 16,256 65,290 Z. 1 3 10 31 91 252 680 1787

A N E V O L U T I O N A R Y A P P R O A C H T O E N Z Y M E K I N E T I C S 71

E t ( S q - P) v - N ' (12)

with

and

and

N = D t + D E S + D a P (12a)

1 n-1 1 f i kj m l = k -t- ~ ~ k ' (12b)

--n 1 - - j r = j = r + l

DE k-n i = 1 j = l i = 1 s = l r = i + l j = r + l '

.-i 1 .-Ii i 1 IJ kj

j = l i = 2 r = l j = r + l

(12c)

(12d)

ki q = II (12e)

i = 1 k-i

(cf. Peller and Alberty, 1959). The kinetic equation (12) can be rewritten in the form:

V + V- --S- P

v = S P , (13) I+L+ where V ÷ and V- are the maximal activities of the forward and the backward reaction and K s and Kp the Michaelis constants of the substrate and the product, respectively. A comparison of formulae (12) and (13) yields:

V + Etq V - Et Ks D~ D1 (14a-d) - - O 2 , - - Ds, - D 2 , Kp - Da.

It is convenient to scale the kinetic constants with respect to their upper limits:

k , *k , , ki *k i (15a c) --* kx ' k d k~£- '

for i - - 1 , n and i= -t-2, . . . , + ( n - 1 ) . For the sake of simplicity only two types of upper limits are considered, one for the first order rate constants (k,,)


and one for the second order rate constants (kd). Normalizat ion of substrate concentrations and reaction rate yields:

v kaS ~ kaP k,-~t-,v, ~ S, krupP. (16a-c)

After normalization one gets:

(Sq - P) V-- ~ , O~ki, k i~l . (17)

The Michaelis constants and the maximal activities are normalized in the same way as S, P and v, respectively.

3.1. Two-step mechanism. For the reaction mechanism

Skl

E X

k2 Scheme 2.

one obtains from equation (12):

1 k 2 Sk 1 P

N = ~ z + k _ l k ~ + k lk_2 +k_l (18)

According to the formulae (9)-(11) there are for n = 2 three optimal solutions for the rate constants maximizing the absolute amount I v ] of the reaction rate. The results are given in Table 2. Two solutions are of type To, 1 and one solution

Table 2. Optimal solutions for the rate constants for the enzymic reaction depicted in Scheme 2 as functions of the

concentration of the product for q >/l

Solution Type k 1 k 1 k2 k-2

1 1 To, 1 1 - 1 1

q

1 2 To, 1 1 1 1 -

q

3 To, 2 1 1


3

P 2

..'" R2

. . ' \

," ~2

." R 3

R 1

0 I 3 z~

S Figure 1. Subdivision of the (S, P)-plane into subregions R i corresponding to the three solutions for optimal rate constants of the reversible two-step kinetic mechanism depicted in Scheme 2 for q=2 . The points ~1 and f~2 have the coordinates (1/q 2, 1/q) and (1, q), respectively_ The dotted line indicates Sq = P_

is of type T o , 2 , It is worth mentioning that the optimal kinetic constants are functions of the product concentration P but not dependent on the substrate concentration S. The optimal denominators N~' of the rate equation, for these solutions, are:

N* = 1 + q + Sq + Pq, (19a)

N* = 2q + Sq + P, (19b)

N ~ = q + S q + 2 x / ~ . (19c)

From equations (19a-c) it is easy to see that solution 1 is valid for P < 1/q, solution 2 for P > q, and solution 3 for 1/q < P < q. These conditions for P define three regions R i within the (S, P)-plane (Fig. 1). At the transitions from R 3 to R 1 and from R 3 to R 2 the kinetic constants k _ 2 and k_ x, respectively, become maxima at the corresponding boundaries.

Michaelis Constants and Maximal Activities. Optimal values for these quantities are obtained by introducing the solutions listed in Table 2 into the expressions resulting from equations (14a-d):

V + = k 2, V - = k _ ~ , K s - ( k 2 + k - t ) , G - ( k 2 + k - x ) (20a-d) kl k_ 2

The results are listed in Table 3. The optimal Michaelis constants fulfil the relations:


Table 3. Optimal Michaelis constants and optimal maximal activities (normalized values) for the enzymic reaction

depicted in Scheme 2 as functions of the concentration of the product for q ~> 1

Solution K s Kp V + V

1 1 1 1 1 + - 1 + - 1 -

q q q

2 2 2q 1 1

3 1 + ~ P + x / / ~ 1 X/~

~Ks ~K,, ~--g/> 0, ,~- >/0. (21)

Furthermore, the optimal Michaelis constants are related in the following way:

Kp = qKs(K s - 1), (22)

where the points K s = l + l / q and Ks=2 represent solutions 1 and 2, respectively, and the line with 1 + 1/q < K s < 2, solution 3.

3.2. Three-step mechanism. The case n = 3 with the reaction scheme

E

k S X ~ - k, -" X~

k2 Scheme 3.

and with

1 k k3 s ( k,k k,k N = ~ + k_ ~k_~ + k_ ,k_ ~k ~ + \k_ ,k_ 3 + k_ ,k ~k_ ~ + k_ ,k_ ~k_~J

p ( k 2 1 + 1 ) + \ k - l k 2 t-~_~ ~ (23)

has been treated in detail previously (Heinrich and Hoffmann, 1991). The results can be summarized in the following way.


The optimization principle ]V[=Vm, x yields 10 solutions (cf. equations (9)-(11)) which are listed in Table 4. There are in each case three solutions of type To, 1, type To, 2 and type 7'1,1, and one solution of type To, 3 . The solutions depend on S and P and divide the (S, P)-plane into 10 regions R i (Fig. 2). At the transition between two neighboured regions one non-maximal rate constant becomes maximum or vice versa. The solution (10) with k i = 1, k_ i < 1 (i = 1, 2, 3) which is determined by a fourth-order equation (Table 4) is of special importance because of the central location of the corresponding region R10 within the concentration space. R lo has a "triangular" shape and is bounded by the three lines:

S= 2q-- 1, S= P-, S=P(Pzq+Pq-1), (24a-c) P q

where k_ 1-= 1, k_2= 1 and k _ a = 1, respectively. For the concentrations S = P- - 1 which always belong to Rio the fourth order equation has an explicit solution:

k_l=k_2=k_3=3~/!. (25)

From equation (24) it follows that for q ~ oo solution (10) becomes valid for all positive (S, P)-values. For q = 1, region Rio reduces to the point (1,1).

From Table 4 and Fig. 2 the following properties of the optimal solutions may be derived:

(a) At very low substrate and product concentrations an optimal enzymic activity is achieved by improving the binding of S and P to the enzyme (solution (9): high (S, P)-affinity solution).

(b) At high concentrations of S or P the substrate or product are weakly bound to the enzyme (solution (7): low S-affinity solution, and solution (8): low P-affinity solution).

(C) k 2 is always maximum except for region R 8 where the reaction proceeds backwards.

(d) In contrast to previous assumptions (e.g. Albery and Knowles, 1976; Pettersson, 1989), an optimal enzymic activity is not compulsorily achieved by maximal values of the second-order rate constants. As to k 1 this is the case for solution (7) and as to k 3 for solutions (3), (4), (6), (8), (10).

(e) The internal equilibrium c o n s t a n t / ~ i n t - - k 2 / k _ 2 equals unity for solutions (1), (3), (4), (9). Kint-~l is valid for all near-equilibrium enzymes (Sq " P).


O

t-¢3 ©

E ,.1=

=.

r,.) o

o

© ,4

e~

r~ q

[-

O

O

el


4 R~ ."

P2

0 I 2 3 L~

S

Figure 2. Subdivision of the (S, P)-plane into subregions R~ corresponding to the 10 solutions for optimal rate constants of the reversible three-step kinetic mechanism depicted in Scheme 3 for q = 2. The corners ~i, ~2 and ~3 of region Rio have the coordinates (1/q z, 1/q), ( 2 q - 1, 1), and (1, q), respectively. Along the dotted line

Sq = P holds.

Phenomenological Parameters. Optimal values for V + , V- , K s and Kp are obtained by introducing k i and k i from Table 4 into the following equations resulting from (14a-d):

V + - k2k3 V - = k - l k - 2 (26a,b) k 2 + k 3 + k _ 2' k z + k _ 1 + k _ 2'

k2k3+k l k 3 + k - 1 k 2 k z k 3 + k - l k 3 + k - l k - 2 K s - , Kp = (26c,d)

kl(k/+k3+k_2) k_3(k2+k_~+k_2)

The results may be visualized in a space where the coordinates are the phenomenological parameters. For example, the relationships between the optimal Michaelis constants are shown in Fig. 3. The solutions (1)-(3) are represented within the (Ks, Kp)-plane by points, while solutions (4)-(6) and (7)-(9) are represented by lines. The Michaelis constants for solution (10) with k_ 1, k 2, k_ 3 < 1 can be expressed under consideration of q by two kinetic constants. Therefore, for solution (10) one obtains a two-dimensional manifold.

It is seen from Fig. 3 that solution (9) which is valid for low concentrations of S and P is characterized by low values of K s and K1,. In the limiting case S ~ 0 and P ~ 0 one gets k 1, k3~0 (Table 4) and from equations (26c,d) Ks~0 and Kp~0. Solutions (7) and (8) which are applicable for high concentrations of S and P have high K s- and Kp-values, respectively. For all solutions the following relations hold:


K~ 10

z0 f: ~.~ .6

"2

0 10

/7

20

Ks Figure 3. Michaelis constants K s and Kp in optimal states of the three-step mechanism for the 10 optimal solutions listed in Table 3. The three corners of the "triangle" represent the solutions (1)-(3) (type To,l) with only one submaximal backward-constant, the three edges the solutions (4)-(6) (type To,2), the three lines outside the triangle the solutions (7)-(9) (type T~,a), and the hatched interior of the

triangle solution 10 (type To,3).

OKs ~?KP >- O. (27a,b) ~ > ~ 0 , tp ~-

For the special case S = P = 1 equations (26a-d) lead to:

V + ql/3 1 - V- - (28a,b) 2q 1/3 + 1' q 1/3(2 + q 1/3),

1 + q 1/3 At_ q z/3 1 + q 1/3 + q 2/3 K s = ql/3(1 +2q l /3 ) , Kp - 2 + q l / 3 (28c,d)

For q = 1 one obtains from equat ions (28a-d):

V +=~,1 V - = ½ , K s = l , K p = I , (29a-d)

while for q ~> 1 the phenomenological kinetic parameters may be approximated as follows:

V +'~½, V - ~ - q - Z / 3 , K, ,,.,1 s - z , Kp~-q 1/3" (30a-d)

Equat ions (28c), (29c), (30c) and the results shown in Fig. 3 suppor t the previous conclusion that there is a matching of the Michaelis constant K s to the substrate concentrat ion S (e.g. Crowley, 1975; Cornish-Bowden, 1976).


Effect of P a r a m e t e r Opt imiza t ion . I t m a y be interest ing to analyse the sensitivity of the reac t ion ra te with respect to p a r a m e t e r opt imiza t ion . F o r this pu rpose op t imal reac t ion rates at given values of S and P are c o m p a r e d with

that of a reference state. As the reference state is chosen to be kg = 1, k _ ~ = 1/3~/q which is op t ima l only at S = P = 1. F igure 4 shows the c o n t o u r lines of the funct ion:

A v (opt,ref) /)(opt) __/) (ref)

D (ref) F (re f) (31)

which by defini t ion is zero at S = P = 1 and for S q = P .

It is seen tha t the percen tage change of p a r a m e t e r op t imiza t ion at vary ing values of S and P is no t ve ry large within the concen t r a t i on range considered. Howeve r , it has to be t aken into accoun t tha t all v-values cons idered refer to op t imized states. F o r example , nonop t im ized states where k~ and k_ i or k~ and k_~i_l) s imul taneous ly assume submaximal values would yield very low reac t ion rates and, therefore , m u c h higher devia t ions f rom an opt imized reference state.

5

4

3

P

2

1

0 1 2 3 4

5

Figure 4. Comparison of the optimal reaction rate v ~°pt) (S, P) with that in the reference state S= P = 1 by means of contour lines (v (°pt)- v~rm)/v ~r°f) = const. The contour lines are calculated for: 0_0004; 0.0025; 0.005; 0.01; 0.02; 0.03; 0.05; 0.1 (v > 0) and for - 0.01; - 0.03; - 0.05; - 0.1 (v < 0); dotted line: v = 0. Parameter

values: n=3; q=2.


3.3. Four-step mechanism. For the case n = 4

Sk I E ~ -" X~

k_1

k-3 X~- k~ X2

Scheme 4.

the denominator N of the kinetic equation (12) reads:

1 k 4 k3k 4 k2k3k 4 - - F

N=-~_4+k 3k 4 ~-k_2k_3k_4 k_tk_2k_3k_ 4

( k ~ l k + ktk4 + k'k3k4 4 ktkz +S,,k-1 -4 k-tk-3k-4 k-tk-2k-3k-4 k-tk-2k-4

ktk2k3 ktk2k4 ) q k t k _ z k _ 3 k 4 ]-k_tk_2k_3k_al

( L 1 1 k 3 k 2 k 2 k 3 ) + e + ~ _ + ~ 4 ~ .

- - - 3 k - 2 k - 3 k - l k - 2 k - t k - 2 k - 3 (32)

According to equations (9)-(11) one gets 31 combinations of nonmaximal kinetic constants which lead to optimal solutions (Table 5).

As for the three-step mechanism explicit expressions can only be found for solutions of types To, t, To, 2 and Tt, 1. If k j= 1 ( j = 1, . . . , 4) and only one k i < l (type To, 0, then it follows from equation (12e) that k i= 1/q. In all other cases one rate constant can be expressed in terms of the others with equation (12e). For example, to obtain solution (17) (type Tt,x) with nonmaximal values of k 4 and k_ t = kg/q the denominator N is written in the form:

3Sq + 3Pq Nt7 = 2k 4 + + 1 + q + 3Sq + 3P. (33)

k4

The optimality condition ONtT/Ok 4 ~-0 can easily be solved and yields:

k4=x/3q(S+ P) ' / ~ s + P) k_ t = ~/ Tq (34a,b)


If one puts these optimal rate constants into equation (33), one obtains the corresponding optimal denominator for solution (17):

U~' 7 = 2~/6q(S+ P) + 1 + q + 3Sq + 3P. (35)

In a similar way the other solutions of types To, 2 and T1,1 can be obtained. The remaining solutions are determined numerically. For solutions (19)-(30) with

Table 5. C o m b i n a t i o n s of n o n m a x i m a l kinetic p a r a m e t e r s of the different types of so lu t ions for ordered enzymic react ions wi th four e lementary

steps

Solu t ion Type Pa rame te r s

1 To, 1 k_ 1 2 k 2 3 k_ 3 4 k 4

5 ~,2 k l , k - 2 6 k _ t , k 3 7 k ~ , k 4 8 k_2, k_ 3 9 k_2, k 4

10 k a , k _ 4

11 Ti, i k i , k_ 2 12 k 1 , k_ 3 13 k 2 , k_ 3 14 k2, k 4 15 k3, k_ 1 16 k3, k 4 17 k 4, k_ 1 18 k 4, k_ 2

19 To,a k - l , k 2, k 3 20 k t , k_2 , k_4 21 k i , k 3, k 4 22 k-2, k 3, k 4

23 Ti, 2 k 1 , k_ 2 , k_ 3 24 k 2 , k_ 3 , k_ 4 25 k3, k _ l , k 4 26 k 4, k_ 1 , k_ 2

27 T2,1 k 1 , k 2 , k_ 3 28 kl , k4, k 2 29 k 2 , k 3 , k_ 4 30 k3, k4, k i

31 To, 4 k_t , k_2, k_3, k_ 4


three submaximal rate constants the optimality conditions can be written in the form:

= F l ( k ±j), =F2(k±,), (36a,b)

where k_+i and k_+j are two different submaximal rate constants. The functions F~ and F 2 are different for each solution. For example, solution (19) with k 1, k_ 2 < 1 and k_ 3 = 1/(qk l k_ 2)< 1 is determined by the equations:

_ S + P k2 = S + P ( I + k _ , ) (37a,b) k2-~ q(l+P)k_2' -2 qk_l((l+e)k_~+S)

The equation system of solution (31) with k i = 1 and k_ i< 1 (i = 1 . . . . ,4) reads

k 2 = P ( k - 2 k - 3 + l W k - 3 )

-1 qk_2k_3( l_ t_k_2+k_2k_3 ),

k 2 - = P ( k - * k - 2 + l + k - 1 )

3 q k _ l k _ 2 ( S + S k _ 2 W k _ l k 2),

k22 = P ( k - l + 1) (38a,b) q k _ l k _ 3 ( S + k _ l ) '

1 k 4 - (38c,d)

q k - l k - E k _ 3

Both equations (37) and (38) can be solved numerically in an iterative manner. For given values of the reactant concentrations S and P the optimal kinetic design is given by that solution m with the smallest denominator (i.e. N* <N*(mCk) ) .

This procedure for the determination of the optimal solutions is applicable for all points of the concentration space (S, P). Figures 5(a-c) show the regions of the optimal solutions of the four-step model for q = 1, q = 1.2, and for q = 2, respectively. It is seen that for the case n = 4 not all 31 possible solutions are obtained at variations of S and P. For example, solution (13) with:

k /2q(I+S+P) --Xl gT-F '

= / 2 ( I + S + P ) k - 3 ~] q (S+P)

(39a,b)

cannot be obtained since the condition k 2 ~ 1 cannot be fulfilled for S, P > 0 as long as q >~ 1.

Comparison With Case n = 3. In contrast to the case n--3 the number of solutions found for positive values of S and P depends on the equilibrium constant q. However, the optimal solutions of the three- and four-step mechanisms have much in common. For example, for very low values of S and P one obtains for n = 3: k ~ < 1, k 3 < 1 (solution (9)) and for n = 4 : k_ 1 < 1 and k 4 < 1 (solution (17), given in equation (34)), i.e. a kinetic design with a strong binding of both reactants to the enzyme (high (S, P)-affinity solution). For high values of S or P the optimal kinetic design is characterized by a weak binding of


(a) ' ,..-"

4 29 ""

3

P 0 - ' "

2 • '"" 12

1

~ , 5 ~ y - 1 1 ( 0 1 2 3 4

S

3

P

2

(b) / /

: / /

]571 ~!' \~2 ,23

1 2 3 4 S

F i g u r e 5 (a) a n d (b).


/~ 16 .-"

0 q 2 3 g S

Figure 5 (c).

Figure 5. Subdivisions of the (S, P)-plane into subregions R~ corresponding to the solutions for optimal rate constants (Table 5) of a one-substrate-one-product reaction (n = 4, Scheme 4) for various values of q (a: q = 1, b: q = 1.2, c: q = 2). The numbers within the regions are the indices ofR v The corners f~t, f~2 and f~3 of the region R31 have the coordinates (1/q 2, I/q), ( 2 ~ - 1, 1)0 and (1, q), respectively_ For q = 1 the region R o corresponds to the solutions where all rate constants assume

their maximal values. Dotted line: Sq=P.

S ( n = 3 , 4: k l < l ) and P ( n = 3 : k _ 3 < l ; n = 4 : k _ 4 < l ) , respect ively.

F u r t h e r m o r e , in b o t h mode l s a "cen t ra l reg ion" (n = 3: R lO ; n = 4:t{31) exists for which the op t i m a l so lu t ion is cha rac te r i zed by m a x i m a l values of all f o r w a r d ra te cons tan t s and by s u b m a x i m a l ra te cons tan t s of all b a c k w a r d reac t ions . F o r n = 3 it is b o u n d e d by the lines given by equa t ions (24a~c) a n d for n = 4 by the lines:

S = 2 - 1 , S = -P, S = x / / ~ ( I + P ) P - P , (40a-c) q

where k _ 1 = 1, k_ z - k _ 3 = 1 and k _ 4 = 1, respect ively. I t is w o r t h m e n t i o n i n g tha t the line def ined by equa t i on (40b) is degenera ted , i.e. at t rans i t ions f r o m

reg ion R 31 to region R 7 be tween the corners f~l a n d f~3 two ra te cons tan t s (k_ 2, k 3) tu rn f r o m n o n m a x i m a l to m a x i m a l values.

The po in t S = P = 1 which is a lways loca ted wi th in the cent ra l region has the solut ion:


k , - S 1 ( i = 1 , . . . n), - - v q

which resembles that for the three-step mechanism (equation (25)).

(41)

3.4. Uni-uni-reactions with an arbitrary number of elementary steps. According to formulae (9)-(11) the number of possible solutions for the kinetic parameters in states of maximal activity becomes very large with an increasing number of elementary steps (Table 1). Therefore it seems not to be worthwhile to consider in detail all possible solutions. However, at the determination of optimal rate constants for reaction mechanisms with n ~< 4 two types of solutions were of particular interest: (1) Solutions with maximal forward and submaximal backward constants since for high values of the equilibrium constant q the corresponding regions (n = 2: R3; n = 3: Rio; n = 4: R31) in the concentration space of the reactants dominate all the others. (2) Solutions characterized by a strong binding of the reactants obtained at very low concentrations of S and P (n= 3: solution (9); n=4: solution (17): In the following we investigate whether for the ordered n-step mechanism described by the kinetic equation (12) solutions of those types can also be found.

Central Solutions. Solutions with k i = l ; k _ i < l (i=1, . . , , n) can be calculated using the method of Lagrange multipliers at the minimization of N (equations (12a-d)) in order to take into account the constraint:

1 q - n - const. (42)

[I k_i i = 1

Optimal values of the kinetic parameters k_, have to minimize the function:

2V(k_j, 2 )=N(k j= 1, k_ j )+2 k_~- , (43) i

where 2 denotes a Lagrange multiplier. After some algebra it can be shown that for S=P= 1 the condition:

0 ~ _ ON 1 ~- 2 - ~ = 0 (44)

Ok_j Ok_j q~_~

is fulfilled with:

k , = ~ q , (45)

86 T. W I L H E L M et al,

2 = l - q + n("x/q- 1) (46)

The comparison of formula (45) with the solutions for n = 2, 3, 4 given in Table 2 and in equations (25), (41) shows that at normalized reactant concentrations equal to unity (S=P---1) the optimal solution for enzyme reactions retains its simple structure for an arbitrary number of elementary steps. Using equations (43), (44) it is also possible to determine the boundaries of that region within the two-dimensional space of reactant concentrations where solutions with k i = 1 and k_ i < 1 are found. One obtains a region which is bounded like a triangle by three lines. The first one where k_ 1 = 1 is characterized by the equations:

q S-= 2 . ~ - - 1, (47a)

~ P 1 (i=2, , n - - l ) , (47b,c) k - i = , - , k_, = ~ , . . .

and the line k . = 1 by:

S="-2w/-Pq (I + P ) P - P , (48a)

1 k _ l = P , k_, , - 2 ~ ( i = 2 , . . . , n - 1 ) . (48b,c)

The lines k_i= 1 for i=2, . . . ,n-1 and n>2 are all the same:

S-- P/q, (49a)

k P 1 (49b,c) _,=]q, The line defined by equation (49) separates the region for the solution with n submaximal backward constants from the region of the solution with only two submaximal backward constants (kl, k , ) .

Figure 6 shows that region for n--2, 3, 4 and for n~oe . This region has a "triangular" shape except for n = 2 where it extends from S = 0 to infinite values of S. The corners f~i of the "triangle" have the following (S, P)-coordinates:

( 1 1~ f~2:(2"-2x/q - 1 , 1 ) ; ~3:(1, q). (50a-c) fll" \~' q/ ;


3 ":":::" /n 2

P 2 .n=3

0 1 ,,. 2 3 4 5 S

Figure 6. Regions of the "central" optimal solutions (k~ = 1, k i< 1 (i ~- 1 , . . . , n)) for uni-uni-reactions with n elementary steps for n=2, 3, 4 and n~oe with the equilibrium constant q = 3. The hatched area indicates the region for the central solution for n ~ oo. The boundaries and corners ~i of these regions are determined

by equations (47)-(49) and equation (50), respectively.

The formula for f~2 is also formally valid for n = 2 where one obtains (S~ oo, 1). It may be concluded from equations (50a-c) that not only for n = 2, 3 and 4

but for all values of n the solutions with maximal forward and submaximal backward constants are the optimal ones in the whole S-P-plane if the equilibrium constant tends to infinity.

Introducing (45) into (12) one gets with (14) by means of the formula for geometric progressions for the maximal activities and the Michaelis constants the following expressions:

1-- 1-- V + = , V- = , (51a,b)

n(1 ~ ) 1 q-l+nx/~ - + - - 1 . ~ n q

Ks = ( , ~ q - 1 ) ( ~ - 1) ( ~ - 1) ( l - q ) , K p =

( 2 ) 1 n l - + - - 1 , ~ n q

(51c,d)

In the limit q ~ l one gets:


2 2 V+=V- n ( n - 1 ) ' Ks=Kr" n-1 (52a-d)

and for q >> 1

V+ ~ 1 . V_ 1 ~ ; _ Ks~ 1 . - n - l ' - q n - l ' KP-~ "x/q (53a-d)

For n>> 1 equations (53a-d) can be simplified by using the well-known approximation:

(1 ,S4, Inserting x = ln q into (54) yields:

1 In q -x//q - 1 n (55)

With equations (51), (55) one obtains as approximations for the maximal activities:

V+ 1 ( q ( l n _ q ) 2 "~ 1 ( ( l n q ) 2 ) =n2 q lnq -q+lJ ' V - = ~ q - l - - l n q '

(56a,b)

lnq(q--1) Ks=n \q ln q-q+ l]'

1 (lnq(q--1)'] = \ q - l - in q]" (56c,d)

From formulae (56a-d) one may conclude that for n>> 1 the reaction rate becomes proportional to 1/n 2.

Figure 7 shows the ratios S/K s and P/Kr, as functions of q for different values of n calculated with equations (51c,d). According to formulae (52c), (53c) the ratio S/K s varies between the values ( n - 1)/2 for q = 1 and n - 1 for q ~ , i.e. for low values of n there is an adaptation of K s to S. In contrast P/Kp varies between ( n - 1)/2 and zero (equations (52d), (53d)), i.e. for enzymes catalyzing irreversible reactions no adaptation of Kp to P can be expected from the present model.

High-affinity Solution. In the case S, P ~ 1 a high-affinity solution with k 1, k , < l , (n>2) and k l = k _ . = k + i = 1 ( i = 2 , . . . , n - 1 ) c a n b e found in the following way. With:


S/Ks P/Kp

2

1 1 f / /n=6

ii i f . l /n=5

3 // .---~~ ....

IN=3

I

I

50 q

100

Figure 7. Relat ions of the reac tan t concent ra t ions and their cor responding opt imal Michaelis cons tants P/Kp and S/K s for different n for S = P = 1 as functions of the

equi l ibr ium cons tan t q according to equa t ion (51c,d).

k .=qk , (57)

one gets for the terms D i which enter the denominator of the kinetic equation (12):

k?l D 1 = 1 + ~ + k . ( n - 2 ) , (58a)

n - 1 k. ( n - l ) ( n - 2 ) D 2 - + , (58b)

k_ 1 k_l 2

n - 1 (n - -1) (n -- 2) D3 = k_--~- + 2 (58c)

N becomes minimum for:

k =Tq(n-1)(S+P) , , - - 5 ' k-1 ~/ q ( n - 2 ) (59a,b)


(cf. solutions (9) and (17) for the cases n = 3 and n = 4, respectively). That means for the optimal values of D t, D 2 and D3:

D'~ = 1 + q + ~ / q ( n - - 1 ) ( n - - 2 ) ( S + P ) ,

"V S + P

(60a)

!

/ q ( n - 1) (n-- 2) D~ 4

q(n-1) (n- 2 ) + , (60b)

2

( n - l ) ( n - 2 ) + (60c)

2

The expression for the optimal denominator can be written as:

N * = l + q + ( n - 1) ( n - 2 )

2 ( S q + P ) + 2 x / q ( n - 1 ) ( n - 2 ) ( S + P ) . (61)

F rom equations (60a-c) and (14c,d) it is to be seen that for S ~ 0 and P ~ 0 the Michaelis constants K s and Kp tend to zero, as do the maximal activities V + and V- . Furthermore, one may conclude from equations (55), (57) that for n >> 1 and 0 < S + P < (n - 2 ) / ( q ( n - 1)) the reaction rate decreases at an increase of n proportional to 1/n 2.

This solution as well as other possible solutions for very low normalized reactant concentrations may be of special importance if km/k d (equations (16b,c)) is very high compared to the non-normalized concentrations.

Reaction with One Elementary Step. It may be of theoretical interest that some of the results also apply for the case n --- 1 which is in a strict sense not an enzymic reaction since no enzyme intermediate complex is formed. According to formulae (9)-(11) one gets one optimal solution of type To, ~ which reads k~ = 1 and k_ 1 = 1/q. That is also a "central solution" for which formula (45) holds true.

Equilibrium Constants Smaller than Unity. In Section 3 all optimal solutions refer to the case q >t 1. Solutions for q < 1 are obtained from those of q > 1 using the transformations: v ~ - v, q-~ 1/q, k i ~ k_ (, _ i + 1), k _ i ~ k n - i + 1 and S~--~P.

4. Bi-uni Reaction with Four Elementary Steps. A general kinetic equation for an ordered reaction bireactant in the forward direction and unireactant in the reverse with the substrates A and B and the product P can be derived from equation (4) by replacing kl, k 2 and k . by A k l , B k 2 and P k , , respectively. One gets:


v ABq -- P E, U

(62a)

with

r = - j = r + j = r + l - r = 2 -

- " = - i = 3 r = 2 - "= 1

+AB Z + Z 1 ] Z k_~ - i = 2 j = l i = 2 j = l - r = i + l j = r + l

. - 1 1 1LI k j .

i = 2 j = 2

(62b)

In the following let us consider the special case n = 4 which is characterized by reaction scheme 4 replacing Sk 1 with Akl and k 2 with B k 2 . The denominator of the kinetic equation for this reaction reads:

N= 1 k 4 k 3 k 4

k 4 q k-3k- 4 ~k_ 2 k - 3 k- 4

A ( - k l k l k 4

+ \k l k_4+k lk 3k_4 klk3k 4 "~

4 k_lk_~f-_3k_4j

/ k2k3k 4 \ [ klk 2

klk2k3 k_lk 2k_3k_4

k lk2k'* _~ + k_lk_2k_3k 4 j

1 1 k3 .~+BP(- k2 + (63) + P + _ _ k 2k_3} \k_lk_ 2 k l k z k 3 } "

Compared with the uni-uni-mechanism with four elementary steps not only k 1 and k 4 but also k 2 is a second order rate constant which, therefore, is normalized by k a, too.

Optimal Rate Constants. F rom the results of Section 2 it can be concluded that not only for a uni-uni-reaction but also for a bi-uni-reaction with four


elementary steps 31 optimal solutions are obtained (cf. Table 5). Obviously, these solutions depend for bi-uni-reactions on the concentrations of substrates A and B and the product P. The procedure for the determination of the optimal solutions described in Section 3.3 is also applicable for all points of the concentration space (A, B, P). Explicit solutions can be obtained for types To,~, To, 2 and Tt, 1 . For example, solution (17) (TI,1) reads:

k4= / ~ - ~ + P) {l + k_ + P)-q + 2a) 1 ~ / 2q '

(64)

i.e. it is a high (A, P)-affinity solution. Due to the upper limits for the rate constants it is necessary but not sufficient for the validity of solution (17) that:

2 A + P < q(1 + 2 B ) (65)

Relation (65) and corresponding relations for the other solutions divide the three-dimensional concentration space of the reactants uniquely into 31 regions R i . In contrast to uni-uni-reactions all 31 solutions exist and changes of the equilibrium constant only influence their size.

Figures 8(a-c) show intersection planes P = 1 of the three-dimensional (A, B, P)-space for various values of the equilibrium constant where the solid lines indicate bounderies of different regions Ri. The dotted lines divide the parameter space into regions of v < 0 and v> 0. Intersections for B = 1 are identical with those shown in Figs 5(a-c) (where S is to be replaced by A) since for this special concentration of the second substrate equation (63) reduces to equation (12) of the uni-uni-reaction. The following conclusions can be derived:

(1) For q--*l all regions of solutions with more than one submaximal backward constant and all ki= 1 disappear while the regions of solutions (1)-(4) with one submaximal rate constant merge into one region R o where all kinetic constants become unity (Figs 5(a), 8(a)).

(2) Solution 31 with four submaximal backward constants is described by the following equation system:

k 2 = P(k-2k 3-/-B+Bk_3) k ' qk-2k-3( l + k - 2 + k - 2 -3)

k 2 - = P ( k - l k - 2 + B + k - 1 ) 3 qk lk 2 (AB+Ak_z+k_ lk_2 ),

k2 2 - e ( k - l + B) qk , k_3(A+k_l ) , (66a,b)

1 k , - q k l k_2k_3 (66c,d)

It is found as the central one for values of normalized concentrations of the three reactants not very different from unity. F rom Figs 5 and 8 it is seen that


~I,o,/ I \? / 4 13 27

B 23 ! 1

~L. ° \ ~ ~ ~ - - - ~ . . . . . . ~ ~

17 18 28 . . . . . . . . . . . . . . . . . . . .

0 1 2 3 Z~ A

i/,~,/

3 ~ 12

B / 23 -...'19 2 / L ~ . :1.1 ...... , . . . . . . . . .

1 2 3 4 A Figure 8 (a) and (b).


!a4 4 : ( c )

7 3 ~4 10

B !

2 ::

7" q

0 I 2 ? 4 A

Figure 8 (c)_

Figure 8, Subdivision of the concentration space of the bi-uni-reaction depicted in Scheme 5 for P = t into regions R i corresponding to the different optimal solutions (Table 5) for various values of the equilibrium constant (a: q = 1; b: q = 1.2; c: q = 2). For q = 1 region R o corresponds to solutions where all rate constants assume their maximal values. The numbers within the regions are the indices of R~. Dotted line

A B q = P.

the region R31 becomes larger with increasing values of q. For q = 2 the three-dimensional representation of R 31 is given in Fig. 9. The solid lines show edges of this region while the dotted lines and broken lines indicate intersections with the planes P = 1 and B = 1, respectively. Obviously, R 31 has the topology of a tetraeder. While inside the "tetraeder" all optimal backward constants are nonmaximum on the four lateral faces only three k_ i < 1. On the six edges0 i,_ j one has k_i , k j < l . At the four corners ®_i one gets k i < l . The coordinates of the corners ®_ ~ within the (A,B,P)-space and the equations of the edges 0 _ i _ j are given in the legend of Fig. 9. The equations for the location of these faces, edges and corners may be derived by inserting the relevant conditions into equation (66). It can be shown that the face hatched by fine lines with k I , k z , k _ 4 < 1, k 3 - - 1 is located on the surface:

A B q - - P = O , (67)

where v -- 0. These lines indicate the curves originating from intersections of the surface (67) with planes P = const. Obviously, for A = B = P = 1 the optimality condition is fulfilled by the same solution as for the uni-uni-reaction at S = P - - 1 (equation (41)). It follows from equation (66) that the point


~ ~ 3.0

2.0 P

13IO '' ~ - 2 . . 0 ~ ~-,~'- ~, 1 ~ ~ 0 10

/ V3.0 Figure 9. Region R31 of the central optimal solution with k i~< 1 (i= 1 . . . . , 4) within the space of normalized reactant concentrations for q = 2. On the four lateral faces of the "tetraeder" in each case one backward constant assumes its maximum. The corners have the coordinates ® - l : (l/q 2, 1, 1/q), ® - 2 : (q2, 1/q2, q), ® 3: (1, 2q-1, 1/q) and ® - 4 : (1, 1, q). The hatched area where k _ 3 = 1 is located on the plane ABq=P. The solid lines 0 _ i _ J represent the edges of R31 where k i and k_.i have nonmaximal values. They have the following (A, B, P)-coordinates: 0 1,-2: (1/(qZk42) k22, 1/(qk22)) 0 - 1 - 3 : ((l+q+qk-3)/(qak-3), 2/k 3 - 1 , l/q); 0 1-4: (k{a, 1, qk2_l); 0 2 3:'((qk22-k_2-1-1)/(qk3_2), qka_2-t-qk2_2-k_2, 1/(qk2-2)) 0-2, 4:(1/k22 k2-'2, q) '9-3 -4 : (1 2/k_3-1, qk{3), where the rate constant used for the parameter representation of an edge varies within 1/q ~< k i~< 1. Broken and dotted lines: Intersections of the "tetraeder" with the

planes B = 1 and P = 1, respectively.

A = B = P = 1 is always located within the interior of the "tetraeder", that means it belongs to the region of solution (31).

(3) For very low reactant concentrations one obtains as optimal kinetic design k 4, k_ 1, k_ 2 • 1 and maximal values of the other rate constants which has the physical interpretation of strong binding of all reactants to the enzyme. In particular, one gets k4,/c _ 1, k _ 2 ----~0 for A, B, P ~ 0 . As for uni-uni-reactions (Section 3) low normalized reactant concentrations have to be considered if the normalization factor km/k a is very large.

(4) Solutions with submaximal second order rate constants k I , k 2 or k 4 are only obtained in regions with high concentrations of reactants A, B or P, respectively.

At P = 1 the subdivision of the reactant space into regions R i displays a symmetry with respect to the line A = B (see Figs 8(a-c)) corresponding to the fact that for that product concentration the rate equation (63) is invariant with respect to the transformation: A+-~B, kt+-*k2, ka*--~k, ~ and k_z+-~k_ 4.

The topology of the subdivision of the space of reactant concentrations into the 31 regions R i can be understood in the following way. All regions for the 15


solutions with only submaximal backward rate constants are related to each other in the same way as regions following from the subdivision of a three- dimensional space by four planes bounding a tetraeder. The four regions for the solutions of type To, 3 with three submaximal backward constants are adjacent to the four surfaces of the "tetraeder" R 31. The regions for the solutions of type To. 2 are adjacent to the six edges of R31. Each of these regions has two surfaces to regions for solutions of type To, 3 and To, 1 . Regions R 1 , . . . , R 4 are adjacent to the four corners of R31 (points ® - i in Fig. 9). The 16 regions for solutions with submaximal forward constants have no direct contact to R31. The four regions of type T1, 2 are in each case completely located in the interior of that region of type To, 2 which is characterized by the same submaximal backward rate constants. Inside R 6 and R 9 (To,z) there are no other regions. Each of the four regions R 1 , . . . , R 4 contains two regions for solutions of type T1,1 and one region for a solution of T2,1 (e.g. region R 1 contains Rls , R17 and R3o ). All regions except of R 31 are either open or bounded by the planes A = 0, B---- 0 or P = 0. Figure 10 illustrates the connections between the regions R 1 , . . . , R3o. By folding the given two-dimensional network a three-dimensional structure results which gives a correct representation of the subdivision of the reactant space into all these regions R i .

Phenomenological Parameters. Equation (62) can be rewritten in the following form:

V + V- - - A B - - P KAB Kp

v = (68) A B P A B B P

1 +K +K.

with the maximal activities:

V + = k3k4 k 3 + k 4 + k _ 3 '

and the parameters:

V - _ k_lk_2k_ 3

k_lk_2q-k_lk_3q-k_2k 3q-k3k_l

k _ 1 K A - kl ,

k - l + k - l k - 2 k - l k - 2 k 3

(69a,b)

(69c)

(69d)


k - l ( k 3 k 4 + k 4 k - 2 + k - 2 k a) K P = k - 4 ( k 3 k - l + k - l k - 2 + k l k - 3 + k - 2 k 3)'

(69e)

k _ 1 (k3k4 + k4k- 2 + k_ 2 k _ 3) KAB = , (69f)

kx(k2k- 3 + k2k4 + k2k3)

k 1(k3k4+k4 k 2 + k _ 2 k _ 3 ) KBp - (69g)

k2k-4(k3 + k 3)

F r o m equa t ions (69a g) the op t imal phenornenolog ica l pa ramete r s can be calculated using the op t imal e l emen ta ry rate constants . Since the eight opt imal rate cons tants are funct ions of the three reac tan t concen t ra t ions , the seven op t imal phenomeno log i ca l pa ramete r s canno t be cons idered as i ndependen t f rom each other . F o r all solut ions the pa ramete r s V ÷ , V and KAB, KBp can be

Figure 10. Surface of the schematized body which contains the 31 optimal solutions. Since all regions but the central one are open (in other cases condition (5) would be broken), the remaining 30 solutions are visible_ The numbers are the indices of the

nonmaximal kinetic constants.


expressed as functions of KA, K n and Kp. The various solutions j are characterized by different functions KA,j[ki(A, B, P)], KB,j[ki(A , B, P)] and Kp,j[ki(A , B, P)] (i = __ 1 . . . . , _+ 4) which can be graphically represented in a three-dimensional space where the coordinates are the Michaelis constants, This structure shown in Fig. l l(a) resembles that of a three-step uni-uni- reaction in Fig. 3 taking into account the different dimensionality of the problem as well as the different number of possible solutions. Solutions of type T~,~ yield within the (KA, KB, Kp)-space (0c + fl - 1)-dimensional manifolds. The interior of the "tetraeder" belongs to solution (31), the four faces to solutions of type To, 3 and the six edges 7 - i , - j to solutions of type To, 2. The four corners F 1 F-4 with the coordinates given in the legend to Fig. 11 correspond to solutions (1)-(4). The K A, K n, Kp-values of solutions T1,1 are located on the edges 7i,-j, originating from the corners F ~ . The four faces between two of these edges and one edge of the "tetraeder" correspond to T1, 2. The four faces originating from corners F j and bounded by two edges 7i,-j belong to solutions of type T2,1. The three solid lines a, b and c in Fig. 1 l(b) are different routes through the (K A, K B, Kv)-space calculated for reactant concentrations (A = B = P), (A = B, P = 0) and (A = B--0, P), respectively. The arrows on the lines indicate how the Michaelis constants change at an increase of the reactant concentrations. For example, curve a starts with A = B = P = 0 on the face corresponding to the solution with k 4, k_ a, k 2 < 1. With increasing reactant concentrations curve a runs through the following regions: From the face of solution (26) the curve goes on the edge 7-1, 2 of the "tetraeder" corresponding to solution (5), from there on the face of solution (19) (one surface of the tetraeder with k 1, k_ 2 , k_ 3 < 1), afterwards in the interior of the "tetraeder" (solution (31)). From there the curve runs on the face of solution (22) (again one surface of the tetraeder with k_ 2, k_ 3, k _ 4 < 1), then on the edge 7-3 , -4 (solution (10)) and at last on the face corresponding to solution (24) (k 2, k_3, k 4< 1). A common characteristic of the three lines is that an increase of the reactant concentrations is accompanied by an increase of the Michaelis constants. A more detailed analysis shows that in general the following relations hold:

OKAoA ~> 0, OKBoB >~ 0, --0p ~> 0. (70a,b,c)

One may conclude from relations (70) that theoretically an adaptation of the Michaelis constant towards the corresponding reactant concentration should hold. A similar result was obtained previously by other authors on the basis of simpler models (e.g. Crowley, 1975; Cornish-Bowden, 1976). This theoretical prediction corresponds to the experimental fact that substrate concentrations and Michaelis constants are often in the same order of magnitude (Lowry and


7/ /

20

K~

~ ~ 25 K~

- 0 05 10

KA

U C

j {

t

2.0

Kp "--b

1.0 5.0

KB

o o 0.5 I.O

KA Figure 11. Optimal Michaelis constants (equations (69c~e)) of the reactants A, B and P of a bl-uni-reaction with q = 2 within the (K n, KB, Ke)-space. (a) The corners F i (F - l : (1/q, 3/q, 3/(3+q)),F_2: (1, 1+2/q, (q+2)/(2q+Z)),F 3: (1,2+l/q, (2q+ 1)/(2q+2)), F 4 : (1, 3, 3q/4)) of the "tetraeder" represent solutions (1)-(4) where the Michaelis constants are independent of the reactant concentrations. The edges 7-i,-j of the "tetraeder" represent solutions (5)-(10) and the edges 7~. j solutions (11)-(18). The unhatched faces of the "tetraeder" belong to solutions


Passonneau, 1964; Fersht, 1985). However, it follows from the present model that despite the validity of relations (70) the ratio between the reactant concentration and the corresponding Michaelis constant is not necessarily very close to unity. It is worth mentioning that the optimal Michaelis constant of a certain reactant is also dependent on the concentrations of the other reactants. For example, for P = 0 an increase o f K e is observed with increasing values of A and B (Fig. l l (b) , curve b). Similarly, K a and K B increase with increasing values of P, even if A = B = 0.

5. Discussion. In the present paper kinetic parameters of various ordered enzymic mechanisms in states of maximal activity were calculated. The analysis is based on the assumption that an increase in reaction rates was an important characteristic of enzyme evolution. At the solution of the nonlinear optimization problem Iv[ = Vma ~ tWO different constraints were taken into consideration: (1) The equilibrium constant of an enzymic reaction remains fixed at alterations of the rate constants of the microscopic reactions, and (2) the individual rate constants cannot exceed certain upper limits. In contrast to previous investigations upper limits were introduced not only for the second order rate constants (kd) for the binding of reactants to the enzyme but also for the first order rate constants characterizing the dissociation of the reactants from the enzymes and the velocities of the interconversion of the various enzyme-intermediate complexes. It is important to note that the variety of different solutions obtained for ordered mechanisms with n~> 2 elementary steps is a direct consequence of the fact that upper limits (k,,) were also introduced for the first order rate constants. This may be seen as follows. According to equation (16) one gets infinitesimally small values of the normalized reactant concentrations in the limit k,, ~ oe. Therefore, this limiting case is characterized within the present model by only one solution, namely that obtained for very small normalized concentrations of substrates and products. For all uni-uni-reactions with n > 2 these are the solutions characterized by a strong binding of the substrate (k 1 = 1, k 1 ~ 1) as well as of the product (kn~l , k _ , = l ) . A similar conclusion is derived for enzymic bi-uni-reactions where one obtains k 4, k _ 1, k _ 2 ~ 1 for very small normalized

Continuation of caption to Fig. 11.

(19)-(22). The hatched planes refer to solutions (23)-(30). All points inside the "tetraeder" correspond to the central solution (31) (cf. Table 5). (b) Solid lines represent Michaelis constants for different substrate and product concentrations_ Curve a with 0 ~< A = B = P ~< 10 runs through the regions of solutions (26), (5), (19), (31), (22), (10), (24). Curve b with 0~<A=B~<10 and P = 0 passes the regions of solutions (26), (5), (19), (8), (23), (12). Curve c with A = B = 0 and 0~<P~<10 goes

through the regions of solutions (26), (17), (1), (7), (25), (16).


concentrations of all three reactants while all other kinetic constants attain their maximal values.

The present model gives rather detailed information concerning the functional dependence of the optimal microscopic parameters on the concentrations of the reactants. At the present stage of the experimental analysis of the kinetic properties of enzymes it may be difficult to test these relationships quantitatively. However, this may not be the case for the following general conclusions:

(1) As derived in Section 2 for the general ordered mechanism high enzymic rates are incompatible with parameter constellations where simultaneously one of the forward rate constants (k~) and at least one of the backward rate constants k_ i or k_(i_l) has a low value (taking into account the cyclic notation for the index i (k,+i=ki, k(n+i)=k_i) ).

(2) Independently of the number n of elementary steps one gets optimal solutions for nearly irreversible reactions (q ~> 1) in a very large region of the reactant concentrations which are characterized by maximal values of the forward rate constants and submaximal values of the backward rate constants.

(3) Generally a matching between substrate concentrations and Michaelis constants is predicted. This result which is most concisely expressed by formula (27) for the uni-uni-mechanism and by formula (70) for the bi- uni-mechanism is in accordance with the experimental observations that for many enzymes substrate concentrations and Michaelis constants are in the same order of magnitudes (Lowry and Passoneau, 1964; Crowley, 1975; Cornish-Bowden, 1976; Fersht, 1985). Obviously, this result is independent of the special choice of the normalization parameters k d and k,, since reactant concentrations and Michaelis constants are scaled in the same way.

However, more quantitative comparisons with experimental data necessitate detailed information on the numeric values of the upper limits of the rate constants of enzymic reactions. Theoretical estimates for ke may be obtained taking into account rate limitation by diffusion. A typical value is k d - - 1 0 9 M - 1 s- 1 (Eigen and Hammes, 1963). The exact value of k d depends on the reaction and it may vary within the range 108-1011 M - 1 s- 1 (Fersht, 1985). The latter value may be reached by H ÷ which has a very high diffusion coefficient. The upper limit k,, of first order reactions is much more uncertain. Theoretically, k,, is limited by the frequency of molecular vibrations which is given by the universal frequency kBT/h~-6.21 x 1012 s -1. For enzymic reactions much lower maximal values (~-104-106 s -a) are observed for first order rate constants characterizing the release of reactants from the enzymes which is also limited by diffusion (Eigen and Hammes, 1963; Somogyi and Damjanovich,


1975). For the evaluation of the kinetic parameters for the interconversion of enzyme-intermediate complexes different mechanisms have to be taken into account. The upper limits may vary enourmeously. Values within the range 103-1012 s - t have been reported.

The model presented in this paper may be generalized by taking into account different upper limits for different types of first order elementary processes. As a first step one may use instead of equations (15c), (16) the following normalization for the kinetic constants k _ 1 ~ k_ 1/kr, k,--, k , /kr , k + i ~ k + i/km (i = 2 , . . . , n - 1), the reaction rate v--* v/(krEt) and the reactant concentrations: S ~ k a S / k ~ , P-~keP/k~ where k r denotes the upper limit of the rate constants for the release of the reactants (cf. Heinrich and Hoffmann, 1991 for the case n = 3 where an additional parameter p = km/k ~ was introduced).

Throughout the paper a kinetic description was used without consideration of the thermodynamic basis of enzyme catalysis. Obviously, the model can be reformulated using free energy profiles of enzymic reactions where instead of the elementary rate constants the free energies of the formation of enzyme- intermediates would play the role of the variables for the optimization problem. It is worth noting, for example, that in terms of free energy profiles the "central solution" with maximal values for all forward rate constants and nonmaximal values for all backward rate constants shows some correspondence to the result of the "descendent staircase model" proposed by Stackhouse et al. (1985).

Theoretical investigations of evolutionary optimization of enzymes are often discussed in terms of the internal equilibrium constant Kin t (Albery and Knowles, 1976; Burbaum et al., 1989; Stackhouse et al., 1985; Pettersson, 1991). It was stated that states of maximal enzymic activity are characterized by K~nt-~ 1 (for experimental data cf. Burbaum and Knowles, 1989). Using the result of the present model detailed conclusions concerning optimal values of Kin t may be derived. Generally, Kin t may substantially deviate from unity depending on the substrate and product concentration. However, with regard to uni-uni-reactions as well as to bi-uni-reactions there are several solutions characterized by an internal equilibrium constant equal to unity independent of the overall equilibrium constant q. These are: (1) Optimal solutions for enzymes working under near equilibrium conditions (Nq~-P for uni-uni- reactions, A B q ~ P for bi-uni-reactions), and (2) solutions obtained for very low substrate and product concentrations (high-affinity solutions). The result obtained for the first case corresponds to that derived previously by other authors (Burbaum et al., 1989).

Probably, a more detailed comparison of the theoretical result with experimental data should be based on the phenomenological enzyme parameters, in particular the Michaelis constants, rather than the individual rate constants. Let us consider, for example carbonic anhydrase (EC 4.2.1.1)

AN E V O L U T I O N A R Y A P P R O A C H TO E N Z Y M E KINETICS 103

catalyzing the reaction C O 2 + H 2 0 ~ H 2 C O 3 . Carbonic anhydrase is often considered as a perfectly evolved enzyme. With [ H 2 0 ] ~ -cons t . the kinetic mechanism of this enzyme can be regarded as an ordered uni-uni-mechanism with four elementary steps (Lindskog et al., 1971; Silverman and Lindskog, 1988). Taking into account the concentrations in human erythrocytes [CO2] ~- 1.2 mM and [ H 2 C O 3 ] -~ 24 mM one may conclude that this enzyme works under near equilibrium conditions with the apparent equilibrium constant q= q [ H 2 0 ] = [H2CO3]/[CO2] ~- 20. Using the scaling parameters /ca-----109 M -1 s -1 and k,,= 107 S-1 one gets the following values for the normalized concentrations: S [CO2] =0.12, P [ H 2 C O 3 ] = 2.4. According to the results of Section 3.3 for the uni-uni-reaction with n = 4 one may expect that carbonic anhydrase is characterized by solution (7) with submaximal values of k 1 and k 4 or by solution (31) with submaximal values of all backward rate constants k_ i (i = 1 , . . . , 4). For these solutions the following estimates for the normalized phenomenological parameters may be derived: K s < 1, Kp > K s and V +=1/6, V - < V +. With the given values for k d and k m one gets for non-normalized phenomenological parameters the following predictions: K s < 10 mM, Kp> 10 mM and V+/Et = 1.66 x 106 S- 1 V - / E t < 1.66 x 106 S- 1 These values are in reasonable agreement with the following experimental data for two different types of the enzyme: Human B: K + =4 mM, K- = 16 raM, V+/Et=0.2 × 10 6 s - 1 , V-/Et=0.3 × 1 0 4 s - 1 and Human C: K + =9 mM, K- =22 mM, V+/Et = 1.4 x 106 s - x , V-/Et= 8.1 X 104. S -1 (Lindskog et al., 1971).

Probably, validation of the theoretical results of the present investigation on a broader experimental scale needs the statistical evaluation of the kinetic properties of many enzymes. For that databases on enzymes and metabolic pathways could be of great help (Sel'kov et al., 1989).

Concerning the general conclusions derived in the present paper it is of great interest that there is a relation between the number of steps n on the one hand and the dimensionality of the space of reactant concentrations necessary for a full representation of all possible solutions on the other hand. As shown in Sections 3.1 and 3.2 all optimal solutions for uni-uni-reactions may be found for n = 2 as functions of one reactant concentration (concentration of the product; cf. Table 2) and for n = 3 as functions of the two reactant concentrations S and P (cf. Table 4). For uni-uni-reactions with four elementary steps the number of possible solutions found for S, P > 0 is strongly dependent on the equilibrium constant (Section 3.3). However, the full spectrum of the 31 possible solutions derived for n = 4 can be found within the three-dimensional space of reactant concentrations for bi-uni-reactions. In line with this conclusion is the fact that for the case n = 1 the optimal solution is independent of the concentrations of the reactants (Section 3.4). It may be hypothesized that the case n = 5 where formulae (9)-(11) predict 91 different


optimal solutions can be fully represented only within a four -d imens iona l space of reactant concentrations which would be, for example, that of enzymic reactions with two substrates and two products. For such reactions a special prediction can also be made for a possible "central solution" with maximal values of all forward rate constants and submaximal values of all backward rate constants. For 1 < n ~<4 the concentration regions where such solutions are found are bounded for n = 2 by two points (region R3, cf. Section 3.1), for n = 3 by three lines (region Rio, equation (24)) and for n = 4 (bi-uni-reaction) by four two-dimensional surfaces (R31, Section 4). It can be hypothesized that for a bi-bi-mechanism with n = 5 a region for the solution with k z = 1 and k _ z < 1 is bounded by five three-dimensional subspaces within the four-dimensional space of the substrate concentrations A, B and product concentrations P, Q. From the results obtained for n ~< 4 it may be expected that for very high values of the equilibrium constant q such a "central solution" becomes most important for bi-bi-reactions with n = 5 due to a strong increase in the corresponding region with increasing values of q.

It needs further investigations as to whether the results derived in Section 4 for ordered bi-uni-reactions may be generalized to the case of random binding of the substrates A and B to the enzyme. Another generalization of the present analysis could be the consideration of optimal states of coupled enzymic reactions which meets with greater difficulties since in such cases the concentrations of some of the reactants cannot be considered to be fixed. Previous investigations in such a direction are based on oversimplified kinetic equations for the participating enzymes (Heinrich et al., 1987) or are confined to the case of only two coupled enzymic reactions (Pettersson, 1989; Heinrich and Hoffmann, 1991).

Throughout the paper it was tacitly assumed that evolutionary pressure on enzymes was mainly directed towards maximization of reaction rates. We are well aware that many other properties of enzymes have to be considered to arrive at a deeper understanding of enzyme evolution (cf. Srivastava, 1991). In the future, the analysis could be extended by considering several optimality criteria to be equally important which would necessitate the application of multiobjective programming approaches.

We are grateful to E. Mel6ndez-Hevia (University La Laguna, Spain) and F. Montero (University Madrid, Spain) for very stimulating discussions during the preparation of the manuscript.

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R e c e i v e d 21 F e b r u a r y 1993

an evolutionary approach to enzyme kinetics: optimization of ordered mechanisms

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