J. Phys. Chem. 1995, 99, 6595-6600
6595
New Perspective on the Kinetics of Enzyme Catalysis John Happel" Department of Chemical Engineering, Materials Science and Mining, Columbia University, New York, New York 10027
Peter H.Sellers Laboratory of Mathematics, The Rockefeller University, New York, New York 10021 Received: January 4, 1995@
This is a sequel to our recent paper (J. Phys. Chem. 1992, 96, 2593) in enzyme kinetics on deriving steadystate rate formulas by computer for chemical systems that are too complex for the methods of King and Altman (King, E. L.; Altman, C. J . Phys. Chem. 1956,60,1375) and Hill (Hill, T. L. Free Energy Transduction and Biochemical Cycle Kinetics; Springer-Verlag: New York, 1989). In this paper we illustrate how such computer-derived formulas, which are themselves too complex even to print out, are to be used. With the reaction catalyzed by the enzyme dihydrofolate reductase as an example, we demonstrate how information on the mechanism and kinetics of a reaction can be deduced from computer-stored rate formulas used in conjunction with experimental data. Thus, a new mathematical perspective in enzyme kinetics makes it possible to replace hand calculations with programs that list reaction mechanisms, derive kinetic formulas, and make use of data bases to gain insight as to how mechanisms may be characterized.
Introduction
This study in enzyme kinetics is a continuation of a preceding paper' in which we demonstrated a programmable procedure which would make it possible to generate kinetic formulas by computer. We introduced an algebraic approach to the widely recognized diagram methods used by King and Altman,2 Hill,3 and others4q5 for formula derivation. Our object, now, is to demonstrate how computer-derived rate formulas, which may be exceedingly complex, can be used to elucidate the mechanism of a given enzymatic reaction by combination of appropriate direct mechanisms. We review the initial assumptions of the King, Altman, and Hill (KAH) methods. A chemical system is understood to have one or more overall reactions, as well as a set of elementary reactions, which, taken in certain linear combinations, equal each overall reaction. In this paper we assume there is one overall reaction. The chemical species appearing in it are the substrates and products of the chemical system, referred to here collectively as the terminal species. The elementary reactions of the systems involve, not only these, but also other species in the system, including enzymes and enzyme-substrate complexes, collectively called intermediates. The concept of mechanism is formalized by understanding each elementary reaction to be the result of an irreducible physical process, called a mechanistic step. If a certain linear combination of elementary reactions is equal to the given overall reaction, then the same linear combination of the corresponding mechanistic steps is a mechanism for the reaction. This formalization regards reactions and mechanisms as vectors (linear combinations of species and elementary steps, respectively) and makes it possible to determine algebraically6s7every possible mechanism for a given overall chemical reaction. In our preceding paper' we showed that the same algebraic procedure, as we used to list mechanisms, could be used to derive KAH-type steady-state formulas, provided that the chemical system satisfied the assumptions ordinarily made in @Abstractpublished in Advance ACS Abstracts, April 1, 1995.
0022-365419512099-6595$09.00/0
enzyme chemistry. The procedure is readily programmable, which means that the list of all mechanisms and their steadystate rate formulas can be derived by computer. Starting at this point, this paper demonstrates the use of computer-generated rate formulas to determine which of the mechanisms (also found by computer) advance at significant rates. In order to make such rate estimates, experimental data must be available to be used as parameter values in the rate formulas. This whole approach to mechanism determination, based on programmable algebraic methods, is demonstrated here for the chemical reaction catalyzed by the enzyme dihydrofolate reductase (DHFR).The computer-generated rate formula for this reaction is too complex for there to be any use in printing it out, but it is readily evaluated by computer which is done here for four sets of experimentally determined parameters. The possibility of arriving at KAH-type formulas by computer and using them to explore the networks of mechanisms in biochemical systems opens up a perspective in enzyme kinetics that was not accessible by the older diagram methods.
Direct Mechanisms and Parallel Combinations The mathematical procedure for listing possible mechanisms was fnst applied by us6-* to catalytic systems in general without the additional assumptionsthat can be made in enzyme catalysis. It was shown that given a reaction and a finite set of possible mechanistic steps, there is a unique finite set of direct mechanisms for that reaction. The word direct is defined as it was in our previous p a p e r s l ~ and ~ . ~ as it was earlier by Milne~-.~ Here, our object is to consider all possible mechanisms for a given overall reaction, which means we must include not only all direct mechanisms for that reaction but also certain combinations of them. Since the set of direct mechanisms is not necessarily linearly independent, there will be some combinations of them equal to others. Accordingly, to make a complete list of mechanisms, it is not necessary to employ all combinations. We shall demonstrate that all possible mechanisms are obtainable by combining only those direct mechanisms that are parallel to each other in the following sense: Two or more mechanisms 1995 American Chemical Society
6596 J. Phys. Chem., Vol. 99, No. 17, 1995 are parallel, if they produce the same reaction in the same direction, but not necessarily at the same rate, such that no mechanistic step has a positive direction in one mechanism and a negative direction in another. For a given reaction only two types of mechanisms are possible, those that are direct and those that are combinations of two or more parallel direct mechanisms. No mechanism can be of both types. This method of mechanism classification has been established algebraically by one of us. It was seen by the other of us as having a physical meaning identified by steady-state multiple path velocities, each characterized by a cascade of positive free energy contributions in a series of elementary mechanistic steps.8 The statement that every elementary mechanistic step has the same net direction in every member of a set of direct mechanisms, each of which is contributing positively to the advancement of a given overall reaction, is consistent with recognized transition-state theory, where such a step involves a single activated state or potential energy barrier in configuration space over which a net transfer of species occurs. A computer programlo has been written to list all direct mechanisms for a given reaction. A procedure for selecting the possible sets of parallel direct mechanisms from such a list will be demonstrated in this paper. What we have called a direct mechanism is what is usually simply called a mechanism in the chemical literature, where combinations of direct mechanisms are not given the special attention we give them here. Some writers such as Hill3 use the word cycle for what we would call a direct mechanism, but we prefer to reserve that term for a hypothetical mechanism that recycles all species, so as to produce a zero overall reaction. A direct mechanism has a nonzero overall reaction, which means that it can recycle only intermediates. Calling a direct mechanism a cycle would agree with our usage, if the terminal species were treated as if they were not present, which can be done without affecting the kinetics, if the rate constants are modified appropriately. Systems which become first order, when this device is applied, are said to be pseudo-first order.
Rate Formulas The KAH method^^.^ give a way to derive steady-state concentrations and steady-state reaction rates in first-order or pseudo-first-order chemical systems, in which the rates of the elementary reactions are assumed to conform with the law of mass action. In our previous paper and in this one, we take the KAH methods and context for granted as a starting point. The formulas given by these methods in ordinary cases involve very long summations of terms, each of which is a product of mass action rate constants and terminal species concentrations. Each product is represented by a diagram, and the summations are found by drawing all diagrams of a specified kind. The KAH methods assume that all the appropriate diagrams can be found by simple inspection, but this is not true except in the simplest cases. In our previous paper' we demonstrated a programmable procedure for listing the terms in each of the summations in a KAH type formula, which was identical from a mathematical viewpoint with the programlo for listing direct mechanisms. In a paper by Poland1] on the derivation of steady-state rate formulas he brought attention to the fact that the dihydrofolate reductase (DHFR) system with only eleven mechanistic steps was not directly amenable to the KAH methods, there being no systematic way to generate all diagrams. Accordingly, we returned to this example in our preceding paper, using it to demonstrate our programmable version of the KAH methods and its use in deriving rate formulas by computer.
Happel and Sellers TABLE 1: Mechanistic Steps and Their Reactions in the
DHFR System step
elementary reaction equations E+BzEB EB A S E A B ESAZEA EA+ B zEAB EAB ECD ECZE+C ECD zD EC ECD C + ED EDZD+E EB C ZEBC EC B EBC
+
+
+ +
Rate Constants and the Equilibrium Constant The KAH methods2s3and our algebraic development of them are based on two major assumptions on the character of a chemical system. Since we are concerned here with a system in which there is one designated overall reaction, we state the assumptions this way: (i) The overall reaction is equal to one or more linear combinations of elementary reactions, whose rates are governed by the law of mass action. Accordingly, every chemical species A in the system is assumed to have a numerical concentration [A] and every elementary reaction to have two rate constants. For the ith reaction, ki denotes the forward rate constant and k-i the reverse rate constant. (ii) There is a set of concentrations of the species for which the system is in equilibrium, Le., for which the net rate of every elementary reaction is zero. Accordingly, the overall reaction has a well defined equilibrium constant K . Assumption ii cannot hold unless the rate constants in i are related to each other by equations sometimes called Wegscheider conditions.12 A derivation of these coditions for any system satisfying i and ii is given in a mathematical sequel13 to this paper. These conditions for the DHFR system, which involves eleven elementary reactions and, therefore, has 22 rate constants, are shown in (1). Poland" and Hill3 both employ the same relationships in their treatment that formed the basis of our previous paper.
In a later section we shall see a confirmation of eq 1 under steady-state conditions. Another observation we shall make later is as follows: If experimentallydetermined values for K and all the rate constants were given and it were found that the seven algebraic expressions in (1) were not all equal to K , then it would always be possible to adjust four of the rate constants so as to achieve equality and thus make eq 1 hold exactly. This follows from the fact that four, and only four, of the equations in (1) are independent, in the sense that the remaining ones can be derived algebraically from those four. If K were not specified in advance, it would take three, and only three, changes to make (1) hold exactly.
J. Phys. Chem., Vol. 99, No. 17, 1995 6591
New Perspective on the Kinetics of Enzyme Catalysis The DHFR system has one overall reaction abbreviated by (2), and it has the 11 mechanistic steps listed in Table 1, in which it is understood that the positive direction of each step carries the correspondingreaction from left to right. With overall
(2)
A+BSC+D
reaction 2 and Table 1 as input, we can use the direct mechanism programlo and get the complete list 3 of all direct mechanisms for the given overall reaction. It is immediately evident how the computer-generated list 3 is converted into the rate constznt dependence eq 1.
d , = s3 d2 = $3
+ s4 + s, + s8 + s9 + + + s6 + 84
+ + + + + s3 + s4 + s, + s7 - sl0+ s l l
d6 = 81 d7 = -sl
S7
S5
32
S5
Sg
S7
(3)
Cycle-Free Mechanisms In addition to the relations among rate constants (l), which are mathematical consequences13of the mass-action and equilibrium assumptions i and ii, made in the last section, there is another mathematical consequence needed in this paper, which we shall state as follows: Every mechanism is ~ y c l e - f r e eWe .~~ consider what this means in the context of the DHFR system. The mechanisms in list 4 are cycles. Each entry has a zero s1 '1
s1 - s7
+ s2 - s3 - s4 Sl0
mechanism for the DHFR overall reaction 2 is a positive linear combination of direct mechanisms selected from one group. List 5 is found by noting that all the steps in (3) have consistent signs except s1 and $6. Therefore, the only way to arrive at parallel sets of direct mechanisms is to choose those where s1 and s6 have signs chosen in one of the following four ways: (+,+), (+,-), (-,+), or (-,-). Each one of these gives us one of the four parallel sets (5). When experimental data are taken into account, it will be possible to determine the signs of s1 and S6 in the DHFR system, which means that all cycle-free mechanisms are generated by combining direct mechanisms from only one of the fourparallel groups.
Hill's Formula for the Overall Reaction Rate In our preceding paper1 we demonstrated the derivation by computer of Hill's3 formula for the steady-state rate of the overall reaction 2 in the DHFR system. It breaks into a sum of formulas, one for the rate F, of each direct mechanism d,. In other words, for the DHFR system in a steady-state there are seven numbers F1 through F7 that characterize the kinetics of the system, as displayed in expression 6. The expanded expres-
+ F2d, + F3d3+ F4d4 + F5d5+ F6d6 + F7d7 = + F6 - F7)s1+ ( F , 4-F4 + F, + F6)s24-( F l + F2 + F7)s3+ ( F , + F2 + F7)s4+ ( F , + F2 + F, + F4 + F, + F6 + F7)s5+ (F2 - F4 F6)s6+ (F2 + F3 F6 + F7)s7+ ( F , + F4 + F5)sg+ ( F , + F4 + F5)s9+
Fldl (F,
+ ' 6 + '10 - $11
+ sg + s9 +
with the signs in the cycles, as shown in list 3. All other cyclefree mechanisms can be obtained by combining parallel direct mechanisms. This way of determining all cycle-free mechanisms, which is easy to see in the present case, can be proved in general.13 It gives all possible cycle-free mechanisms for a given overall reaction. To get all cycle-free mechanisms for the overall reaction 2 in the DHFR system, we separate the seven direct mechanisms into four parallel groups, as shown in list 5. Every cycle-free
- SI1
(-F3
overall reaction, as can be seen by combining the elementary reactions from Table 1 in the manner indicated in the six cycles (4).None of these cycles, nor any linear combination of them, can occur under our assumptions. This fact, which is consistent with the principle of detailed balancing, is only part of what is meant by the assertion that all mechanisms are cycle-free. Beyond that, cycle-free means that if we take the steps that appear in any one of the cycles (4),they cannot all recur in a cycle-free mechanism with the same signs as they have in the cycle nor can they all recur with signs exactly the opposite of those they have in the cycle. In other words, there are 12 sign patterns, the 6 patterns in (4)and the 6 with opposite signs, that cannot occur in any cycle-free mechanism. This restricts only the signs, not the absolute values of the coefficients, in a cycle-free mechanism. The cycle-free condition will be important in this paper, as we pare down the set of all mechanisms that are possible for a given overall reaction. It can be applied to the DHFR system as follows: All the direct mechanisms are cycle-free, as can be seen by comparing the signs of their steps, as shown in list 2,
- F4 - F7>s10
+ ( F 3 + F4 + F7)s11
(6)
sion in (6) is obtained by substituting for the values of the d's as given in (3). The coefficient of each step is its rate. Since each d, produces the overall reaction at rate 1, the entire mechanism produces the overall reaction at the rate (F1 FZ ... F7). In our previous paper' we showed how to arrive at a computer-generated formula for F,, which may be symbolized by
+
+ +
F, =
(n,+- II,-)Z,E
For the DHFR system i ranges from 1 to 7; n,+and II,-take the values shown in (7), in which each HI+and H,- is found from the formula in (3) for d, as follows: HI+is the product of the forward rates of each step in d,, and n,- is the product of the reverse rates of each step in d,. For example, the factors of n13+corresponding to its steps s2, s5, s7, -s10, $11 are k2[Al. k5, k7, k-10, kll[B], respectively, whose product equals H3+ as shown in (7). Ill+
= k,k4k,kgk9[Al[B]
Ill- = k-3k-4k-,k-,k_,[Cl[Dl
n2+= k3k&k6k7[A] [B] n2-= k_3k_4k-,k-,k_7[CI[D]
6598 J. Phys. Chem., Vol. 99, No. 17, 1995
Happel and Sellers
TABLE 2: Pseudo-First-Order Rate Constants for the Dihydrofolate Reaction ref 14 ref 15 fwd rev fwd rev 20[B] 20 20[B] 3.5 SI 4[A] 180 sz 40[A] 40 4[A] 180 40[A] 20 s3 5[B] 3.6 1.7 s4 5[B] 165 0.16 0.6 ~5 950 6[C] S6 25[c] 35 1.4 5[D] 185 s7 5[D] 200 6[C] sg 25[C] 35 2.4 13[D] 225 ~9 5[D] 300 12.5 0.5[C] 68 s i 0 2[C] si1 8[B] 85 2[B] 60 The derivation of the formulas for Ci and is as shown in our previous paper' to be an application of the direct mechanism enumeration program.', By means of the relations 1 among rate constants, the left-hand factors in the formulas for F1, F2, etc. reduce to the following simple form:
Only the forward rate constants appear, and there is a constant factor on the right which is zero at equilibrium. This shows that the rate Fi attributed to d, reduces to zero at equilibrium for all i. Conversely, if Fi = 0 for all i, it becomes an immediate consequence of the steady-state rate formula that the relations 1 among the rate constants must hold. This confirms the relations 1 in the present case.
Dihydrofolate Reductase System In this section we use a series of four studies by Benkovic and c o - ~ o r k e r s ' ~ -to ' ~ ascertain as closely as possible the steady-state mechanisms of the reaction catalyzed by four strains of the DHFR enzyme. The reaction is the conversion of 7,8dihydrofolate to 5,6,7,8-tetrahydrofolate.It was denoted above in (2) by A + B t C + D where A = H2F, B = NADPH, C = &F, and D = NADP. In these studies the authors have reported estimated values of the mass-action rate constants for each of the assumed 11 mechanistic steps shown in Table 1. The four sets of published rate constants, corresponding to different strains of DHFR, are listed in Table 2. All second-order rate constants are multiplied by [A], [B], [C], or [D], by means of which they become pseudofirst-order rate constants. We take the rate constants listed in Table 2 at face value and ignore until the next section the fact that they do not quite conform with the thermodynamic restrictions (1). Two of the rate constants would have to be changed by at least 8% and others by lesser amounts to bring the system into complete conformity with (1). Regarding this 8% as being within experimental error, we calculate F1 through F7 which, when added, give the overall reaction rate. The formula for Fi was derived by computer, using the procedure of our previous paper,' and is stored electronically ready to be evaluated for each set of experimental data. The formula in this case is a function of 1216 terms, which is too large for conventional KAH methods to be used, but small by computer standards. Table 3 gives the values found for F1 through F 7 for each strain of DHFR. The first column, based on data from Benkovic et al.,14 was given in Table IX of our previous paper except for slight errors resulting from hand calculations given in the earlier paper.
~~
ref 16 fwd
15[B] 15[A] 15[A] 15[B] 100
rev 0.08
20 3.7 0.5
1.5 15[c] 10[D] 290 15[C] 1.7 10[D] 85 15[C] 40 15[B] 8 0.5
ref 17 fwd rev 2.8[B] 2.4 30[A] 12 33[A] 21 3.3[B] 1.8 9000 90 15 WCI 5.7[D] 450 18 18KI 2.6[D] 93 5[C] 40 1[Bl 26
TABLE 3: Rates of Seven Direct Mechanisms
Corresponding to Four Strains of DHFR wild type, Val 113, E. mutant mechanism E. ~ 0 1 i l 4 colii5 L.caseii6 mouse17 0.0067 0.0500 0.0038 0.0118 0.2804 1.1866 0.0192 0.0817 0.6882 0.1017 0.0170 4.3374 0.0008 0.0014 0.0001 0.0015 0.0166 0.3125 0.1179 0.0149 0.6927 0.7412 0.6000 0.1016 0.0163 0.0035 0.0004 0.0106 total
1.7017
0.7584
4.5595
2.3969
The terminal species concentrationsused in computation were [AI = [B] = 0.9, [C] = [D] = 0.1, and [EO]= 1, as in the studies previously reported on by Poland" and ourselves.' [Eo] denotes the total concentrations of all intermediates, including E (Le., all but the terminal species). Now we exploit the cycle-free concept, discussed earlier, to get a sharper picture of the mechanism structure. We have seen that any mechanism in the DHFR system is expressible as a sum of parallel mechanisms selected from one of the groups in list 5, each advancing at a rate that is a positive fraction of the overall rate of reaction. For the experimental data at hand, we can easily see that the parallel mechanisms must be selected from the group (dl, d2, d3, d5, d6) of parallel direct mechanisms, because the rates of S I and $6 are both positive. Thus, from (6) we get the following: rate Of SI = F5f F6- F7> 0 rateofs,= F2- F4f F6 > 0 The smallness of F4 and F7 in Table 3 shows that these rates are well above zero. The direct mechanisms d4 and d7 are absent from the parallel group we are using. To eliminate d4 and d7 from mechanism 6, we use the dependence relations d4 = dl d2 f d3 and d7 = dl id3 - d5, which can be verified by replacing each d, in the formulas by the value given for it in (3). When d4 and d7 have been eliminated from (6), we get expression 8.
+
+ + F4f F7)d3f
( F , f F4 F7)dl f (F2 - F4)d2 (F3
(F5
- F7)d5
+ F6dfj
(8)
The sum of the coefficients is the overall rate of reaction, and the individual coefficients are the positive partial rates of the parallel direct mechanisms. If the rates F1 through F 7 are those given in Table 3, then the five coefficients above take the values shown in Table 4 for each strain of DHFR. In the next section, when the rate constants are changed to conform with the restrictions (1) the parallel decomposition 8 will still apply.
New Perspective on the Kinetics of Enzyme Catalysis
TABLE 4: Rates of Five Parallel Direct Mechanisms Corresponding to Four Strains of DHFR wild type Val 113 mutant mechanism rate E. E. colP L.c a s e P mouse1’ dl FI F4 F7 0.0238 0.0043 0.0239 0.0549 d2 F2- F4 0.2796 0.0191 0.0802 1.1852 d3 F3 + F4+ Fi 0.7053 0.0175 4.3495 0.1066 d5 F5 - F7 0.0003 0.1175 0.0043 0.3090 0.6927 0.6000 0.1016 0.7412 db Fb total 1.7017 0.7584 4.5595 2.3969
+ +
Without further examination of how closely the thermodynamic constraints (1) are satisfied by the rate constants in Table 2, it is clear that there are substantial differences in rates shown in Table 3 between the four strains of the enzyme, but major contributions are confined to the rates F2, F3, and F6. The predominance of the corresponding direct mechanisms d2, d3, and d6 was noted previously for the system treated by Fierke et al.I4 where wild type E. Coli DHFR was studied. In the treatment of the Val 113 strain of E. coli by Fierke and Benkovic15 the authors suggest using a model that reduces to the mechanism d3, but from our results in Table 3 it would appear that ds is by far the dominant mechanism. Since d6 does not involve steps s10 and s11, the importance of them claimed by the authors is not supported. On the other hand, data in the third study, reported by Andrews et al.,16 show that d3 is indeed predominant for the L. casei DHFR. Surprisingly, the data in the fourth study by Thillet et al.” for mouse DHFR show that the path dZ predominates. This is the only enzyme strain showing this characteristic.
Thermodynamic Restrictions In view of the striking differences between the four sets of results seen in Tables 3 and 4, we consider how they would be affected if the given rate constant data were minimally adjusted to take into account the thermodynamic restrictions (1). Such a procedure is essentially an assessment of the consistency of the rate constant data that we have been accepting at face value. We know mathematically that with three adjustments within any one of the four sets of given rate constants that the data can be made consistent with (1). Insofar as we can achieve consistency within a given set of rate constants with less than three adjustments, we will be relying on a degree of consistency already inherent in the data rather than merely a mathematical adjustment that gives a greater appearance of consistency. If, however, four or more adjustments are to be made, then it is a simple problem in linear programming to minimize the adjustments. In the absence of special information indicating that some rate constants have been more accurately determined than others, we propose the following changes as an indication of how nearly consistent the given rate constants are: ref 14
change k,,[B] from 8[B] to 4.352[B]
ref 15
change k,[A] from 4[A] to 2.88[A] change k, from 185 to 225 change k,,[B] from 2[B] to 2.574[B]
ref 16
change k3[A] from 15[A] to 17.34[A] change k,,[B] from 15[B] to 18.75[B]
ref 17
change k,,[B] from 1[B] to 1.721[B]
In the first, third, and fourth cases we achieve substantial consistency with less than three changes, but in the second case, where consistency was less easily achieved, we decided to
J. Phys. Chem., Vol. 99, No. 17,1995 6599
TABLE 5: Equilibrium Constants mechanism ref 14 ref 15 ref 16 ref 17 Evaluated on the Basis of the Rate Constants in Table 2 di 20 633 8355 7811 10 305 d2 20 862 6869 7838 10 388 d3 37 254 3843 7250 607.2 d4 36 847 4675 7225 605.3 d5 20 044 6016 9031 10 432 d6 20 266 4946 9063 10 466 di 38 350 5339 6270 599.8 Evaluated on the Basis of Rate Constants As Adjusted for Thermodynamic Consistency dl 20 633 6016 9029 10 305 d2 20 863 6016 9061 10 338 d3 20 267 6017 9062 10 451 d4 20 044 6017 9031 10 418 d5 20 044 6016 9031 10 432 db 20 267 6016 9012 10 467 di 20 863 6017 9061 10 323 TABLE 6: Rates for Constituent DHFR Mechanisms Based on Rate Constants Adjusted for Thermodynamic Consistency rate ref 14 ref 15 ref 16 ref 17 Hill’s Decomposition into Mechanisms dl through d7 0.0060 0.0023 0.0137 0.0509 Fi F2 0.2923 0.0142 0.0776 1.1780 F3 0.3902 0.0224 4.7413 0.1737 F4 0.0004 0.0011 0.0015 0.0024 F5 0.0147 0.1001 0.0150 0.0318 F6 0.7220 0.6153 0.0835 0.7359 F7 0.0092 0.0004 0.0126 0.0060 total 1.4348 0.7558 4.9452 2.1787 Parallel Decomposition into dl, d2, d3, d5, and d6 F1 + F4 + F7 0.0156 0.0038 0.0278 0.0593 F2 - F4 0.2919 0.0131 0.0761 1.1756 F3 + F4 + F7 0.3998 0.0239 4.7554 0.1821 F5 - Fi 0.0055 0.0997 0.0024 0.0258 F6 0.7220 0.6153 0.0835 0.7359 total 1.4348 0.7558 4.9452 2.1787 illustrate the fact that absolute consistency can be reached with three changes. The consistency we achieve by the proposed changes is shown in Table 5, where we list the equilibrium constant derived for each direct mechanism both before and after the changes have been made. Formulation 1 is used to calculate these constants for each of the seven direct mechanisms. Using the adjusted rate constants, we can recalculate the rate Tables 3 and 4. The results are shown in Table 6. The major conclusions based on the adjusted rate constants are much the same as for the original values except that in all cases d3 accounts for a slightly greater contribution to the overall rate. Notice that our proposed rate constant adjustments were applied only to forward rate constants. It is clear from (1) that, instead of changing k, by a certain factor, we could change k, and k-, by any amount that would change k,lk-, by the desired factor.
Effects of Hydrogen Ion Concentration All of the mechanistic steps as well as the overall reaction in these four studies omit the hydrogen ion, so it must be assumed that its concentration is constant in the course of each study. As shown in Table 5 after adjustment of the rate constants to achieve thermodynamic consistency, the equilibrium constant is consistent within each study, but different from one study to another. The hydrogen ion enters into the rate constant for step s5 because the actual reaction for this step is
EAB+H+=ECD
6600 J. Phys. Chem., Vol. 99, No. 17, 1995
Happel and Sellers
TABLE 7: Rates for Constituent DHFR Mechanisms Based on Rate Constants Adjusted for Both Thermodynamic and pH Consistency rate ref 14 ref 15 ref 16 ref 17 Hill’s Decomposition into Mechanisms dl through d7
0.0060 0.2923 0.3902 0.0004 0.0147 0.7220 0.0092 1.4348
Fi F2 F3 F4 F5 F6 F7 total
0.0036 0.0218 0.0345 0.0002 0.1542 0.9469 0.0006 1.1618
0.0147 0.0836 5.1086 0.0016 0.0161 0.0899 0.0136 5.3281
0.0509 1.1783 0.1738 0.0024 0.0318 0.7361 0.0060 2.1793
Parallel Decomposition into dl, d2, d3, ds, and d6
Fi F2 F3 Fs F6
+ F4 + F7 - F4
+ F4 + F7 - F7
total
0.0211 0.2919 0.3998 0.0055 0.7220 1.4348
0.0044 0.0216 0.0353 0.1536 0.9469 1.1618
0.0299 0.0820 5.1238 0.0025 0.0899 5.3281
0.0593 1.1759 0.1822 0.0258 0.7361 2.1793
Fierke et al.14 stated that their data, using an MTEN buffer, corresponded to a pH of 6.5. A very small change in pH will alter the apparent constant for this reaction significantly, but if the data fumished in each of the four studies correspond to a fixed equilibrium constant, then it appears that each separate study was conducted at constant pH. Therefore, we assume the pH-dependent rate constant, previously denoted by ks, is actually a pseudo-rate constant containing factor [H+]. Since k5 is a linear factor of the equilibrium constant K, as seen in (l), we conclude that K also contains a factor of [H+]. Accordingly, the actual equilibrium constant Kactis determined by the relationship K = KactHc0l-I Using this formula, we can determine a correction factor Hcorr, which would equalize the equilibrium constants listed in the last seven rows of Table 5:
because the most useful procedures in such studies are programmable. We have shown that the enumeration of all hypothetically possible mechanisms for a given reaction, the derivation of steady-state formulas, and the evaluation of such formulas for various experimentally determined sets of parameters can all be done by computer. The creation of suitable programs to replace hand methods has required theoretical developments based on algebraic characterizations of reactions and mechanisms, as well as subcategories of these including elementary reactions, overall reactions, direct mechanisms, cycle-free mechanisms, and cycles. These vectors, which are related to each other in a way that describes a chemical system mathematically, can be stored electronically in matrix form. Accordingly, it is possible to contemplate systems far more complex than the DHFR example in this paper. The size of chemical system that can be studied is limited more by the absence of an expeirmental data base than by the limit of computer time and memory. As data bases and programs improve, larger and more complete systems can be studied and simplifying assumptions less needed. Our study of the DHFR system by computer for four different sets of data corresponding to four different strains of the enzyme showed somewhat different mechanisms predominating in each case. The activity of the L. casei strain reported by Andrews et al.16 was considerably higher than the others and its activity was 95% due to a single mechanism from among five parallel paths. The high degree of consistency within each set of results speaks well for the quality of the experimental data and also gives a satisfactory indication of the workability of this approach to enzyme kinetics.
Acknowledgment. We are grateful to John Noss and Masood Otarod for help in computer programming and computations reported here. We also appreciate the helpful comments by Miguel A. Hnatow and Huk Y. Cheh as well as their support of this research.
ref 14
H,,, = 1.000, k5 = 950
References and Notes
ref 15
H,,, = 3.395, k5 = 560.2
ref 16
H,,, = 2.258, k, = 225.8
ref 17
H,,, = 1.966, k, = 17 692
(1) Happel, J.; Sellers, P. H. J . Phys. Chem. 1992, 96, 2593. (2) King, E. L.; Altman, C. J . Phys. Chem. 1956, 60, 1375. (3) Hill, T. L. Free Energy Transduction and Biochemical Cycle Kinetics; Springer-Verlag: New York, 1989. (4) Segel, I. H. Enzyme Kinetics; Wiley: New York, 1975. (5) Comish-Bowden, A.; Wharton, C. W. Enzyme Kinetics; I R L Press: Washington, D.C., 1988. (6) Happel, J.; Sellers, P. H. Id.Eng. Chem. Fundam. 1982, 21, 67. (7) Happel, J.; Sellers, P. H. Adv. Catal. 1983, 32, 273. (8) Happel, J. Isotopic Assessment of Heterogeneous Catalysis; Academic: Orlando, FL, 1986. (9) Milner, P. C. J . Electrochem. SOC. 1964, 111, 228. (10) Happel, J.; Sellers, P. H.; Otarod, M. Znd. Eng. Chem. Res. 1990, 29, 1057. (11) Poland, D. J . Phys. Chem. 1989, 93, 3613. (12) Schuster, S . ; Schuster, R. J . Math. Chem. 1989, 3, 25. (13) Sellers, P. H. A Mathematical Model for Enzyme Kinetics and Its
Table 7 is the result of recalculating all the fluxes with these new values of ks to conform with the pH of Fierke et al.14 The rates in the first column of Table 7 are the same as in Table 6. The others depend on the assumption that the pH adjustment does not require any change due to differences in catalyst structure. The dramatic increase in d3 observed for the third set of results is still evident, and the fourth set exhibits the same major contribution of d2 as previously. It is easy to perform other calculations with different assumed values for rate constants to explore the effect of possible catalyst modifications. Thus, we believe that the application offers a promising approach to enzyme studies in general.
Conclusions The principal conclusion of this paper is that studies in enzyme kinetics do not have to be confined to small systems,
Use in the Derivation of Steady-State Formulas by Computer. Manuscript in preparation. (14) Fierke, C. A.; Johnson, K. A,; Benkovic, S. J. Biochemistry 1987, 26, 4085. (15) Fierke, C. A.; Benkovic, S . J. Biochemistry 1989, 28, 478. (16) Andrews, J.; Fierke, C. A,; Birdsall, B.; Ostler, G.; Feeney, J.; Roberts, G. C. K.; Benkovic, S.J. Biochemistry 1989, 28, 5143. (17) Thillet, J.; Adams, J. A,; Benkovic, S. J. Biochemistry 1990, 29,
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