J. Phys. Chem. B 2000, 104, 5209-5212
5209
New Treatment of Enzyme Kinetics Applied to Human Dihydrofolate Reductase John Happel* Department of Chemical Engineering and Applied Chemistry, Columbia UniVersity, New York, New York 10027
Masood Otarod Department of Mathematics, UniVersity of Scranton, Scranton, PennsylVania 18510 ReceiVed: February 2, 2000
This paper presents a new procedure for the derivation of steady-state rate calculations by computer for enzyme kinetics using data on strains of human dihydrofolate reductase (DHFR) reported in the literature. This enzyme continues to command the attention of scientists since its discovery in the late 1950s in connection with applications to cancer therapy. Kinetics of nonhuman strains previously reported (Happel, J.; Sellers, P. H. J. Phys. Chem. 1995, 99, 6595) appears to be much simpler. In the present case it is necessary to introduce the concepts that an elementary step should be treated as a vector possessing the same direction in each submechanism that it occurs.
1. Introduction An important step in the study of catalysts, including enzymes, is the identification of possible mechanisms for a given reaction. For our purposes a mechanism is defined by a list of elementary reactions whose net rates of advancement are specified. The mechanism for a reaction can be reduced to a sum of submechanisms that add up to that of the whole mechanism. A programmable procedure that lists all such submechanisms, called direct mechanisms, was first applied by Happel, Sellers, and Otarod1 to pioneering studies of Poland2 using the method of King, Altman, and Hill (KAH). With a computer program3 to implement our procedure, we were able not only to give a confirmation4,5 of Poland’s results but also to apply the program to cases where the necessary enumerations could not have been done by simple inspection. The chemical reaction catalyzed by DHFR is the reduction of (7,8)-dihydrofolate, which is essential for various metabolic functions. In this paper we report results using a new modification of our program to determine the steady-state kinetics of the reactions catalyzed by two strains of human DHFR that involve more elementary reactions than those forms of DHFR previously considered. 2. Enzyme Kinetics Systems One of the advantages that the investigator of enzyme kinetics enjoys is that it is often possible to determine experimentally the kinetics of the individual elementary reaction steps for a given reaction. In studies of heterogeneous catalysis, the situation is more difficult because elementary reaction steps on solid surfaces usually cannot be readily studied experimentally under conditions corresponding to the overall stoichiometric reaction being observed. Before considering DHFR catalysis, let us review the assumptions on which our procedure for enzyme catalysis is based. * Author to whom correspondence should be addressed.
(i) Steady-State. This assumes that the chemical species that appear in the overall reaction are present in large proportions compared to the intermediates and that the intermediates remain at constant concentrations because the rate at which each one is produced equals the rate at which it is being consumed. (ii) Mass Action. The net rate of reaction of each elementary reaction is the difference between the forward rate, which is directly proportional to the product of the concentrations of the reactants, and the reverse rate, which is directly proportional to the product of the concentrations of the reaction products. (iii) Pseudo-First-Order. It is assumed that each intermediate is either a free enzyme or an enzyme that has been modified by interactions with terminal species, and it is further assumed that all elementary reactions are first-order with respect to intermediates. The rates involving the product of one intermediate and a terminal species are made first-order by the device of replacing some rate constants by pseudo-rate constants with respect to the concentration of the terminal species involved. The basis for this is the steady-state assumption (i), in which it is assumed that the terminal species are present in concentrations well in excess of the concentrations of the intermediate species. Consequently, every forward and reverse elementary reaction rate can be expressed as linear functions of the concentrations of the intermediate species, as the terminal species are omitted from the chemical equations. With these assumptions, we used a program that derives kinetic formulas as a first step, after which it accepts numerical values for the input parameters and substitutes them in the derived formulas to obtain numerical values for the reaction rates of interest. (iv) Transition-State Theory. In the course of using this program, we observed that it implies the assumption that an elementary reaction can proceed in opposite directions in different submechanisms or reaction paths. We believe that this contradicts the widely accepted principle implied in transitionstate theory developed by Eyring and co-workers6 in which each submechanism proceeds along a separate minimum energy path termed the reaction coordinate. Accordingly in this paper we modified our approach to include the concept that it is only
10.1021/jp000412u CCC: $19.00 © 2000 American Chemical Society Published on Web 05/05/2000
5210 J. Phys. Chem. B, Vol. 104, No. 21, 2000
Happel and Otarod
TABLE 1: Elementary Reaction Steps in the DHFR System step
reaction
s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 s12 s13
E + B h EB EB + A h EAB E+ A h EA EA + B h EAB EAB h ECD EC h E + C ECD h D + EC ECD h C + ED ED h D + E EB + C h EBC EC + B h EBC EA + D h EAD ED + A h EAD
possible to consider submechanisms that are parallel, where to be parallel an elementary step should be treated as a vector possessing the same direction in all steps in which it occurs. This distinction is developed in the present paper. 3. Dihydrofolate Reductase (DHFR) To facilitate comparison with our previous studies of DFHR catalysis, we have modified our program to generate the same kind of rate tables as before. Table 1 includes a listing of all the elementary reactions that have been observed to participate in the dihydrofolate reaction E
A+BdC+D
(1)
where A ) H2F, B ) NADPH, C ) H4F, D ) NADP, and E is the dihydrofolate reductase the net concentration of which does not change, since it is a catalyst. The first 11 of these are given in the carefully documented series of studies by Benkovic and co-workers that we used in our previous studies.4,5 The two final elementary reactions listed in the table are the additional ones given by Blakley and co-workers7,8,9 for two strains of human DHFR that are the focus of this paper. The reactants in this listing include not only the terminal species A, B, C, and D but also combinations of E with A, B, C, and D. Table 2 lists all the forward and reverse rate constants for the reactions in Table 1, as reported in the studies mentioned above for each of the six strains of DHFR considered here. The rate constants are first-order for elementary reactions involving only intermediates and pseudo-first-order for those involving terminal species. The factors [A], [B], [C], and [D] in the table are the concentrations of the four terminal species whose values will be set respectively at 0.9, 0.9, 0.1, and 0.1 in this study. We experienced some problems in determining the proper values to use for rate constants involved in the human DHFR enzyme strains that were partially resolved by Dr. Raymond L.
Blakley, who authored a monograph7 in 1995 that involved reference to earlier studies of these enzymes. Rate constants for s1 were not correctly reported in ref 8, but correct values are listed in refs 7 and 9 that were published at later dates. Rate constants for step, 10 of W24F r. Human DHFR are not directly included in the paper by Beard et al.9 but it is simply stated that k-10 . 300. Without information on the exact extent that k-10 exceeds 300, we decided to choose that value as being conservative. k10 is also omitted from the listing for W24F r. Human DHFR by Beard et al.9 We assumed it is to be the same as k10 for Wild r. Human DHFR, equal to 14[C]. Tables 1 and 2 provide the basic information required for treatment of enzyme kinetics used in this study. The next step involves further development of the theory using the computer program presently discussed3 in advance of using the specific values for rate constants listed in Table 2. Accordingly, the complete listing of all possible direct mechanisms for the reaction A + B ) C + D is developed in Table 3, where each mechanism involves some subset of the 13 elementary reaction steps. Table 3 shows that the addition of steps s12 and s13 results in the possibility of five additional mechanisms. Table 4 lists the rates as found by the KAH procedure corresponding to each mechanism. The first four columns in this table correspond to rates reported in our previous paper5 and listed in Table 3 of that paper. In that paper, following the KAH procedure, we simply added rates to the seven direct mechanisms listed to obtain total rates for each of the four nonhuman strains of DHFR reported. In effect, as noted in (iv) above, this assumes that all seven of the direct mechanisms can be combined. But following principles of transition-state theory, we believe that the nonhuman strain rates of mechanisms d4 and d7 should not be added to the combination of d1, d2, d3, d5, and d6. It so happens that the rates of d4 and d7 are very small in these cases so that the total rates listed in Table 3 of the previous paper are only slightly too large. For the cases shown in Table 4 of this paper, a more complicated behavior is observed in which the mechanism d4 is not negligible. As a consequence, it is not possible to combine the rates of all 12 submechanisms shown in Table 4. If we regard it as not possible for an elementary step to proceed simultaneously in two opposite directions in a combination of mechanisms, there are several possible allowable parallel sets with two, three, four, five, and six submechanisms. These sets are generated by sequentially selecting 2-, 3-, 4-, 5-, or 6-submechanism subsets of the overall set of all 12 mechanisms and testing for being parallel. The nonparallel sets are discarded, and the fluxes of the parallel ones are obtained by adding the fluxes of each submechanism in the set. In each category the
TABLE 2: Elementary Rate Constants (Refs 5, 7, 8) Wild E. coli rates step s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 s12 s13
Val 113 E. coli rates
for.
rev.
for.
20[B] 40[A] 40[A] 5[B] 950 1.4 200 2.4 300 2[C] 8[B]
3.5 40 20 1.7 0.6 25[C] 5[D] 25[C] 13[D] 12.5 85
20[B] 4[A] 4[A] 5[B] 165 35 185 35 225 0.5[C] 2[B]
rev. 20 180 180 3.6 0.16 6[C] 5[D] 6[C] 5[D] 68 60
L. Casei rates for.
rev.
15[B] 15[A] 15[A] 15[B] 100 0.5 290 1.7 85 15[C] 15[B]
0.08 20 3.70 0.5 1.5 15[C] 10[D] 15[C] 10[D] 40 8
mutant mouse rates for.
rev.
2.7[B] 30[A] 35[A] 3.3[B] 9000 15 450 18 93 5[C] 1[B]
2.4 12 21 1.8 90 33[C] 5.7[D] 18[C] 2.6[D] 40 26
Wild r. Human for.
rev.
38[B] 98[A] 264[A] 24[B] 1360 5.1 84 46 32 14[C] 4.4[B] 20[D] 110[A]
1.7 94 14 19 37 117[C] 0.7[D] 24[C] 17[D] 225 100 4.6 1.3
W24F r. Human rates for.
rev.
28[B] 97[A] 100[A] 33[B] 89 47 130 125 250 14[C] 30[B] 17[D] 58[A]
3.2 350 22 60 4 44[C] 2.9[D] 34[C] 19[D] 300 32 50 4.9
New Treatment of Enzyme Kinetics Applied to DHFR
J. Phys. Chem. B, Vol. 104, No. 21, 2000 5211 Studies in which the required data are presented are rare. Our earlier paper5 considered a series of studies by Benkovic and co-workers for various nonhuman strains of catalysis by dihydrofolate. In the present paper later studies of strains for the more complicated cases of Wild r. Human DHFR and W24F Human DHFR have been considered.8,9 For the human strains studied, somewhat higher rates and different submechanisms are indicated. In the case of Wild Human DHFR, the data appear to be consistent and reasonable demonstrating the value of the structural modifications involved. There are some inconsistencies in the data for W24F Human DHFR that should be resolved. These relate to step 10 elementary rate constants that appear in Table 2 and subsequent tables in which these values are analyzed. It would be desirable to obtain a value for the rate constant k10. This constant is the only one completely missing from the listing of 26 constants proposed to characterize the W24F Human strain of DHFR. The high rates reported in which k-10 . 300 should also be more accurately established if possible to determine more definitely W24F Human strain rating as compared with Wild r. Human strain in Table 5.
TABLE 3: Possible Mechanisms Based on Steps Listing in Table 2 steps mechanism
s1
s2
d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12
0 0 0 0 1 1 -1 0 0 -1 0 1
0 1 1 0 1 1 1 0 0 1 0 0 1 0 0 1 0 0 0 1 1 0 0 1 0 0 1 0 0 1 1 -1 0 1 -1 0
s3
s4 s5
s6
s7 s8
1 0 0 1 1 1 1 0 1 1 -1 0 1 0 0 1 1 1 1 0 1 1 0 0 1 1 1 1 0 1 1 -1 0 1 0 0
s9
s10 s11 s12 s13
1 1 0 0 0 0 0 0 -1 1 1 -1 1 1 0 0 0 0 0 0 -1 1 0 0 0 -1 0 0 -1 -1 1 0 -1 1 0 0
0 0 1 1 0 0 1 0 0 1 1 0
0 0 0 0 0 0 0 -1 -1 -1 -1 -1
0 0 0 0 0 0 0 1 1 1 1 1
TABLE 4: KAH Rates of Direct Mechanisms mechanism d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12
Wild E. coli
Val 113 E. coli
L. Casei
mutant mouse
Wild r. Human
W24F r. Human
0.007 0.280 0.688 0.001 0.017 0.693 0.016
0.004 0.019 0.017 0.000 0.118 0.600 0.000
0.012 0.082 4.337 0.002 0.015 0.102 0.011
0.050 1.187 0.102 0.001 0.313 0.741 0.004
1.294 0.009 0.461 1.884 1.317 0.013 0.012 0.000 0.019 0.108 0.440 0.101
0.656 0.007 2.639 2.167 1.710 0.074 0.006 0.000 0.040 1.243 1.020 0.100
5. Conclusion The procedure that we have presented should be of more general applicability than the examples that we have considered for the enzyme dihydrofolate reductase. Studies reported in the literature have been most complete for this enzyme, which has continued to commend the attention of scientists since its discovery in the late 1930s as discussed by Huennekens10 in connection with cancer therapy. In order to test the computer models, it would of course be desirable to conduct experiments to determine overall reaction rates. Such rate data should be obtained sufficiently far from equilibrium that the linear relationship postulate by the KAH treatment is applicable, as discussed in a recent treatment by Schulz.11 Results presented here demonstrate a method for following the influence of individual elementary steps that should be useful in providing guidance for enzyme comparison and possible improvement of pharmaceuticals based on DHFR modifications. A recent example of how our methodology might be applied relates to a recent study by Sawaya and Kraut.12 These authors introduce a procedure that employs crystallographic studies to elucidate the behavior of Escherichia coli DHFR treated by Fierke et al.13 This amounts to the assumption of a single mechanism, namely, d6 in Table 4 of this paper. As can be seen from Table 4, EC-DHFR cycles through four intermediate states. The authors have illustrated graphically a detailed picture of this mechanism, providing a useful tool for the study of enzyme dynamics. As we have discussed here, the mechanism that they chose embraced one of the five possible paths along a reaction coordinate. Development of the geometry along other coordinates would be of interest. We believe that enzyme design using combinatorial chemistry can be assisted by information concerning rates of individual mechanistic steps as detailed in this paper.
parallel set with the maximum flux is recorded. The number of the possible parallel sets and their maximal fluxes are reported in Table 5. Only one of these sets, corresponding to the path of least resistance, will result when appropriate reaction velocity constants from Table 4 are applied. Using these reactions, we computed the total fluxes corresponding to allowable sets listed in Table 5. It is of interest that the maximum number of combined mechanisms corresponds to six for W24F r. Human and five for Wild r. Human. The maximum flux equals 7.636 for W24F r. Human DHFR and 4.956 for Wild r. Human DHFR. The latter is slightly larger than 4.548, the corrected value omitting fluxes corresponding to d4 and d7, reported for the L. Casei DHFR in a previous paper.5 Note that as the flux of mechanism d8 is 0.000, the maximum fluxes for W24F r. Human DHFR corresponding to {d3, d4, d5, d8, d11, d12} and {d3, d4, d5, d11, d12} are equal. And, for Wild r. Human DHFR, the fluxes of the two sets {d1, d3, d4, d5, d8} and {d1, d3, d4, d5} are also equal. 4. Discussion We believe that the procedure presented in this paper furnishes a method for the application of modern computer techniques to the study of enzyme kinetics. Given a complete listing of the rate constants for elementary steps, it shows how various interaction effects can be modeled. TABLE 5: Number of Possible Sets of Parallel Submechanisms
Wild r. Human no. submechs. i 2 3 4 5 6
max no. sets, ( 66 220 495 792 924
12 i
)
W24F r. Human
no. parallel sets
max flux set
flux
max flux set
flux
46 78 64 24 3
{d4,d5} {d1, d4, d5} {d1, d3,d4,d5} {d1, d3,d4,d5,d8} {d3,d4, d5, d8, d11, d12}
3.201 4.495 4.956 4.956 4.203
{d3, d4} {d3,d4, d5} {d3, d4, d5, d11} {d3, d4, d5, d11, d12} {d3,d4, d5, d8, d11, d12}
4.806 6.516 7.536 7.636 7.636
5212 J. Phys. Chem. B, Vol. 104, No. 21, 2000 Acknowledgment. We are grateful for the help of Dr. Peter Sellers of Rockefeller University on many aspects of this work over a number of years. References and Notes (1) Happel, J.; Sellers, P. H.; Otarod, M. Ind. Eng. Chem. Res. 1990, 29, 1057. (2) Poland, D. J. Phys. Chem. 1989, 93, 3613. (3) Otarod, M.; Happel, J. Chem. Eng. Commun. 1996, 148-150, 221. (4) Happel J.; Sellers, P. H. J. Phys. Chem. 1992, 96, 2593. (5) Happel J.; Sellers, P. H. J. Phys. Chem. 1995, 99, 6595. (6) Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of Rate Processes; McGraw Hill Book Co., New York, 1941.
Happel and Otarod (7) Blakley, R. L. Eukaryotic Dihydrofolate Reductase; Advances in Enzymology and Related Areas of Molecular Biology, Vol. 70; Meister, Alton, Ed.; John Wiley and Sons, Inc.: New York, 1995. (8) Appleman, J. R.; Beard, W. D.; Delcamp, T. J.; Prendergast, N. J.; Freisheim, J. H.; Blakley, R. L. J. Biol. Chem. 1990, 265, 2740. (9) Beard, W. A.; Appleman, J. R.; Huang, S.; Delcamp, T. J.; Freisheim, J. H.; Blakley, R. L. Biochemistry 1991, 30, 1432. (10) Huennekens, F. M. Protein Sci. 1996, 5, 1201. (11) Schulz, A. R. Enzyme Kinetics; Cambridge University Press: New York, 1994. (12) Sawaya, M. R.; Kraut, J. Biochemistry 1997, 36, 586. (13) Fierke, C. A.; Johnson, K. A.; Benkovic, S. J. Biochemistry 1987, 26, 4085.
