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# How important are entropic contributions to enzyme catalysis?

Communicated by Thomas C. Bruice, University of California, Santa Barbara, CA (received for review April 3, 2000)

## Abstract

The idea that enzymes accelerate their reactions by entropic
effects has played a major role in many prominent proposals about the
origin of enzyme catalysis. This idea implies that the binding to an
enzyme active site freezes the motion of the reacting fragments and
eliminates their entropic contributions,
(Δ*S*_{cat}^{‡})′, to the
activation energy. It is also implied that the binding entropy is equal
to the activation entropy,
(Δ*S*_{w}^{‡})′, of the
corresponding solution reaction. It is, however, difficult to examine
this idea by experimental approaches. The present paper defines the
entropic proposal in a rigorous way and develops a computer simulation
approach that determines (Δ*S*^{‡})′. This
approach allows us to evaluate the differences between
(Δ*S*^{‡})′ of an enzymatic reaction and of the
corresponding reference reaction in solution. Our approach is used in a
study of the entropic contribution to the catalytic reaction of
subtilisin. It is found that this contribution is much smaller than
previously thought. This result is due to the following:
(*i*) Many of the motions that are free in the
reactants state of the reference solution reaction are also free at the
transition state. (*ii*) The binding to the enzyme
does not completely freeze the motion of the reacting fragments so that
(Δ*S*^{‡})′ in the enzymes is not zero.
(*iii*) The binding entropy is not necessarily equal
to (Δ*S*_{w}^{‡})′.

Many prominent proposals (e.g., see refs. 1 and 2) and textbooks that consider biochemical systems (e.g., refs. 3 and 4) invoke entropic contributions as major factors in enzyme catalysis. These proposals, which are intuitively very appealing (e.g., see ref. 5), have assumed that the large configurational space available for the reacting fragments in water would be drastically restricted in the enzyme active site. It has been thus deduced that this should lead to large entropic contributions to the difference between the activation barrier in the enzyme and in the reference solution reaction. However, the validity of these proposals is far from being obvious (6, 7). For example, the very influential proposal introduced by Page and Jencks (1) reflects the assumption that the formation of the transition state in a bimolecular reaction in solution involves complete loss of three translational and three rotational degrees of freedom. However, two or more of these degrees of freedom are usually almost free in the transition state (see below). More serious is the implicit assumption that the entropic contribution to catalysis is given approximately by the negative of the binding entropy (see below). Other problems with simple estimates of the entropic contribution will be mentioned in the next section.

The main stumbling block for determining the validity of the entropic proposal is the absence of direct experimental information about the corresponding contribution of the reacting fragments to the activation entropy in the enzyme and in solution. In this respect it is interesting to note the recent analysis of cytidine deaminase by Wolfenden and co-workers (8). This study found that the entropies of activation in the enzyme and in water are very similar and that the overall catalysis is due to enthalpic effects. Interestingly, it was found that the activation entropy in water and the binding entropy are significant. However, it is not known what parts of these entropies are associated with the restriction of the motion of the reacting fragments and what parts are associated with solvation entropies.

Intramolecular cyclizations of model compounds were used as “proofs” that enzyme catalysis involves large entropic contributions (e.g., ref. 2). However, recent simulations of Bruice and co-workers (9) indicated that the rate acceleration associated with smaller ring size is not necessarily associated with entropic contributions. Furthermore, our studies (6) have challenged the direct relevance of these model compounds to enzyme catalysis.

The entropic contributions to enzyme catalysis can be determined in principle by computer simulation approaches, but the development of the proper method is far from trivial because of enormous convergence problems. Nevertheless, some encouraging progress has been made in studies of related problems. This includes the progress in evaluating solvation entropies (10) and in evaluating binding entropies (11). Yet, no attempt has been made in evaluating activation entropies by simulation approaches. Note in this respect that the recent interesting studies reported by Kollman and co-workers (12, 13) have not provided ways for consistent evaluation of activation entropies or binding entropies.† At any rate, it is clear that calculations of activation entropies in enzyme active sites and in the corresponding reference solution reactions are essential for a quantitative understanding of entropic contributions to enzyme catalysis (6).

The present work introduces an approach for evaluating the contribution from the motion of the reacting fragments to the activation entropies of enzymatic reactions and uses this approach in assessing the importance of entropic effects to enzyme catalysis.

## Defining the Problem

To evaluate the importance of a specific contribution to catalysis
it is essential to define the relevant thermodynamic cycle. This is
particularly important in considering entropy contributions, whose
definition and estimates involve in many cases incomplete thermodynamic
cycles. The starting point of the present study is the free energy
diagram of Fig. 1. This diagram compares
the activation free energies
Δ*g*_{p}^{‡} and
Δ*g*_{w}^{‡} for a given reaction in
a protein (p) and in water (w) and also compares the corresponding
activation free energies (Δ*g*^{‡})′ when the
reacting fragments are frozen (see below). As shown in Fig. 1,
Δ*g*_{p}^{‡} is given by
Δ*G*_{bind} +
Δ*g*_{cat}^{‡}, and thus reflects the
binding energy Δ*G*_{bind}, whose nature is quite
clear and is not involved in the catalytic puzzle (see also ref. 7).
The main open question is the relationship between
Δ*g*_{cat}^{‡} (which corresponds to
*k*_{cat}) and the activation barrier of the reaction
in water, Δ*g*_{w}^{‡}. It is also
interesting, although not essential, to compare
Δ*g*_{cat}^{‡} to the barrier
Δ*g*_{cage}^{‡} obtained by
correcting Δ*g*_{w}^{‡} for the free
energy associated with bringing the reactants to a solvent cage, where
each of the (*n* − 1) reactants (from an overall
*n* reactants) is at a relatively small distance from the
central reactant (the *n*th reactant). At this distance the
interaction with the central ligand is small and each reactant moves
almost freely. In a case of small reactant molecules we can define the
cage volume by requiring that the central reactant will have a standard
1 M concentration and the other reactants will have a concentration
that is less than or equal to 55 M. In other words, taking into account
the free energy contribution associated with confining a molecule that
occupies a molar volume *v*_{0} (1,660
Å^{3} at 300 K) (6) to a volume *v*_{cage}
we may write
1
where Δ*g*_{w}^{‡} is the
activation barrier in solution at a standard state concentration, and
*v*_{cage} is the volume of the solvent cage, which
is taken to be larger than 1,660 Å^{3}/55. Here
*n*_{r} is the number of the ligands that become
bound to the substrate at the transition state. This number is equal to
(*n* − 1) in a fully concerted reaction, whereas it
reduces to 2 in a stepwise reaction. The seemingly arbitrary definition
of Δ*g*_{cage}^{‡} (which will not
change our final evaluation of
Δ*g*_{w}^{‡}) reflects two
considerations. First, early workers (*e.g.*, refs. 1 and 2)
implied or stated correctly that the free energy price associated with
bringing the reacting fragments to a distance of a weak interaction is
given by simple concentration considerations and is given approximately
by −*RT*ln 55. Thus the real puzzle was (and is) why
Δ*g*_{cat}^{‡} is much smaller than
the barrier obtained by considering the small volume correction to
Δ*g*_{w}^{‡}. And second, comparing
Δ*g*_{cat}^{‡} and
Δ*g*_{cage}^{‡} provides conceptually
and computationally closely related quantities, the activation barrier
in the enzyme active site and the activation barrier in a solvent cage,
where the reactants are in close proximity. It should also be stated
that our cage system is very different than the system used by Jencks
and others, where the reactants in solution are frozen in a space
similar to that available in the enzyme active site. This traditional
definition makes it rather difficult to define a simple thermodynamics
cycle for the comparison of
Δ*g*_{cat}^{‡} and
Δ*g*_{w}^{‡}.

Now, the major question one has to address is related to the entropy
contribution associated with the restriction of the motions of the
reacting fragments by the enzyme active site, relative to the
corresponding contribution in the solvent cage. This effect does not
include the contribution from the change in the entropy of the
environment (*e.g.*, solvent), since the early proposals were
defined clearly in terms of the substrate entropy rather than the
entropy of the surroundings. Thus, we are not addressing here the
change in the so called cratic entropy (11) that includes the overall
change in entropy upon binding, but rather the changes in the
contribution, Δ*S*′, from the configurational space of the
reacting fragments. The factors involved in the catalytic effect of
Δ*S*′ are illustrated in Fig. 1. The figure compares the
free energy profile Δ*g*_{w} and the corresponding
(Δ*g*_{w})′, obtained by freezing the
degrees of freedom of the substrate that are perpendicular to the
reaction coordinate. Similarly, the figure compares
Δ*g*_{p} and
(Δ*g*_{p})′. As clarified above, we are
interested in the contribution of Δ*S*′ to
(Δ*g*_{cat}^{‡} −
Δ*g*_{w}^{‡}) or to
(Δ*g*_{cat}^{‡} −
Δ*g*_{cage}^{‡}). Using Fig. 1, we obtain:
2
where RS and TS, respectively, designate the corresponding
contributions from the reactants state and transition state in the
protein or in the solvent cage. Of course we can also obtain easily
(ΔΔ*S*_{w → cat}^{‡})′.

The first point to note from Eq. 2 is that one must
consider the entropy contributions from the RS and TS both in the
enzyme and in the solvent cage. To the best of our knowledge all
previous estimates, with the exception of ref. 6, have not considered
both the RS and TS and have not defined the catalytic effect in a
rigorous way. Probably the most systematic attempt was made by Jencks
(page 720 of ref. 2), who defined the problem by considering the
hypothetical case with identical Δ*H*^{‡} in the
enzyme and in water. He assumed that
Δ*S*_{cat}^{‡} is zero and estimated
Δ*S*_{w}^{‡} by considering the
process of bringing the reactants in water to the same frozen position
that they have in the enzyme. Furthermore, the TS in water was also
assumed to be frozen in the same way. Basically, it was assumed
implicitly that *S*_{w}^{TS} =
*S*_{p}^{TS} ≃ 0, and that the activation
entropy in water can be approximated by the binding entropy of the
protein. In other words, Jencks as well as most other workers implied
that we can freeze the motion of the reactants in solution in the same
restricted space available for these motions in the protein and that
the free energy that should be invested in this configurational
restriction is the entropic contribution to the difference between
Δ*g*_{w}^{‡} and
Δ*g*_{cat}^{‡}. This (apparently
incorrect assumption) can be formulated by the more accurate terms of
Fig. 1 as:
3
where *S*_{cage}^{RS} is the hypothetical
frozen RS of the solution reaction. Here we subtract
−*T*(*S*_{EL}^{RS})′ from
−*T*(*S*_{cage}^{RS})′, because the
substrate is not completely frozen in the protein. However, as seen
from Fig. 1, this is a problematic assumption in large part because of
the fact that *S*′ is not zero at the TS of the solution
reaction. Or in other words because of the fact that many of the
motions that are free in the RS in the solvent cage are also free in
the TS in this cage.

To further clarify this point we can consider the hypothetical case
when all the motions are frozen in the enzyme in both the RS and the TS
(so that (Δ*S*_{cat}^{‡})′ = 0) and
all the motions are free in solution in the RS and in the TS (so that
(Δ*S*_{cage}^{‡})′ = 0). In this case
we have (ΔΔ*S*_{cage → cat}^{‡})′ =
0 and ΔΔ*S*_{w → cat}^{‡} ≃
8 entropy units. On the other hand, if we only consider the RS in
water and in the protein site we may obtain a large
(ΔΔ*S*_{cage → cat}^{‡})′.

Finally, it might be useful to comment on the inherent assumption of
Page and Jencks (1) that the formation of a bimolecular complex
involves the loss of three translational and three rotational degrees
of freedom. This model was based on considering two spherical
fragments. However, in actual formation of the cage complex we still
retain between one to three of these degrees of freedom. For example,
in the attack of a CH_{3}O^{−} group on an amide we
do not lose the rotation around the C–O axis, and two of the
translational motions of the CH_{3} group are almost free (see
Fig. 2).

The above discussion should not be viewed as a list of problems with previous estimates but as a reminder that the problem of evaluating entropic contributions to enzyme catalysis has not been solved. It seems to us that this challenging problem cannot be resolved without some form of computer simulation approach.

## Methods

As clarified in the previous section, our task is to evaluate the
entropic contribution of the reacting fragments to
ΔΔ*g*_{w → cat}^{‡}. To do so
we developed a restrain–release approach (RRA) which is related in
some aspects to the approach developed by Hermans and co-workers (11)
for studies of binding entropies. Our approach has been described in
detail in ref. 14, where it was used for studies of the activation
entropy of amide hydrolysis in water. Here we discuss only the main
points of this approach, which is based in the thermodynamic cycle of
Fig. 3. This cycle considers the
activation free energy, Δ*g*^{‡}, for the given
reaction in the given system (the enzyme or solvent cage) in two
limiting conditions. In the first case (the upper part of the cycle)
the reacting system is transformed from the RS to the TS along a
(unspecified) reaction coordinate while a restraint is used to minimize
the available configuration space in the direction perpendicular to
this reaction coordinate. In this case the activation free energy,
Δ*g*_{1}^{‡}, does not involve the entropic
contributions of the solute, (Δ*S*_{1}^{‡})′ =
0, since the corresponding motions are frozen. In the second case
(the lower part of the cycle) the reacting fragments are free to move
so that the corresponding activation barrier,
Δ*g*_{2}^{‡}, includes the entropic
contributions of the solute. Thus, the difference between the two
Δ*g*^{‡} values gives the desired
−*T*(Δ*S*^{‡})′. The possible enthalpic
contribution can be minimized with the proper selection of the initial
conditions for the simulation (see below). Using the thermodynamic
cycle, we can obtain (Δ*g*_{2}^{‡} −
Δ*g*_{1}^{‡}) from
Δ*G*′_{RS} and
Δ*G*′_{TS}. This can be done by imposing a
strong position restraint in both states I′ and II′ and evaluating the
corresponding free energies
Δ*G*′_{RS} =
Δ*G*′_{I′ → I} and
Δ*G*′_{TS} =
Δ*G*′_{II′ → II} associated with the release
of these constraints (see below). In this way we can write
4
The enthalpic contribution to Δ*G*′ will be discussed
below. Now, the practical evaluation of the Δ*G*′ values
involves the introduction of restrain potentials of the form
5
where *i* runs over the substrate coordinates and
**R̄**^{N} are reference coordinates that
define the minimum of the restrain potential (see below) at the given
state (*N* = *I*, or *N* = II for the RS and
TS, respectively). The reference coordinates
**R̄**^{N} are evaluated by running
molecular dynamics (MD) relaxation runs on the RS and the TS with
*K* = 0 (see also below). Different conditions for the MD
runs can generate different values of
**R̄**^{N}, and the selection of the optimal
coordinates as well as the implications of this selection will be
discussed below.

The constraint release free energies (Δ*G*′) are evaluated
by a free energy perturbation (FEP) approach, where we use a mapping
potential of the form
6
where λ_{m} is changed from 0 to 1
in *n* increments and *E* designates the
unconstrained potential surface of the system.
*R*_{c}^{F} is the distance between an
atom of the central fragment *F* = 1 and an atom of the
*F*th fragment. The *K*_{cage} term is
needed to prevent divergence when *K*_{2} → 0. The
value of *R*_{c}^{F} is chosen in a way
that the *F*th fragment can move freely around the central
ligand while still being close to this ligand.
*K*_{cage} is chosen so that
*v*_{cage} will correspond to 55 M concentration.

Now we can write
7
where
(Δ*S*^{‡})′_{0} designates the
entropic contribution in the 1 M standard state and the last term in
Eq. 7 gives the free energy associated with the change of
the cage volume from *v*_{cage} to
*v*_{0}.

The results of the thermodynamic cycle of Fig. 3 depend in principle on
the chosen **R̄** and may involve, in addition to
Δ*S*′, contributions from the enthalpy of the reacting
fragments and from their solvation by the surrounding environment. To
extract Δ*S*′ from the total Δ*G*′ we have
to formulate the cycle of Fig. 3 in a more rigorous way. This is done
by the approach presented in the appendix of ref. 14, which considers
the free energy Δ*G*′(**R̄**) of the system as
a function of the restraint coordinates of the solute. Using the
quadratic expansion of this potential of mean force (PMF) and
evaluating the corresponding contributions has established that
Δ*G*′(**R̄**_{0}) (where
**R̄** = **R̄**_{0} is the
**R̄** that minimizes the PMF) gives the desired
Δ*S*′ without additional enthalpic contributions. Although
the minimization of this PMF is impractical at present, we can estimate
the exact result for Δ*S*′ by performing several random
simulations with different **R̄**^{N} and
selecting the one that gives the minimum value of
|Δ*G*′|. In other words, we can use the lower bound (14)
8
and estimate the value of
|−*T*Δ*S*′_{N}(**R̄**_{0}^{N})|
by finding the smallest
|Δ*G*′_{N}(**R̄**^{N})|.
With
Δ*S*′_{N}(**R̄**_{0}^{N})
we can obtain (Δ*S*^{‡})′ by using (see appendix
of ref. 14)
9
Eqs. 8 and 9 and the treatment in ref.
14 outline the formal requirements for rigorous evaluation of
(Δ*S*^{‡})′. However, because the evaluation of
Δ*S*′ with random **R̄** is very expensive we
confine the present study to the evaluation of the values of
Δ*S*′_{N} at several values of
**R̄**, obtained by running MD simulations on the
corresponding potential surface with zero restrain potential. We then
use the **R̄** values that give the lowest
|Δ*G*′| in evaluating Δ*S*′.

To evaluate activation entropies it is essential to perform long MD or
Monte Carlo (MC) samplings of the corresponding potential surfaces.
Such potential surfaces should provide a reliable representation of the
relevant reactions and also allow for sufficient sampling at a
reasonable computer time. At present, it is impossible to satisfy this
requirement with a high-level *ab initio* potential surface
because this would require an enormous amount of computer time.
Approaches that obtain the solute potential surface by fitting it to
the corresponding *ab initio* gas phase surface and then
consider the interaction of the gas phase charges with the solvent can
provide reasonable estimates of the solvent contribution to the
activation entropies in reactions that involve relatively small charge
separation. However, it is hard to use such approaches for evaluation
of the solute contribution to the activation entropy. This is
particularly true when the solute is constrained to move along the gas
phase reaction coordinate (e.g. ref. 15) and the substrate
fluctuations in the directions perpendicular to the reaction coordinate
are not taken into account. In our opinion the optimal strategy is to
use empirical valence bond (EVB) potential surfaces (6). These surfaces
can be fitted to reliable *ab initio* surfaces and to relevant
experimental information and then to provide a consistent and reliable
description of the solution reaction. This description includes a
consistent incorporation of the solvent in the solute Hamiltonian,
consistent description of the motion of the reacting fragments, and an
analytical representation of the given surface that allows one to
perform very extensive MD simulations. The EVB method was described
extensively elsewhere (6) and has been used by many other groups (see
ref. 14 and references therein). Thus we will mention here only several
crucial points.

The EVB approach describes the reacting system by mixing several
diabatic states that represent the reactants, products, and crucial
intermediates. The energy of each diabatic state *i*
(ɛ_{i}) is described by a force-field like potential
function and the off-diagonal terms (*H*_{ij}) are
described by simple analytical functions. The parameters in the
ɛ_{i} and *H*_{ij} values are
determined by using both experimental information and *ab
initio* calculations. To perform our RRA calculations we have to
confine the reactants to the specified region of the potential surface
(*e.g.*, RS and TS). Here we limit our discussion to the
common case where the system is represented by three diabatic states.
In the case of a stepwise mechanism it is reasonable to use (14)
10
where the λs are the values of the FEP mapping parameters that
bring the system to the TS region.

## Results and Discussion

The present study considered as a test case the catalytic
reaction of subtilisin. The potential surface of this reaction was
evaluated by the previously developed EVB potential surface (16). The
three resonance structures that roughly correspond to the three states
in the mechanism of action of subtilisin (reactants, intermediate, and
products) are described in figure 6 of
ref. 16. The parameters for this EVB potential energy surface are
similar to those used in ref. 16. The rate-limiting step of the
reaction is the attack of the O_{γ} of Ser-221 on the
carbonyl carbon of the substrate. Thus, the entropy calculations have
been done considering the RS and the TS, whose ground state energies
*E*_{g}^{RS} and
*E*_{g}^{TS} were obtained, respectively,
by using Eq. 10 and λ_{m}^{TS} = (0.0, 0.4, 0.6), as described above. Within this model the
system is divided into several regions (see ref. 17 for a detailed
discussion). The reaction region (region I) is surrounded by a
protein + solvent sphere (region II, with a radius
*r*_{a}) that is surrounded by external regions with
special polarization and position boundary conditions (17). The protein
atoms in these external regions are held by strong constraints. All the
calculations were done with the enzymix package (17).

The simulations involved the release of the position restrains in 5 FEP
stages, changing *K*_{j} from 100 to 30, from 30 to
3, from 3 to 0.3, from 0.3 to 0.03, and from 0.03 to 0.003, where all
the values of *K* are given in
kcal/(mol⋅Å^{2}). Each of these stages involved
11 mapping steps, and each of these steps followed an initial
relaxation of the given system. The total simulation time for these
five stages was 1,100 ps for both the water and protein runs.
*K*_{cage} was taken as 0.4 kcal/mol (which
corresponds to *v*_{cage} of 1,660/55
Å^{3}) and *R*_{c}^{F} were
taken as 3.0 and 3.1 for the O_{γ}–C distance and the
O_{γ}–N_{ɛ2} distance, respectively. To generate
the different **R̄**s needed for obtaining the optimal
Δ*G*′ we use different relaxation runs with different
constraints on region I and II. Overall, we generated nine different
**R̄**s for the RS and nine for the TS. Next we performed
RRA calculations for the different **R̄**s. Table
1 gives the results of the simulations
with *r*_{a} = 14 Å. Additionally,
we depict in Fig. 4 the values of the
Δ*G*′ in an increasing order. As is apparent from the
figure, we obtain a reasonable convergence in the selection of minimal
values of |Δ*G*′|. The differences between
Δ*G*′_{TS} and
Δ*G*′_{RS} (plus the volume correction of
Eq. 7) yield the final values of
−*T*(Δ*S*^{‡})′_{0} that are given
in the last column of Table 1. As seen from the table, the entropic
contribution of the reacting fragments is not so different in the
enzyme and in the solvent cage. More specifically, the simulations give
a difference of (2.5 − (2.6 + 2.4)) = −2.5
for −*T*((Δ*S*_{cat}^{‡})′ −
(Δ*S*_{w}^{‡})′). This estimate already
includes the effect of *K*_{cage} (≈−2.4
kcal/mol). Note that in this volume correction
*n*_{r} is taken as 1, because the solution reaction
has been found to involve a stepwise mechanism (18). The present
estimate of −*T*(ΔΔ*S*^{‡})′ involves a few
kcal/mol error and is likely to change with a more complete
convergence (see footnote §). The main
point is, however, that
|*T*(ΔΔ*S*^{‡})′_{0}| is much
smaller than previously thought. The reason for this exciting finding
is rationalized in Fig. 5, which shows
snapshots of the last FEP frame of the simulation where *K*
changes from 0.003 to 0.0003 kcal/(mol·Å^{2}). As
seen in the figure the degree of movement for each system (cage and
protein) is similar in RS and in TS. If this conclusion will hold in
other systems, which is likely (see below), then entropic effects do
not contribute in a major way to the reduction of
Δ*g*_{cat}^{‡}.

## Concluding Remarks

This work examined the importance of the entropic contribution of
the reacting fragments in enzyme catalysis. This was done by developing
a computer simulation method capable of evaluating
(Δ*S*^{‡})′ in enzyme and in solution, and using
this method in calculations of (ΔΔ*S*^{‡})′ for
the catalytic reaction of subtilisin. It was found that
(ΔΔ*S*^{‡})′ is much smaller than commonly
assumed and should not be approximated by
*T*(Δ*S*_{bind})′.

Our finding of a rather small (ΔΔ*S*^{‡})′ is
consistent with a study (19) of nucleophilic displacements on phenyl
esters in water. It was shown that dividing
−*T*Δ*S*_{w}^{‡} by the kinetic order
of the reaction (first, second, and third) gives 4 to 5
kcal/mol. This finding was interpreted to indicate that bringing
each species to the TS does not involve more than 5 kcal/mol
entropic contribution (19, 20). Since the solvent contributions (14) to
both −*T*Δ*S*_{w}^{‡} and
−*T*(Δ*S*_{cat}^{‡})′ are positive,
|−*T*(ΔΔ*S*^{‡})′| should be quite small.

Obviously one may wonder what is the origin for the difference between
the present results and previous estimates. One of the reasons that
were alluded to in the Introduction is the fact that many of the
motions that are free at the RS are also free at the TS. Another
important factor is associated with the fact that
(Δ*S*_{cat}^{‡})′ is not zero in the
protein. Finally, as was noted in the Introduction (see Fig. 2), the
early estimate of Page and Jencks (1) provides an overestimate of the
entropy loss upon binding, where some rotational and translational
motions remain free even in the solvent cage.

The importance of entropic effects in enzyme catalysis has been
frequently deduced from the trend in the rates of intramolecular
cyclization reaction, where the formation of smaller rings is
associated with larger rate constants. Recent studies (9) have
suggested, however, that this trend may be due to RS destabilization
enthalpic effects. This does not mean that such enthalpic effects play
a significant role in enzyme catalysis. In fact, it is not clear that
intramolecular cyclization reactions are directly relevant to most
enzymatic reactions (6). At any rate, the present approach may provide
a powerful way of exploring RS enthalpic contributions. That is, if
instead of using the optimal **R̄** for both the enzyme and
the solution reaction we use the **R̄** of the enzyme
reaction for the water reaction we should be able to evaluate the
enthalpic contribution associated with the distortion of the structure
of the substrate by the enzyme.

The present work is an attempt to obtain a quantitative estimate of the
entropic contribution of the reacting fragments to enzyme catalysis.
The finding that these contributions are much smaller than previous
estimates as well as simulations of many enzymatic reactions is
consistent with the view that electrostatic effects are the major
factor in enzyme catalysis. As argued repeatedly before
(*e.g.*, refs. 6 and 7), the electrostatic effects are due to
the preorganized polar environment of the enzyme active site and to the
relatively small reorganization free energy of this environment. The
ability of the enzyme to provide smaller reorganization free energy
than water does involves, of course, some entropic effects. These
effects are *not*, however, the entropic effects introduced in
all early proposals of entropic contributions to enzyme catalysis. That
is, previous proposals were based on the assumption that the enzyme
decreases the entropy of the reacting fragments. On the other hand, our
electrostatic proposal involves the reduction of the reorganization
free energy (which includes some entropic contributions) by folding the
enzyme to its catalytic configuration. This contribution of the
enzyme–enzyme rather than the enzyme–substrate interaction is the
most important factor in enzyme catalysis.

## Acknowledgments

We thank Dr. Chienyu Jen and Dr. Jan Florián for insightful discussions. J.V. acknowledges European Molecular Biology Organization Fellowship ALTF 509-1998. This work was supported by National Institutes of Health Grant GM24492.

## Footnotes

↵* To whom reprint requests should be addressed at: E-mail: warshel{at}usc.edu.

↵† Ref. 12 used molecular dynamics (MD) runs to estimate the force constants necessary for keeping the system in the reactant conformation with a rather arbitrary standard deviation of 0.2 Å and 20° for distances and angular dependence, respectively. These force constants were used for deriving the entropy contributions by using equations derived by Hermans and Wang (11). However, this approach does not amount to the evaluation of the ground state entropy but rather to assuming the corresponding effective volume. Ref. 13 used gas phase entropy calculations to estimate the entropy cost of bringing the reactants in water to the corresponding configuration in the ES complex. However, the gas phase entropy may be drastically modified in solution. More importantly, the assumption (2) that the estimated entropy is equal to −ΔΔ

*S*_{w → cat}^{‡}is not justified (see text).↵§ To examine the convergence we performed additional calculations with

*r*^{a}of 13 Å and 16 Å. For*r*_{a}= 13 Å we obtained Δ*G*′_{TS}(in kcal/mol) −86 and −85 for the water case and −89 and −87 in the protein case. These results are similar to those obtained with*r*_{a}= 15 Å. On the other hand, in the 16-Å case we obtained for −*T*(ΔΔ*S*^{‡})′ more positive value (by about 9 kcal/mol) than for the smaller radius. This is not a major problem, because it represents a well-understood convergence problem and because it still leads to the same conclusion as obtained from the small*r*_{a}. That is, while the natural tendency is to assume that the results obtained with larger*r*_{a}are more reliable, we are limited here by enormous convergence problems that are reduced significantly for smaller*r*_{a}. Moreover, the incomplete convergence does not present a serious problem, because our main point is that | (ΔΔ*g*_{w → cat}^{‡})′| is small. By reducing the motions in the protein we obtain a lower limit for −*T*(Δ*S*_{cage}^{‡})′ and an upper limit for the entropic contribution to catalysis. Thus, we can focus on the results obtained with*r*_{a}≤ 14 Å. We also examined the convergence of the simulations (rather than the search of the optimal**R̄**). This was done by examining the change in the results in the last 550 ps. It was found that the results become more negative for both the RS and TS. However, (Δ*S*_{w}^{‡})′ and (Δ*S*_{cat}^{‡}) changed by only 0.5 and 0.2 kcal/mol, respectively. In general, we expect that with more complete convergence Δ*S*_{w}^{‡}will become negative by a few kcal/mol.

## Abbreviations

- EVB,
- empirical valence bond;
- MD,
- molecular dynamics;
- FEP,
- free energy perturbation;
- RS,
- reactant state;
- TS,
- transition state;
- RRA,
- restrain–release approach

- Received April 3, 2000.
- Accepted August 21, 2000.

- Copyright © 2000, The National Academy of Sciences

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- Štrajbl M,
- Sham Y Y,
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- Warshel A

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- Chandrasekhar J,
- Jorgensen W

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- Bruice T C

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