Multidimensional tunneling, recrossing, and the transmission coefficient for enzymatic reactions.

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Multidimensional Tunneling, Recrossing, and the
Transmission Coefficient for Enzymatic Reactions
Jingzhi Pu, Jiali Gao, and Donald G. Truhlar
Chem. Rev., 2006, 106 (8), 3140-3169 • DOI: 10.1021/cr050308e
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Multidimensional Tunneling, Recrossing, and the Transmission Coefficient for
Enzymatic Reactions
Jingzhi Pu, Jiali Gao,* and Donald G. Truhlar*
Department of Chemistry and Supercomputer Institute, University of Minnesota, 207 Pleasant Street S.E., Minneapolis, Minnesota 55455-0431
Received January 20, 2006
Contents
1. Introduction
2. Previous Reviews
3. Experimental Manifestations of Hydrogen
Tunneling in Enzymes
3.1. Kinetic Isotope Effects and Swain−Schaad
Exponents
3.2. Isotope Effects on Arrhenius Pre-exponential
Factors
3.3. Temperature Dependence of KIEs
3.4. Survey of Tunneling Systems
3.4.1. Soybean Lipoxygenase
3.4.2. Methylamine Dehydrogenase and Related
Enzymes
3.4.3. Aromatic Amine Dehydrogenase
3.4.4. Methylmalonyl-CoA Mutase
3.4.5. Dihydrofolate Reductase
3.4.6. Other Systems
4. Models
5. Quantitative Computational Methods
5.1. Ensemble-Averaged Variational Transition
State Theory with Multidimensional Tunneling
(EA-VTST/MT)
5.2. Mixed Quantum/Classical Molecular
Dynamics
5.3. Quantized Classical Path Method
5.4. Methods Based on a Single Reaction Path
6. Recrossing
6.1. EA-VTST Recrossing Transmission
Coefficients
6.2. Reactive Flux Method
6.3. Model Theories
6.4. Survey
7. Applications
7.1. Yeast Enolase
7.2. Triosephosphate Isomerase
7.3. Methylamine Dehydrogenase
7.4. Alcohol Dehydrogenase
7.5. Thermophilic Alcohol Dehydrogenase
7.6. Haloalkane Dehalogenase
7.7. Dihydrofolate Reductase from E. Coli
7.8. Hyperthermophilic DHFR from Thermotoga
Maritima
7.9. Xylose Isomerase
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7.10. Short-Chain Acyl-CoA Dehydrogenase
7.11. Catechol O-Methyltransferase
7.12. Glyoxalase I
8. Further Discussion of Ensembles
9. Concluding Remarks
10. Acknowledgments
11. Appendix. Glossary
12. References
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1. Introduction
Although many aspects of enzyme catalysis have been
constructively analyzed,1-52there are still many aspects that
are imperfectly understood. Of particular interest in this
regard are the roles that nonstatistical dynamical effects play,
as manifested in quantum mechanical nuclear tunnel-
ing,16,25,27,30,31,39,42,44,49,53-163
ing,24,27,46,97,104,113,115,121,127,131,133,137,138,144,153,163-173and non-
equilibrium effects.24,27,46The language of enzyme dynamical
effects has also been used in various contexts to refer to
protein fluctuations (protein dynamics, protein motion,
protein vibrations),13,33,37,38,49,99,109,174-188conformational
changes,46,49,189-197motionsofindividualvibrationalmodes,119,125
ensemble-averaged collective geometry changes along the
reaction coordinate,179,180,198-200and many more aspects of
enzyme kinetics. However, these kinds of effects can often
be included in rate calculations by a proper treatment of the
free energy of activation, which is a statistical quantity. The
separation of effects into statistical and dynamical is not
unambiguous since, from the one point of view, the statistical
free energy of activation is derived from the dynamical flux
through a hypersurface in phase space and, from the other
point of view, the dynamical effects of tunneling and
recrossing must be statistically averaged. We prefer a division
into “quasithermodynamic” and “nonsubstantial” effects, as
will be explained below. This too is not unique, but it
provides a clear framework for discussion and understanding
in terms of generalized transition state theory, which will
simply be called transition state theory (TST) in the rest of
this article. (The appendix contains a glossary of acronyms
and terms with a special usage.)
As a fundamental approach to describing the reaction rate
in enzyme-catalyzed reactions, as well as reactions in the
gas phase and the solution phase, transition state theory1,201-203
(TST) provides an important language for interpreting
chemically activated processes. In fact, the very existence
of a transmission coefficient is tied to TST since the
transmission coefficient is defined as the factor that accounts
for all effects not included in the TST rate constant. The
dynamicalrecross-
* To whom correspondence should be addressed. E-mail addresses:
gao@chem.umn.edu and truhlar@umn.edu.
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Chem. Rev. 2006, 106, 3140−3169
10.1021/cr050308e CCC: $59.00© 2006 American Chemical Society
Published on Web 07/26/2006

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effects included in the TST rate constant are called quasi-
thermodynamic, because the TST treatment of the transition
state, which is not a real molecule, may be expressed in a
language analogous to that used by thermodynamics for
treating real substances,204-207and the effects included in
the transmission coefficient may therefore be labeled as
nonsubstantial. [We use the word “substance” in the tradi-
tional chemical sense of “The Equilibrium of the Heteroge-
neous Substances”204or “Application of the First Law to a
Pure Homogeneous Substance”.206] Both the quasithermo-
dynamic parameters and the transmission coefficient derive
from dynamics, and both quantities must be treated statisti-
cally.205The ambiguity in partitioning the factors affecting
chemical reaction rates into quasithermodynamic and non-
substrantial, and hence the ambiguity in defining the
transmission coefficient, derives from the fact that there is
more than one way to apply transition state theory because
there is more than one way to define the transition state.
When the calculation is well defined though, the concept of
a transmission coefficient is very useful. A major objective
of the present review is to elaborate on this issue.
To discuss the application of TST to enzyme reactions,
we start from the well-known Michaelis-Menten model,22
in which enzymatic reactions are described by the scheme
where E, S, and P denote the enzyme, substrate, and product,
respectively, and ES is a Michaelis complex. In Michaelis-
Menton kinetics, one associates the macroscopic rate constant
kcatwith the final step of eq 1, where this step represents all
the microscopic rate constants from ES to the release of
product. A more explicit and widely used generalization of
eq 1 is
We focus on the catalytic step that converts ES to EP, which
is associated with a microscopic rate constant to be denoted
as k. The actual rate-determining step of the enzymatic
reaction may occur at any of the three arrows in eq 2 (or the
mechanism may be more complicated, such as involving a
ternary complex with a coenzyme or involving one or more
additional intermediates), but we assume that the mechanism
has been sorted out (e.g., by analyzing intrinsic kinetic
isotope effects (KIEs)158,208-210or other specially designed
experiments) and that kcatis known. In particular, we will
be concerned here especially with reactions where kcathas
also been simulated.
Jingzhi Pu was born in Beijing, China, in 1976. He received a B.A. in
Chemistry from Peking University in 1999 and a Ph.D. from the University
of Minnesota in 2004 under the direction of Donald G. Truhlar. After
finishing a short postdoc with Jiali Gao, he moved to Boston. He is currently
a postdoctoral associate at Harvard with Martin Karplus. His recent
research interests include quantum effects in enzymatic reactions and
protein dynamics. He has been married to Yan He since 2002.
Jiali Gao was born in China in 1962. He received a B.S. in Chemistry
from Beijing University in 1982, which marked the first collegial graduation
after a decade of political turmoil, called “the Cultural Revolution”, when
millions of high school graduates were sent to the remote countryside for
re-education. Subsequently, he came to Purdue University, under a
Graduate Program arranged by William von E. Doering, and obtained a
Ph.D. in 1987 under the guidance of William L. Jorgensen. After
postdoctoral research with Martin Karplus at Harvard, he joined the faculty
of the State University of New York at Buffalo in 1990 and was promoted
to full Professor in 1997. Since 2000, he has been on the faculty of the
University of Minnesota, where he is Professor of Chemistry. His research
interests include computational studies of chemical and biological problems.
He is a recipient of the 2000 Dirac Medal from the World Association for
Theoretically Oriented Chemists.
Donald G. Truhlar was born in Chicago in 1944. He received a B.A. in
Chemistry from St. Mary’s College of Minnesota in 1965 and a Ph.D.
from Caltech in 1970 under the direction of Aron Kuppermannn. He has
been on the faculty of the University of Minnesota since 1969, where he
is currently Lloyd H. Reyerson Professor of Chemistry, Chemical Physics,
Scientific Computation, and Nanoparticle Science and Engineering. His
research interests are theoretical and computational chemical dynamics
and molecular structure and energetics. He is the author of over 800
scientific publications, and he has received several awards for his research,
including a Sloan Fellowship, Fellowships in the American Physical Society
and the American Association for the Advancement of Science, an NSF
Creativity Award, the ACS Award for Computers in Chemical and
Pharmaceutical Research, the Minnesota Award, the National Academy
of Sciences Award for Scientific Reviewing, the ACS Peter Debye Award
for Physical Chemistry, and the Schro ¨dinger Medal of The World
Association of Theoretical and Computational Chemists. He has been
married to Jane Truhlar since 1965, and he has two children, Sara
Elizabeth Truhlar and Stephanie Marie Eaton Truhlar.
E + S f ES f E + P (1)
E + S f ES f EP f E + P (2)
Tunneling and Recrossing for Enzymatic ReactionsChemical Reviews, 2006, Vol. 106, No. 8
3141

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The fundamental assumption of transition state theory is
that there exists a hypersurface in coordinate space or phase
space (the space of coordinates and momenta) that divides
the reactant region from the product region. This hypersur-
face is usually called the transition state dividing surface or
simply the transition state (TS). Typically, the transition state
is chosen as a (3N - 1)-dimensional dividing surface in the
coordinate space for a system that contains N atoms. (For
gas-phase reactions, which are not of interest here, one
usually separates out 3 degrees of overall translation and 3
degrees of overall orientation, and 3N - 1 becomes 3N -
7.) As in thermodynamics, one key issue in using TST is
specifying the “system”. In general, the system can exchange
energy with its surroundings, but it is usually convenient
for discussing enzyme kinetics to refer to “closed sys-
tems”,206,207by which we mean that all atoms are definitely
assigned to either the system or its surroundings. The system
must contain at least a part of the substrate, and it may
contain all or part of the coenzyme(s), the apoenzyme, and
the solvent. The partition into a system and its surroundings
is very familiar in the theory of molecular solutions where
one often uses the language of “solute” and “solvent”. For
enzyme kinetics, it is usually more appropriate to refer to a
“system/bath” or “system/environment” separation. We also
sometimes use the language “primary zone/secondary zone”.
The flexibility in how the system is defined is the first
example of the fact mentioned above that there is more than
one way to apply TST to a given problem.
TST also makes the assumption of local equilibrium,
namely that the internal states of the reactant and the
transition state are in a Boltzmann distribution. This is also
called the quasiequilibrium assumption, and it should be well
satisfied for most reactions in solution and most enzyme
reactions.46As a consequence of local equilibrium, the TST
approximation for kcatcan be written
where γ(T) is the transmission coefficient, kBis Boltzmann’s
constant, h is Planck’s constant, T is temperature, R is the
gas constant, and ∆GT
free energy of activation for the reaction of interest at the
given temperature T and is given by
qis the quasithermodynamic molar
In eq 4, GT
a quasithermodynamic quantity used to describe the free
energy of the transition state, which is an imaginary species
in that one degree of freedom corresponding to the reaction
coordinate is missing; thus GT
ermodynamic quantities to distinguish them from the quanti-
ties such as GT
When comparing eq 3 to experiment, it is important to
compare it to the phenomenological expression often used
to interpret experimental data, namely
exp[-∆GT
Ris the molar free energy of reactants, and GT
qis
qand ∆GT
qare called quasith-
Rthat correspond to true physical substances.
where ∆GT
energy of activation. Comparing eq 5 to eq 3 yields
actwill be called the phenomenological free
where ∆Gextrais the extrathermodynamic contribution to the
free energy of activation and is given by
One often sees discussions of “the validity of TST”. Since
γ may be defined211to make eq 3 exact (assuming that a
phenomenological rate constant even exists, which is another
matter24,212-214), TST is always valid. In general though, the
meaning of such discussions of the validity of TST is “How
accurate is TST with γ ) 1?” or “How accurate is TST with
some particular model for γ?” One source of ambiguity is
that the model used for γ is not always specified, although
it should be.
Precise discussions of the validity of TST are ultimately
rooted in classical mechanics because TST can be derived
rigorously only in a classical mechanical world. For eq 3 to
be valid in classical mechanics with γ(T) ) 1, the transition
state dividing surface must provide a dynamical bottleneck
for the flux from the reactant to the product region of phase
space; that is, once trajectories originating on the reactant
side of the transition state dividing surface cross it in the
forward direction (i.e., toward the product side), they do not
return to the reactant side via recrossing through the dividing
surface. Furthermore, all such forward crossing trajectories
must have originated on the reactant side. Under this
assumption, the one-way forward flux of the reactive
trajectories is equivalent to the net flux through the transition
state dividing surface that corresponds to the phenomeno-
logical reaction rate constants, and TST is exact, at least in
classical mechanics.
There would be no recrossing if the reaction coordinate
were separable. When the nonrecrossing assumption is not
satisfied, a transmission coefficient may be used to account
for its breakdown. Equation 3 then provides the basis for
partitioning the phenomenological reaction rate into a
“substantial” part and a “nonsubstantial” part,215where the
former involves the use of equilibrium thermodynamic
variables such as free energies [e.g., the exponential part in
eq 3] for describing the transition state as a substance and
the latter involves the transmission coefficient γ. (The
“substantial/nonsubstantial” language is based on the de-
scription “substances” by thermodynamics, and “nonsub-
stantial” does not mean “unimportant” in this context.)
When hydrogen motion is involved, nuclear quantum
effects, in particular quantized vibrations and tunneling,
become important. In classical mechanics (and hence in most
molecular dynamics simulations that have been carried out
on proteins), vibrational energies can take on a continuous
distribution of values, and the averaged vibrational energy
per mode is often well approximated by the classical
harmonic-oscillator value, which is given in molar energy
units by RT. In quantum mechanics, though, vibrational
energies of bound states are limited to a discrete set of values;
this is called quantization. The lowest allowed value is called
the zero point energy. For a harmonic oscillator, the zero
point energy in molar energy units is1/2NAhcυ j, where NAis
Avogadro’s number, c is the speed of light, and υ j is the
vibrational frequency of the mode in wavenumbers (cm-1).
For a mode with υ j ) 3000 cm-1(a typical value for a C-H
stretch), the zero point energy is 4.3 kcal/mol, whereas RT
at 300 K is 0.6 kcal/mol. Thus, the quantized energy
kTST) γ(T)kBT
h
exp[-∆GT
q
RT]
(3)
∆GT
q) GT
q- GT
R
(4)
k )kBT
h
act
RT]
(5)
∆GT
act) ∆GT
q+ ∆Gextra(T)(6)
∆Gextra) -RT ln γ(T) (7)
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requirements can be very important. Since the transition state
is a metastable state, strictly speaking, it does not have
quantized energy levels. However, to a good approximation,
one may assume that all the bound modes still have a
quantized energy requirement,216and this assumption has
been used since the early days of transition state theory.217
It is also well validated by more recent studies employing
accurate quantum dynamics.218,219In fact, most workers
usually discuss the quantized energy levels of the transition
state without even mentioning that such quantization is an
approximation. Since the transition state has an unbound
mode (the reaction coordinate, which corresponds to motion
with a barrier potential rather than a Hooke’s Law potential),
it has a finite lifetime (∼5-30 fs), and the quantized energy
levels are broadened. Thus, some systems pass through the
transition state with less than the quantized energy that would
be calculated if the transition state had an infinite lifetime;
this is tunneling.219Usually, though, tunneling is visualized
in a different way (one of the beauties of quantum mechanics,
sometimes dizzying to newcomers, is that there is more than
one correct way to understand nonclassical phenomena220-223);
in particular, one uses an effective barrier model. In this kind
of model, one identifies a tunneling coordinate (a nuclear-
motion coordinate that may be the same as the classical
reaction coordinate but need not be and, in multidimensional
tunneling models, usually is not). The effective potential
along this tunneling coordinate consists of the potential
energy surface (which, by the Born-Oppenheimer ap-
proximation, ultimately represents the quantized electronic
energy requirement plus nuclear repulsion) plus the energy
requirement of the quantized vibrations in the other nuclear
coordinates, computed as if they are not lifetime broadened.
(In multidimensional tunneling models the effective barriers
may also contain internal centrifugal terms due to the fact
that the tunneling coordinate is curvilinear. In one-dimen-
sional tunneling models, one neglects the variation of the
quantized vibrational energies as a function of progress along
the tunneling coordinate.) Now one has reduced the tunneling
problem to an effectiVe one-dimensional problem with an
effective potential and effective reduced mass for one-
dimensional motion, and tunneling shows up as the ability
of a quantum wave packet to pass a barrier even when its
average energy is less than the barrier top. Semiclassically
speaking (we always use “semiclassical” to refer to ap-
proximate ways to carry out quantum mechanical calculations
that are based on classical concepts,224-227such as the
Wentzel-Kramers-Brillouin (WKB) approximation;228-230
we never use “semiclassical” to mean neglect of tunneling,
which is a widespread usage in the isotope effect com-
munity209,231), tunneling in this picture is passage through a
barrier with negative kinetic energy and, hence, imaginary
momentum and imaginary action226(here we use “action”
in the sense in which it occurs in Hamilton’s principle in
classical mechanics).
The effect of quantizing vibrations is usually included in
∆GT
more than one way to include quantum effects in TST.
Independently of how individual effects are partitioned
between γ and ∆GT
tunneling are included in ∆GT
The goal of the present article is to provide an overview
of all these issues, with a special focus on tunneling and
recrossing effects in enzymatic reactions and with recent
theoretical developments highlighted. The organization of
qwith tunneling contributions included in γ, but there is
q, both quantized vibrational effects and
act.
the article is as follows. Section 2 summarizes the previous
reviews covering tunneling and recrossing in enzymes. In
section 3, we open our discussion by providing a historical
overview of the existing experimental evidence suggesting
the importance of quantum tunneling in enzymatic reactions.
Conceptual models that have been proposed and widely used
to interpret these experimental kinetics data in terms of
tunneling are discussed in section 4. Following that, sophis-
ticated quantitative models that are capable of revealing
detailed tunneling mechanisms at the atomic level are
explained and compared in section 5; section 6 considers
recrossing. Section 7 provides a survey of important enzyme
systems that have been studied with the most complete
theories and summarizes the tunneling and dynamical
recrossing effects in these systems. Section 8 gives the
concluding remarks.
This review is primarily concerned with tunneling and
recrossing. Tunneling is most important for reactions involv-
ing the transfer of a proton, hydride ion, hydrogen atom,
deuteron, deuterium atom, deuteride ion, triton, tritium atom,
or tritide ion. Rather than repeat the litany of charge states
and isotopes, we will often just say hydrogen or H nucleus
to refer to all nine of these cases. Similarly, when we say
kinetic isotope effect (KIE) without specifying the isotopes,
it means a deuterium KIE, that is, kH/kD, where k is a rate
constant.
This review does not consider electron transfer reactions
or pressure effects on reaction rates.
2. Previous Reviews
The investigation of quantum mechanical tunneling effects
in enzymes and the question of nonstatistical dynamical
effects in enzyme reactions have attracted increasing attention
during the past 15 years, marked by an intensive interplay
between experiment and theory. A number of reviews of
enzyme kinetics are available, with emphasis on one or more
aspects.1,7,8,11-14,18,23-42,44,46,48,49,63,84,94,95,124,143,144,179,232-236In
this section, we provide a few remarks about the most
relevant previous reviews.
The application of dynamical simulation techniques to
enzyme reactions requires quantum mechanical treatment of
the potential energy surface (PES) because of the electronic
delocalization that accompanies chemical bond rearrange-
ment. Because of the large size of the protein-substrate-
cofactor complex and because of the importance of its
interaction with solvent, the field has been greatly advanced
by the development of new techniques for the efficient and
more accurate treatment of such PESs. An earlier review31
entitled Quantum Mechanical Methods for Enzyme Kinetics
overviewed practical methods for incorporating electronic
quantum mechanics into PESs for enzyme reactions as well
as methods for incorporating nuclear quantum effects that
affect enzyme dynamics, with a special emphasis on the
combined quantum mechanical and molecular mechanical237
(QM/MM) approach for PESs, approximate quantal methods
for tunneling dynamics, and quantized TST with semiclas-
sical nuclear dynamics. Two other reviews30,144focused more
specifically on ensemble-averaged variational transition state
theory with multidimensional tunneling (EA-VTST/MT)
including practical procedures for applying the method to
enzyme kinetics, along with summaries of applications of
the method to rate constants and KIEs, and a related review236
focuses on this kind of treatment for KIEs of both enzymatic
and nonenzymatic reactions. Another review46provided a
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