Geometry optimization and transition state search in enzymes: Different options in the microiterative method

Abstract

In the current article we present a systematic analysis of the different options in the so-called “microiterative method” used to locate minima and transition-state (TS) structures of big systems on quantum mechanics/molecular mechanics potential energy surfaces. The method splits the system in two parts: a core zone in which accurate second-order search is carried out, and an environment that is kept minimized with a cheap first-order algorithm. The different options studied here are: the alternating frequency between the environment minimization and the TS search in the core, the number of atoms included in each zone, and two alternative ways to reduce the computational cost in the calculation of core–environment interactions. The tests have been done at two different steps of the enzymatic mechanism of mandelate racemase: a proton transfer and a carbon configuration inversion step. The two selected TS structures differ in the number of atoms involved in their associated transition vectors; the proton transfer TS is an example of a local motion, whereas the carbon configuration inversion TS corresponds to a more global movement of several groups and residues, including an important number of atoms. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

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Geometry Optimization and Transition
State Search in Enzymes: Different
Options in the Microiterative Method
XAVIER PRAT-RESINA,1JOSEP MARIA BOFILL,2
A`NGELS GONZA´LEZ-LAFONT,1JOSE´ M. LLUCH1
1Departament de Quı ´mica, Universitat Auto `noma de Barcelona, 08193 Bellaterra, Barcelona, Spain
2Departament de Quı ´mica Orga `nica and Centre Especial de Recerca en Quı ´mica Teo `rica, Universitat
de Barcelona, Martı ´ i Franque `s 1, 08028 Barcelona, Spain
Received 13 June 2003; revised 3 September 2003; accepted 15 December 2003
Published online 22 March 2004 in Wiley InterScience (www.interscience.wiley.com).
DOI 10.1002/qua.20072
ABSTRACT: In the current article we present a systematic analysis of the different
options in the so-called “microiterative method” used to locate minima and transition-
state (TS) structures of big systems on quantum mechanics/molecular mechanics
potential energy surfaces. The method splits the system in two parts: a core zone in
which accurate second-order search is carried out, and an environment that is kept
minimized with a cheap first-order algorithm. The different options studied here are:
the alternating frequency between the environment minimization and the TS search in
the core, the number of atoms included in each zone, and two alternative ways to
reduce the computational cost in the calculation of core–environment interactions. The
tests have been done at two different steps of the enzymatic mechanism of mandelate
racemase: a proton transfer and a carbon configuration inversion step. The two selected
TS structures differ in the number of atoms involved in their associated transition
vectors; the proton transfer TS is an example of a local motion, whereas the carbon
configuration inversion TS corresponds to a more global movement of several groups
and residues, including an important number of atoms.
Int J Quantum Chem 98: 367–377, 2004
© 2004 Wiley Periodicals, Inc.
Key words: enzyme catalysis; Mandelate Racemase reaction mechanisms;
microiterative method; QM/MM transition state structures; second derivatives direct
optimization method
Correspondence to: J. M. Lluch; e-mail: lluch@klingon.uab.es
Contract grant sponsors: Ministerio de Ciencia y Tecnologı ´a
and Fondo Europeo de Desarrollo Regional.
Contract grant numbers: BQU2002-00301 and BQU2002-
00293.
International Journal of Quantum Chemistry, Vol 98, 367–377 (2004)
© 2004 Wiley Periodicals, Inc.

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Introduction
T
reactive systems constituted by thousands of atoms.
Condensed phase reactions, enzymatic reactions,
and solid state catalysis are some of the most suc-
cessful areas. When so many degrees of freedom
are taken into account, the eternal problem of de-
scribing the system with an adequate potential is
especially challenging. In addition, an appropriate
exploration of the configurational space is crucial to
calculate thermodynamic and kinetic magnitudes
and to understand the mechanism of a particular
reactive process. Some work was done on the ac-
celeration of that exploration of the configurational
space by molecular dynamics techniques [1, 2].
Another important issue has to do with the need
of building up a path connecting reactants to prod-
ucts before performing any dynamics calculations.
In this way one tries to avoid trajectories getting
trapped in regions far away from any of the reac-
tion pathways. To that end some strategies were
proposed to elucidate the reaction pathways, that
is, the classical minimum energy path (MEP) [3],
the recently developed temperature-depending
MEP [4, 5], or an ensemble of transition paths that
take into account dynamical effects [6].
In any case, the procedures mentioned above
require the availability of very cheap potential en-
ergy expressions. However, the state of the art in
enzymatic catalysis makes use of quantum mechan-
ical/molecularmechanics
[7–9] or, as an alternative, of linear scaling methods
[10, 11]. Although these potentials were originally
developed to obtain a fast energy evaluation in big
systems, their computational costs are still too high
to afford the full exploration of the configurational
space or even to completely locate a temperature-
depending path. In that sense, the search of local
saddle points (from here on, we refer to them as
“transition states” [TS]) of a given reaction step in
the potential energy surface (PES) is still a good
strategy to verify the reliability of the reaction path-
way. Moreover, when QM/MM potentials are so
expensive as to render a posterior dynamics study
prohibitive, the exploration of a local valley to lo-
cate stationary points might be the only way to
obtain information about the reactivity of a chemi-
cal system.
hese last years, theoretical chemistry has fo-
cused its efforts on a deeper understanding of
(QM/MM)methods
The location of stationary points in chemical sys-
tems involving up to few tens of atoms is a very
well stablished field [3]. The most effective algo-
rithms are based on the Newton–Raphson equation
that uses the second derivatives matrix (Hessian)
[12]. On the other hand, when hundreds of atoms
need to be moved, several problems emerge, due
to the manipulation of these very big matrices,
mainly, their computation, diagonalization, and
storage. Some solutions were proposed [13, 14];
however, most avoid the usage of the Hessian and
so increase the number of energy and gradient eval-
uations.
In high-dimensioned systems, there are many
available parallel reaction paths at a given finite
temperature. As a matter of fact, in systems such as
enzymes, the PES becomes so shallow that special
care must be taken to always perform the search in
the same local valley. Therefore, we need efficient
and tight algorithms. Furthermore, when expensive
potentials such as the QM/MM ones are used, the
process used should contain as few steps as possi-
ble. For all these reasons, we decided to take ad-
vantage of the efficiency of the microiterative
method as a possible solution to locate not only TS
but also stationary points in general in systems with
many degrees of freedom.
The microiterative method is a strategy first used
by Maseras and Morokuma [15] in the Integrated
Molecular Orbital Molecular Mechanics (IMOMM)
scheme and then applied by several groups to en-
zymatic systems [16–21]. The method splits the sys-
tem in two parts: a core zone, in which an accurate
second-order search is done; and an environment
that is kept minimized with a cheap first-order
method. The separation makes the sum of the ex-
penses of the two processes considerably lower
than that of a single global search.
In the current work, we study some of the
possible strategies that can be adopted in the
microiterative method. To our knowledge, no
similar systematic study has been published so
far. The article is presented in two main parts.
First we present the microiterative method as a
possible solution to our problem, discuss the dif-
ferent options that this method can offer, and
mention the big Hessian manipulation problems.
Second we present the results, briefly describe the
enzymatic system that we have used to perform
our systematic tests, discuss the results obtained
with the different options of the method, and,
finally, present the conclusions.
PRAT-RESINA ET AL.
368
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Method
LOCATION OF STATIONARY POINTS IN BIG
SYSTEMS
Most of the program packages designed to study
biological systems have several standard minimiz-
ers. The algorithms were conceived for cases in
which potential energy, usually molecular mechan-
ics, is computationally cheap and the main com-
puter demand is the storage of enormous vector
arrays and the slow matrix diagonalizations. Exam-
ples of the methods are steepest descent, conjugate
gradient, or the more accurate Adopted Basis New-
ton–Raphson (ABNR) [22], Truncated Newton [23],
and Limited Memory Broyden–Fletcher–Goldfarb–
Shanno (L-BFGS) [24] (see Ref. [25] for a recent
comparison of some of the methods). Usually they
are applied to minimize an experimental Protein
Data Bank (PDB) structure before running molecu-
lar dynamics trajectories or a normal mode analy-
sis. However, none of the methods were originally
designed to search for TS structures. We need effi-
cient methods to locate directly TS structures in
enzymatic catalysis using the information provided
by the second derivatives. In addition, the methods
must be developed to use the rather expensive
QM/MM potentials but with a demand of a rea-
sonable computer cost.
In a recent article [26], we pointed out how im-
portant the accurate location of TS is. Sometimes,
when the appropriate reaction coordinate is more
complicated than a unique distance or an angle, the
direct location of the stationary point making use of
second derivatives is a good strategy. This prevents
wrong conclusions when both MEP analysis and
posterior free energy calculation along the reaction
path are made.
A compromise between the need for using a
second derivative method and the drawbacks of
moving many degrees of freedom is the microitera-
tive method. In this case, a Newton-like method
[12] is used to locate a stationary point in a small
core zone where the number of atoms to be moved
is small enough to avoid computational problems.
The rest of the atoms of the system belong to the
environment zone, which is minimized with an
inexpensive method. Both processes are alternated
until consistency is reached (see Fig. 1). In our
particular case, we make use of the rational func-
tion optimization method (RFO) [27–29] as a sec-
ond-order method for the search in the core,
whereas for the environment, the L-BFGS method is
applied. Although the scheme can be applied to
locate any kind of stationary points, from here on
we refer only to TS structures.
The core and environment zones do not have to
match the QM and MM regions, respectively. The
quantum region is selected in a QM/MM system as
the set of atoms whose interactions need to be
described by a QM potential, usually when bond
breaking or charge transfer is involved. This selec-
tion defines a certain PES. Conversely, the selection
of the core zone is a strategy used to locate station-
ary points, and it can change when looking for
different stationary points in a reaction mechanism.
The criterium to select the core zone is not based on
the interaction energy but on geometrical criteria,
that is, the core zone must include the atoms whose
movement may play an important role when look-
ing for the stationary point at the current chemical
step. Depending on the reaction type, a big core
(even bigger than the QM zone) must be chosen,
whereas in some other cases, a core with few atoms
is good enough to easily reach the stationary point.
POSSIBLE OPTIONS IN THE
MICROITERATIVE METHOD
Different possible options can be chosen in the
general microiterative scheme, some of which, as
described below, may be crucial in order to reach
the accurate convergence as quickly as possible.
Although some authors [15, 16] perform a full
minimization of the environment at every TS search
step, it is not evident that this is the best way to
proceed. We could run a full TS search until con-
vergence before minimizing the environment again.
As a matter of fact, there is as yet no agreement as
to which is the most suitable alternating frequency
FIGURE 1. Scheme of the microiterative method.
GEOMETRY OPTIMIZATION AND TRANSITION STATE
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
369

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between the two processes: minimization of the
environment, and the Newton step in the core.
Another important aspect is how big the core
zone must be in order to reach convergence as
quickly as possible. When the core is big, the cou-
pling between the two zones is minimized. This
would tend to reduce the number of steps required
to converge. On the other hand, a big core region
implies a bigger Hessian matrix and, as a conse-
quence, a more expensive initial Hessian calcula-
tion (see below for a discussion of other problems
related to big Hessian matrices).
CORE–ENVIRONMENT INTERACTION
Another option in the microiterative method that
deserves more discussion is how to handle the in-
teraction between the core and the environment.
The environment zone is usually bigger than the
core zone, and it needs to be optimized with a
cheap minimizer, such as conjugate gradient,
ABNR, or L-BFGS. Although the methods imply
low memory requirements, their poor efficiency
consequently requires many steps to reach conver-
gence. Depending on the QM level, the minimiza-
tion could demand a high computational effort if a
full QM/MM calculation is performed at every
step.
Different strategies were proposed to perform
faster QM/MM energy evaluations [19, 20, 30–33].
The strategies are based on the idea that an exact
self-consistent field (SCF) evaluation of the QM
zone might not be needed at each minimization
step in order to calculate the energy of the system.
The methods were not only applied to the location
of stationary points [19, 20], but some work had
already been done in QM/MM Monte Carlo to
explore the environment configurational space,
avoiding a full SCF evaluation at every step of the
simulation [30–33].
To summarize the different approximations
adopted to accelerate the QM/MM calculation, we
have to keep in mind the general QM/MM elec-
tronic embedding scheme [7]. The full Hamiltonian
can be expressed as:
H ? HQM? HMM? HQM/MM
(1)
where HQMis the Hamiltonian describing the set of
atoms whose interactions are computed using QM
and HMMis the MM Hamiltonian. The crucial as-
pect is the calculation of HQM/MMthat describes the
interaction between the two regions treated at the
two different levels,
HQM/MM? VQM/MM
van der Waals? ?
i
electrons?
C
classicalQC
riC
? ?
K
nuclei?
C
classicalZKQC
RKC
, (2)
where QCare the MM charges, ZKare the effective
nuclear charges of quantum atoms, and riCand RKC
are the distances from the MM charges to the elec-
trons and quantum nuclei, respectively. The second
term of the right side of Eq. (2) describes the inter-
action between the MM charges QCand the elec-
trons. The QCcharges will polarize the wavefunc-
tion. So when MM atoms are moved, all the terms
in Eq. (1) must be recomputed in order to consider
the effect of quantum atoms on the MM zone. In-
deed, when the QM level is highly accurate, the
energy computation becomes unaffordable.
A possible attempt to speed up the full QM/MM
calculation is to consider a quantum atom as a
classical atom during the MM atoms movement.
This can be carried out associating point charges to
the quantum atoms, for example, computing the
electrostatic potential (ESP) fitted charges [34] from
an electron density obtained by means of a full
QM/MM calculation. So, in Eq. (2) we can join the
second and third term, giving the expression shown
in Eq. (3):
HQM/MM? VESP/MM
van der Waals? ?
K
quantum?
C
classicalQESPKQC
RKC
. (3)
The ESP charges for the quantum atoms are con-
stant during the environment minimization and
they will be recomputed at the next core step or at
the next full QM/MM evaluation. In this case it is
evident that the core zone must always include all
the quantum atoms.
In the current work, the ESP charges are taken
from a parametric method 3 (PM3) wavefunction
instead of using an ab initio one [19], but this does
not include any systematic error in the obtained
results.
It is important to note that the ESP/MM strategy
reaches a geometry that is not a stationary point on
the real QM/MM surface. It is not evident that the
interaction between classical charges and ESP
PRAT-RESINA ET AL.
370
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charges is equivalent to the second and the third
term in Eq. (2). A better approximation is a method
that is based on the original QM/MM expression,
hereafter called “1SCF method” in the current pa-
per.
If we take a look at the QM/MM Eq. (2), we
realize that the QM/MM interaction Hamiltonian
only contributes to the total one-electron core quan-
tum Hamiltonian but not to the two-electron inte-
grals. This means that for a fixed geometry of the
quantum atoms, if we move the MM atoms, the
two-electron integrals do not have to be recom-
puted and the one-electron integrals can be easily
updated at the new QCconfiguration. If we save the
last converged wavefunction (?frozen) and perform
only one SCF cycle, the Fock matrix will not be
diagonal, but we will obtain a good and cheap
approximation to the exact QM/MM energy [see
Eq. (4)]:
E1SCF? ??frozen?HQM? ?
i
electrons?
C
classicalQC
riC??frozen?
? HMM? VQM/MM
van der Waals? ?
K
nuclei?
C
classicalZKQC
RKC
. (4)
Our proposal is to use the method to save com-
puting time during the minimization of the envi-
ronment in the microiterative scheme. The effi-
ciency of the strategy was evaluated by Evans and
Truong [35] in a Monte Carlo sampling of the en-
vironment. Keeping the wavefunction frozen gives
a good approximation as long as the perturbation of
the classical charges QCis small. That is, if the
distribution of charges QCduring the minimization
does not change too much or they are not too close
to the QM part to polarize the real ??? in a very
intense manner, the 1SCF method works well.
SOLUTIONS TO THE MANIPULATION OF A
BIG HESSIAN
As mentioned above, the reason standard New-
ton methods are not used in many systems with a
great number of atoms is the computational diffi-
culties encountered in a big Hessian manipulation.
Microiterative methods overcome this problem, but
we must still take into account that an expensive
QM method or an increase in the core size provokes
these computational problems to reappear.
Three main problems arise when working with a
big Hessian, that is, the initial Hessian calculation,
its diagonalization at every step of the process, and
its storage. The initial Hessian computational effort
is related to the QM level, whereas the two other
issues are related only to the size. In any case, as
described in the results section, a core size bigger
than a thousand atoms is not useful, so that the
storage problem is not a real problem in modern
computers. Moreover, the full diagonalization can
be avoided with a standard iterative diagonaliza-
tion that provides the two lowest eigenpairs needed
to calculate the displacement in the RFO scheme
[29, 36].
Depending on the QM level, a compromise must
be found between the quality of the initial Hessian
and the steps required to converge. That is, a very
cheap second-derivatives matrix could be used, but
this would imply a larger number of steps. Because
the number of steps required to converge with a
bad initial Hessian can increase very rapidly, our
recommendation would be to use the highest qual-
ity Hessian possible. Of course, in a TS search, the
last statement is crucial. It was also shown [18, 37]
that an adequate initial Hessian eliminates the
known problems of coupling between Cartesian
coordinates.
Any of the approximations proposed above to
save computing time during the minimization of
the environment in a microiterative scheme also can
be used when calculating the energy derivatives
with respect to the coordinates of the classical at-
oms that belong to the core zone. In this sense, we
have seen that the 1SCF approximation is a good
compromise between computational cost and accu-
racy.
Results and Discussion: Tests on
Mandelate Racemase
MANDELATE RACEMASE ENZYME
We have performed several series of tests on a
model of mandelate racemase enzyme, which we
had already studied in previous works [26, 38, 39].
The enzyme catalyzes the reversible interconver-
sion of the (S)- and (R)-enantiomers of mandelate
and other, ?,?-unsaturated ?-hydroxy carboxylates.
Thus, propargylglycolate was found to be a mod-
erately good substrate for racemization. In Figure 2,
the active site of the enzyme shows that many
residues can interact directly with the substrate.
GEOMETRY OPTIMIZATION AND TRANSITION STATE
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
371