Accelerated Superposition State Molecular Dynamics for Condensed Phase Systems
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Department of Chemistry and Center for Biophysica
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Department Of Chemistry And Center For Biophysica
Introduction
Molecular dynamics (MD) simulation models the time
evolution of a given atomistic-level system by integrating
Newton’s equations of motion.1–3 Extensions to MD employ
dynamics described by a Langevin equation4–6 or variations
of the previous differential equations with addit...
Accelerated Superposition State Molecular Dynamics for
Condensed Phase Systems
Michele Ceotto
Dipartimento di Chimica Fisica ed Elettrochimica, UniVersità degli Studi di Milano,
Via Golgi 19, 20133 Milano, Italy
Gary S. Ayton and Gregory A. Voth*
Department of Chemistry and Center for Biophysical Modeling and Simulation,
UniVersity of Utah, Salt Lake City, Utah 84112
Received December 1, 2007
Abstract: An extension of superposition state molecular dynamics (SSMD) [Venkatnathan and
Voth J. Chem. Theory Comput. 2005, 1, 36] is presented with the goal to accelerate timescales
and enable the study of “long-time” phenomena for condensed phase systems. It does not require
any a priori knowledge about final and transition state configurations, or specific topologies.
The system is induced to explore new configurations by virtue of a fictitious (free-particle-like)
accelerating potential. The acceleration method can be applied to all degrees of freedom in the
system and can be applied to condensed phases and fluids.
I. Introduction sure) calculated via accumulated time averages differ (some-
times greatly) from the ensemble average (where all points
Molecular dynamics (MD) simulation models the time
in phase space are considered). In other words, the timescale
evolution of a given atomistic-level system by integrating
used for the measurement is much shorter than the actual
Newton’s equations of motion.1–3 Extensions to MD employ
relaxation time for the system. As a result, in this particular
dynamics described by a Langevin equation4–6 or variations
of the previous differential equations with additional param- case, the MD simulation performs a sum of subaverages of
eters (e.g., friction coefficients) or degrees of freedom (e.g., isolated phase space subsets and misses some others. As an
Nosé-Hoover thermostats1,7–9). The integration of the example of an approach to overcome these issues, Andri-
underlying equations of motion are limited in time; in fact, cioaei and Straub16 have introduced new generalized Monte
most MD simulations are far too limited in duration to Carlo (MC) and MD algorithms inspired by Tsallis statistics
examine many important biomolecular processes which can (q-jumping) and later, “smart walking” MC.17 Other ap-
occur on timescales longer than tens of nanoseconds. For proaches include the multicanonical algorithm,18,19 where
example, lateral diffusion in lipid bilayers occurs on the extensive macrovariables are added, simulated tempering,20,21
second scale,10 while protein folding occurs on the mil- and replica exchange (REX),22–25 where intensive thermo-
lisecond scale.11–15 On the other hand, relying on the rapid dynamic state quantities (pressure, temperature, chemical
growth in computer technology employing a “brute force” potentials, etc.) are varied. All the above methods have been
MD approach is also not feasible as a speedup of 6 orders implemented both in MC and MD simulation algorithms. A
of magnitude will be required in order to access the relevant main limitation involves the number of replicas; these can
timescales. grow unmanageably large when many macrostates are
In some complex systems, the multiple time- and length- required for the simulation to satisfy ergodicity. Fenwick et
scales can lead to the so-called effect of “broken ergodicity”, al.26 have combined MC replica methods with biased force-
where mechanical observables (e.g., internal energy, pres- field parameters in order to directly modulate the specific
molecular interaction responsible for the kinetic traps and,
* Corresponding author. E-mail: voth@chem.utah.edu. in the same spirit, others27,28 have modified force-field
10.1021/ct7003275 CCC: $40.75 2008 American Chemical Society
Published on Web 03/04/2008
, SSMD for Condensed Phase Systems J. Chem. Theory Comput., Vol. 4, No. 4, 2008 561
parameters (except those related to solvent-solvent interac- subsequently performed while the rest of the solid lattice is
tions) in an MD protocol. kept static. This is done by the use of a thermal activity
Another approach for timescale acceleration is the parallel function, which decays to zero in a sigmoid fashion once
replica method (PRD),29–31 where the system is replicated out of the active region.
in parallel and independent MD trajectories are generated Although the above accelerating potential procedures
via different initial velocity distributions. Whenever a partially removes the problem of broken ergodicity and
successful trajectory is obtained, all processors are stopped. accelerates the crossing of barriers, it is not suitable for
This state is then replicated over all processors, and the whole calculating equilibrium thermodynamic properties as it
process is restarted. A further implementation of this method undersamples low-energy states. Self-guided molecular
is the parallel sequential synchronization (PSS)32 which dynamics approaches57–61 can avoid this problem and can
shows how PRD is easy to combine with other techniques. be implemented without any a priori knowledge of the
To accelerate timescales and enable the study of “long- system; rather they are based on the cumulative history of a
time” phenomena within the MD framework, one can system’s trajectory. In particular, when this idea is combined
perform simulations at higher temperatures. Voter33 has with replica exchange,61 the copies are self-regulating and
developed, and later improved,29 a method called temperature compete during the simulation to overcome to any under-
accelerated dynamics (TAD) which raises the temperature sampled region. Along the same line of an adaptive
and corrects for this bias by filtering out transitions that would algorithm, Laio and Parrinello62–65 introduced the so-called
not occur at the original temperature. A completely different “metadynamics” method in which a history-dependent
approach for “rare events” dynamics (especially for passages potential, given as the sum of Gaussians centered on the
over high barriers) employs accelerating potentials.34–45 Here, trajectory in a reduced “collective variable” space, fills the
the potential energy surface is modified for a small set of free energy surfaces minima and drives the systems to
degrees of freedom and the original state is recovered on- explore new wells.
the-fly by means of non-Boltzmann weights. Conformational In condensed phases, the complexity of the system is such
flooding46 first selects a subconformational space and then that it is not possible to isolate a priori a subset of degrees
destabilizes the initial conformation and consequently lowers of freedom responsible for the long-time properties. For
the free energy barrier of structural transitions. Similarly, example, a simple picture of a double well coupled to a bath
the hyperdynamics method29,35,36 focuses on infrequent with a starting and ending configuration is, most of the time,
transition events from one potential energy basin to another an inadequate description of the system. Consequently, a
and then constructs a bias potential such that the original state-to-state transition model with a transition state config-
potential energy surface changes are done without affecting uration, which follows a first order kinetic picture, is often
the transition state regions where the rate is calculated using out of the question. An example problem that enters into
the harmonic limit of transition state theory. Significant this category is the bilateral diffusion of phospholipids in
boosts have been found for surface diffusion dynamics;29,34–36,42 membranes; the possible configurations are so numerous that
however, for more complex systems, the construction of such trying to isolate any subsets of degrees of freedom will
a bias potential is not trivial and may not solve the low barrier mostly turn out to be impossible, if not counterproductive.
problem when the system is trapped by a set of states In fact, the transition state that separates two multidimen-
connected by low barriers. A simpler recipe is one offered sional basins in a double well picture is misleading for such
by Tully and co-workers,37,41 where the bias potential is systems since no specific event and reaction subset of
chosen so that the system evolves in a flat “puddle region” coordinates can be identified. To deal with these issues, a
instead of sinking into a local minimum. This method has more generalized version of enhancing sampling must be
been applied to dihedral degrees of freedom in small peptide adopted, which is not restricted to first order kinetics. Instead,
dynamics37–39,41,47 and has been implemented in MC simula- the system should enchance its ergodic properties without
tion with a bias in momentum space.37,48,49 Recent work has any specific instructions or a priori topological constraints,
employed a “boost potential” to modify the original potentials and should be left to explore new configurations in an
that govern the system.44,50–52 The scheme raises the wells unrestricted fashion.
depths in a continuous manner and has been successfully In light of the above considerations, the goal of the present
applied, for example, to alanine dipeptide in an explicit work is to develop an approach within the MD framework
solvent. In conjunction with a quasiharmonic analysis, this that can deal with all the degrees of freedom of complex
approach has been used to calculate the entropy for an eight condensed phase systems, for example fluids, by not requir-
residue peptide in explicit water.44 ing any a priori knowledge of the system (such as the final
A different solution to the direct kinetic dynamics of rare state), and by guaranteeing a full recovery of the unbiased
events is transition path sampling.53–55 In this method potential statistics. The starting point of the present work is
reactants and products are known a priori, and path en- the superposition state molecular dynamics (SSMD)
sembles between these states are generated by constructing method.45 The main idea underlying SSMD is to create
a random walk in path space with a MC algorithm. fictitious potentials whose dynamics are accelerated with
In solid-state simulations, a popular choice to accelerate respect to the original (physical) one. These potentials are
MD is feature activated molecular dynamics (FAMD),56 built up as a superposition of the upper and lower states with
which creates localized regions around a defect atom or respect the physical one and they are coupled in such a way
cluster (active region) sites. A full MD simulation is that the overall dynamics smoothly and continuously switch
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