Simulated Tempering Distributed Replica Sampling,
Virtual Replica Exchange, and Other
Generalized-Ensemble Methods for Conformational
Sampling
Sarah Rauscher,
†,‡
Chris Neale,
†,‡
and Re´gis Pome`s*
,†,‡
Molecular Structure and Function, Hospital for Sick Children, 555 UniVersity AVenue,
Toronto, ON, Canada M5G 1X8 and Department of Biochemistry, UniVersity of
Toronto, 1 King’s College Circle, Toronto, ON, Canada M5S 1A8
Received June 11, 2009
Abstract: Generalized-ensemble algorithms in temperature space have become popular tools
to enhance conformational sampling in biomolecular simulations. A random walk in temperature
leads to a corresponding random walk in potential energy, which can be used to cross over
energetic barriers and overcome the problem of quasi-nonergodicity. In this paper, we introduce
two novel methods: simulated tempering distributed replica sampling (STDR) and virtual replica
exchange (VREX). These methods are designed to address the practical issues inherent in the
replica exchange (RE), simulated tempering (ST), and serial replica exchange (SREM)
algorithms. RE requires a large, dedicated, and homogeneous cluster of CPUs to function
efficiently when applied to complex systems. ST and SREM both have the drawback of requiring
extensive initial simulations, possibly adaptive, for the calculation of weight factors or potential
energy distribution functions. STDR and VREX alleviate the need for lengthy initial simulations,
and for synchronization and extensive communication between replicas. Both methods are
therefore suitable for distributed or heterogeneous computing platforms. We perform an objective
comparison of all five algorithms in terms of both implementation issues and sampling efficiency.
We use disordered peptides in explicit water as test systems, for a total simulation time of over
42 µs. Efficiency is defined in terms of both structural convergence and temperature diffusion,
and we show that these definitions of efficiency are in fact correlated. Importantly, we find that
ST-based methods exhibit faster temperature diffusion and correspondingly faster convergence
of structural properties compared to RE-based methods. Within the RE-based methods, VREX
is superior to both SREM and RE. On the basis of our observations, we conclude that ST is
ideal for simple systems, while STDR is well-suited for complex systems.
Introduction
Achieving complete (or even adequate) conformational
sampling is one of the key challenges in biomolecular
simulations.
1
The energy landscape of most biomolecules is
“rugged”, and the source of this ruggedness is two-fold. The
energetic barriers separating accessible states are often larger
than the available thermal energy, and there are typically a
large number of states to be sampled. The time scales of
many biomolecular processes, such as protein folding, are
still far beyond the reach of our current computational
capability, which is generally limited to the 10
-8
to 10
-7
s
time scale for continuous simulations. For example, even
the folding of small domains or secondary structure elements,
such as
-hairpins and mini-proteins, occur on the 1-10 µs
time scale.
1
Consequently, conventional or “brute force”
molecular dynamics (MD) alone is often insufficient to
* Corresponding author e-mail: pomes@sickkids.ca.
†
Hospital for Sick Children.
‡
University of Toronto.
J. Chem. Theory Comput. 2009, 5, 2640–26622640
10.1021/ct900302n CCC: $40.75  2009 American Chemical Society
Published on Web 09/16/2009

achieve complete Boltzmann sampling of the important states
of many biologically relevant systems. For this reason,
generalized-ensemble algorithms have become popular tools
for conformational sampling.
A variety of generalized-ensemble algorithms have been
developed with the common intention of overcoming ener-
getic barriers in order to enhance sampling of conformational
space. These methods use a generalized Hamiltonian for the
purpose of achieving uniform sampling along a reaction
coordinate of interest. Practically, one is faced with choosing
the most appropriate method and reaction coordinate for a
particular application. While the optimal reaction coordinate
is not known a priori, it may be possible to make generaliza-
tions regarding the optimal methodology. To this end, we
consider the following important question: given limited
computational resources, which algorithm is most efficient
at sampling a complex energy landscape? Some generalized-
ensemble methods employ a random walk in potential
energy, while others use different parameters which are
relevant to the system of interest.
2
In this article, we compare
the efficiency of a set of algorithms which make use of a
random walk in temperature to enhance conformational
sampling of biomolecules. We focus on the following five
methods: simulated tempering (ST),
3,4
replica exchange
(RE),
5-9
the serial replica exchange method (SREM),
10
and
two novel methods, virtual replica exchange (VREX) and
simulated tempering distributed replica sampling (STDR),
which is a combination of ST and distributed replica
sampling (DR).
11-13
The generalized-ensemble algorithms compared in this
paper all rely on the fact that the free energy surface becomes
less rugged at high temperatures, increasing the frequency
of interconversion between conformational states.
14
Simula-
tions performed at low temperatures often require a relatively
long time to cross the energetic barriers between states and
appear to be trapped. Transitions between regions separated
by barriers may not be observed over time scales accessible
to simulation. In this case, multiple simulations initiated in
different conformational basins may sample different subsets
of phase space. The result is that an ergodic system appears
nonergodic, a phenomenon known as quasi-nonergodicity.
15
Utilizing generalized-ensemble algorithms that induce a
random walk in temperature may alleviate this source of
error.
The sampling enhancement of generalized-ensemble meth-
ods relative to canonical MD or Monte Carlo (MC) simula-
tions has been demonstrated for several systems,
3,7,16,17
including peptides.
6,14,18-24
Conversely, there have also been
studies that question the relative sampling efficiency of RE
compared to brute force MD,
25
highlighting the importance
of a rigorous definition of efficiency which accounts for the
total computer time required for all temperatures.
26-28
It is
important to note that data obtained at multiple temperatures
in generalized-ensemble simulations may be of interest in
some studies, such as protein folding.
21,22
In general,
however, the data at high temperatures are not useful.
Furthermore, the observed speedup also strongly depends
on the lowest temperature.
26
It is essential to assess the
convergence of both the conventional MD simulations as
well as the generalized-ensemble simulations in order to
perform a meaningful comparison, in addition to identifying
a meaningful quantity on which to base the comparison. Any
evaluation of sampling enhancement compared to single-
temperature MD is also likely to depend heavily on the
molecular system under study (depending on the number of
basins in the landscape and the heights of barriers). It is
therefore quite difficult to accurately quantify the sampling
enhancement due to the introduction of a random walk in
temperature.
We begin with a brief introduction of each of the
generalized-ensemble methods, including the presentation of
our two novel methods, STDR and VREX. We then perform
a thorough comparison of the algorithms in terms of both
practical implementation limitations and sampling efficiency
for a disordered octapeptide in explicit water, a molecular
system combining high relevance to protein folding and
moderate complexity. In addition to providing a comparison
between generalized-ensemble algorithms, we also provide
a comparison to conventional MD. We discuss efficiency in
terms of both convergence of structural properties and
temperature diffusion, and we show that these definitions of
efficiency are correlated. Finally, we compare the efficiency
of STDR and conventional MD for a 35-residue peptide with
a complex conformational landscape.
Theory and Methods
Simulated Tempering (ST). Simulated tempering was
originally introduced to enhance sampling of a random field
Ising model.
3
This system has a rough energy landscape for
which spin-flips from the state favored by the magnetic field
to the opposite state are statistically rare events. ST facilitates
exchanges between these states, whereas the MC algorithm
remains trapped.
3
ST has also been shown to be effective in
exploring the energy landscapes of biomolecules, which
similarly have multiple energy minima separated by barri-
ers.
29
In the ST algorithm, temperature becomes a dynamic
variable
3,4
that can take on discrete values labeled by an
index m (m ) 1, ..., M). ST makes use of a generalized
Hamiltonian, H(X,m), which depends on all configurational
degrees of freedom (X), in addition to temperature:
where

m
is the inverse temperature, H(X) is the system’s
original Hamiltonian, and a
m
is a constant which depends
on temperature.
3
The generalized ensemble has a corre-
sponding generalized partition function, Z, given by:
where Z
m
is the partition function corresponding to the
temperature T
m
.
30
The partition function of the generalized
ensemble, Z, is the weighted sum of the partition functions
of the canonical ensembles at each temperature, Z
m
.We
H(X,m) )

m
Η(X) - a
m
(1)
Z )
∑
m
∫
dX [e
-H(X,m)
] )
∑
m
∫
dX [e
-

m
Η(X)+a
m
] )
∑
m
Z
m
e
a
m
(2)
Generalized-Ensemble Methods J. Chem. Theory Comput., Vol. 5, No. 10, 2009 2641