Efficient Use of Nonequilibrium Measurement to Estimate
Free Energy Differences for Molecular Systems
F. MARTY YTREBERG and DANIEL M. ZUCKERMAN
Center for Computational Biology and Bioinformatics, University of Pittsburgh, 200 Lothrop St.,
Pittsburgh, Pennsylvania 15261
Received 26 February 2004; Accepted 4 June 2004
DOI 10.1002/jcc.20103
Published online in Wiley InterScience (www.interscience.wiley.com).
Abstract: A promising method for calculating free energy differences ⌬F is to generate nonequilibrium data via
“fast-growth” simulations or by experiments—and then use Jarzynski’s equality. However, a difficulty with using
Jarzynski’s equality is that ⌬F estimates converge very slowly and unreliably due to the nonlinear nature of the
calculation—thus requiring large, costly data sets. The purpose of the work presented here is to determine the best
estimate for ⌬F given a (finite) set of work values previously generated by simulation or experiment. Exploiting
statistical properties of Jarzynski’s equality, we present two fully automated analyses of nonequilibrium data from a toy
model, and various simulated molecular systems. Both schemes remove at least several k B T of bias from ⌬F estimates,
compared to direct application of Jarzynski’s equality, for modest sized data sets (100 work values), in all tested
systems. Results from one of the new methods suggest that good estimates of ⌬F can be obtained using 5– 40-fold less
data than was previously possible. Extending previous work, the new results exploit the systematic behavior of bias due
to finite sample size. A key innovation is better use of the more statistically reliable information available from the raw
data.
© 2004 Wiley Periodicals, Inc.
J Comput Chem 25: 1749 –1759, 2004
Key words: free energy; Jarzynski relation; extrapolation; non-equilibrium; block averages
Introduction
The calculation of free energy differences ⌬F plays an essential
role in many fields of physics, chemistry, and biology.1–20
Examples include determination of the solubility of small molecules, and binding affinities of ligands to proteins. Rapid and
reliable estimates of ⌬F would be particularly valuable to
structure-based drug design, where current approaches to virtual screening of candidate compounds rely primarily on ad hoc
methods.21,22 Free energy estimates are also critical for protein
engineering.23,24
The focus of this report is nonequilibrium “fast-growth” free
energy calculations.17–20,25 These methods hold promise—yet to
be fully realized—for very rapid estimation of ⌬F. The central
idea behind the nonequilibrium methods is to calculate the irreversible work during a very rapid (thus nonequilibrium) switch
between the two systems or states of interest. Multiple switches are
performed, and the resulting set of work values can be used to
estimate ⌬F via Jarzynski’s equality (detailed later).17,18 Recent
path sampling approaches to nonequilibrium ⌬F calculations are,
we note, extremely encouraging,26,27 although currently unproven
for large molecular systems.
Somewhat surprisingly, nonequilibrium ⌬F calculations are
critical for analyzing single-molecule pulling experiments.1,8 In
essence, these experiments generate nonequilibrium work values,
as pointed out by Hummer and Szabo, so the only way to estimate
the free energy profile is to use Jarzynski’s equality.8,28 The
methods that we develop in this report should be equally useful for
analyzing such experiments.
It has been accepted for some time that there are three sources
of error29 for nonequilibrium ⌬F calculation via molecular mechanics simulation: (1) inaccuracy of the force field,6,30 (2) inadequate sampling of the configurational space,31–35 and (3) bias due
to finite sample size.18,25,36 –39 Indeed, errors in free energy calculations have been of long-standing interest.40 – 44
Correspondence to: F. M. Ytreberg; e-mail: [email protected]; D. M.
Zuckerman; e-mail: [email protected]
Contract/grant sponsor: Department of Environmental and Occupational
Health, and the Center for Computational Biology and Bioinformatics at
the University of Pittsburgh.
Contract/grant sponsor: National Institutes of Health; Contract/Grant
number: T32 ES007318.
© 2004 Wiley Periodicals, Inc.
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Ytreberg and Zuckerman
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Vol. 25, No. 14
Journal of Computational Chemistry
•
Table 1. Features of the Test Systems Used in the Current Study, as Well as the Sources of the Independent
Estimates ⌬F ind.
Test system/source of ⌬F ind
Switching between two one-dimensional harmonic
potentials in 100 ␭-steps with 1 relaxation step
at each value of ␭. ⌬F ind is the exact ⌬F
obtained numerically.
Chemical potential calculation for a Lennard–
Jones fluid in 1 ␭-step. This corresponds to
instantaneous switching or single-stage free
energy perturbation. ⌬F ind is a slow-growth
estimate.20
Growing a chloride ion by 1.0 Å in water in 10
␭-steps with 1 relaxation step at each value of
␭. ⌬F ind is a slow-growth estimate.
Methanol-to-ethane mutation in water using 200
␭-steps with 1 relaxation step at each value of
␭. ⌬F ind is from multistage free energy
perturbation.25
Palmitic-to-stearic acid mutation in water using 55
␭-steps with 10 relaxation steps at each value
of ␭.67 ⌬F ind is from another fast-growth data
set using 55 ␭-steps with 40 relaxation steps.
␴W
具W典
N tot
Skew
⌬F all
⌬F ind
30,000
29.5 k B T
14.4 k B T
0.52
0.7 k B T
0.7 k B T
100,000
305.1 k B T
83.5 k B T
⫺0.59
0.7 k B T
1.2 k B T
40,000
40.1 kcal/mol
8.6 kcal/mol
0.37
18.4 kcal/mol
16.9 kcal/mol
9,600
37.0 kcal/mol
12.3 kcal/mol
0.38
7.4 kcal/mol
5.3 kcal/mol
20,000
28.6 kcal/mol
7.5 kcal/mol
1.41
15.2 kcal/mol
13.6 kcal/mol
The first column gives simulation details of the systems under consideration. The next three columns show statistical
features of each data set: total number of work values N tot, the mean work 具W典, the standard deviation ␴ W , and the skew.
The fifth column is the standard Jarzynski estimate for a data set, ⌬F all, obtained by using Equation (5) with all N tot work
values. The last column is an independent measure of the free energy ⌬F ind using a non-fast-growth method, in all cases
but one.
The present study addresses only source (3), and attempts to
determine the most efficient use of fast-growth work values. In
other words, given a (finite) set of work values generated by
simulation or experiment, what is the best estimate for ⌬F? We do
not here attempt to prescribe the best method for generating
nonequilibrium work values. Methods to lessen the effect of bias
due to finite sample size have been proposed for the case when
switching between two states is performed in both directions,45– 47
and for the idealized case in which the nonequilibrium work values
follow a nearly Gaussian distribution.7,48 Hummer also considered
errors in nonequilibrium ⌬F calculations.19 To our knowledge,
however, other workers have not addressed unidirectional switching in highly non-Gaussian systems.
The techniques outlined in the following sections offer rapid
estimates for ⌬F for the systems we studied—namely, a onedimensional “toy” system switched between two harmonic potentials, the chemical potential for a Lennard–Jones fluid, “growing”
a chloride ion in water, and the relative solubilities of methanol
and ethane, as well as of palmitic and stearic acids. Work values
for these systems follow highly non-Gaussian distributions; (see
Table 1). We compare our extrapolated results to ⌬F obtained by
using Jarzynski’s equality, finding a 5– 40-fold decrease in the the
number of work values needed to estimate ⌬F for the test systems
considered here.
We proceed by first introducing a modified block averaging
technique, building on the original proposal by Wood et al.49
Block averaging provides well-behaved, but biased, ⌬F estimates.
We then discuss two distinct schemes for extrapolating to the
“infinite data limit.” Both methods combine rigorous and ad hoc
approaches, but are guaranteed to converge to the correct ⌬F as
the size/quality of the data set increases. Our work systematizes
and extends previous work by Zuckerman and Woolf,25 who
originally proposed the use of extrapolation via block averages.
Fast-Growth
Nonequilibrium, fast-growth techniques have been described in
detail elsewhere,17–19,25 so we will simply outline the method.
Consider two systems defined by potential energy functions U 0
and U 1 . A typical case is when the two potentials describe distinct
chemical species, as for solubility or binding affinity calculations,
via an appropriate thermodynamic cycle.50 To calculate the free
energy difference ⌬F between these two systems, one must
“switch” the system from U 0 to U 1 , which is most simply done via
a switching parameter ␭ in the expression
U ␭共x兲 ⫽ U0 共x兲 ⫹ ␭关U1 共x兲 ⫺ U0 共x兲兴,
(1)
where x is a set of configurational coordinates, and U ␭ (x) describes the “hybrid” potential energy function for all values of ␭
from 0 to 1. We note that nonlinear scaling with ␭ is also possible,5,6,14,15,51 resulting in hybrid potentials differing from eq. (1).
Calculating Free Energy Differences
Figure 1. Distribution of work values for the palmitic to stearic acid
mutation test system described in the Test Systems section. Also
included in this plot are the estimators given by eqs. (2) and (3) shown
by the green dotted and blue dot-dash lines, respectively. The dashed
black line shows the ⌬F estimate obtained by using Jarzynski’s
equality in eq. (5) for all available data. Note that 1.0 kcal/mol ⬇ 1.7
k B T.
Our approach here also applies, in principle, to other such choices.
Essentially, the idea behind fast-growth methods is to perform
rapid switches from ␭ ⫽ 0 3 1, where each switch is generated
starting from coordinates drawn from the equilibrium ensemble for
␭ ⫽ 0. During each switching simulation, the irreversible work is
accumulated, generating a single work value W. 17,25 Multiple
Figure 2. The running Jarzynski estimate, given by eq. (5), as a
function of the number of work values used in the estimate, N is shown
as a solid blue line. The dashed red line shows the (subsampled) block
averaged free energy estimate given by eq. (9), plotted as a function of
the number of work values in each block, n. Data used for these
estimates were obtained from the chemical potential calculation test
system. The Jarzynski estimate displays erratic convergence behavior,
while the block averaged free energy estimate displays a smooth
monotonically decreasing estimate. Such behavior occurs in all systems for which ␴ W ⬎ k B T.
1751
Figure 3. The subsampled block-averaged free energy ⌬F n , given by
eq. (9), obtained by generating 500 subsets of 100 work values each,
drawn at random from the palmitic to stearic acid data set. The red
squares with errorbars show the averages of the ⌬F n estimates with
standard deviations obtained by using the 500 subsets. The blue curve
shows the ⌬F n data from a single representative subset of 100 work
values, and ⌬F all from Table 1 is included as the dashed black line. In
this plot, extrapolating to the infinite data limit corresponds to continuing the single subset ⌬F n data (blue curve) to ␹ ⫽ 0 to obtain the
intercept. This plot also demonstrates that the large ␹ (small n) data are
more reliable as shown by the errorbars.
Figure 4. Examples of the cumulative integral, CI( ␹ ) are shown for
two subsets of 100 work values drawn at random from the palmitic to
stearic acid mutation test system. The first subset is represented by
open symbols and the second by closed symbols. The dashed black
line shows the estimate using all available data ⌬F all, the blue squares
are the subsampled ⌬F n and the red triangles are the CI( ␹ ). The
strength of using CI( ␹ ) for extrapolation is its explicit use of all the
⌬F n values. For each subset, the value of ␶ was chosen by a fully
automated process to minimize the slope of the small-␹ tail of CI( ␹ ),
as described in the text. For the open symbols it was found that ␶ ⫽
0.75, and for the closed symbols ␶ ⫽ 0.39. In this example, the subset
represented by the open symbols slightly overestimates ⌬F all, while
the subset represented by the closed symbols slightly underestimates
⌬F all.
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Vol. 25, No. 14
switches are performed to generate a distribution of these work
values ␳ (W).
Simple Estimators
It has been appreciated for some time that the average work
obtained over many such switches provides a rigorous upper
bound for the free energy difference,49,52
⌬F ⱕ 具W典 0,
(2)
where the 具. . .典0 represents an average over many switches starting
from the equilibrium ensemble for ␭ ⫽ 0 and ending at ␭ ⫽ 1.
Equality occurs only in the limit of infinitely slow switches,
because the work values generated in this limit are reversible.
Tighter bounds are also known.2,18,29 Further, if the distribution of
work values ␳ (W) is Gaussian (this will always be the case for
sufficiently slow switching,17,19,53 but also may occur in certain
nonequilibrium situations7), then the first term of the high temperature expansion of Zwanzig54 gives the exact result
⌬F ⫽ 具W典 0 ⫺
1
2
␤␴ W
,
2
(3)
where ␴ W is the standard deviation of ␳ (W), and ␤ ⫽ 1/k B T
where T is the temperature of the system and k B is the Boltzmann
constant.
However, for the fast-growth work values under consideration
here, the distribution of work values can be very broad and
non-Gaussian (see Table 1). Thus, eqs. (2) and (3) will not provide
reasonable estimates of ⌬F; (see Fig. 1). It is possible to use higher
order moments to estimate ⌬F (see, for example, refs. 7, 48, 54,
and 55), but these estimators are only considered useful in the
near-equilibrium regime.17,19
Jarzynski Equality
Due to recent work by Jarzynski,17,18,20,56 it is possible to estimate
⌬F using fast-growth work values W via
e ⫺␤⌬F ⫽ 具e ⫺␤W典 0.
(4)
This remarkable relationship is valid for arbitrary switching speed,
implying that switches may be performed as rapidly as desired.
The Jarzynski equality thus provides an estimate for ⌬F for a set
of N work values given by
⌬F ⬟ ⌬F Jarz ⫽ ⫺
冋冘 册
1
1
ln
␤
N
N
e⫺ ␤ W i ,
(5)
i⫽1
where the “⬟” denotes a computational estimate.
The ⌬F estimates given by eq. (5), however, are very sensitive
to the distribution of work values ␳ (W). 7,18,25 If the width of the
work distribution is large, i.e., ␴ W ⬎⬎ k B T (this implies a very
rapid switch and/or a complex system), then often thousands, or
even tens of thousands of work values are needed to reliably
•
Journal of Computational Chemistry
estimate ⌬F. An example of this can be seen in Figure 1, where a
histogram of work values is shown for the palmitic to stearic acid
system (described later). The value of ⌬F all given by the Jarzynski
1
2
equality, as well as estimators 具W典 0 and 具W典 0 ⫺ 2 ␤␴ W
, are
shown on this plot. This graphically demonstrates why eqs. (2) and
(3) are often poor estimates of the free energy for fast-growth work
values.
In principle, nonequilibrium switching can be performed at
arbitrary speeds. Hendrix and Jarzynski explored varying switch
speeds for a Lennard–Jones fluid.20 Their work suggested that the
determining factor in the accurate calculation of ⌬F was physical
CPU time spent during the calculation. Rapid switches appeared to
have no advantage over doing fewer slower switches, based upon
using eq. (5) for all ⌬F estimates.
If the switch is performed “instantaneously,” then the work
performed is simply the difference in potential functions, and eq.
(4) becomes
e ⫺␤⌬F ⫽ 具e ⫺␤关U1共x兲⫺U 0共x兲兴 典0 ,
(6)
often called single-stage free energy perturbation.54,57 In this limit,
the system is not allowed to relax at any intermediate values of ␭.
Instead U 1 (x) is evaluated at values of x drawn from the equilibrium ensemble for ␭ ⫽ 0. The advantage of this method is that data
can be generated very quickly. However, in practice, unless there
is sufficient overlap between the states described by U 0 and U 1 ,
the estimate of ⌬F will be biased, often by many k B T. 58,59 The
problem of attaining overlap of states can be improved by drawing
from the equilibrium ensemble for an unphysical “soft-core” state
(such as for ␭ ⫽ 0.5).33,51
Other Methods
Use of the Jarzynski equality for free energy calculations has only
been described recently, and a comparison to other approaches is
informative. Equilibrium approaches, for instance, are better
known procedures for free energy calculations.6,9,12,15,35,60 – 62
One quasi-equilibrium approach to estimating ⌬F using fewer
work values is “slow-growth”—i.e., performing the switching
process more slowly, leading to a narrower ␳ (W). 38,52,53,63,64
However, slower switching speed means that more computational
time will be spent to generate each work value— offsetting some
of the advantage gained by doing fewer switches. If the switch is
performed so slowly that the system remains near equilibrium
during the switch, then the width of the distribution will be very
small ( ␴ W ⬍ k B T), and thus only a few work values are required
for accurate estimation of ⌬F. 38,53 Slow-growth approaches have
also made use of the bounds provided by eq. (2) in several
cases.52,64
Thermodynamic integration (TI) is probably the most common
fully equilibrium approach. In TI, ⌬F is calculated by allowing the
system to attain equilibrium at each value of ␭. Then ⌬F is found
using
冕 冓
1
⌬F ⫽
d␭
0
冔
⭸U ␭共x兲
.
⭸␭ ␭
(7)
Calculating Free Energy Differences
Thermodynamic integration can provide very accurate ⌬F calculations, but is also computationally expensive.6,15,35,60
Bidirectional switching in nonequilibrium calculations is also
possible. When using the Jarzynski equality eq. (5), the equilibrium ensemble is generated for ␭ ⫽ 0. Then ␳ (W) is generated by
doing switches from ␭ ⫽ 0 3 1 (forward switches) with configurations drawn from the ␭ ⫽ 0 ensemble. It is also possible to
generate another equilibrium ensemble for ␭ ⫽ 1 and then perform
reverse switches from ␭ ⫽ 1 3 0. This bidirectional nonequilibrium approach was originally used by Reinhardt and coworkers for
bounds-based ⌬F estimates.52,65 It has been shown that, if one
combines the use of the forward and reverse work values, accurate
free energy estimates can be obtained using fewer work values
than with forward switches only.45– 47,66 It has been recently
demonstrated that most efficient use of forward and reverse work
values is for Bennett’s method.3,45,46
There is, however, a distinct motivation for using Jarzynski
estimates with only forward switches, when one considers the
eventual goal of predicting relative binding affinities for application in drug design. In this situation, one need only generate a
single high-quality equilibrium ensemble for a particular ligandreceptor or reference complex, echoing the strategy of van Gunsteren and coworkers.33 Then relative binding affinities can be
determined for other ligands without generating another equilibrium ensemble—potentially a significant decrease in computational expense.
Test Systems
To show the generality of the methods proposed in this study we
consider five test systems of varying complexity: a one-dimensional “toy” system switched between two harmonic potentials, a
chemical potential calculation for a Lennard–Jones fluid, “growing” a chloride ion in water, methanol-to-ethane mutation in water,
and stearic-to-palmitic acid mutation in water. The latter two
systems are examples of components of relative solubility calculations and consist of alchemical mutations of fully solvated molecules (see refs. 25, 67 for simulation details). The second system
is a chemical potential calculation done by the particle insertion
method (see ref. 20 for details).
Table 1 explains important features of the data sets for each of
the test systems used here. Statistical features of the data sets
include the total number of work values N tot, the mean work 具W典,
the standard deviation ␴ W , and skew.68 These data sets are all
considered difficult to use for ⌬F calculations owing to their
broadness ( ␴ W Ⰷ k B T; see Fig. 1) and the fact that 具W典 ⫺ ⌬F ⬎
10 k B T; see eqs. (2) and (3). The values for the skew demonstrate
that these data sets are non-Gaussian.
Table 1 also includes estimates of ⌬F. The first estimate ⌬F all
is obtained by using eq. (5) with all available data, and thus
represents the best estimate for a given data set using the “standard” Jarzynski approach. The second estimate ⌬F ind is an independent measure of the free energy difference, obtained by a
non-fast-growth approach, in all cases but one, and is specified in
Table 1 for each case.
As will be seen, some of the fast-growth data sets may not be
fully converged, even considering all the work values; apparently
1753
these “full” data sets still suffer from finite-sampling bias. Therefore, it is valuable to consider independent estimates ⌬F ind. Below, we compare our results to both ⌬F all and ⌬F ind from Table
1.
Of the five data sets analyzed in this report, three were obtained
in other studies noted in Table 1, and simulation details are not
given here. For the present report, we performed simulations of a
toy model and a chloride system.
The one-dimensional toy data were obtained by switching the
system from H 0 ( x) ⫽ 0.5x 2 to H 1 ( x) ⫽ 2.0( x ⫺ 4.0) 2. The
free-energy difference for this system can be solved exactly, giving
⌬F ⫽ 0.7 k B T. Fast-growth work values were obtained by
Brownian dynamics simulation with a time step of 0.001. Mass,
friction coefficient and k B T values were all set to 1.0, and starting
configurations for each fast-growth trajectory were generated every 3000 time steps.
The growing chloride system was studied using TINKER version 4.1,69 with the simulation conditions chosen to closely match
those of Lybrand et al. in ref. 60. Stochastic dynamics simulations
were carried out in the canonical ensemble (constant N, V, T) in
a cubic box of edge length 18.6216 Å. The temperature was held
at 300 K by a Berendsen thermostat with a time constant of 0.1
ps.70 The chloride ion was modeled with Lennard–Jones parameters ␴ ⫽ 4.4463 Å and ␧ ⫽ 0.1070 kcal/mol, and was solvated by
214 SPC water molecules. Ewald summation approximated charge
interactions and RATTLE was used to hold the water molecules
rigid.71 For this test system, the Lennard–Jones “size” was increased by 1.0 Å, from ␴ ⫽ 4.4463 Å at ␭ ⫽ 0 to ␴ ⫽ 5.4463 Å
at ␭ ⫽ 1. To obtain fast-growth work values, a time step of 1.0 fs
was used. The system was equilibrated for at least 10 ps, after
which starting configurations for each fast-growth trajectory were
generated every 100 time steps.
The slow-growth estimate ⌬F ind for the chloride system was
obtained by generating fifty work values using 500 ␭-steps with
1000 relaxation steps at each value of ␭. The resulting distribution
had ␴ W ⬇ 0.2 k B T. ⌬F ind was then generated from this distribution using eq. (5).
Block Averaging
Block averages are the key to the present method. The motivation
for using block averages can be seen in Figure 2 (see also refs. 2
and 49). The solid blue line is a running estimate for ⌬F obtained
by using eq. (5) and the dashed red line is obtained by block
averaging. Both curves were obtained using the chemical potential
calculation test system. The running Jarzynski estimate exhibits
very poor convergence behavior, making it very difficult to establish when a reliable estimate of ⌬F has been obtained. The block
averaged free energy, however, displays a monotonically decreasing ⌬F estimate, which smoothly approaches ⌬F all.
Each block averaged free energy (⌬F n ) estimate was obtained
from a set of N work values (W 1 , W 2 , . . . , W N ) using the steps
below. The present procedure represents a slight modification of
earlier schemes,12,25,49 described by the following steps.
1. Choose n different work values at random from the set, generating a subset with relabeled indices (W 1 , W 2 , . . . , W n ). Let j
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Vol. 25, No. 14
denote the index for the particular draw performed, so that j ⫽
1 for the first subset (“block”) of n values, j ⫽ 2 for the second,
and so on.
2. Use Jarzynski’s equality, (5), to obtain a free energy estimate
Ᏺ n, j for this block
Ᏺ n, j ⫽ ⫺
冉
1
1
ln
␤
n
冘
冊
e⫺ ␤ W i .
i僆block j
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Journal of Computational Chemistry
Jarzynski estimate using all N data points—i.e., given a data set
with N values, ⌬F N is approximated by ⌬F all for subsampling.
This indeed makes sense since ⌬F all is the only estimate available
for ⌬F N given N work values. Bootstrapping, on the other hand,
always overestimates ⌬F N by comparison. Therefore, all results
shown below are obtained using the subsampled estimates for ⌬F n
only.
(8)
Extrapolation Methods
3. Repeat steps 1 and 2 until m blocks have been obtained, each
containing n values. The value of m will be discussed below.
Now the average ⌬F n and standard deviation ␴ n can be calculated using
⌬F n ⬟
1
m
冘
m
Ᏺ n, j ⫽
j⫽1
␴ n2 ⬟
1
m
1
m
冘冋
m
j⫽1
冘
⫺
冉
1
1
ln
␤
n
冘
i僆block j
e⫺ ␤ W i
冊册
,
(9)
m
共Ᏺ n, j ⫺ ⌬F n兲 2,
(10)
j⫽1
where the “⬟” denotes a computational estimate. This process is
carried out for every possible value of n (i.e. n ⫽ 1, 2, 3, . . . , N).
In previous work,25,49 the value for the number of blocks was
chosen to be m ⫽ N/n. The weakness of this choice is that a
reshuffling of the data set gives a new (generally different) set of
⌬F n approximations. Yet the estimates of ⌬F n should be insensitive to reshuffling, as the work values have no intrinsic order.
Therefore, we choose m large enough that the resulting ⌬F n
estimates do not depend upon the value of m. This is typically
accomplished with m ⬃ 100 ⫻ N/n. Note that our scheme uses
work values drawn at random without replacement. We call this a
“subsampled” ⌬F n following ref. 72.
It is worth mentioning that the “bootstrap” analysis strategy
provides yet another route to block averaging. For this scheme,
work values are drawn from (W 1 , W 2 , . . . , W N ) at random with
replacement—i.e., it is possible to draw a particular work value
more than once in a given block.73
The difference between the bootstrapped and subsampled
methods can be illustrated by considering a data set of N work
values where N ⫺ 1 values are large, and one value is very small.
Due to the highly nonlinear nature of the Jarzynski equality, the
single small work value will dominate eq. (9). Suppose estimates
for ⌬F n are generated for n ⫽ N using both of these methods. The
subsampled method will only have one ⌬F N estimate since reshuffling the work values has no effect when n ⫽ N. However, the
bootstrapped method calculates a ⌬F N estimate that is larger than
the subsampled ⌬F N due to the fact that it will draw the small
work value only a fraction of the time. A generalization of this
argument shows that the bootstrapped ⌬F n will exceed the subsampled ⌬F n for every value of n.
Although we have successfully used both bootstrapped and
subsampled ⌬F n estimates (data not shown), there is a distinct
advantage to using the subsampled estimates for the approaches
outlined below. The subsampled estimate for n ⫽ N is just the
Now that a smooth function has been obtained in the block
averaged free energy ⌬F n , shown in Figure 2, extrapolation to the
infinite data limit becomes feasible, as originally suggested by
Zuckerman and Woolf.25 The basic idea is to plot ⌬F n as a
function of some variable and then extrapolate to the infinite data
limit (n 3 ⬁). It is useful to plot ⌬F n as a function of ␹ ⫽ 1/n ␶ ,
with ␶ ⬍ 1.0, as shown in Figure 3.25 The plot was obtained by
generating 500 subsets of 100 work values each, drawn at random
from the palmitic to stearic acid mutation data set. Subsampled
⌬F n estimates were then computed for each subset of 100 work
values following the steps outlined later. The red squares with
errorbars show the averages of the ⌬F n estimates with standard
deviations obtained by using the 500 subsets. In other words,
Figure 3 shows the range of ⌬F n estimates expected from a set of
100 work values. The blue curve shows the ⌬F n data from a single
representative subset of 100 work values, and ⌬F all from Table 1
is included as the dashed black line. A value of ␶ ⫽ 0.5 was chosen
as discussed below. Note that the smallest uncertainty occurs for
␹ ⫽ n ⫽ 1 due to the fact that ⌬F n ( ␹ ⫽ 1) is simply the average
work and is not expected to vary substantially between each
subset.
Plotting ⌬F n estimates as a function of ␹ ⫽ 1/n ␶ as in Figure
3 (rather than vs. n) allows us to develop two simple extrapolation
schemes as explained in the following sections. The infinite data
limit (n 3 ⬁) now corresponds to ␹ ⫽ 0, and in addition, this
simple form gives a bounded interval ␹ 僆 (0, 1], rather than an
infinite one (such as ⌬F n as a function of n, shown in Fig. 2).
Below, two extrapolation schemes are outlined. The second,
based on cumulative integrals, appears to perform significantly
better than the first method, which relies on a linear extrapolation.
Linear Extrapolation
Because the block-averaged free energy ⌬F n in eq. (9) decreases
monotonically,2,25,29 one can hope to obtain a reasonable estimate
of ⌬F by simply continuing the curve in Figure 3 with a straight
line. Such a linear extrapolation (as opposed to higher order)
guarantees that our extrapolated results will not exceed ⌬F n for the
maximum available n, which estimates a rigorous upper bound for
the true ⌬F. 29
We test this extrapolation method using the test systems described earlier. This fully automated process contains the following steps: (1) draw a subset containing N work values (W 1 ,
W 2 , . . . , W N ) at random from the full data set. (2) Generate the
subsampled ⌬F n as a function of ␹ ⫽ 1/n ␶ for n ⬍ N for ␶ ⫽ 1/2.
(There is a range of valid choices 0.3 ⬍ ␶ ⬍ 0.7 that produce
similar results and are discussed below.) (3) Extrapolate to ␹ ⫽ 0
Calculating Free Energy Differences
using a straight line from the small-␹ tail of the ⌬F n data. The
intercept at ␹ ⫽ 0 is our extrapolated free energy ⌬F lin. (4) Using
these same N work values, calculate the Jarzynski free energy
estimate ⌬F Jarz using eq. (5). This process is repeated 500 times to
obtain the average and standard deviation of ⌬F lin and ⌬F Jarz.
The results from this simple extrapolation scheme are guaranteed to converge to the correct ⌬F as N 3 ⬁—that is, as the
size/quality of a data set is increased. This is simply due to the fact
that the leftmost ⌬F n value (Fig. 3) will approach ␹ ⫽ 0, leaving
less “room” for error. Of course, for N ⬍ ⬁, it cannot be shown
rigorously whether an extrapolated result will approach ⌬F from
above or below as N grows.
Our choice of ␶ ⫽ 0.5 for the linear extrapolation method is
arbitrary, yet there are several practical considerations that have
been taken into account. In general, smaller values of ␶ generate
estimates with greater uncertainty. This occurs because, as ␶ is
decreased, the ⌬F n data in Figure 3 are confined to a very small
region in the rightmost half of the figure. Thus, the extrapolation
must span a larger “distance,” and very small differences in the
slope of the tail will cause large differences in the extrapolated
estimate. This reasoning also provides the heuristic lower limit on
␶, since values of ␶ ⬍ 0.3 tend to “overshoot” ⌬F. On the other
hand, for larger ␶, the slope of the tail typically becomes very
large, also leading to uncertain extrapolations. (We note that ␶ ⫽
0.5 with subsampled block-averages, appears to be similar to ␶ ⬇
0.3 for the “naive” averaging scheme pursued previously25.)
In the linear method, there is some leeway in the amount of data
one considers to be the small-␹ tail of the ⌬F n data. For our results
we have chosen the tail to be 1/5 of the data. We found that
including more than a 1/5 of the data (say, 1/3 or 1/4) introduced
bias in the estimate, and that including much less data (say, 1/10 or
1/20) resulted in the same estimate as that for 1/5 but with larger
uncertainty.
A simple extension of the linear method shown here, is to fit
⌬F n to a nonlinear function, such as quadratic in ␹, as in previous
work by Zuckerman and Woolf.25 These nonlinear extrapolation
methods offer little, if any, improvement in the average accuracy
of ⌬F extrapolations. Furthermore, due to the inherent instability
of high order fits, the standard deviations for the nonlinear extrapolated results are much larger than those obtained for linear extrapolation (data not shown).
Cumulative Integral Extrapolation
As previously mentioned (see Fig. 3), the most precise ⌬F n values
are obtained for larger ␹ ⬇ 1 (i.e., smaller n), yet the previous
linear extrapolation scheme relies exclusively on small values of ␹.
Thus, in an effort to use the more precise large-␹ data to extrapolate ⌬F, we now formulate an integration scheme, which explicitly includes all values of ␹.
Consider ⌬F n in Figure 3 to be a smooth function ⌬F n ( ␹ ),
from ␹ ⫽ 0 to 1. We are free to consider the area under this
function, rewritten using integration by parts,
冕
1
0
d ␹ ⌬F n共 ␹ 兲 ⫽
冕
1
0
d ␹ 共1 ⫺ ␹ 兲
d⌬F n共 ␹ 兲
⫹ ⌬F n共 ␹ ⫽ 0兲.
d␹
(11)
1755
But ⌬F n ( ␹ ⫽ 0) ⫽ ⌬F is the desired free energy, and so one can
consider a cumulative integral estimate, defined by
冕 冉
1
CI共 ␹ 兲 ⬅
d ␹ ⬘ ⌬F n共 ␹ ⬘兲 ⫺ 共1 ⫺ ␹ ⬘兲
␹
冊
d⌬F n共 ␹ ⬘兲
,
d␹⬘
(12)
where CI 3 ⌬F as ␹ 3 0. We evaluate this integral for decreasing
␹—i.e., starting from CI(␹ ⫽ 1) ⫽ 0 and “accumulating” CI(␹) from
right to left in Figure 3. The derivative is evaluated numerically.
A sample plot of the cumulative integral is shown in Figure 4.
This plot was generated using two subsets (represented by open
and closed symbols) of 100 work values drawn at random from the
palmitic to stearic acid mutation test system. The dashed black line
shows the estimate using all available data ⌬F all, the blue squares
are the subsampled ⌬F n and the red triangles are CI( ␹ ). For each
of the two subsets, the value of ␶ was chosen to minimize the slope
of the tail of CI( ␹ ), as discussed below. For the open symbols it
was found that ␶ ⫽ 0.75 and for the closed symbols ␶ ⫽ 0.39. The
subset represented by the open symbols slightly overestimates
⌬F all, while the subset represented by the closed symbols slightly
underestimates ⌬F all— both as a result of statistical fluctuation.
To motivate our approach to choosing ␶, consider the case
where N is large enough to obtain ⌬F exactly. In this situation, if
␶ is chosen carefully, CI( ␹ ) will have nearly zero slope for small
␹, because accumulating more ␹ values will not change the estimate. Thus, while there is no guarantee, one can hope to extrapolate ⌬F by simply finding a value of ␶ where the slope
dCI( ␹ )/d ␹ is minimized for small ␹. The extrapolated free energy
⌬F ci will be the value of CI( ␹ ) for the smallest value of ␹
available, ␹min ⫽ 1/N ␶ .
Our fully automated test of this new extrapolation method is
very similar to that described in the previous section, with only
minor differences: (1) the value of ␶ is not fixed, rather ␶ is
automatically chosen to minimize the slope of the small-␹ tail of
CI( ␹ )—see Figure 4, and (2) once the value of ␶ is determined, the
free energy is estimated by ⌬F ci ⫽ CI( ␹ min). Once repeated CI
estimates ⌬F ci are obtained, comparison is made to the Jarzynski
estimate ⌬F Jarz using the same procedure as in the last section.
The CI approach, like the linear method, is guaranteed to yield
the true ⌬F for large N. However, also as with the linear approach,
the CI extrapolations are neither upper nor lower bounds. Yet, as
will be seen in the next section, the CI estimates appear to decrease
monotonically, suggesting a heuristic upper bound.
In implementing the CI method, as for the linear method we
chose the small-␹ tail of CI( ␹ ) to be 1/5 of the data. Including
more than a 1/5 of the data (1/4, 1/3) introduced bias in the
estimate, and including much less data (1/10, 1/20) resulted in the
same estimate as that for 1/5 but with larger uncertainty.
Results
The results of this initial study are very positive, as shown by the
rapid convergence of our extrapolated ⌬F estimates (Fig. 5).
Compared to ⌬F Jarz, estimates of ⌬F can be made with 5– 40-fold
fewer work values, i.e., less computational expense. Furthermore,
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Ytreberg and Zuckerman
•
Vol. 25, No. 14
the cumulative integral (CI) extrapolations compare favorably to
the independent estimates ⌬F ind, hinting that they “see” beyond
the poorly-sampled fast-growth data.
Figure 5 demonstrates how the linear and cumulative integral
(CI) methods described above compare to the Jarzynski estimate
of eq. (5), for each of the five test systems. For all of the plots
shown, the dashed black horizontal line corresponds to the Jarzynski estimate using all available work values, ⌬F all from Table 1.
The solid black horizontal line is an independent estimate ⌬F ind
described in Table 1. The red squares are averages of ⌬F Jarz using
eq. (5). The blue triangles are averages of ⌬F lin from Sec. VIA for
␶ ⫽ 0.5, and the vertical blue bars indicate the range of ⌬F lin
obtained for 0.4 ⬍ ␶ ⬍ 0.6 (i.e., not statistical uncertainty), with
lower estimates given by ␶ ⫽ 0.4. The green circles are averages
of ⌬F ci from the previous section. The inset for each plot shows
standard deviations for each of the estimates ␴ ⌬F . Averages and
standard deviations were obtained by performing 500 independent
trials for each estimate (⌬F Jarz, ⌬F lin , ⌬F ci) for every value of N.
Thus ␴ ⌬F indicates the expected statistical uncertainty—that is the
range of values one would expect if the calculation was performed
de novo.
A glance at Figure 5 reveals that the linearly extrapolated ⌬F lin
estimates converge to the estimate ⌬F all more quickly than the
Jarzynski estimate ⌬F Jarz. Remarkably, the linear estimates have
similar uncertainty to the Jarzynski estimates for all the test systems. Thus, with the possible exception of the Lennard–Jones fluid
test system, the linear extrapolation significantly outperforms the
Jarzynski estimates. The weakness of the linear method is the
arbitrary choice of ␶ ⫽ 0.5, and its reliance on the less certain
(large-n) ⌬F n values as explained earlier (see Fig. 3). Thus, the
linear method can never be expected to “see beyond the data”
analyzed.
The CI approach overcomes some of the disadvantages of the
linearly extrapolated estimates. The value of ␶ is chosen by an
automated process, and the CI method explicitly utilizes the more
certain (small-n) ⌬F n values. It should be noted that, while the
uncertainty is larger for the CI method than the linear and Jarzynski approaches in the last three test systems, these three systems
are believed to have biased data. (Here we use “biased” to indicate
兩⌬F ind ⫺ ⌬F all兩 ⬎ k B T, with ⌬F ind believed to be a reliable result
for the true ⌬F. The bias in these ⌬F all values can be independently verified using a self-consistency criterion (data not shown).
For the first two systems, where the fast-growth data sets are
known to be converged, the CI method has comparable uncertainty
to the linear and Jarzynski approaches.
Remarkably, the CI method appears to have the ability to partly
overcome bias due to incomplete sampling of the nonequilibrium
data. This can be seen by comparison to independent estimates
⌬F ind, obtained by methods other than fast-growth in four of the
five cases; (see Table 1). In all five of these systems, the CI method
converges to ⌬F ind while the linear and Jarzynski methods are
converging to ⌬F all. This implies that the CI method may have the
ability to “see beyond the data”—at least beyond the poorly
sampled estimates of ⌬F n for large n.
The reason that the CI method may be able to partially overcome the bias in the data is due to the fact that the CI method
explicitly uses all the ⌬F n data (as discussed earlier). By design,
the reliable small-n (large-␹) data is explicitly included. This is
•
Journal of Computational Chemistry
important because, as the size/quality of a data set is increased, the
small-n ⌬F n data will remain essentially unchanged, while the
large-n (small-␹) tail will change significantly. Thus, the CI
method is expected to be less sensitive to the bias in the data due
to a small number of work values.
To obtain a quantitative comparison between CI extrapolated
estimates ⌬F ci and Jarzynski estimates ⌬F Jarz, we ask the following question: how many work values are necessary to obtain a ⌬F
estimate that falls within approximately 2.0 kcal/mol of the independent estimate ⌬F ind? (Note that a comparison using ⌬F all
likely is not valid, because ⌬F all appears to be biased for the
growing chloride and methanol to ethane test systems.) Table 2
summarizes the results of this comparison. The CI estimates offer
a significant improvement over the Jarzynski estimates in all of the
test systems, with a 5– 40-fold decrease in the number of work
values needed to estimate ⌬F ind within 2.0 kcal/mol.
A comparison for the palmitic to stearic acid test system is not
attempted since the independent estimate ⌬F ind is obtained from a
fast-growth data set, and thus may suffer from bias. However,
because the data used for ⌬F ind was obtained for a slower switching speed than ⌬F all, it is encouraging that the CI estimates
overshoot ⌬F all, as it is expected that the actual value of ⌬F lies
closer to ⌬F ind than ⌬F all.
Conclusions
We have described two methods that improve standard nonequilibrium estimates of free energy differences, ⌬F: linear extrapolation, and cumulative integral (CI) extrapolation. These approaches apply to data obtained either in experiments or
computation. Both of the methods rely on block averaged free
energies ⌬F n , which are extrapolated to the infinite data limit, and
offer more rapid estimates of ⌬F than using the Jarzynski equality
alone. Five test systems were used in this study: a one-dimensional
“toy” system switched between two harmonic potentials, a chemical potential calculation for a Lennard–Jones fluid, growing a
chloride ion in water, a methanol-to-ethane mutation in water, and
a palmitic-to-stearic acid mutation in water.
Previous work by Zuckerman and Woolf25 used a quadratic
extrapolation method to estimate ⌬F. The present study offers
several improvements: (1) fully automated methods; (2) two new
schemes for calculating block averaged free energies, ⌬F n ; (3) a
key innovation in CI extrapolation’s use of the more reliable
(small-n) ⌬F n data; (4) a quantitative comparison between the CI
and Jarzynski ⌬F estimates, showing a 5– 40-fold decrease in the
number of work values needed for the CI estimates; (5) tests of our
extrapolation methods on five systems of varying complexity.
Bootstrapped and subsampled block averaged free energies are
described, which are key to reliable extrapolations. These approaches to estimating ⌬F n offer very smooth convergence properties, and hence, statistically reliable extrapolation.
The linearly extrapolated estimates have uncertainties similar
to the “standard” Jarzynski estimates, and produce results that
converge more rapidly. The success of the “simple-minded” linear
extrapolation illustrates the power of the underlying idea: systematic behavior in bias can be exploited.
Calculating Free Energy Differences
Figure 5. A comparison between ⌬F estimates for linear extrapolation, cumulative integral (CI) extrapolation,
and the Jarzynski equality for all of the test systems. The red squares are averages of Jarzynski estimates given
by eq. (5), the blue triangles are averages of linearly extrapolated estimates from the Extrapolation Methods
section for ␶ ⫽ 0.5, and the blue bars indicate the range of ⌬Flin for 0.4 ⬍ ␶ ⬍ 0.6 (i.e., not statistical
uncertainty). The green circles are averages of CI extrapolated estimates from the Extrapolation Methods
section. For each plot, the dashed black horizontal line indicates the Jarzynski estimate using all available data
⌬Fall, and the solid black horizontal line is a reliable independent estimate ⌬Find— both from Table 1. Note that
⌬Fall ⬇ ⌬Find for the first two plots. The inset in each plot shows the standard deviation for each of the
estimates. Averages and standard deviations were obtained by performing 500 independent trials for each
estimate for each value of N. Note that the vertical scales for each plot are different. [Color figure can be viewed
in the online issue, which is available at www.interscience.wiley.com.]
1757
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Ytreberg and Zuckerman
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Vol. 25, No. 14
Table 2. A Quantitative Comparison between the Cumulative Integral
(CI) Estimates ⌬F ci and Jarzynski Estimates ⌬F Jarz Shown in Figure 5.
Test system
N ci
N Jarz
CI ratio
One-dimensional to system
Chemical potential for a Lennard–Jones fluid
Growing chloride by 1.0 Å in water
Methanol-to-ethane mutation in water
40
700
150
400
200
3500
6000
9600
5
5
40
24
The first column shows the test system used in the comparison. The second
and third columns are the number of work values needed to obtain estimates that falls within 2.0 kcal/mol of ⌬F ind for the CI (N ci) and Jarzynski
(N Jarz) estimates. The last column is the ratios of these values—i.e., the
approximate improvement of the linear and CI estimates over the Jarzynski
estimates. Note that the palmitic to stearic acid test system is not included
since ⌬F ind is obtained from another (slower) fast-growth data set, and thus
may not represent the actual ⌬F.
A quantitative comparison between the CI extrapolated ⌬F
estimates and those using Jarzynski’s equality show a marked
decrease in the number of work values needed to estimate ⌬F
when using the CI estimates. The CI extrapolation method
obtains accurate ⌬F estimates using 5– 40 times less data than
the traditional Jarzynski approach. It was also found that the CI
method appears to have the ability to “see beyond” the statistically uncertain large-n block averages, ⌬Fn. The large n data
are dominated by small work values which, typically, are
poorly sampled. By relying on the full range of n, the CI
results compare very well with reliable, independent estimates
of ⌬F.
Other similar extrapolation methods could be developed that
may offer improvement over those presented here. Such methods
are currently under investigation by the authors. Future work by
the authors will use extrapolation methods, such as those described
here, to generate ⌬F estimates for large molecular systems such as
relative protein–ligand binding affinities.
Software which automatically performs the analyses described
here is available from the authors (www.ccbb.pitt.edu/zuckerman).
Acknowledgments
We gratefully acknowledge Jay Ponder for granting F.M.Y. permission to write code for TINKER to perform fast-growth free
energy calculations, and to Alan Grossfield for his assistance in
writing and implementing this code. We would like to thank Arun
Setty for fruitful discussion and valuable suggestions. Special
thanks are due to Hirsh Nanda and Thomas Woolf for permission
to use the raw data for the palmitic to stearic mutation, and to
David Hendrix and Chris Jarzynski for permission to use the raw
data for the chemical potential calculation.
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