Re: error estimate in FEP calculation

From: Chris Chipot (
Date: Sun Oct 29 2006 - 09:51:43 CST


When is your trajectory long enough to assume safely that the free
energy calculation has converged? Sources of errors likely to be at
play are diverse, and, hence, will modulate the results differently.
The choice of the force field parameters, which correspond to a
systematic error, undoubtedly affects the results of the simulation,
but this contribution can be largely concealed by the statistical
error arising from insufficient sampling. Paradoxically, exceedingly
short free energy calculations employing inadequate force field
parameters may, nonetheless, yield the correct answer. Under the
hypothetical assumption of an optimally designed potential energy
function, quasi non-ergodicity scenarios constitute a common pitfall
towards fully converged simulations.

Appreciation of the statistical error has been devised following
different schemes. Historically, the free energy changes for the
lambda -> lambda + delta lambda and the lambda -> lambda - delta
lambda perturbations were computed simultaneously to provide the
hysteresis between the forward and the reverse transformations. In
practice, it can be shown that when delta lambda is sufficiently
small, the hysteresis of such "double-wide sampling" (Jorgensen,
1985) simulation becomes negligible, irrespective of the amount of
sampling generated in each window - as would be the case in a
"slow-growth" calculation. A somewhat less arguable point of view
consists in performing the transformation in the forward, a -> b,
and in the reverse, b -> a, directions. Micro-reversibility imposes
that, in principle, Delta A(b -> a) = -Delta A(a -> b). Unfortunately,
forward and reverse transformations do not necessarily share the
same convergence properties. Case in point, the Widom's insertion
and deletion of a particle: Whereas the former simulation converges
rapidly towards the expected excess chemical potential, the latter
never does. This shortcoming can be ascribed to the fact that
configurations in which a cavity does not exist where a real atom
is present are never sampled. In terms of density of states, this
scenario would translate into the distribution of a embracing that
of b entirely, thereby ensuring a proper convergence of the forward
simulation, whereas the same cannot be said for the reciprocal,
reverse transformation. Estimation of errors based on forward and
reverse simulations should, therefore, be considered with great
care. In fact, appropriate combination of the two can be used
profitably to improve the accuracy of free energy calculations
(Bennett's acceptance ratio, simple overlap sampling, etc... - Lu,

In FEP, convergence may be probed by monitoring the time-evolution
of the ensemble average. This is, however, a necessary, but not
sufficient condition for convergence, because apparent plateaus of
the ensemble average often conceal anomalous overlap of the density
of states characterizing the initial and the final states. The latter
should be the key-criterion to ascertain the local convergence of the
simulation for those degrees of freedom that are effectively sampled.
Statistical errors in FEP calculations may be estimated by means of
a first-order expansion of the free energy, which involves an
estimation of the sampling ratio of the latter of the calculation
(Straatsma, 1986).

In the idealistic cases where a thermodynamic cycle can be defined,
closure of the latter imposes that the sum of individual free energy
contributions sum up to zero. In principle, any deviation from this
target should provide a valuable guidance to improve sampling
efficiency. In practice, discrimination of the faulty transformation,
or transformations, is cumbersome on account of possible mutual
compensation or cancellation of errors.

As I mentioned it previously, visual inspection of the density of
states indicates whether the free energy calculation has converged.
Deficiencies in the overlap of the two distributions is also
suggestive of possible errors, but it should be kept in mind that
approximations like a first-order expansion of the free energy only
reflect the statistical precision of the computation, and evidently
do not account for fluctuations in the system occurring over long
time scales. In sharp contrast, the statistical accuracy is expected
o yield a more faithful picture of the degrees of freedom that have
been actually sampled. The safest route to estimate this quantity
consists in performing the same free energy calculation, starting
from different regions of the phase space - the error is defined as
the root mean square deviation over the different simulations.
Semantically speaking, the error measured from one individual run
yields the statistical precision of the free energy calculation,
whereas that derived from the ensemble of simulations provides its
statistical accuracy.

Chris Chipot a écrit :
> Dear all,
> Can someone please recommend references for the estimate of error
> (including sampling error) for the free energy perturbation calculation?
> I want to know how reliable my delt_delt_G is.
> Thanks so much!
> Jeny


Chris Chipot, Ph.D.
Equipe de dynamique des assemblages membranaires
Unité mixte de recherche CNRS/UHP No 7565
Université Henri Poincaré - Nancy 1 Phone: (33) 3-83-68-40-97
B.P. 239 Fax: (33) 3-83-68-43-87
54506 Vandœuvre-lès-Nancy Cedex, France


       To sin by silence when we should protest makes cowards out of men
                                                     Ella Wheeler Wilcox

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