being collected. Nonetheless, it is important to try to exclude
nonequilibrated regions of the simulations from the analysis and
two main approaches have been proposed for this task. One
approach is to plot the convergence of the free energy results in
both the “forward” and “reverse” directions, that is, plotting the
free energy estimate as a function of the simulated time by includ-
ing more data while going both fromt 0 totf,and fromtftot 0 ,
where tf is the time of the final snapshot in the simulations
[30]. When all data is included, the forward and reverse estimates
are the same. If the data have been collected at equilibrium and
from the same distributions, and if the calculations have converged,
the forward and reverse plots should agree within error. If, when
considering the data from the two ends of the simulations, the
“forward” and “reverse” free energy estimates converge to two
separate and well-defines values, this indicate that nonequilibrated
samples have likely been included in the analysis [30]. A different
approach, which detects the equilibrated region automatically, has
instead recently been proposed by Chodera [83]. In this automated
approach, the autocorrelation time is calculated while removing
larger portions of the simulation data, and the equilibration time
is chosen as the time that maximizes the number of effective uncor-
related samples [83].
Phase space overlap is another property that should be checked
when analyzing the results. Poor overlap in specific regions can be
resolved by using more, or differently spaced,λwindows. Little
overlap does not necessarily lead to wrong free energy estimates,
but does result in increased uncertainty; the user should decide
what level of uncertainty is acceptable. However, when using per-
turbation approaches, very poor overlap can result in an underesti-
mate of the variance and an inaccurate free energy estimate [30]. If
the MBAR estimator is used, it is possible to obtain an overlap
matrix that provides a quantitative estimate of the phase space
overlap between simulations [24]. This matrix shows the probabil-
ity of a sample from stateihaving been generated in statej, thus
providing an indication of the degree of phase space overlap. An
example of such an overlap matrix is shown in Fig.5. The overlap
matrix should be tridiagonal, which means that all elements of the
main diagonal, as well as the diagonal below and above it, should be
nonzero. A value of 0.03 for the tridiagonal elements has been
suggested as a threshold to highlight potential phase space overlap
issues between two windows [30]. If phase space overlap issues are
identified during testing of the free energy protocol for a specific
system, it might be possible to adjust the spacing of theλwindows
so to increase overlap and reduce the uncertainty of the free energy
estimate. If phase space overlap issues are identified instead after
running extensive simulations, it is still possible to run additionalλ
windows in the problematic area of the path, which will be more
cost-effective than rerunning the whole calculation with a different
Absolute Alchemical Free Energy 219