  Computing the equilibrium properties of complex systems, such as free energy
differences, is often hampered by rare events in the dynamics. Enhanced
sampling methods may be used in order to speed up sampling by, for example,
using high temperatures, as in parallel tempering, or simulating with a biasing
potential such as in the case of umbrella sampling. The equilibrium properties
of the thermodynamic state of interest (e.g., lowest temperature or unbiased
potential) can be computed using reweighting estimators such as the weighted
histogram analysis method or the multistate Bennett acceptance ratio (MBAR).
weighted histogram analysis method and MBAR produce unbiased estimates, the
simulation samples from the global equilibria at their respective thermodynamic
state--a requirement that can be prohibitively expensive for some simulations
such as a large parallel tempering ensemble of an explicitly solvated
biomolecule. Here, we introduce the transition-based reweighting analysis
method (TRAM)--a class of estimators that exploit ideas from Markov modeling
and only require the simulation data to be in local equilibrium within subsets
of the configuration space. We formulate the expanded TRAM (xTRAM) estimator
that is shown to be asymptotically unbiased and a generalization of MBAR. Using
four exemplary systems of varying complexity, we demonstrate the improved
convergence (ranging from a twofold improvement to several orders of magnitude)
of xTRAM in comparison to a direct counting estimator and MBAR, with respect to
the invested simulation effort. Lastly, we introduce a random-swapping
simulation protocol that can be used with xTRAM, gaining orders-of-magnitude
advantages over simulation protocols that require the constraint of sampling
from a global equilibrium.
