xTRAM: Estimating equilibrium expectations from time-correlated simulation data at multiple thermodynamic states

Antonia S.J.S. Mey, Hao Wu, Frank Noé*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 states-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 randomswapping 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.

Original languageEnglish
Article number041018
Number of pages1
JournalPhysical Review X
Issue number4
Publication statusPublished - 29 Oct 2014


  • Biological physics
  • Computational physics
  • Statistical physics


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