Reconstructing Free Energy Profiles from Nonequilibrium Relaxation Trajectories |
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Authors: | Qi Zhang Jasna Bruji? Eric Vanden-Eijnden |
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Institution: | 1.Courant Institute of Mathematical Sciences,New York University,New York,USA;2.Center for Soft Matter Research, Physics Department,New York University,New York,USA;3.Mechanobiology Institute,National University of Singapore,Singapore,Singapore |
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Abstract: | Reconstructing free energy profiles is an important problem in bimolecular reactions, protein folding or allosteric conformational
changes. Nonequilibrium trajectories are readily measured experimentally, but their statistical significance and relation
to equilibrium system properties still call for rigorous methods of assessment and interpretation. Here we introduce methods
to compute the equilibrium free energy profile of a given variable from a set of short nonequilibrium trajectories, obtained
by externally driving a system out of equilibrium and subsequently observing its relaxation. This protocol is not suitable
for the Jarzynski equality since the irreversible work on the system is instantaneous. Assuming that the variable of interest
satisfies an overdamped Langevin equation, which is frequently used for modeling biomolecular processes, we show that the
trajectories sample a nonequilibrium stationary distribution that can be calculated in closed form. This allows for the estimation
of the free energy via an inversion procedure that is analogous to that used in equilibrium and bypasses more complicated
path integral methods, which we derive for comparison. We generalize the inversion procedure to systems with a diffusion constant
that depends on the reaction coordinate, as is the case in protein folding, as well as to protocols in which the trajectories
are initiated at random points. Using only a statistical pool of tens of synthetic trajectories, we demonstrate the versatility
of these methods by reconstructing double and multi-well potentials, as well as a proposed profile for the hydrophobic collapse
of a protein. |
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