TY - JOUR

T1 - Reconstructing Free Energy Profiles from Nonequilibrium Relaxation Trajectories

AU - Zhang, Qi

AU - Brujić, Jasna

AU - Vanden-Eijnden, Eric

N1 - Funding Information:
Acknowledgements We thank Jean-Philippe Bouchaud for inspiration. We also thank Claudia de Rham and Maxime Clusel for discussions about path integral methods. JB holds a Career Award at the Scientific Interface from the Burroughs Wellcome Fund and was supported in part by New York University Materials Research Science and Engineering Center Award DMR-0820341 and a Career Award 0955621. The research of EVE was supported in part by NSF grants DMS-0718172 and DMS-0708140, and ONR grant N00014-04-1-6046.

PY - 2011/7

Y1 - 2011/7

N2 - 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.

AB - 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.

KW - Bayesian sampling

KW - Free energy calculation

KW - Maximum likelihood

KW - Nonequilibrium sampling

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U2 - 10.1007/s10955-011-0242-7

DO - 10.1007/s10955-011-0242-7

M3 - Article

AN - SCOPUS:79960978673

VL - 144

SP - 344

EP - 366

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 2

ER -