Unbiased free energy estimates in fast nonequilibrium transformations using Gaussian mixtures

Abstract

In this paper, we present an improved method for obtaining unbiased estimates of the free energy difference between two thermodynamic states using the work distribution measured in nonequilibrium driven experiments connecting these states. The method is based on the assumption that any observed work distribution is given by a mixture of Gaussian distributions, whose normal components are identical in either direction of the nonequilibrium process, with weights regulated by the Crooks theorem. Using the prototypical example for the driven unfolding/folding of deca-alanine, we show that the predicted behavior of the forward and reverse work distributions, assuming a combination of only two Gaussian components with Crooks derived weights, explains surprisingly well the striking asymmetry in the observed distributions at fast pulling speeds. The proposed methodology opens the way for a perfectly parallel implementation of Jarzynski-based free energy calculations in complex systems.

I. Dissociation free energies of drug–receptor systems via non-equilibrium alchemical simulations: a theoretical framework

“…While the decoupling free energy of the ligand in the bulk, ∆G L , bears no dependence on the reference state, when alchemically decoupling the ligand in the complex, the computed free energy ∆G RL depends on the effective reference concentration of (or volume available to) the ligand implied in the simulation. [8,31,43,45] For example, when the RL complex is unrestrained except for periodic boundary conditions, the volume available to the ligand is apparently that of the simulation box [21,31,32,45].…”

“…imposes that the underlying reverse work distribution P BA (−W 0→1 ) for the fast growth process must also be Gaussian with the same variance σ and with mean work given by [32,52,54,56,59,60] hence providing an unbiased unidirectional estimate of the annihilation free energy based on the forward process alone of the form…”

“…The second simplifying assumption implies no loss of generality [32] and is based on the reasonable expectation that the mean dissipation, βσ 2 /2, depends in essence on the particle density in the given thermodynamic conditions and that therefore σ should be weakly dependent on the environment surrounding the ligand. Given the forward distribution Eq.…”

“…[28][29][30] The absolute standard binding free energy can be finally computed as the free energy difference between the two decoupling process [18] using a correction [8,31,32] to account for the reversible work needed to bring the ligand volume from that imposed in the MD simulation to that of the standard state. The alchemical procedure can be merged with GE approaches by letting λ hopping between neighboring λ states so as to favor conformational sampling of the ligand.…”

“…[31,34,37] We further show that most of pitfalls and entanglements in the equilibrium approach can be resolved by using the recently proposed non-equilibrium variant of the alchemical method, named Fast Switching Double Annihilation Method (FS-DAM) [38] relying on the production of many independent non-equilibrium trajectories with a continuous dynamical evolution of an externally driven alchemical coordinate, [39] completing the alchemical decoupling of the ligand in a matter of few tens of picoseconds rather than nanoseconds. The absolute binding free energy is recovered from the annihilation work distributions by applying an unidirectional free energy estimate, on the assumption that any observed work distribution is given by a mixture of Gaussian distributions, [32] whose normal components are identical in either direction of the non-equilibrium process, with weights regulated by the Crooks theorem. [40] In FS-DAM, the sampling issue at intermediate λ state is eliminated altogether.…”