Valid and efficient formula for free energy difference from nonequilibrium work

Abstract

Atomic force microscopes and optical tweezers afford direct probe into the inner working of single biomolecules by mechanically unfolding them.^1-15^ Critical to the success of this type of probe is to correctly extract the free energy differences between the various conformations of a protein/nucleic acid along its forced unfolding pathways. Current studies rely on the Jarzynski equality^16^ (JE) or its undergirding Crooks fluctuation theorem^17^ (CFT), even though questions remain on its validity^17-19^ and on its accuracy.^13,20-21^ The validity of JE relies on the assumption of microscopic reversibility.^17,18^ The dynamics of biomolecules, however, is Langevin stochastic in nature. The frictional force in the Langevin equation breaks the time reversal symmetry and renders the dynamics microscopically irreversible even though detailed balance holds true. The inaccuracy of JE has largely been attributed to the fact that one cannot sample a large enough number of unfolding paths in a given study, experimental or computational.^13,15^ Here I show that both of these questions can be answered with a new equation relating the nonequilibrium work to the equilibrium free energy difference. The validity of this new equation requires detailed balance but not microscopic reversibility. Taking into the new equation equal number of unfolding and refolding paths, the accuracy is enhanced ten folds in comparison to a JE study based on a similar but larger number of unfolding paths.