Theory of a Systematic Computational Error in Free Energy Differences

Abstract

Systematic inaccuracy is inherent in any computational estimate of a non-linear average, due to the availability of only a finite number of data values, N . Free energy differences ∆F between two states or systems are critically important examples of such averages in physical, chemical and biological settings. Previous work has demonstrated, empirically, that the "finite-sampling error" can be very large -many times kBT -in ∆F estimates for simple molecular systems. Here, we present a theoretical description of the inaccuracy, including the exact solution of a sample problem, the precise asymptotic behavior in terms of 1/N for large N , the identification of universal law, and numerical illustrations. The theory relies on corrections to the central and other limit theorems, and thus a role is played by stable (Lévy) probability distributions.Introduction. Free energy difference calculations have a tremendous range of applications in physical, chemical, and biological systems; examples include computations relating magnetic phases, estimates of chemical potentials, and of binding affinities of ligands to proteins (e.g., [1][2][3][4][5][6]). Since the work of Kirkwood [7], it has been appreciated that the free energy difference, ∆F ≡ ∆F 0→1 , of switching from a Hamiltonian H 0 to H 1 is given by a non-linear average,

Phase transition in the Jarzynski estimator of free energy differences

“…In Refs. [36,37] E[ F M ] is referred to as the finite-data average free energy difference. This quantity is an estimator of F , albeit a biased one, in general.…”

Probing Molecular Free Energy Landscapes by Periodic Loading

“…This frequency should not be too large in order to enable the system to follow the external drive. Moreover, the Jarzynski procedure converges the slower the further one is away from equilibrium [25]. For too small ω, on the other hand, one does not generate enough crossings under the constraint of a finite total measuring time.…”

Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies

“…Experimental testing of Jarzynski's equality in single-molecule force experiments 16 used the P5ab RNA hairpin 7,8 , which can be folded and unfolded quasi-reversibly. But for processes that occur far from equilibrium, the applicability of Jarzynski's equality is hampered by large statistical uncertainties that arise from the sensitivity of the exponential average to rare events 17,18 (low values of W). Moreover, although the equality 〈W〉 = ΔG holds for processes occurring near equilibrium, spatial drift in the experimental system usually makes it difficult in practice to extract unfolding free energies using small loading rates (below a few pN s −1 ).…”