Calculation of the potential of mean force from nonequilibrium measurements via maximum likelihood estimators

We present an approach to the estimate of the potential of mean force along a generic reaction coordinate based on maximum likelihood methods and path-ensemble averages in systems driven far from equilibrium. Following similar arguments, various free energy estimators can be recovered, all providing comparable computational accuracy. The method, applied to the unfolding process of the $\alpha$-helix form of an alanine decapeptide, gives results in good agreement with thermodynamic integration.

Figure 1

Schematic representation of forward and reverse realizations with the notation used in the text.

Figure 2

PMF of ${\mathrm{A}}_{10}$ as a function of the reaction coordinate $\zeta$ (end-to-end distance). Squares: Eq. (8); triangles: Eq. (9); circles: Eq. (16); solid lines: thermodynamic integration (TI). The PMF profiles from ML estimators are calculated using $F$ and $R$ realizations performed with pulling speed of $80\mathrm{\AA }{\mathrm{ns}}^{-1}$. For the sake of clarity the curves are upshifted.

Figure 3

$\sigma$ value [Eq. (18)] as a function of the pulling speed for various ML estimators. Squares: Eq. (8); triangles: Eq. (9); circles: Eq. (16). The lines are drawn as a guide for eyes.