Metadynamics convergence law in a multidimensional system

Abstract

Metadynamics is a powerful sampling technique that uses a nonequilibrium history-dependent process to reconstruct the free-energy surface as a function of the relevant collective variables s . In Bussi [Phys. Rev. Lett. 96, 090601 (2006)] it is proved that, in a Langevin process, metadynamics provides an unbiased estimate of the free energy F(s) . We here study the convergence properties of this approach in a multidimensional system, with a Hamiltonian depending on several variables. Specifically, we show that in a Monte Carlo metadynamics simulation of an Ising model the time average of the history-dependent potential converge to F(s) with the same law of an umbrella sampling performed in optimal conditions (i.e., with a bias exactly equal to the negative of the free energy). Remarkably, after a short transient, the error becomes approximately independent on the filling speed, showing that even in out-of-equilibrium conditions metadynamics allows recovering an accurate estimate of F(s) . These results have been obtained introducing a functional form of the history-dependent potential that avoids the onset of systematic errors near the boundaries of the free-energy landscape.

Two-state dynamics of the SH3–SH2 tandem of Abl kinase and the allosteric role of the N-cap

“…Boundary quadratic potentials were used to restrict S 1 and S 3 to values between 1 and 2 and S 3 to values between 0 and 3 nm 2 . To avoid systematic errors in the calculated free en-ergies, a boundary correction scheme was also used at these values, as described previously (51). In the metadynamics simulations that included the Ncap region or the SH2-KL, distance restraints were applied to a subset of residue pairs (or centers-of-mass) to ensure that these remained bound to the SH3-SH2 tandem.…”

Sampling free energy surfaces as slices by combining umbrella sampling and metadynamics

“…Using WHAM based on Equation (14) and Equation (15), reweight the umbrella potential as well as combine the M distribution functions to get P(s α , s β ). Note that P u h (s α , s β ) can be the input for any standard WHAM programs, but by setting the bias along the s β coordinates to zero.…”

Molecular Dynamics in the Multicanonical Ensemble: Equivalence of Wang–Landau Sampling, Statistical Temperature Molecular Dynamics, and Metadynamics

“…The focus of this paper is on this explicit equivalence; discussions concerning the overall convergence behavior and analogies in that regard between different strategies in Wang-Landau sampling and metadynamics can be found in the literature, see Refs. [14,[33][34][35]. We believe that a consistent view of flat-histogram methods as presented here is beneficial to foster transfer of ideas between the respective communities.…”