TY - JOUR A1 - Donati, Luca A1 - Weber, Marcus A1 - Keller, Bettina G. T1 - Markov models from the square root approximation of the Fokker–Planck equation: calculating the grid-dependent flux JF - Journal of Physics: Condensed Matter N2 - Molecular dynamics (MD) are extremely complex, yet understanding the slow components of their dynamics is essential to understanding their macroscopic properties. To achieve this, one models the MD as a stochastic process and analyses the dominant eigenfunctions of the associated Fokker–Planck operator, or of closely related transfer operators. So far, the calculation of the discretized operators requires extensive MD simulations. The square-root approximation of the Fokker–Planck equation is a method to calculate transition rates as a ratio of the Boltzmann densities of neighboring grid cells times a flux, and can in principle be calculated without a simulation. In a previous work we still used MD simulations to determine the flux. Here, we propose several methods to calculate the exact or approximate flux for various grid types, and thus estimate the rate matrix without a simulation. Using model potentials we test computational efficiency of the methods, and the accuracy with which they reproduce the dominant eigenfunctions and eigenvalues. For these model potentials, rate matrices with up to $\mathcal{O}\left(1{0}^{6}\right)$ states can be obtained within seconds on a single high-performance compute server if regular grids are used. Y1 - 2021 U6 - https://doi.org/10.1088/1361-648X/abd5f7 VL - 33 IS - 11 SP - 115902 ER - TY - JOUR A1 - Kieninger, Stefanie A1 - Donati, Luca A1 - Keller, Bettina G. T1 - Dynamical reweighting methods for Markov models JF - Current Opinion in Structural Biology N2 - Conformational dynamics is essential to biomolecular processes. Markov State Models (MSMs) are widely used to elucidate dynamic properties of molecular systems from unbiased Molecular Dynamics (MD). However, the implementation of reweighting schemes for MSMs to analyze biased simulations is still at an early stage of development. Several dynamical reweighing approaches have been proposed, which can be classified as approaches based on (i) Kramers rate theory, (ii) rescaling of the probability density flux, (iii) reweighting by formulating a likelihood function, (iv) path reweighting. We present the state-of-the-art and discuss the methodological differences of these methods, their limitations and recent applications. Y1 - 2020 U6 - https://doi.org/10.1016/j.sbi.2019.12.018 VL - 61 SP - 124 EP - 131 ER - TY - JOUR A1 - Donati, Luca A1 - Keller, Bettina G. T1 - Girsanov reweighting for metadynamics simulations JF - The Journal of Chemical Physics N2 - Metadynamics is a computational method to explore the phase space of a molecular system. Gaussian functions are added along relevant coordinates on the fly during a molecular-dynamics simulation to force the system to escape from minima in the potential energy function. The dynamics in the resulting trajectory are however unphysical and cannot be used directly to estimate dynamical properties of the system. Girsanov reweighting is a recent method used to construct the Markov State Model (MSM) of a system subjected to an external perturbation. With the combination of these two techniques—metadynamics/Girsanov-reweighting—the unphysical dynamics in a metadynamics simulation can be reweighted to obtain the MSM of the unbiased system. We demonstrate the method on a one-dimensional diffusion process, alanine dipeptide, and the hexapeptide Val-Gly-Val-Ala-Pro-Gly (VGVAPG). The results are in excellent agreement with the MSMs obtained from direct unbiased simulations of these systems. We also apply metadynamics/Girsanov-reweighting to a β-hairpin peptide, whose dynamics is too slow to efficiently explore its phase space by direct simulation Y1 - 2018 U6 - https://doi.org/10.1063/1.5027728 VL - 149 IS - 7 SP - 072335 ER -