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We have combined infrared (IR) experiments with molecular dynamics (MD) simulations in solution at finite temperature to analyse the vibrational signature of the small floppy peptide Alanine-Leucine. IR spectra computed from first-principles MD simulations exhibit no distinct differences between conformational clusters of -helix or -sheet-like folds with different orientations of the bulky leucine side chain. All computed spectra show two prominent bands, in good agreement with the experiment, that are assigned to the stretch vibrations of the carbonyl and carboxyl group, respectively. Variations in band widths and exact maxima are likely due to small fluctuations in the backbone torsion angles.
We have investigated how Langevin dynamics is affected by the friction
coefficient using the novel algorithm ISOKANN, which combines the transfer
operator approach with modern machine learning techniques. ISOKANN describes
the dynamics in terms of an invariant subspace projection of the Koopman
operator defined in the entire state space, avoiding approximations due to
dimensionality reduction and discretization. Our results are consistent with
the Kramers turnover and show that in the low and moderate friction regimes,
metastable macro-states and transition rates are defined in phase space, not
only in position space.
Estimating the rate of rare conformational changes in molecular systems is one of the goals of molecular dynamics simulations. In the past few decades, a lot of progress has been done in data-based approaches toward this problem. In contrast, model-based methods, such as the Square Root Approximation (SqRA), directly derive these quantities from the potential energy functions. In this article, we demonstrate how the SqRA formalism naturally blends with the tensor structure obtained by coupling multiple systems, resulting in the tensor-based Square Root Approximation (tSqRA). It enables efficient treatment of high-dimensional systems using the SqRA and provides an algebraic expression of the impact of coupling energies between molecular subsystems. Based on the tSqRA, we also develop the projected rate estimation, a hybrid data-model-based algorithm that efficiently estimates the slowest rates for coupled systems. In addition, we investigate the possibility of integrating low-rank approximations within this framework to maximize the potential of the tSqRA.
The dynamical response of molecular systems, when the potential energy function is perturbed at a microscopic level, is difficult to predict without a numerical or laboratory experiment. This is due to the non-linearity and high-dimensionality of molecular systems. An efficient investigation of such a behaviour is necessary to better understand the nature of molecules and to improve the predictability of Molecular Dynamics simulations. In this thesis we propose a reweighting scheme for Markov State Models (MSMs), based on the Girsanov theorem, that permits to reduce the computational cost of the analysis when the potential energy function of a molecule is perturbed. The method has been successfully extended and implemented with metadynamics, in order to build the MSM of a molecular system in a significantly shorter computational time compared to a standard unbiased MD simulation. We also propose a new method to discretize the infinitesimal generator into a rate matrix, that could be used to efficiently study Hamiltonian perturbations as well.
This chapter explains the different ways in which Markov State Models (MSMs) can be helpful in structure-based drug design. MSMs are constructed from the time series of molecular dynamics (MD), which can be generated by classical MD simulations. Several features of the MSMs can be utilized for rational drug design. The discretization of a validated MSM is particularly suited to extract meaningful representatives from the conformational ensemble, because the discretization yields a small number of microstates and mirrors the features of the free energy landscape. Long-lived conformations consist of a set of microstates which show high transition rates within the set and low transition rates to microstates outside of the set. The Bayesian agglomerative clustering engine (BACE) algorithm uses the observed transition counts to extract long-lived conformations from an MSM. By iteratively merging microstates according to the Bayes factor and recalculating the Bayes-factor matrix, the algorithm yields an aggregation of the microstates into long-lived conformations.
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.
The sensitivity of molecular dynamics on changes in the potential energy function plays an important role in understanding the dynamics and function of complex molecules. We present a method to obtain path ensemble averages of a perturbed dynamics from a set of paths generated by a reference dynamics. It is based on the concept of path probability measure and the Girsanov theorem, a result from stochastic analysis to estimate a change of measure of a path ensemble. Since Markov state models (MSMs) of the molecular dynamics can be formulated as a combined phase-space and path ensemble average, the method can be extended to reweight MSMs by combining it with a reweighting of the Boltzmann distribution. We demonstrate how to efficiently implement the Girsanov reweighting in a molecular dynamics simulation program by calculating parts of the reweighting factor “on the fly” during the simulation, and we benchmark the method on test systems ranging from a two-dimensional diffusion process and an artificial many-body system to alanine dipeptide and valine dipeptide in implicit and explicit water. The method can be used to study the sensitivity of molecular dynamics on external perturbations as well as to reweight trajectories generated by enhanced sampling schemes to the original dynamics.
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
In recent years, for the analysis of molecular processes, the estimation of time-scales and transition rates has become fundamental. Estimating the transition rates between molecular conformations is—from a mathematical point of view—an invariant subspace projection problem. We present a method to project the infinitesimal generator acting on function space to a low-dimensional rate matrix. This projection can be performed in two steps. First, we discretize the conformational space in a Voronoi tessellation, then the transition rates between adjacent cells is approximated by the geometric average of the Boltzmann weights of the Voronoi cells. This method demonstrates that there is a direct relation between the potential energy surface of molecular structures and the transition rates of conformational changes. We will show also that this approximation is correct and converges to the generator of the Smoluchowski equation in the limit of infinitely small Voronoi cells. We present results for a two dimensional diffusion process and alanine dipeptide as a high-dimensional system.
Extracting the kinetic properties of a system whose dynamics depend on the pH of the environment with which it exchanges energy and atoms requires sampling the Grand Canonical Ensemble. As an alternative, we present a novel strategy that requires simulating only the most recurrent Canonical Ensembles that compose the Grand Canonical Ensemble. The simulations are used to estimate the Gran Canonical distribution for a specific pH value by reweighting and to construct the transition rate matrix by discretizing the Fokker-Planck equation by Square Root Approximation and robust Perron Cluster Cluster Analysis. As an application, we have studied the tripeptide Ala-Asp-Ala.