Estimation of the infinitesimal generator by square-root approximation

For the analysis of molecular processes, the estimation of time-scales, i.e., transition rates, is very important. Estimating the transition rates between molecular conformations is -- from a mathematical point of view -- an invariant subspace projection problem. A certain infinitesimal generator acting on function space is projected to a low-dimensional rate matrix. This projection can be performed in two steps. First, the infinitesimal generator is discretized, then the invariant subspace is approxi-mated and used for the subspace projection. In our approach, the discretization will be based on a Voronoi tessellation of the conformational space. We will show that the discretized infinitesimal generator can simply be approximated by the geometric average of the Boltzmann weights of the Voronoi cells. Thus, there is a direct correla-tion between the potential energy surface of molecular structures and the transition rates of conformational changes. We present results for a 2d-diffusion process and Alanine dipeptide

Publisher’s Note: “Density-based cluster algorithms for the identification of core sets” [J. Chem. Phys. 145, 164104 (2016)]

Original Article: J. Chem. Phys. 145, 164104 (2016) This article was originally published online on 26 October 2016 with an error in the second author’s name. “Bettina G. Lemke” should be “Bettina G. Keller.” AIP Publishing apologizes for this error. All online versions of the article were corrected on 27 October 2016; the article is correct as it appears in the printed version of the journal

GROMACS Stochastic Dynamics and BAOAB are equivalent configurational sampling algorithms

Two of the most widely used Langevin integrators for molecular dynamics simulations are the GROMACS Stochastic Dynamics (GSD) integrator and the splitting method BAOAB. We show that the GROMACS Stochastic Dynamics integrator is equal to the less frequently used splitting method BAOA. It immediately follows that GSD and BAOAB sample the same configurations and have the same high configurational accuracy. Our numerical results indicate that GSD/BAOA has higher kinetic accuracy than BAOAB

Path probability ratios for Langevin dynamics -- exact and approximate

Path reweighting is a principally exact method to estimate dynamic properties from biased simulations - provided that the path probability ratio matches the stochastic integrator used in the simulation. Previously reported path probability ratios match the Euler-Maruyama scheme for overdamped Langevin dynamics. Since MD simulations use Langevin dynamics rather than overdamped Langevin dynamics, this severely impedes the application of path reweighting methods. Here, we derive the path probability ratio $M_L$ for Langevin dynamics propagated by a variant of the Langevin Leapfrog integrator. This new path probability ratio allows for exact reweighting of Langevin dynamics propagated by this integrator. We also show that a previously derived approximate path probability ratio $M_{\mathrm{approx}}$ differs from the exact $M_L$ only by $\mathcal{O}(\xi^4\Delta t^4)$, and thus yields highly accurate dynamic reweighting results. ($\Delta t$ is the integration time step, $\xi$ is the collision rate.) The results are tested and the efficiency of path-reweighting is explored using butane as an example

Grid-based state space exploration for molecular binding

Binding processes are difficult to sample with molecular-dynamics (MD) simulations. In particular, the state space exploration is often incomplete. Evaluating the molecular interaction energy on a grid circumvents this problem but is heavily limited by state space dimensionality. Here, we make the first steps towards a low-dimensional grid-based model of molecular binding. We discretise the state space of relative positions and orientations of the two molecules under the rigid body assumption.The corresponding program is published as the Python package molgri. For the rotational component of the grids, we test algorithms based on Euler angles, polyhedra and quaternions, of which the polyhedra-based are the most uniform. The program outputs a sequence of molecular structures that can be easily processed by standard MD programs to calculate grid point energies. We demonstrate the grid-based approach on two molecular systems: a water dimer and a coiled-coil protein interacting with a chloride anion. For the second system we relax the rigid-body assumption and improve the accuracy of the grid point energies by an energy minimisation. In both cases, oriented bonding patterns and energies confirm expectations from chemical intuition and MD simulations. We also demonstrate how analysis of energy contributions on a grid can be performed and demonstrate that electrostatically-driven association is sufficiently resolved by point-energy calculations. Overall, grid-based models of molecular binding are potentially a powerful complement to molecular sampling approaches, and we see the potential to expand the method to quantum chemistry and flexible docking applications.Comment: 13 pages, 7 figure

A review of Girsanov Reweighting and of Square Root Approximation for building molecular Markov State Models

Dynamical reweighting methods permit to estimate kinetic observables of a stochastic process governed by a target potential $\tilde{V}(x)$ from trajectories that have been generated at a different potential $V(x)$. In this article, we present Girsanov reweighting and Square Root Approximation (SqRA): the first method reweights path probabilities exploiting the Girsanov theorem and can be applied to Markov State Models (MSMs) to reweight transition probabilities; the second method was originally developed to discretize the Fokker-Planck operator into a transition rate matrix, but here we implement it into a reweighting scheme for transition rates. We begin by reviewing the theoretical background of the methods, then present two applications relevant to Molecular Dynamics (MD), highlighting their strengths and weaknesses

Markov models from the square root approximation of the Fokker–Planck equation: calculating the grid-dependent flux

Abstract 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 O(106) states can be obtained within seconds on a single high-performance compute server if regular grids are used

How chromophore labels shape the structure and dynamics of a peptide hydrogel

Biocompatible and functionalizable hydrogels have a wide range of (potential) medicinal applications. In contrast to conventional hydrogels formed by interconnected or interlocked polymer chains, self-assembled hydrogels form from small building blocks like short peptide chains. This has the advantage that the building blocks can be functionalized separately and then mixed to obtain the desired properties. However, the hydrogelation process for these systems, especially those with very low polymer weight percentage (< 1 wt%), is not well understood, and therefore it is hard to predict whether a given molecular building block will self-assemble into a hydrogel. This severely hinders the rational design of self-assembled hydrogels. In this study, we demonstrate the impact of an N-terminal chromophore label amino-benzoic acid on the self-assembly and rheology of hydrogel hFF03 (hydrogelating, fibril forming) using molecular dynamics simulations, which self-assembles into {\alpha}-helical coiled-coils. We find that the chromophore and even its specific regioisomers have a significant influence on the microscopic structure and dynamics of the self-assembled fibril, and on the macroscopic mechanical properties. This is because the chromophore influences the possible salt-bridges which form and stabilize the fibril formation. Furthermore we find that the solvation shell fibrils by itself cannot explain the viscoelasticity of hFF03 hydrogels. Our atomistic model of the hFF03 fibril formation enables a more rational design of these hydrogels. In particular, altering the N-terminal chromophore emergesas a design strategy to tune the mechanic properties of these self-assembled peptide hydrogels.Comment: 15 pages, 15 including appendi

The occurrence of ansamers in the synthesis of cyclic peptides

α-Amanitin is a bicyclic octapeptide composed of a macrolactam with a tryptathionine cross-link forming a handle. Previously, the occurrence of isomers of amanitin, termed atropisomers has been postulated. Although the total synthesis of α-amanitin has been accomplished this aspect still remains unsolved. We perform the synthesis of amanitin analogs, accompanied by in-depth spectroscopic, crystallographic and molecular dynamics studies. The data unambiguously confirms the synthesis of two amatoxin-type isomers, for which we propose the term ansamers. The natural structure of the P-ansamer can be ansa-selectively synthesized using an optimized synthetic strategy. We believe that the here described terminology does also have implications for many other peptide structures, e.g. norbornapeptides, lasso peptides, tryptorubins and others, and helps to unambiguously describe conformational isomerism of cyclic peptides

A Formylglycine-Peptide for the Site-Directed Identification of Phosphotyrosine-Mimetic Fragments

Discovery of protein-binding fragments for precisely defined binding sites is an unmet challenge to date. Herein, formylglycine is investigated as a molecular probe for the sensitive detection of fragments binding to a spatially defined protein site . Formylglycine peptide 3 was derived from a phosphotyrosine-containing peptide substrate of protein tyrosine phosphatase PTP1B by replacing the phosphorylated amino acid with the reactive electrophile. Fragment ligation with formylglycine occurred in situ in aqueous physiological buffer. Structures and kinetics were validated by NMR spectroscopy. Screening and hit validation revealed fluorinated and non-fluorinated hit fragments being able to replace the native phosphotyrosine residue. The formylglycine probe identified low-affinity fragments with high spatial resolution as substantiated by molecular modelling. The best fragment hit, 4-amino-phenyl-acetic acid, was converted into a cellularly active, nanomolar inhibitor of the protein tyrosine phosphatase SHP2