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

Biomolecular structure refinement based on adaptive restraints using local-elevation simulation

Introducing experimental values as restraints into molecular dynamics (MD) simulation to bias the values of particular molecular properties, such as nuclear Overhauser effect intensities or distances, dipolar couplings, 3 J-coupling constants, chemical shifts or crystallographic structure factors, towards experimental values is a widely used structure refinement method. Because multiple torsion angle values ϕ correspond to the same 3 J-coupling constant and high-energy barriers are separating those, restraining 3 J-coupling constants remains difficult. A method to adaptively enforce restraints using a local elevation (LE) potential energy function is presented and applied to 3 J-coupling constant restraining in an MD simulation of hen egg-white lysozyme (HEWL). The method succesfully enhances sampling of the restrained torsion angles until the 37 experimental 3 J-coupling constant values are reached, thereby also improving the agreement with the 1,630 experimental NOE atom-atom distance upper bounds. Afterwards the torsional angles ϕ are kept restrained by the built-up local-elevation potential energie

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

Estimation of the infinitesimal generator by square-root approximation

For the analysis of molecular processes, the estimation of time-scales, i.e., tran- sition rates, is very important. Estimating the transition rates between molecular conformations is - from a mathematical point of view - an invariant subspace projec- tion 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 ap- proximated 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 ge- ometric average of the Boltzmann weights of the Voronoi cells. Thus, there is a direct correlation 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

On using oscillating time-dependent restraints in MD simulation

The use of time-dependent restraints in molecular simulation in order to generate a conformational ensemble for molecules that is in accordance with measured ensemble averages for particular observable quantities is investigated. Using a model system consisting of liquid butane and the cyclic peptide antamanide the reproduction of particular average 3 J-coupling constant values in a molecular dynamics simulation is analysed. It is shown that the multiple-valuedness and the sizeable gradients of the Karplus curve relating 3 J-coupling constants measured in NMR experiments to the corresponding torsional-angle values cause severe problems when trying to restrain a 3 J-coupling constant to a value close to the extrema of the Karplus curve. The introduction of a factor oscillating with time into the restraining penalty function alleviates this problem and enhances the restrained conformational samplin

Transdisciplinary transformative change: an analysis of some best practices and barriers, and the potential of critical social science in getting us there

Biodiversity experts now widely acknowledge that transformative change is best supported through transdisciplinary collaborations. Yet, such collaborations rarely successfully occur in major biodiversity research institutions and those that do rarely achieve the paradigmatic effects they aim to deliver. To gain some insight into this global phenomenon, we surveyed Swiss-based researchers and non-academic stakeholders addressing global change and biodiversity. In this article, we connect our findings to global patterns in transdisciplinary transformative change initiatives (TTCIs) and heuristically divide collaboration barriers into two categories: lack of resources and lack of vital functional elements. Two of the major themes that emerged from this research were the continued difficulties with (1) establishing a common ‘language’, understanding, and goals, and (2) meaningful pluralization of knowledge in transdisciplinary collaborations aimed at addressing global change and biodiversity loss. The former is widely cited in the literature as contributing to the failure of TTCIs in the form of incoherent problem-framing, while the latter is often identified as contributing to the lack of structural transformative change (e.g., paradigmatic shifts) in completed initiatives. Another major theme reflected in TTCI literature was limited time. Moreover, based on our own extensive inter- and transdisciplinary experience, we agree with other experts that there is a persistent lack of understanding of the potential contributions of critical social science (CSS) to TTCIs. We thus argue that enhancing resource availability for TTCIs, especially tools for improving CSS literacy, could save time and support both problem-framing alignment and delivery of the structural/paradigmatic changes we aspire to