Non-Markovian Modeling of Molecular Many-Body Dynamics

Starting from the Hamiltonian equations of motion of an arbitrary molecular many-body system, we first derive non-Markovian models in the form of various generalized Langevin equations (GLEs) using projection operators. The derived GLEs are integrodifferential equations for observables that are arbitrary functions of atomistic positions. We construct the projection operators to include nonlinear potentials and nonlinear memory functions in the GLEs. The primary motivation to introduce nonlinear GLEs is to move as much information as possible from the part of the GLE that ends up being modeled by a stochastic process to the deterministic part of the GLE. In this way, we ensure that one loses less information through the stochastic modeling of the exact GLE. Following this chapter, we present numerical methods to determine nonlinear memory functions from time series data. We demonstrate the numerical extraction method using a trajectory for the dihedral angle of a butane molecule in water generated by molecular dynamics simulations. From the trajectory, we calculate all previously derived GLEs using our method and compare them. For the dihedral angle dynamics of the butane molecule, we find that a position-dependent mass can lead to nonlinear memory effects. This effect can be eliminated by adjusting the mass dependence of the potential term. In the next part, we focus on the so-called approximate GLE, in which nonlinear memory effects are neglected. We discuss under which assumptions the approximate GLE emerges from a nonlinear GLE. By analytically computing the Kramers-Moyal coefficients of the approximate GLE, we show that the Fokker-Planck equation does not describe the dynamics of a non-Markovian system. We extract the friction kernel of the polypeptide Alanine9 from molecular dynamics simulations to quantify the importance of memory effects in protein folding. After parameterizing our GLE, we use the Markovian embedding method to simulate the GLE. Our GLE model very well reproduces the mean first passage times of both the folding and unfolding dynamics. The Kramers-Moyal coefficients and the mean square displacement, with pronounced anomalous diffusion, are also very well captured by the GLE. On the other hand, Markovian models based on Langevin equations with nonlinear friction cannot reproduce the dynamics in both directions with the same accuracy. From this, we conclude that consistent modeling of protein folding dynamics must take into account memory effects. The last part of the thesis is on the Markovian embedding of nonlinear GLEs. We introduce three different embedding systems that allow computationally efficient simulations of nonlinear GLEs. The first embedding system allows the simulation of nonlinear memory effects for a constant effective mass when the memory function has a nonvanishing component consisting of a delta function in time. The delta component can is not necessary for the second embedding system. We derive the second embedding from the nonlinear Zwanzig model by a perturbation expansion. The third embedding system also allows GLE simulations in the case that, in addition to a nonlinear memory function, the effective mass depends on the reaction coordinate. This embedding is not based on an approximation of the Zwanzig model and, like the first system, assumes a delta component in the memory function

Breakdown of linear dielectric theory for the interaction between hydrated ions and graphene

Many vital processes taking place in electrolytes, such as nanoparticle self-assembly, water purification, and the operation of aqueous supercapacitors, rely on the precise many-body interactions between surfaces and ions in water. Here we study the interaction between a hydrated ion and a charge-neutral graphene layer using atomistic molecular dynamics simulations. For small separations, the ion–graphene repulsion is of nonelectrostatic nature, and for intermediate separations, van der Waals attraction becomes important. Contrary to prevailing theory, we show that nonlinear and tensorial dielectric effects become non-negligible close to surfaces, even for monovalent ions. This breakdown of standard isotropic linear dielectric theory has important consequences for the understanding and modeling of charged objects at surfaces

Universal and Nonuniversal Aspects of Electrostatics in Aqueous Nanoconfinement

Dielectric water properties, which significantly change in confinement, determine electrostatic interactions and thereby influence all molecular forces and chemical reactions. We present comparative simulations of water between graphene sheets, decanol monolayers, and phospholipid and glycolipid bilayers. Generally, dielectric profiles strongly differ in perpendicular and parallel surface directions and for large surface separation decay to the bulk value 1-2 nm away from the surface. Polar surface groups enhance the local interfacial dielectric response and for phospholipid bilayers induce a giant parallel contribution. A mapping on a box model with asymptotically determined effective water layer widths demonstrates that the perpendicular effective dielectric constant for all systems decreases for confinement below a nanometer, while the parallel one stays rather constant. The confinement-dependent perpendicular effective dielectric constant for graphene is in agreement with experimental data only if the effective water layer width is suitably adjusted. The interactions between two charges at small separation depend on the product of parallel and perpendicular effective water dielectric components; for large separation the interactions depend on the confining medium. For metallic confining media the interactions at large separation decay exponentially with a decay length that depends on the ratio of the effective parallel and perpendicular water dielectric components

Fast protein folding is governed by memory-dependent friction

When described by a low-dimensional reaction coordinate, the folding rates of most proteins are determined by a subtle interplay between free-energy barriers, which separate folded and unfolded states, and friction. While it is commonplace to extract free-energy profiles from molecular trajectories, a direct evaluation of friction is far more elusive and typically relies on fits of measured reaction rates to memoryless reaction-rate theories. Here, using memory-kernel extraction methods founded on a generalized Langevin equation (GLE) formalism, we directly calculate the time-dependent friction acting on the fraction of native contacts reaction coordinate Q, evaluated for eight fast-folding proteins, taken from a published set of large-scale molecular dynamics protein simulations. Our results reveal that, across the diverse range of proteins represented in this dataset, friction is more influential than free-energy barriers in determining protein folding rates. We also show that proteins fold in a regime where the finite decay time of friction significantly reduces the folding times, in some instances by as much as a factor of 10, compared to predictions based on memoryless friction

Non-Markovian modeling of protein folding.

We extract the folding free energy landscape and the time-dependent friction function, the two ingredients of the generalized Langevin equation (GLE), from explicit-water molecular dynamics (MD) simulations of the α-helix forming polypeptide [Formula: see text] for a one-dimensional reaction coordinate based on the sum of the native H-bond distances. Folding and unfolding times from numerical integration of the GLE agree accurately with MD results, which demonstrate the robustness of our GLE-based non-Markovian model. In contrast, Markovian models do not accurately describe the peptide kinetics and in particular, cannot reproduce the folding and unfolding kinetics simultaneously, even if a spatially dependent friction profile is used. Analysis of the GLE demonstrates that memory effects in the friction significantly speed up peptide folding and unfolding kinetics, as predicted by the Grote-Hynes theory, and are the cause of anomalous diffusion in configuration space. Our methods are applicable to any reaction coordinate and in principle, also to experimental trajectories from single-molecule experiments. Our results demonstrate that a consistent description of protein-folding dynamics must account for memory friction effects

Non-Markovian Modeling of Protein Folding

We extract the folding free-energy landscape and the time-dependent friction function, the two ingredients of the generalized Langevin equation (GLE), from explicit-water molecular dynamics (MD) simulations of the α-helix forming polypeptide Alanine9 for a one-dimensional reaction coordinate based on the sum of the native H-bond distances. Folding and unfolding times from numerical integration of the GLE for the reaction coordinate agree accurately with MD results, which demonstrates the robustness of our GLE-based non-Markovian model. In contrast, Markovian models do not accurately describe the peptide kinetics and in particular cannot reproduce the folding and unfolding kinetics simultaneously, even if a spatially dependent friction profile is used. Analysis of the GLE demonstrates that memory effects in the friction significantly speed up peptide folding and unfolding kinetics, as predicted by Grote-Hynes theory, and are the cause of anomalous diffusion in configuration space. Our methods are applicable to any reaction coordinate and in principle also to experimental trajectories from single-molecule experiments. Our results demonstrate that a consistent description of protein folding dynamics must account for memory friction effects

Generalized Langevin equation with a nonlinear potential of mean force and nonlinear memory friction from a hybrid projection scheme

We introduce a hybrid projection scheme that combines linear Mori projection and conditional Zwanzig projection techniques and use it to derive a generalized Langevin equation (GLE) for a general interacting many-body system. The resulting GLE includes (i) explicitly the potential of mean force (PMF) that describes the equilibrium distribution of the system in the chosen space of reaction coordinates, (ii) a random force term that explicitly depends on the initial state of the system, and (iii) a memory friction contribution that splits into two parts: a part that is linear in the past reaction-coordinate velocity and a part that is in general nonlinear in the past reaction coordinates but does not depend on velocities. Our hybrid scheme thus combines all desirable properties of the Zwanzig and Mori projection schemes. The nonlinear memory friction contribution is shown to be related to correlations between the reaction-coordinate velocity and the random force. We present a numerical method to compute all parameters of our GLE, in particular the nonlinear memory friction function and the random force distribution, from a trajectory in reaction coordinate space. We apply our method on the dihedral-angle dynamics of a butane molecule in water obtained from atomistic molecular dynamics simulations. For this example, we demonstrate that nonlinear memory friction is present and that the random force exhibits significant non-Gaussian corrections. We also present the derivation of the GLE for multidimensional reaction coordinates that are general functions of all positions in the phase-space of the underlying many-body system; this corresponds to a systematic coarse-graining procedure that preserves not only the correct equilibrium behavior but also the correct dynamics of the coarse-grained system

A new record of macrovipera lebetina obtusa (viperidae) from south-eastern anatolia

The viper Macrovipera lebetina obtusa is recorded from Nusaybin, Mardin province of Turkey. Information on morphological features and the biology of this subspecies is given. © 2002 Taylor & Francis Group, LLC

Markovian embedding of generalized Langevin equations with a nonlinear friction kernel and configuration-dependent mass

We consider a generalized Langevin equation (GLE) in which the deterministic force, the mass and the friction kernel are configuration-dependent, i.e. general nonlinear functions of the reaction coordinate. We introduce a projection operator that allows for a self-consistent Markovian embedding of such GLEs. Selfconsistency means that trajectories generated by the Markovian embedding are described by a GLE with the same configuration-dependent deterministic force, mass and friction kernel. Using the projection operator, we derive a closed-form relation between the parameters of the Markovian embedding Langevin equations and the parameters of the GLE. This is accomplished by applying the projection operator formalism to the system of Markovian embedding stochastic equations

Self-consistent Markovian embedding of generalized Langevin equations with configuration-dependent mass and a nonlinear friction kernel

We consider a generalized Langevin equation (GLE) in which the deterministic force, the mass and the friction kernel are configuration-dependent, i.e. general nonlinear functions of the reaction coordinate. We introduce a projection operator that allows for a self-consistent Markovian embedding of such GLEs. Self-consistency means that trajectories generated by the Markovian embedding are described by a GLE with the same configuration-dependent deterministic force, mass and friction kernel. Using the projection operator, we derive a closed-form relation between the parameters of the Markovian embedding Langevin equations and the parameters of the GLE. This is accomplished by applying the projection operator formalism to the system of Markovian embedding stochastic equations