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.

Butane dihedral angle dynamics in water is dominated by internal friction.

The dihedral dynamics of butane in water is known to be rather insensitive to the water viscosity; possible explanations for this involve inertial effects or Kramers' turnover, the finite memory time of friction, and the presence of so-called internal friction. To disentangle these factors, we introduce a method to directly extract the friction memory function from unconstrained simulations in the presence of an arbitrary free-energy landscape. By analysis of the dihedral friction in butane for varying water viscosity, we demonstrate the existence of an internal friction contribution that does not scale linearly with water viscosity. At normal water viscosity, the internal friction turns out to be eight times larger than the solvent friction and thus completely dominates the effective friction. By comparison with simulations of a constrained butane molecule that has the dihedral as the only degree of freedom, we show that internal friction comes from the six additional degrees of freedom in unconstrained butane that are orthogonal to the dihedral angle reaction coordinate. While the insensitivity of butane's dihedral dynamics to water viscosity is solely due to the presence of internal friction, inertial effects nevertheless crucially influence the resultant transition rates. In contrast, non-Markovian effects due to the finite memory time are present but do not significantly influence the dihedral barrier-crossing rate of butane. These results not only settle the character of dihedral dynamics in small solvated molecular systems such as butane, they also have important implications for the folding of polymers and proteins.

Cyclization and Relaxation Dynamics of Finite-Length Collapsed Self-Avoiding Polymers.

We study the cyclization and relaxation dynamics of ideal as well as interacting polymers as a function of chain length N. For the cyclization time τ_{cyc} of ideal chains we recover the known scaling τ_{cyc}∼N^{2} for different backbone models, for a self-avoiding slightly collapsed chain we obtain from Langevin simulations and scaling theory a modified scaling τ_{cyc}∼N^{5/3}. The cyclization and relaxation dynamics of a finite-length collapsed chain scale differently; this unexpected dynamic multiscale behavior is rationalized by the crossover between swollen and collapsed chain behavior.

Non-Markovian barrier crossing with two-time-scale memory is dominated by the faster memory component.

We investigate non-Markovian barrier-crossing kinetics of a massive particle in one dimension in the presence of a memory function that is the sum of two exponentials with different memory times, [Formula: see text] and [Formula: see text] . Our Langevin simulations for the special case where both exponentials contribute equally to the total friction show that the barrier-crossing time becomes independent of the longer memory time if at least one of the two memory times is larger than the intrinsic diffusion time. When we associate memory effects with coupled degrees of freedom that are orthogonal to a one-dimensional reaction coordinate, this counterintuitive result shows that the faster orthogonal degrees of freedom dominate barrier-crossing kinetics in the non-Markovian limit and that the slower orthogonal degrees become negligible, quite contrary to the standard time-scale separation assumption and with important consequences for the proper setup of coarse-graining procedures in the non-Markovian case. By asymptotic matching and symmetry arguments, we construct a crossover formula for the barrier crossing time that is valid for general multi-exponential memory kernels. This formula can be used to estimate barrier-crossing times for general memory functions for high friction, i.e. in the overdamped regime, as well as for low friction, i.e. in the inertial regime. Typical examples where our results are important include protein folding in the high-friction limit and chemical reactions such as proton-transfer reactions in the low-friction limit.

Memory-induced acceleration and slowdown of barrier crossing.

We study the mean first-passage time τMFP for the barrier crossing of a single massive particle with non-Markovian memory by Langevin simulations in one dimension. In the Markovian limit of short memory time τΓ, the expected Kramers turnover between the overdamped (high-friction) and the inertial (low-friction) limits is recovered. Compared to the Markovian case, we find barrier crossing to be accelerated for intermediate memory time, while for long memory time, barrier crossing is slowed down and τMFP increases with τΓ as a power law τMFP∼τΓ2. Both effects are derived from an asymptotic propagator analysis: while barrier crossing acceleration at intermediate memory can be understood as an effective particle mass reduction, slowing down for long memory is caused by the slow kinetics of energy diffusion. A simple and globally accurate heuristic formula for τMFP in terms of all relevant time scales of the system is presented and used to establish a scaling diagram featuring the Markovian overdamped and the Markovian inertial regimes, as well as the non-Markovian intermediate memory time regime where barrier crossing is accelerated and the non-Markovian long memory time regime where barrier crossing is slowed down.

Hydration Friction in Nanoconfinement: From Bulk via Interfacial to Dry Friction.

The viscous properties of nanoscopically confined water are important when hydrated surfaces in close contact are sheared against each other. Numerous experiments have probed the friction between atomically flat hydrated surfaces in the subnanometer separation regime and suggested an increased water viscosity, but the value of the effective viscosity of ultraconfined water, the mechanism of hydration layer friction, and the crossover to the dry friction limit are unclear. We study the shear friction between polar surfaces by extensive nonequilibrium molecular dynamics simulations in the linear-response regime at low shearing velocity, which is the relevant regime for typical biological applications. With decreasing water film thickness we find three consecutive friction regimes: For thick films friction is governed by bulk water viscosity. At separations of about a nanometer the highly viscous interfacial water layers dominate and increase the surface friction, while at the transition to the dry friction limit interfacial slip sets in. Based on our simulation results, we construct a confinement-dependent friction model which accounts for the additive friction contributions from bulklike water, interfacial water layers, and interfacial slip and which is valid for arbitrary water film thickness.

Measurement of irreversibility and entropy production via the tubular ensemble.

The appealing theoretical measure of irreversibility in a stochastic process, as the ratio of the probabilities of a trajectory and its time reversal, cannot be accessed directly in experiments since the probability of a single trajectory is zero. We regularize this definition by considering, instead, the limiting ratio of probabilities for trajectories to remain in the tubular neighborhood of a smooth path and its time reversal. The resulting pathwise medium entropy production agrees with the formal expression from stochastic thermodynamics and can be obtained from measurable tube probabilities. Estimating the latter from numerically sampled trajectories for Langevin dynamics yields excellent agreement with theory. By combining our measurement of pathwise entropy production with a Markov chain Monte Carlo algorithm, we infer the entropy-production distribution for a transition path ensemble directly from short recorded trajectories. Our work enables the measurement of irreversibility along individual paths and path ensembles in a model-free manner.

Barrier-crossing times for different non-Markovian friction in well and barrier: A numerical study.

We introduce a generalized Langevin model system for different non-Markovian effects in the well and barrier regions of a potential, and use it to numerically study the barrier-crossing time. In the appropriate limits, our model interpolates between the theoretical barrier-crossing-time predictions by Grote and Hynes (GH), as well as by Pollak et al., which for a single barrier memory time can differ by several orders of magnitude. Our model furthermore allows one to test an analytic rate theory for space-inhomogeneous memory, which disagrees with our numerical results in the long well-memory regime. In this regime, we find that short barrier memory decreases the barrier-crossing time as compared to long barrier memory. This is in contrast with the short well-memory regime, where both our numerical results and the GH theory predict an acceleration of the barrier crossing time with increasing barrier memory time. Both effects, the "Markovian-barrier acceleration" and GH "non-Markovian-barrier acceleration," can be understood from a committor analysis. Our model combines finite relaxation times of orthogonal degrees of freedom with a space-inhomogeneous coupling to such degrees and represents a step towards more realistic modeling of reaction coordinates.

Time-Dependent Friction Effects on Vibrational Infrared Frequencies and Line Shapes of Liquid Water.

From ab initio simulations of liquid water, the time-dependent friction functions and time-averaged nonlinear effective bond potentials for the OH stretch and HOH bend vibrations are extracted. The obtained friction exhibits not only adiabatic contributions at and below the vibrational time scales but also much slower nonadiabatic contributions, reflecting homogeneous and inhomogeneous line broadening mechanisms, respectively. Intermolecular interactions in liquid water soften both stretch and bend potentials compared to the gas phase, which by itself would lead to a red-shift of the corresponding vibrational bands. In contrast, nonadiabatic friction contributions cause a spectral blue shift. For the stretch mode, the potential effect dominates, and thus, a significant red shift when going from gas to the liquid phase results. For the bend mode, potential and nonadiabatic friction effects are of comparable magnitude, so that a slight blue shift results, in agreement with well-known but puzzling experimental findings. The observed line broadening is shown to be roughly equally caused by adiabatic and nonadiabatic friction contributions for both the stretch and bend modes in liquid water. Thus, the quantitative analysis of the time-dependent friction that acts on vibrational modes in liquids advances the understanding of infrared vibrational frequencies and line shapes.

Efficient Bayesian inference of fully stochastic epidemiological models with applications to COVID-19.

Epidemiological forecasts are beset by uncertainties about the underlying epidemiological processes, and the surveillance process through which data are acquired. We present a Bayesian inference methodology that quantifies these uncertainties, for epidemics that are modelled by (possibly) non-stationary, continuous-time, Markov population processes. The efficiency of the method derives from a functional central limit theorem approximation of the likelihood, valid for large populations. We demonstrate the methodology by analysing the early stages of the COVID-19 pandemic in the UK, based on age-structured data for the number of deaths. This includes maximum a posteriori estimates, Markov chain Monte Carlo sampling of the posterior, computation of the model evidence, and the determination of parameter sensitivities via the Fisher information matrix. Our methodology is implemented in PyRoss, an open-source platform for analysis of epidemiological compartment models.

Memory-kernel extraction for different molecular solutes in solvents of varying viscosity in confinement.

The friction coefficient of molecular solutes depends on the solute, on the solvent, and on the solute-solvent interactions, but is typically assumed to not depend on an externally applied force that acts on the solute. In this paper we compute the friction memory function from molecular dynamics simulations and show that the friction coefficients of harmonically confined methane, water, Na^{+}, an artificial Na^{-} ion, and glycerol in water in fact increase with confinement strength. The results show that the friction increase with confinement strength is a fundamental effect that occurs for hydrophobic, hydrophilic, as well as charged molecules. We demonstrate that a parameter-free extraction of the running integral over the memory function yields the most robust results when compared to methods based on parametrization or Fourier transforms. In all systems, this friction increase is accompanied by a slowdown of the solvent dynamics in the first hydration shell of the solutes. By simulations of a confined glycerol molecule in water-glycerol mixtures, we furthermore demonstrate that the friction dependence on the confining potential is magnified in more viscous solvents, which suggests that this effect plays an important role for larger molecules in highly viscous solutions like polymer melts, in line with dynamic scaling arguments.