How the diffusivity profile reduces the arbitrariness of protein folding free energies

Spectral Map for Slow Collective Variables, Markovian Dynamics, and Transition State Ensembles

Understanding the behavior of complex molecular systems is a fundamental problem in physical chemistry. To describe the long-time dynamics of such systems, which is responsible for their most informative characteristics, we can identify a few slow collective variables (CVs) while treating the remaining fast variables as thermal noise. This enables us to simplify the dynamics and treat it as diffusion in a free-energy landscape spanned by slow CVs, effectively rendering the dynamics Markovian. Our recent statistical learning technique, spectral map [Rydzewski, J. J. Phys. Chem. Lett. 2023, 14(22), 5216-5220], explores this strategy to learn slow CVs by maximizing a spectral gap of a transition matrix. In this work, we introduce several advancements into our framework, using a high-dimensional reversible folding process of a protein as an example. We implement an algorithm for coarse-graining Markov transition matrices to partition the reduced space of slow CVs kinetically and use it to define a transition state ensemble. We show that slow CVs learned by spectral map closely approach the Markovian limit for an overdamped diffusion. We demonstrate that coordinate-dependent diffusion coefficients only slightly affect the constructed free-energy landscapes. Finally, we present how spectral maps can be used to quantify the importance of features and compare slow CVs with structural descriptors commonly used in protein folding. Overall, we demonstrate that a single slow CV learned by spectral map can be used as a physical reaction coordinate to capture essential characteristics of protein folding.

Data-Driven Learning of the Generalized Langevin Equation with State-Dependent Memory

We present a data-driven method to learn stochastic reduced models of complex systems that retain a state-dependent memory beyond the standard generalized Langevin equation with a homogeneous kernel. The constructed model naturally encodes the heterogeneous energy dissipation by jointly learning a set of state features and the non-Markovian coupling among the features. Numerical results demonstrate the limitation of the standard generalized Langevin equation and the essential role of the broadly overlooked state-dependency nature in predicting molecule kinetics related to conformation relaxation and transition.

Estimating Position-Dependent and Anisotropic Diffusivity Tensors from Molecular Dynamics Trajectories: Existing Methods and Future Outlook

Confinement can substantially alter the physicochemical properties of materials by breaking translational isotropy and rendering all physical properties position-dependent. Molecular dynamics (MD) simulations have proven instrumental in characterizing such spatial heterogeneities and probing the impact of confinement on materials' properties. For static properties, this is a straightforward task and can be achieved via simple spatial binning. Such an approach, however, cannot be readily applied to transport coefficients due to lack of natural extensions of autocorrelations used for their calculation in the bulk. The prime example of this challenge is diffusivity, which, in the bulk, can be readily estimated from the particles' mobility statistics, which satisfy the Fokker-Planck equation. Under confinement, however, such statistics will follow the Smoluchowski equation, which lacks a closed-form analytical solution. This brief review explores the rich history of estimating profiles of the diffusivity tensor from MD simulations and discusses various approximate methods and algorithms developed for this purpose. Besides discussing heuristic extensions of bulk methods, we overview more rigorous algorithms, including kernel-based methods, Bayesian approaches, and operator discretization techniques. Additionally, we outline methods based on applying biasing potentials or imposing constraints on tracer particles. Finally, we discuss approaches that estimate diffusivity from mean first passage time or committor probability profiles, a conceptual framework originally developed in the context of collective variable spaces describing rare events in computational chemistry and biology. In summary, this paper offers a concise survey of diverse approaches for estimating diffusivity from MD trajectories, highlighting challenges and opportunities in this area.

The role of memory-dependent friction and solvent viscosity in isomerization kinetics in viscogenic media

Molecular isomerization kinetics in liquid solvent depends on a complex interplay between the solvent friction acting on the molecule, internal dissipation effects (also known as internal friction), the viscosity of the solvent, and the dihedral free energy profile. Due to the absence of accurate techniques to directly evaluate isomerization friction, it has not been possible to explore these relationships in full. By combining extensive molecular dynamics simulations with friction memory-kernel extraction techniques we consider a variety of small, isomerising molecules under a range of different viscogenic conditions and directly evaluate the viscosity dependence of the friction acting on a rotating dihedral. We reveal that the influence of different viscogenic media on isomerization kinetics can be dramatically different, even when measured at the same viscosity. This is due to the dynamic solute-solvent coupling, mediated by time-dependent friction memory kernels. We also show that deviations from the linear dependence of isomerization rates on solvent viscosity, which are often simply attributed to internal friction effects, are due to the simultaneous violation of two fundamental relationships: the Stokes-Einstein relation and the overdamped Kramers prediction for the barrier-crossing rate, both of which require explicit knowledge of friction.

Derivation of the Invariant Free-Energy Landscape Based on Langevin Dynamics

A conventionally defined free-energy landscape (FEL) exhibits unphysical dependence on the choice of reaction coordinates and hence lacks universal predictive ability. We here show that three physically plausible requirements uniquely determine the FEL formula for a given reaction coordinate. Our FEL is expressed solely in terms of quantities obtained through time-series data analysis, namely, the probability distribution and the diffusion matrix. It is free from any unphysical coordinate dependence and coincides with the conventional FEL in special cases. The uniqueness and robustness of the formula strongly suggest that our FEL has universal predictive power.

Anisotropic molecular diffusion in confinement I: Transport of small particles in potential and density gradients

HYPOTHESIS

Diffusion in confinement is an important fundamental problem with significant implications for applications of supported liquid phases. However, resolving the spatially dependent diffusion coefficient, parallel and perpendicular to interfaces, has been a standing issue. In the vicinity of interfaces, density fluctuations as a consequence of layering locally impose statistical drift, which impedes the analysis of spatially dependent diffusion coefficients even further. We hypothesise, that we can derive a model to spatially resolve interface-perpendicular diffusion coefficients based on local lifetime statistics with an extension to explicitly account for the effect of local drift using the Smoluchowski equation, that allows us to resolve anisotropic and spatially dependent diffusivity landscapes at interfaces.

METHODS AND SIMULATIONS

An analytic relation between local crossing times in system slices and diffusivity as well as an explicit term for calculating drift-induced systematic errors is presented. The method is validated on Molecular Dynamics simulations of bulk water and applied to simulations of water in slit pores.

FINDINGS

After validation on bulk liquids, we clearly demonstrate the anisotropic nature of diffusion coefficients at interfaces. Significant spatial variations in the diffusivities correlate with interface-induced structuring but cannot be solely attributed to the drift induced by local density fluctuations.

Optimal Reaction Coordinates and Kinetic Rates from the Projected Dynamics of Transition Paths

Finding optimal reaction coordinates and predicting accurate kinetic rates for activated processes are two of the foremost challenges of molecular simulations. We introduce an algorithm that tackles the two problems at once: starting from a limited number of reactive molecular dynamics trajectories (transition paths), we automatically generate with a Monte Carlo approach a sequence of different reaction coordinates that progressively reduce the kinetic rate of their projected effective dynamics. Based on a variational principle, the minimal rate accurately approximates the exact one, and it corresponds to the optimal reaction coordinate. After benchmarking the method on an analytic double-well system, we apply it to complex atomistic systems: the interaction of carbon nanoparticles of different sizes in water.

Anisotropic Friction in a Ligand-Protein Complex

The effect of an externally applied directional force on molecular friction is so far poorly understood. Here, we study the force-driven dissociation of the ligand-protein complex biotin-streptavidin and identify anisotropic friction as a not yet described type of molecular friction. Using AFM-based stereographic single molecule force spectroscopy and targeted molecular dynamics simulations, we find that the rupture force and friction for biotin-streptavidin vary with the pulling angle. This observation holds true for friction extracted from Kramers' rate expression and by dissipation-corrected targeted molecular dynamics simulations based on Jarzynski's identity. We rule out ligand solvation and protein-internal friction as sources of the angle-dependent friction. Instead, we observe a heterogeneity in free energy barriers along an experimentally uncontrolled orientation parameter, which increases the rupture force variance and therefore the overall friction. We anticipate that anisotropic friction needs to be accounted for in a complete understanding of friction in biomolecular dynamics and anisotropic mechanical environments.

Myofilament-associated proteins with intrinsic disorder (MAPIDs) and their resolution by computational modeling

The cardiac sarcomere is a cellular structure in the heart that enables muscle cells to contract. Dozens of proteins belong to the cardiac sarcomere, which work in tandem to generate force and adapt to demands on cardiac output. Intriguingly, the majority of these proteins have significant intrinsic disorder that contributes to their functions, yet the biophysics of these intrinsically disordered regions (IDRs) have been characterized in limited detail. In this review, we first enumerate these myofilament-associated proteins with intrinsic disorder (MAPIDs) and recent biophysical studies to characterize their IDRs. We secondly summarize the biophysics governing IDR properties and the state-of-the-art in computational tools toward MAPID identification and characterization of their conformation ensembles. We conclude with an overview of future computational approaches toward broadening the understanding of intrinsic disorder in the cardiac sarcomere.

Coordinate-Dependent Drift-Diffusion Reveals the Kinetic Intermediate Traps of Top7-Based Proteins

The computer-designed Top7 served as a scaffold to produce immunoreactive proteins by grafting of the 2F5 HIV-1 antibody epitope (Top7-2F5) followed by biotinylation (Top7-2F5-biotin). The resulting nonimmunoglobulin affinity proteins were effective in inducing and detecting the HIV-1 antibody. However, the grafted Top7-2F5 design led to protein aggregation, as opposed to the soluble biotinylated Top7-2F5-biotin. The structure-based model predicted that the thermodynamic cooperativity of Top7 increases after grafting and biotin-labeling, reducing their intermediate state populations. In this work, the folding kinetic traps that might contribute to the aggregation propensity are investigated by the diffusion theory. Since the engineered proteins have similar sequence and structural homology, they served as protein models to study the kinetic intermediate traps that were uncovered by characterizing the position-dependent drift-velocity (v(Q)) and the diffusion (D(Q)) coefficients. These coordinate-dependent coefficients were taken into account to obtain the folding and transition path times over the free energy transition states containing the intermediate kinetic traps. This analysis may be useful to predict the aggregated kinetic traps of scaffold-epitope proteins that might compose novel diagnostic and therapeutic platforms.

Pair-Reaction Dynamics in Water: Competition of Memory, Potential Shape, and Inertial Effects

When described by a one-dimensional reaction coordinate, pair-reaction rates in a solvent depend, in addition to the potential barrier height and the friction coefficient, on the potential shape, the effective mass, and the friction relaxation spectrum, but a rate theory that accurately accounts for all of these effects does not exist. After a review of classical reaction-rate theories, we show how to extract all parameters of the generalized Langevin equation (GLE) and, in particular, the friction memory function from molecular dynamics (MD) simulations of two prototypical pair reactions in water, the dissociation of NaCl and of two methane molecules. The memory exhibits multiple time scales and, for NaCl, pronounced oscillatory components. Simulations of the GLE by Markovian embedding techniques accurately reproduce the pair-reaction kinetics from MD simulations without any fitting parameters, which confirms the accuracy of the approximative form of the GLE and of the parameter extraction techniques. By modification of the GLE parameters, we investigate the relative importance of memory, mass, and potential shape effects. Neglect of memory slows down NaCl and methane dissociation by roughly a factor of 2; neglect of mass accelerates reactions by a similar factor, and the harmonic approximation of the potential shape gives rise to slight acceleration. This partial error cancellation explains why Kramers' theory, which neglects memory effects and treats the potential shape in harmonic approximation, describes reaction rates better than more sophisticated theories. In essence, all three effects, friction memory, inertia, and the potential shape nonharmonicity, are important to quantitatively describe pair-reaction kinetics in water.

Assessing transition rates as functions of environmental variables

We present a method to estimate the transition rates of molecular systems under different environmental conditions that cause the formation or the breaking of bonds and require the sampling of the Grand Canonical Ensemble. For this purpose, we model the molecular system in terms of probable "scenarios," governed by different potential energy functions, which are separately sampled by classical MD simulations. Reweighting the canonical distribution of each scenario according to specific environmental variables, we estimate the grand canonical distribution, then use the Square Root Approximation method to discretize the Fokker-Planck operator into a rate matrix and the robust Perron Cluster Cluster Analysis method to coarse-grain the kinetic model. This permits efficiently estimating the transition rates of conformational states as functions of environmental variables, for example, the local pH at a cell membrane. In this work, we formalize the theoretical framework of the procedure, and we present a numerical experiment comparing the results with those provided by a constant-pH method based on non-equilibrium Molecular Dynamics Monte Carlo simulations. The method is relevant for the development of new drug design strategies that take into account how the cellular environment influences biochemical processes.

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.

Relations among Unidirectional Fluxes at Equilibrium, Committors, and First Passage and Transition Path Times

For multidimensional diffusive dynamics, we algebraically derive remarkable analytical expressions that relate the mean first passage and transition path times between two dividing surfaces with the number of unidirectional transitions per unit time (fluxes) at equilibrium between the two surfaces and the committor (the probability of reaching one surface before the other). In one dimension, such relationships can be easily derived because analytical expressions for all the above-mentioned quantities can be found. This is not possible in higher dimensions, and at first sight, the problem seems much harder. We circumvent the difficulty that the equations determining the mean first passage and transition path times cannot be solved analytically by multiplying these equations by the committor, integrating both sides and finally using the divergence theorem. A byproduct of our derivation is an analytical expression for the starting point distribution over which individual first passage and transition path times must be averaged. It turns out that this distribution is not the Boltzmann one, but it has a simple physical interpretation.

Global diffusion of hydrogen molecules in the heterogeneous structure of polymer electrolytes for fuel cells: Dynamic Monte Carlo combined with molecular dynamics calculations

Using our recently developed dynamic Monte Carlo (MC) method [Nagai et al., J. Chem Phys. 156, 154506 (2022)], we investigated global diffusion of hydrogen molecules over structural heterogeneities of polymer electrolyte membranes in fuel cells. The three-dimensional position-dependent free energies and the diffusion constants of the hydrogen molecules, required by the present dynamic MC calculations, were taken from our previous study [Nagai et al., J. Chem. Phys. 156, 044507 (2022)] and newly evaluated in this work, respectively. The calculations enabled evaluating the hydrogen dynamics over long-time scales, including global diffusion constants. Based on the calculated global diffusion constants and free energies, the permeability of hydrogen molecules was estimated via the solubility-diffusion model. The estimated values were in good agreement with reported experimental data, thus validating the present methodology. The analysis of the Monte Carlo trajectories indicated that the main permeation paths are located in the polymer and interfacial phases, although the water phase may make a non-negligible contribution to mass transport.

Statistical physics of inhomogeneous transport: Unification of diffusion laws and inference from first-passage statistics

Characterization of composite materials, whose properties vary in space over microscopic scales, has become a problem of broad interdisciplinary interest. In particular, estimation of the inhomogeneous transport coefficients, e.g., the diffusion coefficient or the heat conductivity, which shape important processes in biology and engineering, is a challenging task. The analysis of such systems is further complicated because two alternative formulations of the inhomogeneous transport equations exist in the literature-the Smoluchowski and Fokker-Planck equations, which are also related to the so-called Ito-Stratonovich dilemma. Using the theory of statistical physics, we show that the two formulations, usually regarded as distinct models, are physically equivalent. From this result we develop efficient estimates for the transverse space-dependent diffusion coefficient in fluids near a phase boundary. Our method requires only measurements of escape probabilities and mean exit times of molecules leaving a narrow spatial region. We test our estimates in three case studies: (i) a Langevin model of a Büttikker-Landauer ratchet; atomistic molecular-dynamics simulations of liquid-water molecules in contact with (ii) vapor, and (iii) soap (surfactant) film which has promising applications in physical chemistry. Our analysis reveals that near the surfactant monolayer the mobility of water molecules is slowed down almost twice with respect to the bulk liquid. Moreover, the diffusion coefficient of water correlates with the transition from hydrophilic to hydrophobic parts of the film.

Dynamic Monte Carlo calculation generating particle trajectories that satisfy the diffusion equation for heterogeneous systems with a position-dependent diffusion coefficient and free energy

A series of new Monte Carlo (MC) transition probabilities was investigated that could produce molecular trajectories statistically satisfying the diffusion equation with a position-dependent diffusion coefficient and potential energy. The MC trajectories were compared with the numerical solution of the diffusion equation by calculating the time evolution of the probability distribution and the mean first passage time, which exhibited excellent agreement. The method is powerful when investigating, for example, the long-distance and long-time global transportation of a molecule in heterogeneous systems by coarse-graining them into one-particle diffusive molecular motion with a position-dependent diffusion coefficient and free energy. The method can also be applied to many-particle dynamics.

Momentum removal to obtain the position-dependent diffusion constant in constrained molecular dynamics simulation

The position-dependent diffusion coefficient along with free energy profile are important parameters needed to study mass transport in heterogeneous systems such as biological and polymer membranes, and molecular dynamics (MD) calculation is a popular tool to obtain them. Among many methodologies, the Marrink-Berendsen (MB) method is often employed to calculate the position-dependent diffusion coefficient, in which the autocorrelation function of the force on a fixed molecule is related to the friction on the molecule. However, the diffusion coefficient is shown to be affected by the period of the removal of the center-of-mass velocity, τ v 0 , which is necessary when performing MD calculations using the Ewald method for Coulombic interaction. We have clarified theoretically in this study how this operation affects the diffusion coefficient calculated by the MB method, and the theoretical predictions are proven by MD calculations. Therefore, we succeeded in providing guidance on how to select an appropriate τ v 0 value in estimating the position-dependent diffusion coefficient by the MB method. This guideline is applicable also to the Woolf-Roux method.

Non-Markovian modeling of protein folding

Protein-folding kinetics is often described as Markovian (i.e., memoryless) diffusion in a one-dimensional free energy landscape, governed by an instantaneous friction coefficient that is fitted to reproduce experimental or simulated folding times. For the α-helix forming polypeptide alanine9 and a specific reaction coordinate that consists of the summed native hydrogen-bond lengths, we demonstrate that the friction extracted from molecular dynamics simulations exhibits significant memory with a decay time that is in the nanosecond range and thus, of the same order as the folding and unfolding times. Our non-Markovian modeling not only reproduces the molecular dynamics simulations accurately but also demonstrates that memory friction effects lead to anomalous and drastically accelerated protein kinetics.

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 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.

Data-Driven Langevin Modeling of Nonequilibrium Processes

Given nonstationary data from molecular dynamics simulations, a Markovian Langevin model is constructed that aims to reproduce the time evolution of the underlying process. While at equilibrium the free energy landscape is sampled, nonequilibrium processes can be associated with a biased energy landscape, which accounts for finite sampling effects and external driving. When the data-driven Langevin equation (dLE) approach [Phys. Rev. Lett. 2015, 115, 050602] is extended to the modeling of nonequilibrium processes, an efficient way to calculate multidimensional Langevin fields is outlined. The dLE is shown to correctly account for various nonequilibrium processes, including the enforced dissociation of sodium chloride in water, the pressure-jump induced nucleation of a liquid of hard spheres, and the conformational dynamics of a helical peptide sampled from nonstationary short trajectories.

Non-Markovian data-driven modeling of single-cell motility

Trajectories of human breast cancer cells moving on one-dimensional circular tracks are modeled by the non-Markovian version of the Langevin equation that includes an arbitrary memory function. When averaged over cells, the velocity distribution exhibits spurious non-Gaussian behavior, while single cells are characterized by Gaussian velocity distributions. Accordingly, the data are described by a linear memory model which includes different random walk models that were previously used to account for various aspects of cell motility such as migratory persistence, non-Markovian effects, colored noise, and anomalous diffusion. The memory function is extracted from the trajectory data without restrictions or assumptions, thus making our approach truly data driven, and is used for unbiased single-cell comparison. The cell memory displays time-delayed single-exponential negative friction, which clearly distinguishes cell motion from the simple persistent random walk model and suggests a regulatory feedback mechanism that controls cell migration. Based on the extracted memory function we formulate a generalized exactly solvable cell migration model which indicates that negative friction generates cell persistence over long timescales. The nonequilibrium character of cell motion is investigated by mapping the non-Markovian Langevin equation with memory onto a Markovian model that involves a hidden degree of freedom and is equivalent to the underdamped active Ornstein-Uhlenbeck process.

Connecting dynamic reweighting Algorithms: Derivation of the dynamic reweighting family tree

Thermally driven processes of molecular systems include transitions of energy barriers on the microsecond timescales and higher. Sufficient sampling of such processes with molecular dynamics simulations is challenging and often requires accelerating slow transitions using external biasing potentials. Different dynamic reweighting algorithms have been proposed in the past few years to recover the unbiased kinetics from biased systems. However, it remains an open question if and how these dynamic reweighting approaches are connected. In this work, we establish the link between the two main reweighting types, i.e., path-based and energy-based reweighting. We derive a path-based correction factor for the energy-based dynamic histogram analysis method, thus connecting the previously separate reweighting types. We show that the correction factor can be used to combine the advantages of path-based and energy-based reweighting algorithms: it is integrator independent, more robust, and at the same time able to reweight time-dependent biases. We can furthermore demonstrate the relationship between two independently derived path-based reweighting algorithms. Our theoretical findings are verified on a one-dimensional four-well system. By connecting different dynamic reweighting algorithms, this work helps to clarify the strengths and limitations of the different methods and enables a more robust usage of the combined types.

First-passage fingerprints of water diffusion near glutamine surfaces

The extent to which biological interfaces affect the dynamics of water plays a key role in the exchange of matter and chemical interactions that are essential for life. The density and the mobility of water molecules depend on their proximity to biological interfaces and can play an important role in processes such as protein folding and aggregation. In this work, we study the dynamics of water near glutamine surfaces-a system of interest in studies of neurodegenerative diseases. Combining molecular-dynamics simulations and stochastic modelling, we study how the mean first-passage time and related statistics of water molecules escaping subnanometer-sized regions vary from the interface to the bulk. Our analysis reveals a dynamical complexity that reflects underlying chemical and geometrical properties of the glutamine surfaces. From the first-passage time statistics of water molecules, we infer their space-dependent diffusion coefficient in directions normal to the surfaces. Interestingly, our results suggest that the mobility of water varies over a longer length scale than the chemical potential associated with the water-protein interactions. The synergy of molecular dynamics and first-passage techniques opens the possibility for extracting space-dependent diffusion coefficients in more complex, inhomogeneous environments that are commonplace in living matter.

From Nonequilibrium Single-Molecule Trajectories to Underlying Dynamics

Single-molecule observations of biomolecular dynamics and folding are commonly rationalized using the model of diffusive dynamics on a free-energy landscape, which is inferred via the Boltzmann inversion of the equilibrium distribution of the experimental observable. Can the same model be applied to high-resolution single-molecule trajectories of molecular machines that lack thermal equilibrium so that the Boltzmann inversion method is inapplicable? In this Letter, we discuss two approaches to reconstructing the underlying free-energy landscape in such nonequilibrium systems and explore the performance of this model in application to trajectories with complex underlying dynamics.

Intracellular RNA delivery by lipid nanoparticles: Diffusion, degradation, and release

Various RNAs (siRNAs, miRNAs, or mRNAs) can be delivered into cells by lipid nanoparticles (LNPs) of 50-150 nm in diameter. The subsequent RNA release from LNPs may occur via various scenarios. Herein, two related kinetic models are proposed. The first model takes into account that LNPs are often porous so that RNA molecules diffuse in and detach from nanopores. The analysis is focused on RNA diffusion from a pore. The analytical expression obtained for the RNA escape rate constant is used to identify the difference in the release of siRNAs, miRNAs, and mRNAs. The key message here is that the mRNA diffusion from pores appears to be too slow, and accordingly the mRNA release seems to occur primarily via degradation of LNPs. The second coarse-grained model describes the diffusion-mediated release of RNA from a LNP in the situation when this process is accompanied by the LNP degradation at the lipid-solution interface. The corresponding kinetics are shown in detail at different relative rates of the RNA diffusion and LNP degradation. Potentially, this can help to interpret drug plasma levels after various dosing regimens.

Probing Position-Dependent Diffusion in Folding Reactions Using Single-Molecule Force Spectroscopy

Folding of proteins and nucleic acids involves a diffusive search over a multidimensional conformational energy landscape for the minimal-energy structure. When examining the projection of conformational motions onto a one-dimensional reaction coordinate, as done in most experiments, the diffusion coefficient D is generally position dependent. However, it has proven challenging to measure such position-dependence experimentally. We investigated the position-dependence of D in the folding of DNA hairpins as a simple model system in two ways: first, by analyzing the round-trip time to return to a given extension in constant-force extension trajectories measured by force spectroscopy, and second, by analyzing the fall time required to reach a given extension in force jump measurements. These methods yielded conflicting results: the fall time implied a fairly constant D, but the round-trip time implied variations of over an order of magnitude. Comparison of experiments with computational simulations revealed that both methods were strongly affected by experimental artifacts inherent to force spectroscopy measurements, which obscured the intrinsic position-dependence of D. Lastly, we applied Kramers's theory to the kinetics of hairpins with energy barriers located at different positions along the hairpin stem, as a crude probe of D at different stem positions, and we found that D did not vary much as the barrier was moved along the reaction coordinate. This work underlines the difficulties faced when trying to deduce position-dependent diffusion coefficients from experimental folding trajectories.

Committors, first-passage times, fluxes, Markov states, milestones, and all that

Milestoning on a one-dimensional potential starts by choosing a set of points, called milestones, and initiating short trajectories from each milestone, which are terminated when they reach an adjacent milestone for the first time. From the average duration of these trajectories and the probabilities of where they terminate, a rate matrix can be constructed and then used to calculate the mean first-passage time (MFPT) between any two milestones. All these MFPT's turn out to be exact. Here we adopt a point of view from which this remarkable result is not unexpected. In addition, we clarify the nature of the "states" whose interconversion is described by the rate matrix constructed using information obtained from short trajectories and provide a microscopic expression for the "equilibrium population" of these states in terms of equilibrium averages of the committors.

Stochastic diffusion framework determines the free-energy landscape and rate from single-molecule trajectory

A theoretical stochastic diffusion framework is developed that characterizes the position-dependent diffusion coefficient [D(Q)] and drift velocity [ v (Q)] by analysing single-molecule time traces [Q(t)]. The free-energy landscape [F(Q)] that governs the dynamics is reconstructed with the calculated D and v . There are many computational tools that perform this task in which some are computationaly demanding, difficult to run, and, most of the time, not directly available to the community. This is a first attempt to implement the simplified stochastic diffusion framework that is fast, easy to run in a Python environment, and available to be extended as needed. It does not require adjustable parameters, inference methods, or sampling bias such as Monte Carlo Bayesian estimators or umbrella samplings. The stochastic framework was applied in the protein-like lattice model with Monte Carlo simulations, which accurately predicted the folding rates with the coordinate-dependent D and F plugged into Kramers' theory. The results were compared with two other independently developed methodologies (the Bayesian analysis and fep1D algorithm) presenting a good match, which confirms its validity. This theoretical framework might be useful in determining the free-energy and rates by providing time series only from biological or condensed-phase systems. The code is freely available at https://github.com/ronaldolab/stochastic_diffusion.

Perspective: Identification of collective variables and metastable states of protein dynamics

The statistical analysis of molecular dynamics simulations requires dimensionality reduction techniques, which yield a low-dimensional set of collective variables (CVs) {x _i } = x that in some sense describe the essential dynamics of the system. Considering the distribution P( x ) of the CVs, the primal goal of a statistical analysis is to detect the characteristic features of P( x ), in particular, its maxima and their connection paths. This is because these features characterize the low-energy regions and the energy barriers of the corresponding free energy landscape ΔG( x ) = -k _B T ln P( x ), and therefore amount to the metastable states and transition regions of the system. In this perspective, we outline a systematic strategy to identify CVs and metastable states, which subsequently can be employed to construct a Langevin or a Markov state model of the dynamics. In particular, we account for the still limited sampling typically achieved by molecular dynamics simulations, which in practice seriously limits the applicability of theories (e.g., assuming ergodicity) and black-box software tools (e.g., using redundant input coordinates). We show that it is essential to use internal (rather than Cartesian) input coordinates, employ dimensionality reduction methods that avoid rescaling errors (such as principal component analysis), and perform density based (rather than k-means-type) clustering. Finally, we briefly discuss a machine learning approach to dimensionality reduction, which highlights the essential internal coordinates of a system and may reveal hidden reaction mechanisms.

Dynamics of Disordered Proteins under Confinement: Memory Effects and Internal Friction

Many proteins are disordered under physiological conditions. How efficiently they can search for their cellular targets and how fast they can fold upon target binding is determined by their intrinsic dynamics, which have thus attracted much recent attention. Experiments and molecular simulations show that the inherent reconfiguration timescale for unfolded proteins has a solvent friction component and an internal friction component, and the microscopic origin of the latter, along with its proper mathematical description, has been a topic of considerable debate. Internal friction varies across different proteins of comparable length and increases with decreasing denaturant concentration, showing that it depends on how compact the protein is. Here we report on a systematic atomistic simulation study of how confinement, which induces a more compact unfolded state, affects dynamics and friction in disordered peptides. We find that the average reconfiguration timescales increase exponentially as the peptide's spatial dimensions are reduced; at the same time, confinement broadens the spectrum of relaxation timescales exhibited by the peptide. There are two important implications of this broadening: First, it limits applicability of the common Rouse and Zimm models with internal friction, as those models attempt to capture internal friction effects by introducing a single internal friction timescale. Second, the long-tailed distribution of relaxation times leads to anomalous diffusion effects in the dynamics of intramolecular distances. Analysis and interpretation of experimental signals from various measurements that probe intramolecular protein dynamics (such as single-molecule fluorescence correlation spectroscopy and single-molecule force spectroscopy) rely on the assumption of diffusive dynamics along the distances being probed; hence, our results suggest the need for more general models allowing for anomalous diffusion effects.

Accurate Protein-Folding Transition-Path Statistics from a Simple Free-Energy Landscape

A central goal of protein-folding theory is to predict the stochastic dynamics of transition paths-the rare trajectories that transit between the folded and unfolded ensembles-using only thermodynamic information, such as a low-dimensional equilibrium free-energy landscape. However, commonly used one-dimensional landscapes typically fall short of this aim, because an empirical coordinate-dependent diffusion coefficient has to be fit to transition-path trajectory data in order to reproduce the transition-path dynamics. We show that an alternative, first-principles free-energy landscape predicts transition-path statistics that agree well with simulations and single-molecule experiments without requiring dynamical data as an input. This "topological configuration" model assumes that distinct, native-like substructures assemble on a time scale that is slower than native-contact formation but faster than the folding of the entire protein. Using only equilibrium simulation data to determine the free energies of these coarse-grained intermediate states, we predict a broad distribution of transition-path transit times that agrees well with the transition-path durations observed in simulations. We further show that both the distribution of finite-time displacements on a one-dimensional order parameter and the ensemble of transition-path trajectories generated by the model are consistent with the simulated transition paths. These results indicate that a landscape based on transient folding intermediates, which are often hidden by one-dimensional projections, can form the basis of a predictive model of protein-folding transition-path dynamics.

Segmentation and the Entropic Elasticity of Modular Proteins

Single-molecule force spectroscopy utilizes polyproteins, which are composed of tandem modular domains, to study their mechanical and structural properties. Under the application of external load, the polyproteins respond by unfolding and refolding domains to acquire the most favored extensibility. However, unlike single-domain proteins, the sequential unfolding of the each domain modifies the free energy landscape (FEL) of the polyprotein nonlinearly. Here we use force-clamp (FC) spectroscopy to measure unfolding and collapse-refolding dynamics of polyubiquitin and poly(I91). Their reconstructed unfolding FEL involves hundreds of k_B T in accumulating work performed against conformational entropy, which dwarfs the ∼30 k_B T that is typically required to overcome the free energy difference of unfolding. We speculate that the additional entropic energy caused by segmentation of the polyprotein to individual proteins plays a crucial role in defining the "shock absorber" properties of elastic proteins such as the giant muscle protein titin.

Communication: Coordinate-dependent diffusivity from single molecule trajectories

Single-molecule observations of biomolecular folding are commonly interpreted using the model of one-dimensional diffusion along a reaction coordinate, with a coordinate-independent diffusion coefficient. Recent analysis, however, suggests that more general models are required to account for single-molecule measurements performed with high temporal resolution. Here, we consider one such generalization: a model where the diffusion coefficient can be an arbitrary function of the reaction coordinate. Assuming Brownian dynamics along this coordinate, we derive an exact expression for the coordinate-dependent diffusivity in terms of the splitting probability within an arbitrarily chosen interval and the mean transition path time between the interval boundaries. This formula can be used to estimate the effective diffusion coefficient along a reaction coordinate directly from single-molecule trajectories.

Position-Dependent Dynamics Explain Pore-Averaged Diffusion in Strongly Attractive Adsorptive Systems

Using molecular simulations, we investigate the relationship between the pore-averaged and position-dependent self-diffusivity of a fluid adsorbed in a strongly attractive pore as a function of loading. Previous work (Krekelberg, W. P.; Siderius, D. W.; Shen, V. K.; Truskett, T. M.; Errington, J. R. Connection between thermodynamics and dynamics of simple fluids in highly attractive pores. Langmuir 2013, 29, 14527-14535, doi: 10.1021/la4037327) established that pore-averaged self-diffusivity in the multilayer adsorption regime, where the fluid exhibits a dense film at the pore surface and a lower density interior pore region, is nearly constant as a function of loading. Here we show that this puzzling behavior can be understood in terms of how loading affects the fraction of particles that reside in the film and interior pore regions as well as their distinct dynamics. Specifically, the insensitivity of pore-averaged diffusivity to loading arises from the approximate cancellation of two factors: an increase in the fraction of particles in the higher diffusivity interior pore region with loading and a corresponding decrease in the particle diffusivity in that region. We also find that the position-dependent self-diffusivities scale with the position-dependent density. We present a model for predicting the pore-average self-diffusivity based on the position-dependent self-diffusivity, which captures the unusual characteristics of pore-averaged self-diffusivity in strongly attractive pores over several orders of magnitude.

Data-based modeling of drug penetration relates human skin barrier function to the interplay of diffusivity and free-energy profiles

Based on experimental concentration depth profiles of the antiinflammatory drug dexamethasone in human skin, we model the time-dependent drug penetration by the 1D general diffusion equation that accounts for spatial variations in the diffusivity and free energy. For this, we numerically invert the diffusion equation and thereby obtain the diffusivity and the free-energy profiles of the drug as a function of skin depth without further model assumptions. As the only input, drug concentration profiles derived from X-ray microscopy at three consecutive times are used. For dexamethasone, skin barrier function is shown to rely on the combination of a substantially reduced drug diffusivity in the stratum corneum (the outermost epidermal layer), dominant at short times, and a pronounced free-energy barrier at the transition from the epidermis to the dermis underneath, which determines the drug distribution in the long-time limit. Our modeling approach, which is generally applicable to all kinds of barriers and diffusors, allows us to disentangle diffusivity from free-energetic effects. Thereby we can predict short-time drug penetration, where experimental measurements are not feasible, as well as long-time permeation, where ex vivo samples deteriorate, and thus span the entire timescales of biological barrier functioning.

Impact of solvent granularity and layering on tracer hydrodynamics in confinement

Classic hydrodynamic arguments establish that when a spherical tracer particle is suspended between parallel walls, tracer-wall coupling mediated by the solvent will cause the tracer to exhibit position-dependent diffusivity. We investigate how the diffusivity profiles of confined tracers are impacted by the diameter size-ratio of the tracer to solvent: starting from the classic limit of infinite size-ratio (i.e., continuum solvent), we consider size-ratios of four or less to examine how hydrodynamic predictions are disrupted for systems where the tracer and solvent are of similar scale. We use computer simulations and techniques based on the Fokker-Planck formalism to calculate the diffusivity profiles of hard-sphere tracer particles in hard-sphere solvents, focusing on the dynamics perpendicular to the walls. Given wall separations of several tracer diameters, we first consider confinement between hard walls, where anisotropic structuring at the solvent lengthscale generates inhomogeneity in the tracer free-energy landscape and undermines hydrodynamic predictions locally. We then introduce confining planes that we term transparent walls, which restrict tracer and solvent center-accessibilities while completely eliminating static anisotropy, and reveal position-dependent signatures in tracer diffusivity solely attributable to confinement. With or without suppressing static heterogeneity, we find that tracer diffusivity increasingly deviates on a local basis from hydrodynamic predictions at smaller size-ratios. However, hydrodynamic theory still approximately captures spatially-averaged dynamics across the pores even for very small tracer-solvent size-ratios over a wide range of solvent densities and wall separations.

The Effect of Tethers on Artificial Cell Membranes: A Coarse-Grained Molecular Dynamics Study

Tethered bilayer lipid membranes (tBLMs) provide a stable platform for modeling the dynamics and order of biological membranes where the tethers mimic the cytoskeletal supports present in biological cell membranes. In this paper coarse-grained molecular dynamics (CGMD) is applied to study the effects of tethers on lipid membrane properties. Using results from the CGMD model and the overdamped Fokker-Planck equation, we show that the diffusion tensor and particle density of water in the tBLM is spatially dependent. Further, it is shown that the membrane thickness, lipid diffusion, defect density, free energy of lipid flip-flop, and membrane dielectric permittivity are all dependent on the tether density. The numerically computed results from the CGMD model are in agreement with the experimentally measured results from tBLMs containing different tether densities and lipids derived from Archaebacteria. Additionally, using experimental measurements from Escherichia coli bacteria and Saccharomyces Cerevisiae yeast tethered membranes, we illustrate how previous molecular dynamics results can be combined with the proposed model to estimate the dielectric permittivity and defect density of these membranes as a function of tether density.

Characterization of step-edge barrier crossing of para-sexiphenyl on the ZnO (101[combining macron]0) surface

Mass transport processes of conjugated organic molecules (COMs) on inorganic surfaces are essential elements in thin film deposition for hybrid optoelectronic devices. Defects and in particular surface step-edges dictate the molecular nucleation and growth morphology, which itself determine many physical properties of the resulting hybrid interface. Here, we explore the detailed molecular kinetics and transport rates of a single physisorbed para-sexiphenyl (p-6P) molecule crossing a step-edge (a "hetero-Ehrlich-Schwoebel barrier") on the inorganic ZnO (101[combining macron]0) surface by a combination of all-atom molecular dynamics simulations and passage time theory. We determine temperature- and charge-dependent (free) energy landscapes, position-dependent diffusion coefficients, and ultimately the mean first passage time over the step-edges. We find two completely different step-edge crossing mechanisms, the occurrence and rates of which simultaneously depend on both electrostatic and thermal molecule-surface coupling. In weakly coupled systems, the molecule crosses the step relatively quickly (in nanoseconds) by log-roll mechanisms while for strongly coupled systems, it crosses relatively slowly (in microseconds) in a strictly perpendicular fashion. In the latter process, "internal friction" from intramolecular bending and torsional degrees of freedom contribute a significant corrugation to the overall crossing barrier. Furthermore, we show that crossing pathways can also change qualitatively with step-edge height. The great complexity in hetero-barrier crossing of COMs (in contrast to simple atoms) revealed in this study has implications on the interpretation and possible control of nucleation and growth mechanisms at surface defects in hybrid systems.

Confined Water Determines Transport Properties of Guest Molecules in Narrow Pores

We computed the transport of methane through 1 nm wide slit-shaped pores carved out of solid substrates. Models for silica, magnesium oxide, and alumina were used as solid substrates. The pores were filled with water. The results show that the methane permeability through the hydrated pores is strongly dependent on the solid substrate. Detailed analysis of the simulated systems reveals that local properties of confined water, including its structure, and more importantly, evolution of solvation free energy and hydrogen bond structure are responsible for the pronounced differences observed. The simulations are extended to multicomponent systems representative of natural gas, containing methane, ethane, and H2S. The results show that all pores considered have high affinity for H2S, moderate affinity for methane, and low affinity for ethane. The H2S/methane transport selectivity through the hydrated alumina pore is comparable, or superior, to that reported for existing commercial membranes. A multiscale approach was then implemented to demonstrate that a Smoluchowski one-dimensional model is able to reproduce the molecular-level results for short pores when appropriate values for the local self-diffusion coefficients are used as input parameters. We propose that the model can be extended to predict methane transport through uniform hydrated pores of macroscopic length. When verified by experiments, our simulation results could have important implications in applications such as natural gas sweetening and predictions of methane migration through hydraulically fractured shale formations.

Capturing transition paths and transition states for conformational rearrangements in the ribosome

To reveal the molecular determinants of biological function, one seeks to characterize the interactions that are formed in conformational and chemical transition states. In other words, what interactions govern the molecule's energy landscape? To accomplish this, it is necessary to determine which degrees of freedom can unambiguously identify each transition state. Here, we perform simulations of large-scale aminoacyl-transfer RNA (aa-tRNA) rearrangements during accommodation on the ribosome and project the dynamics along experimentally accessible atomic distances. From this analysis, we obtain evidence for which coordinates capture the correct number of barrier-crossing events and accurately indicate when the aa-tRNA is on a transition path. Although a commonly used coordinate in single-molecule experiments performs poorly, this study implicates alternative coordinates along which rearrangements are accurately described as diffusive movements across a one-dimensional free-energy profile. From this, we provide the theoretical foundation required for single-molecule techniques to uncover the energy landscape governing aa-tRNA selection by the ribosome.

Connecting thermal and mechanical protein (un)folding landscapes

Molecular dynamics simulations supplement single-molecule pulling experiments by providing the possibility of examining the full free energy landscape using many coordinates. Here, we use an all-atom structure-based model to study the force and temperature dependence of the unfolding of the protein filamin by applying force at both termini. The unfolding time-force relation τ(F) indicates that the force-induced unfolding behavior of filamin can be characterized into three regimes: barrier-limited low- and intermediate-force regimes, and a barrierless high-force regime. Slope changes of τ(F) separate the three regimes. We show that the behavior of τ(F) can be understood from a two-dimensional free energy landscape projected onto the extension X and the fraction of native contacts Q. In the low-force regime, the unfolding rate is roughly force-independent due to the small (even negative) separation in X between the native ensemble and transition state ensemble (TSE). In the intermediate-force regime, force sufficiently separates the TSE from the native ensemble such that τ(F) roughly follows an exponential relation. This regime is typically explored by pulling experiments. While X may fail to resolve the TSE due to overlap with the unfolded ensemble just below the folding temperature, the overlap is minimal at lower temperatures where experiments are likely to be conducted. The TSE becomes increasingly structured with force, whereas the average order of structural events during unfolding remains roughly unchanged. The high-force regime is characterized by barrierless unfolding, and the unfolding time approaches a limit of ∼10 μs for the highest forces we studied. Finally, a combination of X and Q is shown to be a good reaction coordinate for almost the entire force range.