Dynamical Reweighting for Biased Rare Event Simulations.

Dynamical reweighting techniques aim to recover the correct molecular dynamics from a simulation at a modified potential energy surface. They are important for unbiasing enhanced sampling simulations of molecular rare events. Here, we review the theoretical frameworks of dynamical reweighting for modified potentials. Based on an overview of kinetic models with increasing level of detail, we discuss techniques to reweight two-state dynamics, multistate dynamics, and path integrals. We explore the natural link to transition path sampling and how the effect of nonequilibrium forces can be reweighted. We end by providing an outlook on how dynamical reweighting integrates with techniques for optimizing collective variables and with modern potential energy surfaces.

Decoding dissociation of sequence-specific protein-DNA complexes with non-equilibrium simulations.

Sequence-specific protein-DNA interactions are crucial in processes such as DNA organization, gene regulation and DNA replication. Obtaining detailed insights into the recognition mechanisms of protein-DNA complexes through experiments is hampered by a lack of resolution in both space and time. Here, we present a molecular simulation approach to quantify the sequence specificity of protein-DNA complexes, that yields results fast, and is generally applicable to any protein-DNA complex. The approach is based on molecular dynamics simulations in combination with a sophisticated steering potential and results in an estimate of the free energy difference of dissociation. We provide predictions of the nucleotide specific binding affinity of the minor groove binding Histone-like Nucleoid Structuring (H-NS) protein, that are in agreement with experimental data. Furthermore, our approach offers mechanistic insight into the process of dissociation. Applying our approach to the major groove binding ETS domain in complex with three different nucleotide sequences identified the high affinity consensus sequence, quantitatively in agreement with experiments. Our protocol facilitates quantitative prediction of protein-DNA complex stability, while also providing high resolution insights into recognition mechanisms. As such, our simulation approach has the potential to yield detailed and quantitative insights into biological processes involving sequence-specific protein-DNA interactions.

Data-driven dynamical coarse-graining for condensed matter systems.

Simulations of condensed matter systems often focus on the dynamics of a few distinguished components but require integrating the full system. A prime example is a molecular dynamics simulation of a (macro)molecule in a solution, where the molecule(s) and the solvent dynamics need to be integrated, rendering the simulations computationally costly and often unfeasible for physically/biologically relevant time scales. Standard coarse graining approaches can reproduce equilibrium distributions and structural features but do not properly include the dynamics. In this work, we develop a general data-driven coarse-graining methodology inspired by the Mori-Zwanzig formalism, which shows that macroscopic systems with a large number of degrees of freedom can be described by a few relevant variables and additional noise and memory terms. Our coarse-graining method consists of numerical integrators for the distinguished components, where the noise and interaction terms with other system components are substituted by a random variable sampled from a data-driven model. The model is parameterized using data from multiple short-time full-system simulations, and then, it is used to run long-time simulations. Applying our methodology to three systems-a distinguished particle under a harmonic and a bistable potential and a dimer with two metastable configurations-the resulting coarse-grained models are capable of reproducing not only the equilibrium distributions but also the dynamic behavior due to temporal correlations and memory effects. Remarkably, our method even reproduces the transition dynamics between metastable states, which is challenging to capture correctly. Our approach is not constrained to specific dynamics and can be extended to systems beyond Langevin dynamics, and, in principle, even to non-equilibrium dynamics.

A method of incorporating rate constants as kinetic constraints in molecular dynamics simulations.

From the point of view of statistical mechanics, a full characterization of a molecular system requires an accurate determination of its possible states, their populations, and the respective interconversion rates. Toward this goal, well-established methods increase the accuracy of molecular dynamics simulations by incorporating experimental information about states using structural restraints and about populations using thermodynamics restraints. However, it is still unclear how to include experimental knowledge about interconversion rates. Here, we introduce a method of imposing known rate constants as constraints in molecular dynamics simulations, which is based on a combination of the maximum-entropy and maximum-caliber principles. Starting from an existing ensemble of trajectories, obtained from either molecular dynamics or enhanced trajectory sampling, this method provides a minimally perturbed path distribution consistent with the kinetic constraints, as well as modified free energy and committor landscapes. We illustrate the application of the method to a series of model systems, including all-atom molecular simulations of protein folding. Our results show that by combining experimental rate constants and molecular dynamics simulations, this approach enables the determination of transition states, reaction mechanisms, and free energies. We anticipate that this method will extend the applicability of molecular simulations to kinetic studies in structural biology and that it will assist the development of force fields to reproduce kinetic and thermodynamic observables. Furthermore, this approach is generally applicable to a wide range of systems in biology, physics, chemistry, and material science.

Homogeneous nucleation of crystalline methane hydrate in molecular dynamics transition paths sampled under realistic conditions.

Methane hydrates are important from a scientific and industrial perspective, and form by nucleation and growth from a supersaturated aqueous solution of methane. Molecular simulation is able to shed light on the process of homogeneous nucleation of hydrates, using straightforward molecular dynamics or rare event enhanced sampling techniques with atomistic and coarse grained force fields. In our previous work [Arjun, T. A. Berendsen, and P. G. Bolhuis, Proc. Natl. Acad. Sci. U. S. A. 116, 19305 (2019)], we performed transition path sampling (TPS) simulations using all atom force fields under moderate driving forces at high pressure, which enabled unbiased atomistic insight into the formation of methane hydrates. The supersaturation in these simulations was influenced by the Laplace pressure induced by the spherical gas reservoir. Here, we investigate the effect of removing this influence. Focusing on the supercooled, supersaturated regime to keep the system size tractable, our TPS simulations indicate that nuclei form amorphous structures below roughly 260 K and crystalline sI structures above 260 K. For these temperatures, the average transition path lengths are significantly longer than in our previous study, pushing the boundaries of what can be achieved with TPS. The temperature to observe a critical nucleus of certain size was roughly 20 K lower compared to a spherical reservoir due to the lower concentration of methane in the solution, yielding a reduced driving force. We analyze the TPS results using a model based on classical nucleation theory. The corresponding free energy barriers are estimated and found to be consistent with previous predictions, thus adding to the overall picture of the hydrate formation process.

Optimizing molecular potential models by imposing kinetic constraints with path reweighting.

Empirical force fields employed in molecular dynamics simulations of complex systems are often optimized to reproduce experimentally determined structural and thermodynamic properties. In contrast, experimental knowledge about the interconversion rates between metastable states in such systems is hardly ever incorporated in a force field due to a lack of an efficient approach. Here, we introduce such a framework based on the relationship between dynamical observables, such as rate constants, and the underlying molecular model parameters using the statistical mechanics of trajectories. Given a prior ensemble of molecular dynamics trajectories produced with imperfect force field parameters, the approach allows for the optimal adaption of these parameters such that the imposed constraint of equally predicted and experimental rate constant is obeyed. To do so, the method combines the continuum path ensemble maximum caliber approach with path reweighting methods for stochastic dynamics. When multiple solutions are found, the method selects automatically the combination that corresponds to the smallest perturbation of the entire path ensemble, as required by the maximum entropy principle. To show the validity of the approach, we illustrate the method on simple test systems undergoing rare event dynamics. Next to simple 2D potentials, we explore particle models representing molecular isomerization reactions and protein-ligand unbinding. Besides optimal interaction parameters, the methodology gives physical insights into what parts of the model are most sensitive to the kinetics. We discuss the generality and broad implications of the methodology.

Self-assembly of microcapsules via colloidal bond hybridization and anisotropy.

Particles with directional interactions are promising building blocks for new functional materials and may serve as models for biological structures. Mutually attractive nanoparticles that are deformable owing to flexible surface groups, for example, may spontaneously order themselves into strings, sheets and large vesicles. Furthermore, anisotropic colloids with attractive patches can self-assemble into open lattices and the colloidal equivalents of molecules and micelles. However, model systems that combine mutual attraction, anisotropy and deformability have not yet been realized. Here we synthesize colloidal particles that combine these three characteristics and obtain self-assembled microcapsules. We propose that mutual attraction and deformability induce directional interactions via colloidal bond hybridization. Our particles contain both mutually attractive and repulsive surface groups that are flexible. Analogously to the simplest chemical bond--in which two isotropic orbitals hybridize into the molecular orbital of H2--these flexible groups redistribute on binding. Via colloidal bond hybridization, isotropic spheres self-assemble into planar monolayers, whereas anisotropic snowman-shaped particles self-assemble into hollow monolayer microcapsules. A modest change in the building blocks thus results in much greater complexity of the self-assembled structures. In other words, these relatively simple building blocks self-assemble into markedly more complex structures than do similar particles that are isotropic or non-deformable.

Unbiased atomistic insight in the competing nucleation mechanisms of methane hydrates.

Methane hydrates have important industrial and climate implications, yet their formation via homogeneous nucleation under natural, moderate conditions is poorly understood. Obtaining such understanding could lead to improved control of crystallization, as well as insight into polymorph selection in general, but is hampered by limited experimental resolution. Direct molecular dynamics simulations using atomistic force fields could provide such insight, but are not feasible for moderate undercooling, due to the rare event nature of nucleation. Instead, we harvest ensembles of the rare unbiased nucleation trajectories by employing transition path sampling. We find that with decreasing undercooling the mechanism shifts from amorphous to crystalline polymorph formation. At intermediate temperature the 2 mechanisms compete. Reaction coordinate analysis reveals the amount of a specific methane cage type is crucial for crystallization, while irrelevant for amorphous solids. Polymorph selection is thus governed by kinetic accessibility of the correct cage type and, moreover, occurs at precritical nucleus sizes, apparently against Ostwald's step rule. We argue that these results are still in line with classical nucleation theory. Our findings illuminate how selection between competing methane hydrate polymorphs occurs and might generalize to other hydrates and molecular crystal formation.

Revealing Polymerization Kinetics with Colloidal Dipatch Particles.

Limited-valency colloidal particles can self-assemble into polymeric structures analogous to molecules. While their structural equilibrium properties have attracted wide attention, insight into their dynamics has proven challenging. Here, we investigate the polymerization dynamics of semiflexible polymers in 2D by direct observation of assembling divalent particles, bonded by critical Casimir forces. The reversible critical Casimir force creates living polymerization conditions with tunable chain dissociation, association, and bending rigidity. We find that unlike dilute polymers that show exponential size distributions in excellent agreement with Flory theory, concentrated samples exhibit arrest of rotational and translational diffusion due to a continuous isotropic-to-nematic transition in 2D, slowing down the growth kinetics. These effects are circumvented by the addition of higher-valency particles, cross linking the polymers into networks. Our results connecting polymer flexibility, polymer interactions, and the peculiar isotropic-nematic transition in 2D offer insight into the polymerization processes of synthetic two-dimensional polymers and biopolymers at membranes and interfaces.

Revealing viscoelastic bending relaxation dynamics of isolated semiflexible colloidal polymers.

The viscoelastic properties of filaments and biopolymers play a crucial role in soft and biological materials from biopolymer networks to novel synthetic metamaterials. Colloidal particles with specific valency allow mimicking polymers and more complex molecular structures at the colloidal scale, offering direct observation of their internal degrees of freedom. Here, we elucidate the time-dependent viscoelastic response in the bending of isolated semi-flexible colloidal polymers, assembled from dipatch colloidal particles by reversible critical Casimir forces. By tuning the patch-patch interaction strength, we adjust the polymers' viscoelastic properties, and follow spontaneous bending modes and their relaxation directly on the particle level. We find that the elastic response is well described by that of a semiflexible rod with persistence length of order 1000 µm, tunable by the critical Casimir interaction strength. We identify the viscous relaxation on longer timescales to be due to internal friction, leading to a wavelength-independent relaxation time similar to single biopolymers, but in the colloidal case arising from the contact mechanics of the bonded patches. These tunable mechanical properties of assembled colloidal filaments open the door to "colloidal architectures", rationally designed (network) structures with desired topology and mechanical properties.

Homogenous nucleation rate of CO2 hydrates using transition interface sampling.

Carbon dioxide and water can form solid clathrate structures in which water cages encapsulate the gas molecules. Such hydrates have sparked much interest due to their possible application in CO2 sequestration. How the solid structure forms exactly from the liquid phase via a homogenous nucleation process is still poorly understood. This nucleation event is rare on the molecular timescale even under moderate undercooling or supersaturation conditions because of the large free energy barrier toward crystallization, rendering a brute force simulation of hydrate nucleation unfeasible for moderate undercooling or supersaturation. Here, we perform transition interface sampling simulations to quantify the homogenous nucleation rate for CO2 hydrate formation using accurate atomistic force fields at 500 bars for three different temperatures between 260 and 273 K. Collecting more than 100 000 pathways comprising roughly two milliseconds of simulation time, we computed a nucleation rate in the amorphous phase of ∼1021 nuclei s-1 cm-3 for a temperature of 260 K and a rate of ∼1012 nuclei s-1 cm-3 for a temperature of 265 K. For a temperature of 273 K, we find that the hydrate forms an sI crystalline phase with a rate of order of ∼101 nuclei s-1 cm-3. We compare these rates to classical nucleation theory estimates as well as experiments, and to nucleation rate estimates for methane hydrates and discuss possible causes of the observed differences. Our findings shed light on the kinetics of this important clathrate and should assist in future hydrate formation investigation.

Atomistic insight into the kinetic pathways for Watson-Crick to Hoogsteen transitions in DNA.

DNA predominantly contains Watson-Crick (WC) base pairs, but a non-negligible fraction of base pairs are in the Hoogsteen (HG) hydrogen bonding motif at any time. In HG, the purine is rotated ∼180° relative to the WC motif. The transitions between WC and HG may play a role in recognition and replication, but are difficult to investigate experimentally because they occur quickly, but only rarely. To gain insight into the mechanisms for this process, we performed transition path sampling simulations on a model nucleotide sequence in which an AT pair changes from WC to HG. This transition can occur in two ways, both starting with loss of hydrogen bonds in the base pair, followed by rotation around the glycosidic bond. In one route the adenine base converts from WC to HG geometry while remaining entirely within the double helix. The other route involves the adenine leaving the confines of the double helix and interacting with water. Our results indicate that this outside route is more probable. We used transition interface sampling to compute rate constants and relative free energies for the transitions between WC and HG. Our results agree with experiments, and provide highly detailed insights into the mechanisms of this important process.

Approximating free energy and committor landscapes in standard transition path sampling using virtual interface exchange.

Transition path sampling is a powerful technique for investigating rare transitions, especially when the mechanism is unknown and one does not have access to the reaction coordinate. Straightforward application of transition path sampling does not directly provide the free energy landscape nor the kinetics. This drawback has motivated the development of path sampling extensions able to simultaneously access both kinetics and thermodynamics, such as transition interface sampling, and the reweighted path ensemble. However, performing transition interface sampling is more involved than standard two-state transition path sampling and still requires (some) insight into the reaction to define interfaces. While packages that can efficiently compute path ensembles for transition interface sampling are now available, it would be useful to directly compute the free energy from a single standard transition path sampling simulation. To achieve this, we present here an approximate method, denoted virtual interface exchange transition path sampling, that makes use of the rejected pathways in a form of waste recycling. The method yields an approximate reweighted path ensemble that allows an immediate view of the free energy landscape from a standard single transition path sampling simulation, as well as enables a committor analysis.

Nested Transition Path Sampling.

We introduce a novel transition path (TPS) sampling scheme employing nested sampling. Analogous to how nested sampling explores the entire configurational phase space for atomistic systems, nested TPS samples the entire available trajectory space in one simulation. Thermodynamic and path observables can be constructed a posteriori for all temperatures simultaneously. We exploit this to compute the rate of rare processes at arbitrarily low temperature through the coupling to easily accessible rates at high temperature. We illustrate the method on several model systems.

Water structure and dynamics in the hydration layer of a type III anti-freeze protein.

We report on a molecular dynamics study on the relation between the structure and the orientational (and hydrogen bond) dynamics of hydration water around the ocean pout AFP III anti-freeze protein. We find evidence for an increasing tetrahedral structure from the area opposite to the ice binding site (IBS) towards the protein IBS, with the strongest signal of tetrahedral structure around the THR-18 residue of the IBS. The tetrahedral structural parameter mostly positively correlates with increased reorientation decay times. Interestingly, for several key (polar) residues that are not part of the IBS but are in its vicinity, we observe a decrease of the reorientation time with increasing tetrahedral structure. A similar anti-correlation is observed for the hydrogen-bonded water molecules. These effects are enhanced at a lower temperature. We interpret these results in terms of the structure-making and structure-breaking residues. Moreover, we investigate the tetrahedral structure and dynamics of waters at a partially dehydrated IBS, and for the protein adsorbed at the air-water interface. We find that the mutation changes the preferred protein orientation upon adsorption at an air-water interface. These results are in agreement with the water-air Vibration Sum Frequency Generation spectroscopic experiments showing a strongly reduced tetrahedral signal upon mutation at the IBS.

Rate constants for proteins binding to substrates with multiple binding sites using a generalized forward flux sampling expression.

To predict the response of a biochemical system, knowledge of the intrinsic and effective rate constants of proteins is crucial. The experimentally accessible effective rate constant for association can be decomposed in a diffusion-limited rate at which proteins come into contact and an intrinsic association rate at which the proteins in contact truly bind. Reversely, when dissociating, bound proteins first separate into a contact pair with an intrinsic dissociation rate, before moving away by diffusion. While microscopic expressions exist that enable the calculation of the intrinsic and effective rate constants by conducting a single rare event simulation of the protein dissociation reaction, these expressions are only valid when the substrate has just one binding site. If the substrate has multiple binding sites, a bound enzyme can, besides dissociating into the bulk, also hop to another binding site. Calculating transition rate constants between multiple states with forward flux sampling requires a generalized rate expression. We present this expression here and use it to derive explicit expressions for all intrinsic and effective rate constants involving binding to multiple states, including rebinding. We illustrate our approach by computing the intrinsic and effective association, dissociation, and hopping rate constants for a system in which a patchy particle model enzyme binds to a substrate with two binding sites. We find that these rate constants increase as a function of the rotational diffusion constant of the particles. The hopping rate constant decreases as a function of the distance between the binding sites. Finally, we find that blocking one of the binding sites enhances both association and dissociation rate constants. Our approach and results are important for understanding and modeling association reactions in enzyme-substrate systems and other patchy particle systems and open the way for large multiscale simulations of such systems.

Ratchet-induced variations in bulk states of an active ideal gas.

We study the distribution of active, noninteracting particles over two bulk states separated by a ratchet potential. By solving the steady-state Smoluchowski equations in a flux-free setting, we show that the ratchet potential affects the distribution of particles over the bulks and thus exerts an influence of infinitely long range. As we show, an external potential that is nonlinear is crucial for having such a long-range influence. We characterize how the difference in bulk densities depends on activity and on the ratchet potential, and we identify power law dependencies on system parameters in several limiting cases. While weakly active systems are often understood in terms of an effective temperature, we present an analytical solution that explicitly shows that this is not possible in the current setting. Instead, we rationalize our results by a simple transition state model that presumes particles to cross the potential barrier by Arrhenius rates modified for activity. While this model does not quantitatively describe the difference in bulk densities for feasible parameter values, it does reproduce-in its regime of applicability-the complete power law behavior correctly.

Rotational diffusion affects the dynamical self-assembly pathways of patchy particles.

Predicting the self-assembly kinetics of particles with anisotropic interactions, such as colloidal patchy particles or proteins with multiple binding sites, is important for the design of novel high-tech materials, as well as for understanding biological systems, e.g., viruses or regulatory networks. Often stochastic in nature, such self-assembly processes are fundamentally governed by rotational and translational diffusion. Whereas the rotational diffusion constant of particles is usually considered to be coupled to the translational diffusion via the Stokes-Einstein relation, in the past decade it has become clear that they can be independently altered by molecular crowding agents or via external fields. Because virus capsids naturally assemble in crowded environments such as the cell cytoplasm but also in aqueous solution in vitro, it is important to investigate how varying the rotational diffusion with respect to transitional diffusion alters the kinetic pathways of self-assembly. Kinetic trapping in malformed or intermediate structures often impedes a direct simulation approach of a kinetic network by dramatically slowing down the relaxation to the designed ground state. However, using recently developed path-sampling techniques, we can sample and analyze the entire self-assembly kinetic network of simple patchy particle systems. For assembly of a designed cluster of patchy particles we find that changing the rotational diffusion does not change the equilibrium constants, but significantly affects the dynamical pathways, and enhances (suppresses) the overall relaxation process and the yield of the target structure, by avoiding (encountering) frustrated states. Besides insight, this finding provides a design principle for improved control of nanoparticle self-assembly.

Modelling critical Casimir force induced self-assembly experiments on patchy colloidal dumbbells.

Colloidal particles suspended in a binary liquid mixture can interact via solvent mediated interactions, known as critical Casimir forces. For anisotropic colloids this interaction becomes directional, which leads to rich phase behavior. While experimental imaging and particle tracking techniques allow determination of isotropic effective potentials via Boltzmann inversion, the modeling of effective interaction in anisotropic systems is non-trivial precisely because of this directionality. Here we extract effective interaction potentials for non-spherical dumbbell particles from observed radial and angular distributions, by employing reference interaction site model (RISM) theory and direct Monte Carlo simulations. For colloidal dumbbell particles dispersed in a binary liquid mixture and interacting via induced critical Casimir forces, we determine the effective site-site potentials for a range of experimental temperatures. Using these potentials to simulate the system for strong Casimir forces, we reproduce the experimentally observed collapse, and provide a qualitative explanation for this behavior.

Stability and growth mechanism of self-assembling putative antifreeze cyclic peptides.

Cyclic peptides (CPs) that self-assemble in nanotubes can be candidates for use as antifreeze proteins. Based on the cyclic peptide sequence cyclo-[(l-LYS-d-ALA-l-LEU-d-ALA)2], which can stack into nanotubes, we propose a putative antifreeze cyclic peptide (AFCP) sequence, cyclo-[(l-LYS-d-ALA)2-(l-THR-d-ALA)2], containing THR-ALA-THR ice binding motifs. Using molecular dynamics simulations we investigate the stability of these cyclic peptides and their growth mechanism. Both nanotube sequences get more stable as a function of size. The relative stability of the AFCP sequence CPNT increases at sizes greater than a dimer by forming intermolecular THR side chain H-bonds. We find that, like the naturally occurring AF protein from spruce budworm (Choristoneura fumiferana), the THR distances of the AFCP's ice binding motif match the ice prism plane O-O distances, thus making the AFCP a suitable AF candidate. In addition, we investigated the nanotube growth process, i.e. the association/dissociation of a single CP to an existing AFCP nanotube, by Transition Path Sampling. We found a general dock-lock mechanism, in which a single CP first docks loosely before locking into place. Moreover, we identified several qualitatively different mechanisms for association, involving different metastable intermediates, including a state in which the peptide was misfolded inside the hydrophobic core of the tube. Finally, we find evidence for a mechanism involving non-specific association followed by 1D diffusion. Under most conditions, this will be the dominant pathway. The results yield insights into the mechanisms of peptide assembly, and might lead to an improved design of self-assembling antifreeze proteins.