Rotational Brownian Dynamics simulations of clathrin cage formation

Observing reorientation dynamics with Time-Resolved fluorescence and molecular dynamics in varying periodic boundary conditions

This work presents a combined study of time-resolved fluorescence spectroscopy and all-atom molecular dynamics simulation to investigate periodic boundary conditions' and water models' influence on the orientation dynamics and translational and rotational diffusion of peptides in solution. We have characterized the effects of solvent box size and water model choice on the dynamics of two peptide systems, NATA and WK5. Computationally, translational, and rotational diffusion and internal fluctuations are investigated through all-atom molecular dynamics simulation with two water models and different box sizes. These results are compared with time-resolved fluorescence anisotropy decay (FAD) measurements. The associated time constant and orientation dynamics from FAD measurement along the ¹L_b axis provided baseline data to validate molecular dynamics simulation. The modeling results show that diffusion rates vary roughly in inverse proportion to water model viscosity, as one would expect. Corrections for finite box size are significant for translational diffusion and insignificant for rotational diffusion. This study also finds that internal dynamics described by autocorrelation functions and kinetic network models are relatively insensitive to both box size and water model properties. Our observation suggests that different peptide properties respond differently to a change in simulation conditions.Communicated by Ramaswamy H. Sarma.

Large self-assembled clathrin lattices spontaneously disassemble without sufficient adaptor proteins

Clathrin-coated structures must assemble on cell membranes to internalize receptors, with the clathrin protein only linked to the membrane via adaptor proteins. These structures can grow surprisingly large, containing over 20 clathrin, yet they often fail to form productive vesicles, instead aborting and disassembling. We show that clathrin structures of this size can both form and disassemble spontaneously when adaptor protein availability is low, despite high abundance of clathrin. Here, we combine recent in vitro kinetic measurements with microscopic reaction-diffusion simulations and theory to differentiate mechanisms of stable vs unstable clathrin assembly on membranes. While in vitro conditions drive assembly of robust, stable lattices, we show that concentrations, geometry, and dimensional reduction in physiologic-like conditions do not support nucleation if only the key adaptor AP-2 is included, due to its insufficient abundance. Nucleation requires a stoichiometry of adaptor to clathrin that exceeds 1:1, meaning additional adaptor types are necessary to form lattices successfully and efficiently. We show that the critical nucleus contains ~25 clathrin, remarkably similar to sizes of the transient and abortive structures observed in vivo. Lastly, we quantify the cost of bending the membrane under our curved clathrin lattices using a continuum membrane model. We find that the cost of bending the membrane could be largely offset by the energetic benefit of forming curved rather than flat structures, with numbers comparable to experiments. Our model predicts how adaptor density can tune clathrin-coated structures from the transient to the stable, showing that active energy consumption is therefore not required for lattice disassembly or remodeling during growth, which is a critical advance towards predicting productive vesicle formation.

NERDSS: A Nonequilibrium Simulator for Multibody Self-Assembly at the Cellular Scale

Currently, a significant barrier to building predictive models of cellular self-assembly processes is that molecular models cannot capture minutes-long dynamics that couple distinct components with active processes, whereas reaction-diffusion models cannot capture structures of molecular assembly. Here, we introduce the nonequilibrium reaction-diffusion self-assembly simulator (NERDSS), which addresses this spatiotemporal resolution gap. NERDSS integrates efficient reaction-diffusion algorithms into generalized software that operates on user-defined molecules through diffusion, binding and orientation, unbinding, chemical transformations, and spatial localization. By connecting the fast processes of binding with the slow timescales of large-scale assembly, NERDSS integrates molecular resolution with reversible formation of ordered, multisubunit complexes. NERDSS encodes models using rule-based formatting languages to facilitate model portability, usability, and reproducibility. Applying NERDSS to steps in clathrin-mediated endocytosis, we design multicomponent systems that can form lattices in solution or on the membrane, and we predict how stochastic but localized dephosphorylation of membrane lipids can drive lattice disassembly. The NERDSS simulations reveal the spatial constraints on lattice growth and the role of membrane localization and cooperativity in nucleating assembly. By modeling viral lattice assembly and recapitulating oscillations in protein expression levels for a circadian clock model, we illustrate the adaptability of NERDSS. NERDSS simulates user-defined assembly models that were previously inaccessible to existing software tools, with broad applications to predicting self-assembly in vivo and designing high-yield assemblies in vitro.

Modeling the Self-Assembly of Protein Complexes through a Rigid-Body Rotational Reaction-Diffusion Algorithm

The reaction-diffusion equations provide a powerful framework for modeling nonequilibrium, cell-scale dynamics over the long time scales that are inaccessible by traditional molecular modeling approaches. Single-particle reaction-diffusion offers the highest resolution technique for tracking such dynamics, but it has not been applied to the study of protein self-assembly due to its treatment of reactive species as single-point particles. Here, we develop a relatively simple but accurate approach for building rigid structure and rotation into single-particle reaction-diffusion methods, providing a rate-based method for studying protein self-assembly. Our simplifying assumption is that reactive collisions can be evaluated purely on the basis of the separations between the sites, and not their orientations. The challenge of evaluating reaction probabilities can then be performed using well-known equations based on translational diffusion in both 3D and 2D, by employing an effective diffusion constant we derive here. We show how our approach reproduces both the kinetics of association, which is altered by rotational diffusion, and the equilibrium of reversible association, which is not. Importantly, the macroscopic kinetics of association can be predicted on the basis of the microscopic parameters of our structurally resolved model, allowing for critical comparisons with theory and other rate-based simulations. We demonstrate this method for efficient, rate-based simulations of self-assembly of clathrin trimers, highlighting how formation of regular lattices impacts the kinetics of association.

The role of multivalency in the association kinetics of patchy particle complexes

Association and dissociation of particles are elementary steps in many natural and technological relevant processes. For many such processes, the presence of multiple binding sites is essential. For instance, protein complexes and regular structures such as virus shells are formed from elementary building blocks with multiple binding sites. Here we address a fundamental question concerning the role of multivalency of binding sites in the association kinetics of such complexes. Using single replica transition interface sampling simulations, we investigate the influence of the multivalency on the binding kinetics and the association mechanism of patchy particles that form polyhedral clusters. When the individual bond strength is fixed, the kinetics naturally is very dependent on the multivalency, with dissociation rate constants exponentially decreasing with the number of bonds. In contrast, we find that when the total bond energy per particle is kept constant, association and dissociation rate constants turn out rather independent of multivalency, although of course still very dependent on the total energy. The association and dissociation mechanisms, however, depend on the presence and nature of the intermediate states. For instance, pathways that visit intermediate states are less prevalent for particles with five binding sites compared to the case of particles with only three bonds. The presence of intermediate states can lead to kinetic trapping and malformed aggregates. We discuss implications for natural forming complexes such as virus shells and for the design of artificial colloidal patchy particles.

Mobility and settling rate of agglomerates of polydisperse nanoparticles

Agglomerate settling impacts nanotoxicology and nanomedicine as well as the stability of engineered nanofluids. Here, the mobility of nanostructured fractal-like SiO₂ agglomerates in water is investigated and their settling rate in infinitely dilute suspensions is calculated by a Brownian dynamics algorithm tracking the agglomerate translational and rotational motion. The corresponding friction matrices are obtained using the HYDRO++ algorithm [J. G. de la Torre, G. del Rio Echenique, and A. Ortega, J. Phys. Chem. B 111, 955 (2007)] from the Kirkwood-Riseman theory accounting for hydrodynamic interactions of primary particles (PPs) through the Rotne-Prager-Yamakawa tensor, properly modified for polydisperse PPs. Agglomerates are generated by an event-driven method and have constant mass fractal dimension but varying PP size distribution, mass, and relative shape anisotropy. The calculated diffusion coefficient from HYDRO++ is used to obtain the agglomerate mobility diameter d_m and is compared with that from scaling laws for fractal-like agglomerates. The ratio d_m/d_g of the mobility diameter to the gyration diameter of the agglomerate decreases with increasing relative shape anisotropy. For constant d_m and mean d_p, the agglomerate settling rate, u_s, increases with increasing PP geometric standard deviation σ_p,g (polydispersity). A linear relationship between u_s and agglomerate mass to d_m ratio, m/d_m, is revealed and attributed to the fast Brownian rotation of such small and light nanoparticle agglomerates. An analytical expression for the u_s of agglomerates consisting of polydisperse PPs is then derived, u_s=1-ρ_fρ_pg3πμmd_m (ρ_f is the density of the fluid, ρ_p is the density of PPs, μ is the viscosity of the fluid, and g is the acceleration of gravity), valid for agglomerates for which the characteristic rotational time is considerably shorter than their settling time. Our calculations demonstrate that the commonly made assumption of monodisperse PPs underestimates u_s by a fraction depending on σ_p,g and agglomerate mass mobility exponent. Simulations are in excellent agreement with deposition rate measurements of fumed SiO₂ agglomerates in water.

Clathrin Assembly Regulated by Adaptor Proteins in Coarse-Grained Models

The assembly of clathrin triskelia into polyhedral cages during endocytosis is regulated by adaptor proteins (APs). We explore how APs achieve this by developing coarse-grained models for clathrin and AP2, employing a Monte Carlo click interaction, to simulate their collective aggregation behavior. The phase diagrams indicate that a crucial role is played by the mechanical properties of the disordered linker segment of AP. We also present a statistical-mechanical theory for the assembly behavior of clathrin, yielding good agreement with our simulations and experimental data from the literature. Adaptor proteins are found to regulate the formation of clathrin coats under certain conditions, but can also suppress the formation of cages.

Quantitative computational models of molecular self-assembly in systems biology

Molecular self-assembly is the dominant form of chemical reaction in living systems, yet efforts at systems biology modeling are only beginning to appreciate the need for and challenges to accurate quantitative modeling of self-assembly. Self-assembly reactions are essential to nearly every important process in cell and molecular biology and handling them is thus a necessary step in building comprehensive models of complex cellular systems. They present exceptional challenges, however, to standard methods for simulating complex systems. While the general systems biology world is just beginning to deal with these challenges, there is an extensive literature dealing with them for more specialized self-assembly modeling. This review will examine the challenges of self-assembly modeling, nascent efforts to deal with these challenges in the systems modeling community, and some of the solutions offered in prior work on self-assembly specifically. The review concludes with some consideration of the likely role of self-assembly in the future of complex biological system models more generally.

Coarse-grained simulations for flow of complex soft matter fluids in the bulk and in the presence of solid interfaces

We present a coarse-grained particle-based simulation technique for modeling flow of complex soft matter fluids such as polymer solutions in the presence of solid interfaces. In our coarse-grained description of the system, we track the motion of polymer molecules using their centers-of-mass as our coarse-grain co-ordinates and also keep track of another set of variables that describe the background flow field. The coarse-grain motion is thus influenced not only by the interactions based on appropriate potentials used to model the particular polymer system of interest and the random kicks associated with thermal fluctuations, but also by the motion of the background fluid. In order to couple the motion of the coarse-grain co-ordinates with the background fluid motion, we use a Galilean invariant, first order Brownian dynamics algorithm developed by Padding and Briels [J. Chem. Phys. 141, 244108 (2014)], which on the one hand draws inspiration from smoothed particle hydrodynamics in a way that the motion of the background fluid is efficiently calculated based on a discretization of the Navier-Stokes equation at the positions of the coarse-grain coordinates where it is actually needed, but also differs from it because of the inclusion of thermal fluctuations by having momentum-conserving pairwise stochastic updates. In this paper, we make a few modifications to this algorithm and introduce a new parameter, viz., a friction coefficient associated with the background fluid, and analyze the relationship of the model parameters with the dynamic properties of the system. We also test this algorithm for flow in the presence of solid interfaces to show that appropriate boundary conditions can be imposed at solid-fluid interfaces by using artificial particles embedded in the solid walls which offer friction to the real fluid particles in the vicinity of the wall. We have tested our method using a model system of a star polymer solution at the overlap concentration.

Hydrodynamics and Brownian motions of a spheroid near a rigid wall

In this work, we study in detail the hydrodynamics and the Brownian motions of a spheroidal particle suspended in a Newtonian fluid near a flat rigid wall. We employ 3D Finite Element Method (FEM) simulations to compute how the mobility tensor of the spheroid varies with both the particle-wall separation distance and the particle orientation. We then study the Brownian motion of the spheroid by means of a discretized Langevin equation. We specifically focus on the additional drift terms arising from the position and orientational dependence of the mobility matrix. In this respect, we also propose a numerically convenient approximation of the orientational divergence of the mobility matrix that is required in the solution of the Langevin equation. Our results illustrate that both hydrodynamics and Brownian motions of a spheroidal particle near a confining wall display novel features from those of a sphere in the same type of confinement.