Atomic detail brownian dynamics simulations of concentrated protein solutions with a mean field treatment of hydrodynamic interactions

Computational Screening for mAb Colloidal Stability with Coarse-Grained, Molecular-Scale Simulations

Monoclonal antibodies (mAbs) are an important modality of protein therapeutics with broad applications for numerous diseases. However, colloidal instabilities occurring at high protein concentrations can limit the ability to develop stable, high-concentration liquid dosage forms that are required for patient-centric, device-mediated products. Therefore, it is advantageous to identify colloidally stable mAbs early in the discovery process to ensure that they are selected for development. Experimental screening for colloidal stability can be time- and resource-consuming and is most feasible at the later stages of drug development due to material requirements. Alternatively, computational approaches have emerging potential to provide efficient screening and focus developmental efforts on mAbs with the greatest developability potential, while providing mechanistic relationships for colloidal instability. In this work, coarse-grained, molecular-scale models were fine-tuned to screen for colloidal stability at amino-acid resolution. This model parameterization provides a framework to screen for mAb self-interactions and extrapolate to bulk solution behavior. This approach was applied to a wide array of mAbs under multiple buffer conditions, demonstrating the utility of the presented computational approach to augment early candidate screening and later formulation strategies for protein therapeutics.

What determines sub-diffusive behavior in crowded protein solutions?

The aqueous environment inside cells is densely packed. A typical cell has a macromolecular concentration in the range 90-450 g/L, with 5%-40% of its volume being occupied by macromolecules, resulting in what is known as macromolecular crowding. The space available for the free diffusion of metabolites and other macromolecules is thus greatly reduced, leading to so-called excluded volume effects. The slow diffusion of macromolecules under crowded conditions has been explained using transient complex formation. However, sub-diffusion noted in earlier works is not well characterized, particularly the role played by transient complex formation and excluded volume effects. We have used Brownian dynamics simulations to characterize the diffusion of chymotrypsin inhibitor 2 in protein solutions of bovine serum albumin and lysozyme at concentrations ranging from 50 to 300 g/L. The predicted changes in diffusion coefficient as a function of crowder concentration are consistent with NMR experiments. The sub-diffusive behavior observed in the sub-microsecond timescale can be explained in terms of a so-called cage effect, arising from rattling motion in a local molecular cage as a consequence of excluded volume effects. By selectively manipulating the nature of interactions between protein molecules, we determined that excluded volume effects induce sub-diffusive dynamics at sub-microsecond timescales. These findings may help to explain the diffusion-mediated effects of protein crowding on cellular processes.

Unusual Aspects of Charge Regulation in Flexible Weak Polyelectrolytes

This article reviews the state of the art of the studies on charge regulation (CR) effects in flexible weak polyelectrolytes (FWPE). The characteristic of FWPE is the strong coupling of ionization and conformational degrees of freedom. After introducing the necessary fundamental concepts, some unconventional aspects of the the physical chemistry of FWPE are discussed. These aspects are: (i) the extension of statistical mechanics techniques to include ionization equilibria and, in particular, the use of the recently proposed Site Binding-Rotational Isomeric State (SBRIS) model, which allows the calculation of ionization and conformational properties on the same foot; (ii) the recent progresses in the inclusion of proton equilibria in computer simulations; (iii) the possibility of mechanically induced CR in the stretching of FWPE; (iv) the non-trivial adsorption of FWPE on ionized surfaces with the same charge sign as the PE (the so-called "wrong side" of the isoelectric point); (v) the influence of macromolecular crowding on CR.

Diffusion of small molecule drugs is affected by surface interactions and crowder proteins

Crowded environments are known to affect the diffusion of macromolecules, but their effects on the diffusion of small molecules are largely uncharacterized. We investigate how three protein crowders, bovine serum albumin (BSA), hen egg-white lysozyme, and myoglobin, influence the diffusion rates and interactions of four small molecules: fluorescein, and three drugs, doxorubicin, glycogen synthase kinase-3 inhibitor SB216763, and quinacrine. Using Line-FRAP measurements, Brownian dynamics simulations, and molecular docking, we find that the diffusion rates of the small molecules are highly affected by self-aggregation, interactions with the proteins, and surface adsorption. The diffusion of fluorescein is decreased because of its interactions with the protein crowders and their surface adsorption. Protein crowders increase the diffusion rates of doxorubicin and SB216763 by reducing surface interactions and self-aggregation, respectively. Quinacrine diffusion was not affected by protein crowders. The mechanistic insights gained here may assist in optimization of compounds for higher mobility in complex macromolecular environments.

Computational models for studying physical instabilities in high concentration biotherapeutic formulations

Computational prediction of the behavior of concentrated protein solutions is particularly advantageous in early development stages of biotherapeutics when material availability is limited and a large set of formulation conditions needs to be explored. This review provides an overview of the different computational paradigms that have been successfully used in modeling undesirable physical behaviors of protein solutions with a particular emphasis on high-concentration drug formulations. This includes models ranging from all-atom simulations, coarse-grained representations to macro-scale mathematical descriptions used to study physical instability phenomena of protein solutions such as aggregation, elevated viscosity, and phase separation. These models are compared and summarized in the context of the physical processes and their underlying assumptions and limitations. A detailed analysis is also given for identifying protein interaction processes that are explicitly or implicitly considered in the different modeling approaches and particularly their relations to various formulation parameters. Lastly, many of the shortcomings of existing computational models are discussed, providing perspectives and possible directions toward an efficient computational framework for designing effective protein formulations.

Crowding affects structural dynamics and contributes to membrane association of the NS3/4A complex

Using molecular dynamics simulations, we describe how crowded environments affect the internal dynamics and diffusion of the hepatitis C virus proteases NS3/4A. This protease plays a key role in viral replication and is successfully used as a target for antiviral treatment. The NS3 enzyme requires a peptide cofactor, called NS4A, with its central part interacting with the NS3 β-sheet, and flexible, protruding terminal tails that are unstructured in water solution. The simulations describe the enzyme and water molecules at atomistic resolution, whereas crowders are modeled via either all-atom or coarse-grained models to emphasize different aspects of crowding. Crowders reflect the polyethylene glycol (PEG) molecules used in the experiments to mimic the crowded surrounding. A bead-shell model of folded coarse-grained PEG molecules considers mainly the excluded volume effect, whereas all-atom PEG models afford more protein-like crowder interactions. Circular dichroism spectroscopy experiments of the NS4A N-terminal tail show that a helical structure is formed in the presence of PEG crowders. The simulations suggest that crowding may assist in the formation of an NS4A helical fragment, positioned exactly where a transmembrane helix would fold upon the NS4A contact with the membrane. In addition, partially interactive PEGs help the NS4A N-tail to detach from the protease surface, thus enabling the process of helix insertion and potentially helping the virus establish a replication machinery needed to produce new viruses. Results point to an active role of crowding in assisting structural changes in disordered protein fragments that are necessary for their biological function.

Brownian Dynamics Simulations of Proteins in the Presence of Surfaces: Long-Range Electrostatics and Mean-Field Hydrodynamics

Simulations of macromolecular diffusion and adsorption in confined environments can offer valuable mechanistic insights into numerous biophysical processes. In order to model solutes at atomic detail on relevant time scales, Brownian dynamics simulations can be carried out with the approximation of rigid body solutes moving through a continuum solvent. This allows the precomputation of interaction potential grids for the solutes, thereby allowing the computationally efficient calculation of forces. However, hydrodynamic and long-range electrostatic interactions cannot be fully treated with grid-based approaches alone. Here, we develop a treatment of both hydrodynamic and electrostatic interactions to include the presence of surfaces by modeling grid-based and long-range interactions. We describe its application to simulate the self-association and many-molecule adsorption of the well-characterized protein hen egg-white lysozyme to mica-like and silica-like surfaces. We find that the computational model can recover a number of experimental observables of the adsorption process and provide insights into their determinants. The computational model is implemented in the Simulation of Diffusional Association (SDA) software package.

Influence of macromolecular crowding on the charge regulation of intrinsically disordered proteins

In this work we study the coupling between ionization and conformational properties of two IDPs, histatin-5 and β-amyloid 42, in the presence of neutral and charged crowders. The latter is modeled to resemble bovine serum albumin (BSA). With this aim, semi-grand canonical Monte Carlo simulations are performed, so that the IDP charge is a dynamic property, undergoing protonation/deprotonation processes. Both ionization properties (global and specific amino acid charge and binding capacitance) and radius of gyration are analyzed in a large range of pH values and salt concentrations. Without crowder agents, the titration curve of histatin-5, a polycation, is salt-dependent while that of β-amyloid 42, a polyampholyte, is almost unaffected. The salt concentration is found to be particularly relevant at pH values where the protein binding capacitance (directly linked with charge fluctuation) is larger. Upon addition of neutral crowders, charge regulation is observed in histatin-5, while for β-amyloid 42 this effect is very small. The main mechanism for charge regulation is found to be the effective increase in the ionic strength due to the excluded volume. In the presence of charged crowders, a significant increase in the charge of both IDPs is observed in almost all the pH range. In this case, the IDP charge is altered not only by the increase in the effective ionic strength but also by its direct electrostatic interaction with the charged crowders.

Tracer diffusion in crowded solutions of sticky polymers

Macromolecular diffusion in strongly confined geometries and crowded environments is still to a large extent an open subject in soft matter physics and biology. In this paper, we employ large-scale Langevin dynamics simulations to investigate how the diffusion of a tracer is influenced by the combined action of excluded-volume and weak attractive crowder-tracer interactions. We consider two species of tracers, standard hard-core particles described by the Weeks-Chandler-Andersen (WCA) repulsive potential and core-softened (CS) particles, which model, e.g., globular proteins, charged colloids, and nanoparticles covered by polymeric brushes. These systems are characterized by the presence of two length scales in the interaction and can show waterlike anomalies in their diffusion, stemming from the inherent competition between different length scales. Here we report a comprehensive study of both diffusion and structure of these two tracer species in an environment crowded by quenched configurations of polymers at increasing density. We analyze in detail how the tracer-polymer affinity and the system density affect transport as compared to the emergence of specific static spatial correlations. In particular, we find that, while hardly any differences emerge in the diffusion properties of WCA and CS particles, the propensity to develop structural order for large crowding is strongly frustrated for CS particles. Surprisingly, for large enough affinity for the crowding matrix, the diffusion coefficient of WCA tracers display a nonmonotonic trend as their density is increased when compared to the zero affinity scenario. This waterlike anomaly turns out to be even larger than what observed for CS particle and appears to be rooted in a similar competition between excluded-volume and affinity effects.

Whole-Cell Models and Simulations in Molecular Detail

Comprehensive data about the composition and structure of cellular components have enabled the construction of quantitative whole-cell models. While kinetic network-type models have been established, it is also becoming possible to build physical, molecular-level models of cellular environments. This review outlines challenges in constructing and simulating such models and discusses near- and long-term opportunities for developing physical whole-cell models that can connect molecular structure with biological function.

In-Cell Protein-Protein Contacts: Transient Interactions in the Crowd

Proteins in vivo are immersed in a crowded environment of water, ions, metabolites, and macromolecules. In-cell experiments highlight how transient weak protein-protein interactions promote (via functional "quinary structure") or hinder (via competitive binding or "sticking") complex formation. Computational models of the cytoplasm are expensive. We tackle this challenge with an all-atom model of a small volume of the E. coli cytoplasm to simulate protein-protein contacts up to the 5 μs time scale on the special-purpose supercomputer Anton 2. We use three CHARMM-derived force fields: C22*, C36m, and C36mCU (with CUFIX corrections). We find that both C36m and C36mCU form smaller contact surfaces than C22*. Although CUFIX was developed to reduce protein-protein sticking, larger contacts are observed with C36mCU than C36m. We show that the lifespan Δt of protein-protein contacts obeys a power law distribution between 0.03 and 3 μs, with ∼90% of all contacts lasting <1 μs (similar to the time scale for downhill folding).

Brownian Dynamics Simulations of Biological Molecules

Brownian dynamics (BD) is a technique for carrying out computer simulations of physical systems that are driven by thermal fluctuations. Biological systems at the macromolecular and cellular level, while falling in the gap between well-established atomic-level models and continuum models, are especially suitable for such simulations. We present a brief history, examples of important biological processes that are driven by thermal motion, and those that have been profitably studied by BD. We also present some of the challenges facing developers of algorithms and software, especially in the attempt to simulate larger systems more accurately and for longer times.

Dynamic cluster formation determines viscosity and diffusion in dense protein solutions

We develop a detailed description of protein translational and rotational diffusion in concentrated solution on the basis of all-atom molecular dynamics simulations in explicit solvent. Our systems contain up to 540 fully flexible proteins with 3.6 million atoms. In concentrated protein solutions (100 mg/mL and higher), the proteins ubiquitin and lysozyme, as well as the protein domains third IgG-binding domain of protein G and villin headpiece, diffuse not as isolated particles, but as members of transient clusters between which they constantly exchange. A dynamic cluster model nearly quantitatively explains the increase in viscosity and the decrease in protein diffusivity with protein volume fraction, which both exceed the predictions from widely used colloid models. The Stokes-Einstein relations for translational and rotational diffusion remain valid, but the effective hydrodynamic radius grows linearly with protein volume fraction. This increase follows the observed increase in cluster size and explains the more dramatic slowdown of protein rotation compared with translation. Baxter's sticky-sphere model of colloidal suspensions captures the concentration dependence of cluster size, viscosity, and rotational and translational diffusion. The consistency between simulations and experiments for a diverse set of soluble globular proteins indicates that the cluster model applies broadly to concentrated protein solutions, with equilibrium dissociation constants for nonspecific protein-protein binding in the Kd ≈ 10-mM regime.

Macromolecular diffusion in crowded media beyond the hard-sphere model

The effect of macromolecular crowding on diffusion beyond the hard-core sphere model is studied. A new coarse-grained model is presented, the Chain Entanglement Softened Potential (CESP) model, which takes into account the macromolecular flexibility and chain entanglement. The CESP model uses a shoulder-shaped interaction potential that is implemented in the Brownian Dynamics (BD) computations. The interaction potential contains only one parameter associated with the chain entanglement energetic cost (Ur). The hydrodynamic interactions are included in the BD computations via Tokuyama mean-field equations. The model is used to analyze the diffusion of a streptavidin protein among different sized dextran obstacles. For this system, Ur is obtained by fitting the streptavidin experimental long-time diffusion coefficient Dlongversus the macromolecular concentration for D50 (indicating their molecular weight in kg mol-1) dextran obstacles. The obtained Dlong values show better quantitative agreement with experiments than those obtained with hard-core spheres. Moreover, once parametrized, the CESP model is also able to quantitatively predict Dlong and the anomalous exponent (α) for streptavidin diffusion among D10, D400 and D700 dextran obstacles. Dlong, the short-time diffusion coefficient (Dshort) and α are obtained from the BD simulations by using a new empirical expression, able to describe the full temporal evolution of the diffusion coefficient.

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.

Molecular simulations of cellular processes

It is, nowadays, possible to simulate biological processes in conditions that mimic the different cellular compartments. Several groups have performed these calculations using molecular models that vary in performance and accuracy. In many cases, the atomistic degrees of freedom have been eliminated, sacrificing both structural complexity and chemical specificity to be able to explore slow processes. In this review, we will discuss the insights gained from computer simulations on macromolecule diffusion, nuclear body formation, and processes involving the genetic material inside cell-mimicking spaces. We will also discuss the challenges to generate new models suitable for the simulations of biological processes on a cell scale and for cell-cycle-long times, including non-equilibrium events such as the co-translational folding, misfolding, and aggregation of proteins. A prominent role will be played by the wise choice of the structural simplifications and, simultaneously, of a relatively complex energetic description. These challenging tasks will rely on the integration of experimental and computational methods, achieved through the application of efficient algorithms. Graphical abstract.

Improvements to the APBS biomolecular solvation software suite

The Adaptive Poisson-Boltzmann Solver (APBS) software was developed to solve the equations of continuum electrostatics for large biomolecular assemblages that have provided impact in the study of a broad range of chemical, biological, and biomedical applications. APBS addresses the three key technology challenges for understanding solvation and electrostatics in biomedical applications: accurate and efficient models for biomolecular solvation and electrostatics, robust and scalable software for applying those theories to biomolecular systems, and mechanisms for sharing and analyzing biomolecular electrostatics data in the scientific community. To address new research applications and advancing computational capabilities, we have continually updated APBS and its suite of accompanying software since its release in 2001. In this article, we discuss the models and capabilities that have recently been implemented within the APBS software package including a Poisson-Boltzmann analytical and a semi-analytical solver, an optimized boundary element solver, a geometry-based geometric flow solvation model, a graph theory-based algorithm for determining pK_a values, and an improved web-based visualization tool for viewing electrostatics.

Hydrodynamic Steering in Protein Association Revisited: Surprisingly Minuscule Effects of Considerable Torques

We investigate the previously postulated hydrodynamic steering phenomenon, resulting from complication of molecular shapes, its magnitude and possible relevance for protein-ligand and protein-protein diffusional encounters, and the kinetics of diffusion-controlled association. We consider effects of hydrodynamic interactions in a prototypical model system consisting of a cleft enzyme and an elongated substrate, and real protein-protein complexes, that of barnase and barstar, and human growth hormone and its binding protein. The kinetics of diffusional encounters is evaluated on the basis of rigid-body Brownian dynamics simulations in which hydrodynamic interactions between molecules are modeled using the bead-shell method for detailed description of molecular surfaces, and the first-passage-time approach. We show that magnitudes of steering torques resulting from the hydrodynamic coupling of associating molecules, evaluated for the studied systems on the basis of the Stokes-Einstein type relations for arbitrarily shaped rigid bodies, are comparable with magnitudes of torques resulting from electrostatic interactions of binding partners. Surprisingly, however, unlike in the case of electrostatic torques that strongly affect the diffusional encounter, overall effects of hydrodynamic steering torques on the association kinetics, while clearly discernible in Brownian dynamics simulations, are rather minute. We explain this result as a consequence of the thermal agitation of the binding partners. Our finding is relevant for the general understanding of a wide spectrum of molecular processes in solution but there is also a more practical aspect to it if one considers the low level of shape detail of models that are usually employed to evaluate hydrodynamic interactions in particle-based Stokesian and Brownian dynamics simulations of multicomponent biomolecular systems. Results described in the current work justify, in part at least, such a low-resolution description.

Crowding in Cellular Environments at an Atomistic Level from Computer Simulations

The effects of crowding in biological environments on biomolecular structure, dynamics, and function remain not well understood. Computer simulations of atomistic models of concentrated peptide and protein systems at different levels of complexity are beginning to provide new insights. Crowding, weak interactions with other macromolecules and metabolites, and altered solvent properties within cellular environments appear to remodel the energy landscape of peptides and proteins in significant ways including the possibility of native state destabilization. Crowding is also seen to affect dynamic properties, both conformational dynamics and diffusional properties of macromolecules. Recent simulations that address these questions are reviewed here and discussed in the context of relevant experiments.

Thermodynamics of Macromolecular Association in Heterogeneous Crowding Environments: Theoretical and Simulation Studies with a Simplified Model

The cytoplasm of a cell is crowded with many different kinds of macromolecules. The macromolecular crowding affects the thermodynamics and kinetics of biological reactions in a living cell, such as protein folding, association, and diffusion. Theoretical and simulation studies using simplified models focus on the essential features of the crowding effects and provide a basis for analyzing experimental data. In most of the previous studies on the crowding effects, a uniform crowder size is assumed, which is in contrast to the inhomogeneous size distribution of macromolecules in a living cell. Here, we evaluate the free energy changes upon macromolecular association in a cell-like inhomogeneous crowding system via a theory of hard-sphere fluids and free energy calculations using Brownian dynamics trajectories. The inhomogeneous crowding model based on 41 different types of macromolecules represented by spheres with different radii mimics the physiological concentrations of macromolecules in the cytoplasm of Mycoplasma genitalium. The free energy changes of macromolecular association evaluated by the theory and simulations were in good agreement with each other. The crowder size distribution affects both specific and nonspecific molecular associations, suggesting that not only the volume fraction but also the size distribution of macromolecules are important factors for evaluating in vivo crowding effects. This study relates in vitro experiments on macromolecular crowding to in vivo crowding effects by using the theory of hard-sphere fluids with crowder-size heterogeneity.

Perspective: On the importance of hydrodynamic interactions in the subcellular dynamics of macromolecules

An outstanding challenge in computational biophysics is the simulation of a living cell at molecular detail. Over the past several years, using Stokesian dynamics, progress has been made in simulating coarse grained molecular models of the cytoplasm. Since macromolecules comprise 20%-40% of the volume of a cell, one would expect that steric interactions dominate macromolecular diffusion. However, the reduction in cellular diffusion rates relative to infinite dilution is due, roughly equally, to steric and hydrodynamic interactions, HI, with nonspecific attractive interactions likely playing rather a minor role. HI not only serve to slow down long time diffusion rates but also cause a considerable reduction in the magnitude of the short time diffusion coefficient relative to that at infinite dilution. More importantly, the long range contribution of the Rotne-Prager-Yamakawa diffusion tensor results in temporal and spatial correlations that persist up to microseconds and for intermolecular distances on the order of protein radii. While HI slow down the bimolecular association rate in the early stages of lipid bilayer formation, they accelerate the rate of large scale assembly of lipid aggregates. This is suggestive of an important role for HI in the self-assembly kinetics of large macromolecular complexes such as tubulin. Since HI are important, questions as to whether continuum models of HI are adequate as well as improved simulation methodologies that will make simulations of more complex cellular processes practical need to be addressed. Nevertheless, the stage is set for the molecular simulations of ever more complex subcellular processes.

Coupling and Decoupling of Rotational and Translational Diffusion of Proteins under Crowding Conditions

Molecular motion of biopolymers in vivo is known to be strongly influenced by the high concentration of organic matter inside cells, usually referred to as crowding conditions. To elucidate the effect of intermolecular interactions on Brownian motion of proteins, we performed (1)H pulsed-field gradient NMR and fluorescence correlation spectroscopy (FCS) experiments combined with small-angle X-ray scattering (SAXS) and viscosity measurements for three proteins, αB-crystalline (αBc), bovine serum albumin, and hen egg-white lysozyme (HEWL) in aqueous solution. Our results demonstrate that long-time translational diffusion quantitatively follows the expected increase of macro-viscosity upon increasing the protein concentration in all cases, while rotational diffusion as assessed by polarized FCS and previous multi-frequency (1)H NMR relaxometry experiments reveals protein-specific behavior spanning the full range between the limiting cases of full decoupling from (αBc) and full coupling to (HEWL) the macro-viscosity. SAXS was used to study the interactions between the proteins in solution, whereby it is shown that the three cases cover the range between a weakly interacting hard-sphere system (αBc) and screened Coulomb repulsion combined with short-range attraction (HEWL). Our results, as well as insights from the recent literature, suggest that the unusual rotational-translational coupling may be due to anisotropic interactions originating from hydrodynamic shape effects combined with high charge and possibly a patchy charge distribution.

Effects of Spatially Dependent Mobilities on the Kinetics of the Diffusion-Controlled Association Derived from the First-Passage-Time Approach

Brownian dynamics (BD) simulations and the first-passage-time approach are applied to investigate diffusion-controlled association in a biologically relevant model system consisting of a fixed receptor with an elongated cavity and a capsule-like ligand that fits this cavity precisely. Before the binding at the receptor cavity, the ligand undergoes translational and rotational diffusion, either free or under the influence of electrostatic interactions with the receptor. The spatial dependence of the translational and rotational mobilities of the ligand resulting from its hydrodynamic interactions (HIs) with the receptor is accounted for in BD simulations, and an accurate numerical approach is applied for the evaluation of the spatially dependent mobility tensor of the ligand. Different magnitudes of electrostatic interactions (either attraction or repulsion) between the ligand and receptor are considered. The effective range of receptor-ligand electrostatic interactions is varied to account for their screening under different conditions of ionic strength. The effects of HIs on the kinetics of the diffusion-controlled association, evaluated for different electrostatic properties of binding partners, are thoroughly analyzed and discussed.

Principles and Overview of Sampling Methods for Modeling Macromolecular Structure and Dynamics

Investigation of macromolecular structure and dynamics is fundamental to understanding how macromolecules carry out their functions in the cell. Significant advances have been made toward this end in silico, with a growing number of computational methods proposed yearly to study and simulate various aspects of macromolecular structure and dynamics. This review aims to provide an overview of recent advances, focusing primarily on methods proposed for exploring the structure space of macromolecules in isolation and in assemblies for the purpose of characterizing equilibrium structure and dynamics. In addition to surveying recent applications that showcase current capabilities of computational methods, this review highlights state-of-the-art algorithmic techniques proposed to overcome challenges posed in silico by the disparate spatial and time scales accessed by dynamic macromolecules. This review is not meant to be exhaustive, as such an endeavor is impossible, but rather aims to balance breadth and depth of strategies for modeling macromolecular structure and dynamics for a broad audience of novices and experts.

Molecular basis of high viscosity in concentrated antibody solutions: Strategies for high concentration drug product development

Effective translation of breakthrough discoveries into innovative products in the clinic requires proactive mitigation or elimination of several drug development challenges. These challenges can vary depending upon the type of drug molecule. In the case of therapeutic antibody candidates, a commonly encountered challenge is high viscosity of the concentrated antibody solutions. Concentration-dependent viscosity behaviors of mAbs and other biologic entities may depend on pairwise and higher-order intermolecular interactions, non-native aggregation, and concentration-dependent fluctuations of various antibody regions. This article reviews our current understanding of molecular origins of viscosity behaviors of antibody solutions. We discuss general strategies and guidelines to select low viscosity candidates or optimize lead candidates for lower viscosity at early drug discovery stages. Moreover, strategies for formulation optimization and excipient design are also presented for candidates already in advanced product development stages. Potential future directions for research in this field are also explored.

NMR-detected brownian dynamics of αB-crystallin over a wide range of concentrations

Knowledge about the global translational and rotational motion of proteins under crowded conditions is highly relevant for understanding the function of proteins in vivo. This holds in particular for human αB-crystallin, which is strongly crowded in vivo and inter alia responsible for preventing cataracts. Quantitative information on translational and rotational diffusion is not readily available, and we here demonstrate an approach that combines pulsed-field-gradient NMR for translational diffusion and proton T1ρ/T2 relaxation-time measurements for rotational diffusion, thus overcoming obstacles encountered in previous studies. The relaxation times measured at variable temperature provide a quantitative measure of the correlation function of protein tumbling, which cannot be approximated by a single exponential, because two components are needed for a minimal and adequate description of the data. We find that at high protein concentrations, rotational diffusion is decoupled from translational diffusion, the latter following the macroscopic viscosity change almost quantitatively, resembling the behavior of spherical colloids. Analysis of data reported in the literature shows that well-packed globular proteins follow a scaling relation between the hydrodynamic radius and the molar mass, Rh ∼ M(1/d), with a fractal dimension of d ∼ 2.5 rather than 3. Despite its oligomeric nature, Rh of αB-crystallin as derived from both NMR methods is found to be fully consistent with this relation.

SDA 7: A modular and parallel implementation of the simulation of diffusional association software

The simulation of diffusional association (SDA) Brownian dynamics software package has been widely used in the study of biomacromolecular association. Initially developed to calculate bimolecular protein-protein association rate constants, it has since been extended to study electron transfer rates, to predict the structures of biomacromolecular complexes, to investigate the adsorption of proteins to inorganic surfaces, and to simulate the dynamics of large systems containing many biomacromolecular solutes, allowing the study of concentration-dependent effects. These extensions have led to a number of divergent versions of the software. In this article, we report the development of the latest version of the software (SDA 7). This release was developed to consolidate the existing codes into a single framework, while improving the parallelization of the code to better exploit modern multicore shared memory computer architectures. It is built using a modular object-oriented programming scheme, to allow for easy maintenance and extension of the software, and includes new features, such as adding flexible solute representations. We discuss a number of application examples, which describe some of the methods available in the release, and provide benchmarking data to demonstrate the parallel performance.

Toward an Accurate Modeling of Hydrodynamic Effects on the Translational and Rotational Dynamics of Biomolecules in Many-Body Systems

Proper treatment of hydrodynamic interactions is of importance in evaluation of rigid-body mobility tensors of biomolecules in Stokes flow and in simulations of their folding and solution conformation, as well as in simulations of the translational and rotational dynamics of either flexible or rigid molecules in biological systems at low Reynolds numbers. With macromolecules conveniently modeled in calculations or in dynamic simulations as ensembles of spherical frictional elements, various approximations to hydrodynamic interactions, such as the two-body, far-field Rotne-Prager approach, are commonly used, either without concern or as a compromise between the accuracy and the numerical complexity. Strikingly, even though the analytical Rotne-Prager approach fails to describe (both in the qualitative and quantitative sense) mobilities in the simplest system consisting of two spheres, when the distance between their surfaces is of the order of their size, it is commonly applied to model hydrodynamic effects in macromolecular systems. Here, we closely investigate hydrodynamic effects in two and three-body systems, consisting of bead-shell molecular models, using either the analytical Rotne-Prager approach, or an accurate numerical scheme that correctly accounts for the many-body character of hydrodynamic interactions and their short-range behavior. We analyze mobilities, and translational and rotational velocities of bodies resulting from direct forces acting on them. We show, that with the sufficient number of frictional elements in hydrodynamic models of interacting bodies, the far-field approximation is able to provide a description of hydrodynamic effects that is in a reasonable qualitative as well as quantitative agreement with the description resulting from the application of the virtually exact numerical scheme, even for small separations between bodies.

webSDA: a web server to simulate macromolecular diffusional association

Macromolecular interactions play a crucial role in biological systems. Simulation of diffusional association (SDA) is a software for carrying out Brownian dynamics simulations that can be used to study the interactions between two or more biological macromolecules. webSDA allows users to run Brownian dynamics simulations with SDA to study bimolecular association and encounter complex formation, to compute association rate constants, and to investigate macromolecular crowding using atomically detailed macromolecular structures. webSDA facilitates and automates the use of the SDA software, and offers user-friendly visualization of results. webSDA currently has three modules: 'SDA docking' to generate structures of the diffusional encounter complexes of two macromolecules, 'SDA association' to calculate bimolecular diffusional association rate constants, and 'SDA multiple molecules' to simulate the diffusive motion of hundreds of macromolecules. webSDA is freely available to all users and there is no login requirement. webSDA is available at http://mcm.h-its.org/webSDA/.

Protein Simulations in Fluids: Coupling the OPEP Coarse-Grained Force Field with Hydrodynamics

A novel simulation framework that integrates the OPEP coarse-grained (CG) model for proteins with the Lattice Boltzmann (LB) methodology to account for the fluid solvent at mesoscale level is presented. OPEP is a very efficient, water-free and electrostatic-free force field that reproduces at quasi-atomistic detail processes like peptide folding, structural rearrangements, and aggregation dynamics. The LB method is based on the kinetic description of the solvent in order to solve the fluid mechanics under a wide range of conditions, with the further advantage of being highly scalable on parallel architectures. The capabilities of the approach are presented, and it is shown that the strategy is effective in exploring the role of hydrodynamics on protein relaxation and peptide aggregation. The end result is a strategy for modeling systems of thousands of proteins, such as in the case of dense protein suspensions. The future perspectives of the multiscale approach are also discussed.

Function of metabolic and organelle networks in crowded and organized media

(Macro)molecular crowding and the ability of the ubiquitous cytoskeleton to dynamically polymerize–depolymerize are prevalent cytoplasmic conditions in prokaryotic and eukaryotic cells. Protein interactions, enzymatic or signaling reactions - single, sequential or in complexes - whole metabolic pathways and organelles can be affected by crowding, the type and polymeric status of cytoskeletal proteins (e.g., tubulin, actin), and their imparted organization. The self-organizing capability of the cytoskeleton can orchestrate metabolic fluxes through entire pathways while its fractal organization can frame the scaling of activities in several levels of organization. The intracellular environment dynamics (e.g., biochemical reactions) is dominated by the orderly cytoskeleton and the intrinsic randomness of molecular crowding. Existing evidence underscores the inherent capacity of intracellular organization to generate emergent global behavior. Yet unknown is the relative impact on cell function provided by organelle or functional compartmentation based on transient proteins association driven by weak interactions (quinary structures) under specific environmental challenges or functional conditions (e.g., hypoxia, division, differentiation). We propose a qualitative, integrated structural–functional model of cytoplasmic organization based on a modified version of the Sierspinsky–Menger–Mandelbrot sponge, a 3D representation of a percolation cluster, and examine its capacity to accommodate established experimental facts.

Diffusion within the cytoplasm: a mesoscale model of interacting macromolecules

Recent experiments carried out in the dense cytoplasm of living cells have highlighted the importance of proteome composition and nonspecific intermolecular interactions in regulating macromolecule diffusion and organization. Despite this, the dependence of diffusion-interaction on physicochemical properties such as the degree of poly-dispersity and the balance between steric repulsion and nonspecific attraction among macromolecules was not systematically addressed. In this work, we study the problem of diffusion-interaction in the bacterial cytoplasm, combining theory and experimental data to build a minimal coarse-grained representation of the cytoplasm, which also includes, for the first time to our knowledge, the nucleoid. With stochastic molecular-dynamics simulations of a virtual cytoplasm we are able to track the single biomolecule motion, sizing from 3 to 80 nm, on submillisecond-long trajectories. We demonstrate that the size dependence of diffusion coefficients, anomalous exponents, and the effective viscosity experienced by biomolecules in the cytoplasm is fine-tuned by the intermolecular interactions. Accounting only for excluded volume in these potentials gives a weaker size-dependence than that expected from experimental data. On the contrary, adding nonspecific attraction in the range of 1-10 thermal energy units produces a stronger variation of the transport properties at growing biopolymer sizes. Normal and anomalous diffusive regimes emerge straightforwardly from the combination of high macromolecular concentration, poly-dispersity, stochasticity, and weak nonspecific interactions. As a result, small biopolymers experience a viscous cytoplasm, while the motion of big ones is jammed because the entanglements produced by the network of interactions and the entropic effects caused by poly-dispersity are stronger.

COFFDROP: A Coarse-Grained Nonbonded Force Field for Proteins Derived from All-Atom Explicit-Solvent Molecular Dynamics Simulations of Amino Acids

We describe the derivation of a set of bonded and nonbonded coarse-grained (CG) potential functions for use in implicit-solvent Brownian dynamics (BD) simulations of proteins derived from all-atom explicit-solvent molecular dynamics (MD) simulations of amino acids. Bonded potential functions were derived from 1 μs MD simulations of each of the 20 canonical amino acids, with histidine modeled in both its protonated and neutral forms; nonbonded potential functions were derived from 1 μs MD simulations of every possible pairing of the amino acids (231 different systems). The angle and dihedral probability distributions and radial distribution functions sampled during MD were used to optimize a set of CG potential functions through use of the iterative Boltzmann inversion (IBI) method. The optimized set of potential functions—which we term COFFDROP (COarse-grained Force Field for Dynamic Representation Of Proteins)—quantitatively reproduced all of the “target” MD distributions. In a first test of the force field, it was used to predict the clustering behavior of concentrated amino acid solutions; the predictions were directly compared with the results of corresponding all-atom explicit-solvent MD simulations and found to be in excellent agreement. In a second test, BD simulations of the small protein villin headpiece were carried out at concentrations that have recently been studied in all-atom explicit-solvent MD simulations by Petrov and Zagrovic (PLoS Comput. Biol.2014, 5, e1003638). The anomalously strong intermolecular interactions seen in the MD study were reproduced in the COFFDROP simulations; a simple scaling of COFFDROP’s nonbonded parameters, however, produced results in better accordance with experiment. Overall, our results suggest that potential functions derived from simulations of pairwise amino acid interactions might be of quite broad applicability, with COFFDROP likely to be especially useful for modeling unfolded or intrinsically disordered proteins.

A new coarse-grained model for E. coli cytoplasm: accurate calculation of the diffusion coefficient of proteins and observation of anomalous diffusion

A new coarse-grained model of the E. coli cytoplasm is developed by describing the proteins of the cytoplasm as flexible units consisting of one or more spheres that follow Brownian dynamics (BD), with hydrodynamic interactions (HI) accounted for by a mean-field approach. Extensive BD simulations were performed to calculate the diffusion coefficients of three different proteins in the cellular environment. The results are in close agreement with experimental or previously simulated values, where available. Control simulations without HI showed that use of HI is essential to obtain accurate diffusion coefficients. Anomalous diffusion inside the crowded cellular medium was investigated with Fractional Brownian motion analysis, and found to be present in this model. By running a series of control simulations in which various forces were removed systematically, it was found that repulsive interactions (volume exclusion) are the main cause for anomalous diffusion, with a secondary contribution from HI.

Long range Debye-Hückel correction for computation of grid-based electrostatic forces between biomacromolecules

Background

Brownian dynamics (BD) simulations can be used to study very large molecular systems, such as models of the intracellular environment, using atomic-detail structures. Such simulations require strategies to contain the computational costs, especially for the computation of interaction forces and energies. A common approach is to compute interaction forces between macromolecules by precomputing their interaction potentials on three-dimensional discretized grids. For long-range interactions, such as electrostatics, grid-based methods are subject to finite size errors. We describe here the implementation of a Debye-Hückel correction to the grid-based electrostatic potential used in the SDA BD simulation software that was applied to simulate solutions of bovine serum albumin and of hen egg white lysozyme.

Results

We found that the inclusion of the long-range electrostatic correction increased the accuracy of both the protein-protein interaction profiles and the protein diffusion coefficients at low ionic strength.

Conclusions

An advantage of this method is the low additional computational cost required to treat long-range electrostatic interactions in large biomacromolecular systems. Moreover, the implementation described here for BD simulations of protein solutions can also be applied in implicit solvent molecular dynamics simulations that make use of gridded interaction potentials.

Cellular and molecular structure as a unifying framework for whole-cell modeling

Whole-cell modeling has the potential to play a major role in revolutionizing our understanding of cellular biology over the next few decades. A computational model of the entire cell would allow cellular biologists to integrate data from many disparate sources in a single consistent framework. Such a comprehensive model would be useful both for hypothesis testing and in the discovery of new behaviors that emerge from complex biological networks. Cellular and molecular structure can and should be a key organizing principle in a whole-cell model, connecting models across time and length scales in a multiscale approach. Here I present a summary of recent research centered around using molecular and cellular structure to model the behavior of cells.

Coarse-grain modelling of protein-protein interactions

Here, we review recent advances towards the modelling of protein-protein interactions (PPI) at the coarse-grained (CG) level, a technique that is now widely used to understand protein affinity, aggregation and self-assembly behaviour. PPI models of soluble proteins and membrane proteins are separately described, but we note the parallel development that is present in both research fields with three important themes: firstly, combining CG modelling with knowledge-based approaches to predict and refine protein-protein complexes; secondly, using physics-based CG models for de novo prediction of protein-protein complexes; and thirdly modelling of large scale protein aggregates.

The shape of protein crowders is a major determinant of protein diffusion

As a model for understanding how molecular crowding influences diffusion and transport of proteins in cellular environments, we combined experimental and theoretical approaches to study the diffusion of proteins in highly concentrated protein solutions. Bovine serum albumin and γ-Globulin were chosen as molecular crowders and as tracers. These two proteins are representatives of the main types of plasma protein and have different shapes and sizes. Solutions consisting of one or both proteins were studied. The self-diffusion coefficients of the fluorescently labeled tracer proteins were measured by means of fluorescence correlation spectroscopy at a total protein concentration of up to 400 g/L. γ-Globulin is found to have a stronger influence as a crowder on the tracer self-diffusion coefficient than Bovine serum albumin. Brownian dynamics simulations show that the excluded volume and the shape of the crowding protein have a significantly stronger influence on translational and rotational diffusion coefficients, as well as transient oligomerization, than hydrodynamic or direct interactions. Anomalous subdiffusion, which is not observed at the experimental fluorescence correlation spectroscopy timescales (>100 μs), appears only at very short timescales (<1 μs) in the simulations due to steric effects of the proteins. We envision that the combined experimental and computational approach employed here can be developed to unravel the different biophysical contributions to protein motion and interaction in cellular environments by systematically varying protein properties such as molecular weight, size, shape, and electrostatic interactions.

Molecular dynamics simulations of highly crowded amino acid solutions: comparisons of eight different force field combinations with experiment and with each other

Although it is now commonly accepted that the highly crowded conditions encountered inside biological cells have the potential to significantly alter the thermodynamic properties of biomolecules, it is not known to what extent the thermodynamics of fundamental types of interactions such as salt bridges and hydrophobic interactions are strengthened or weakened by high biomolecular concentrations. As one way of addressing this question we have performed a series of all-atom explicit solvent molecular dynamics (MD) simulations to investigate the effect of increasing solute concentration on the behavior of four types of zwitterionic amino acids in aqueous solution. We have simulated systems containing glycine, valine, phenylalanine or asparagine at concentrations of 50, 100, 200 and 300 mg/ml. Each molecular system has been simulated for 1 μs in order to obtain statistically converged estimates of thermodynamic parameters, and each has been conducted with 8 different force fields and water models; the combined simulation time is 128 μs. The density, viscosity, and dielectric increments of the four amino acids calculated from the simulations have been compared to corresponding experimental measurements. While all of the force fields perform well at reproducing the density increments, discrepancies for the viscosity and dielectric increments raise questions both about the accuracy of the simulation force fields and, in certain cases, the experimental data. We also observe large differences between the various force fields' descriptions of the interaction thermodynamics of salt bridges and, surprisingly, these differences also lead to qualitatively different predictions of their dependences on solute concentration. For the aliphatic interactions of valine sidechains, fewer differences are observed between the force fields, but significant differences are again observed for aromatic interactions of phenylalanine sidechains. Taken together, the results highlight the potential power of using explicit-solvent simulation methods to understand behavior in concentrated systems but also hint at potential difficulties in using these methods to obtain consistent views of behavior in intracellular environments.

Reaching new levels of realism in modeling biological macromolecules in cellular environments

An increasing number of studies are aimed at modeling cellular environments in a comprehensive and realistic fashion. A major challenge in these efforts is how to bridge spatial and temporal scales over many orders of magnitude. Furthermore, there are additional challenges in integrating different aspects ranging from questions about biomolecular stability in crowded environments to the description of reactive processes on cellular scales. In this review, recent studies with models of biomolecules in cellular environments at different levels of detail are discussed in terms of their strengths and weaknesses. In particular, atomistic models, implicit representations of cellular environments, coarse-grained and spheroidal models of biomolecules, as well as the inclusion of reactive processes via reaction-diffusion models are described. Furthermore, strategies for integrating the different models into a comprehensive description of cellular environments are discussed.

Toward optimized potential functions for protein-protein interactions in aqueous solutions: osmotic second virial coefficient calculations using the MARTINI coarse-grained force field

Coarse-grained (CG) simulation methods are now widely used to model the structure and dynamics of large biomolecular systems. One important issue for using such methods - especially with regard to using them to model, for example, intracellular environments - is to demonstrate that they can reproduce experimental data on the thermodynamics of protein-protein interactions in aqueous solutions. To examine this issue, we describe here simulations performed using the popular coarse-grained MARTINI force field, aimed at computing the thermodynamics of lysozyme and chymotrypsinogen self-interactions in aqueous solution. Using molecular dynamics simulations to compute potentials of mean force between a pair of protein molecules, we show that the original parameterization of the MARTINI force field is likely to significantly overestimate the strength of protein-protein interactions to the extent that the computed osmotic second virial coefficients are orders of magnitude more negative than experimental estimates. We then show that a simple down-scaling of the van der Waals parameters that describe the interactions between protein pseudo-atoms can bring the simulated thermodynamics into much closer agreement with experiment. Overall, the work shows that it is feasible to test explicit-solvent CG force fields directly against thermodynamic data for proteins in aqueous solutions, and highlights the potential usefulness of osmotic second virial coefficient measurements for fully parameterizing such force fields.

A molecule-centered method for accelerating the calculation of hydrodynamic interactions in Brownian dynamics simulations containing many flexible biomolecules

Inclusion of hydrodynamic interactions (HIs) is essential in simulations of biological macromolecules that treat the solvent implicitly if the macromolecules are to exhibit correct translational and rotational diffusion. The present work describes the development and testing of a simple approach aimed at allowing more rapid computation of HIs in coarse-grained Brownian dynamics simulations of systems that contain large numbers of flexible macromolecules. The method combines a complete treatment of intramolecular HIs with an approximate treatment of the intermolecular HIs which assumes that the molecules are effectively spherical; all of the HIs are calculated at the Rotne-Prager-Yamakawa level of theory. When combined with Fixman's Chebyshev polynomial method for calculating correlated random displacements, the proposed method provides an approach that is simple to program but sufficiently fast that it makes it computationally viable to include HIs in large-scale simulations. Test calculations performed on very coarse-grained models of the pyruvate dehydrogenase (PDH) E2 complex and on oligomers of ParM (ranging in size from 1 to 20 monomers) indicate that the method reproduces the translational diffusion behavior seen in more complete HI simulations surprisingly well; the method performs less well at capturing rotational diffusion but its discrepancies diminish with increasing size of the simulated assembly. Simulations of residue-level models of two tetrameric protein models demonstrate that the method also works well when more structurally detailed models are used in the simulations. Finally, test simulations of systems containing up to 1024 coarse-grained PDH molecules indicate that the proposed method rapidly becomes more efficient than the conventional BD approach in which correlated random displacements are obtained via a Cholesky decomposition of the complete diffusion tensor.