Coarse-grained molecular simulation of diffusion and reaction kinetics in a crowded virtual cytoplasm

We present a general-purpose model for biomolecular simulations at the molecular level that incorporates stochasticity, spatial dependence, and volume exclusion, using diffusing and reacting particles with physical dimensions. To validate the model, we first established the formal relationship between the microscopic model parameters (timestep, move length, and reaction probabilities) and the macroscopic coefficients for diffusion and reaction rate. We then compared simulation results with Smoluchowski theory for diffusion-limited irreversible reactions and the best available approximation for diffusion-influenced reversible reactions. To simulate the volumetric effects of a crowded intracellular environment, we created a virtual cytoplasm composed of a heterogeneous population of particles diffusing at rates appropriate to their size. The particle-size distribution was estimated from the relative abundance, mass, and stoichiometries of protein complexes using an experimentally derived proteome catalog from Escherichia coli K12. Simulated diffusion constants exhibited anomalous behavior as a function of time and crowding. Although significant, the volumetric impact of crowding on diffusion cannot fully account for retarded protein mobility in vivo, suggesting that other biophysical factors are at play. The simulated effect of crowding on barnase-barstar dimerization, an experimentally characterized example of a bimolecular association reaction, reveals a biphasic time course, indicating that crowding exerts different effects over different timescales. These observations illustrate that quantitative realism in biosimulation will depend to some extent on mesoscale phenomena that are not currently well understood.     

Macromolecular crowding in biological systems: hydrodynamics and NMR methods

Most biologically relevant environments involve highly concentrated macromolecular solutions and most biological processes involve macromolecules that diffuse and interact with other macromolecules. Macromolecular crowding is a general phenomenon that strongly affects the transport properties of macromolecules (rotational and translational diffusion) as well as the position of their equilibria. NMR methods can provide information on molecular interactions, as well as on translational and rotational diffusion. In fact, rotational diffusion, through its determinant role in NMR relaxation, places a practical limit on the systems that can be studied by NMR. While in dilute solutions of non-aggregating macromolecules this limit is set by macromolecular size, in crowded solutions excluded volume effects can have a strong effect on the observed diffusion rates. Hydrodynamic theory offers some insight into the magnitude of crowding effects on NMR observable parameters.     

Concentration sensing in crowded environments

Signal transduction within crowded cellular compartments is essential for the physiological function of cells. Although the accuracy with which receptors can probe the concentration of ligands has been thoroughly investigated in dilute systems, the effect of macromolecular crowding on the inference of concentration remains unclear. In this work, we develop an algorithm to simulate reversible reactions between reacting Brownian particles. Our algorithm facilitates the calculation of reaction rates and correlation times for ligand-receptor systems in the presence of macromolecular crowding. Using this method, we show that it is possible for crowding to increase the accuracy of estimated ligand concentration based on receptor occupancy. In particular, we find that crowding can enhance the effective association rates between small ligands and receptors to a degree sufficient to overcome the increased chance of rebinding due to caging by crowding molecules. For larger ligands, crowding decreases the accuracy of the receptor’s estimate primarily by decreasing the microscopic association and dissociation rates.     

Multi-scale stochastic simulation of diffusion-coupled agents and its application to cell culture simulation

Many biological systems consist of multiple cells that interact by secretion and binding of diffusing molecules, thus coordinating responses across cells. Techniques for simulating systems coupling extracellular and intracellular processes are very limited. Here we present an efficient method to stochastically simulate diffusion processes, which at the same time allows synchronization between internal and external cellular conditions through a modification of Gillespie's chemical reaction algorithm. Individual cells are simulated as independent agents, and each cell accurately reacts to changes in its local environment affected by diffusing molecules. Such a simulation provides time-scale separation between the intra-cellular and extra-cellular processes. We use our methodology to study how human monocyte-derived dendritic cells alert neighboring cells about viral infection using diffusing interferon molecules. A subpopulation of the infected cells reacts early to the infection and secretes interferon into the extra-cellular medium, which helps activate other cells. Findings predicted by our simulation and confirmed by experimental results suggest that the early activation is largely independent of the fraction of infected cells and is thus both sensitive and robust. The concordance with the experimental results supports the value of our method for overcoming the challenges of accurately simulating multiscale biological signaling systems.     

Three-dimensional stochastic off-lattice model of binding chemistry in crowded environments

Molecular crowding is one of the characteristic features of the intracellular environment, defined by a dense mixture of varying kinds of proteins and other molecules. Interaction with these molecules significantly alters the rates and equilibria of chemical reactions in the crowded environment. Numerous fundamental activities of a living cell are strongly influenced by the crowding effect, such as protein folding, protein assembly and disassembly, enzyme activity, and signal transduction. Quantitatively predicting how crowding will affect any particular process is, however, a very challenging problem because many physical and chemical parameters act synergistically in ways that defy easy analysis. To build a more realistic model for this problem, we extend a prior stochastic off-lattice model from two-dimensional (2D) to three-dimensional (3D) space and examine how the 3D results compare to those found in 2D. We show that both models exhibit qualitatively similar crowding effects and similar parameter dependence, particularly with respect to a set of parameters previously shown to act linearly on total reaction equilibrium. There are quantitative differences between 2D and 3D models, although with a generally gradual nonlinear interpolation as a system is extended from 2D to 3D. However, the additional freedom of movement allowed to particles as thickness of the simulation box increases can produce significant quantitative change as a system moves from 2D to 3D. Simulation results over broader parameter ranges further show that the impact of molecular crowding is highly dependent on the specific reaction system examined.     

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.     

Multiple diffusion mechanisms due to nanostructuring in crowded environments

One of the key questions regarding intracellular diffusion is how the environment affects molecular mobility. Mostly, intracellular diffusion has been described as hindered, and the physical reasons for this behavior are: immobile barriers, molecular crowding, and binding interactions with immobile or mobile molecules. Using results from multi-photon fluorescence correlation spectroscopy, we describe how immobile barriers and crowding agents affect translational mobility. To study the hindrance produced by immobile barriers, we used sol-gels (silica nanostructures) that consist of a continuous solid phase and aqueous phase in which fluorescently tagged molecules diffuse. In the case of molecular crowding, translational mobility was assessed in increasing concentrations of 500 kDa dextran solutions. Diffusion of fluorescent tracers in both sol-gels and dextran solutions shows clear evidence of anomalous subdiffusion. In addition, data from the autocorrelation function were analyzed using the maximum entropy method as adapted to fluorescence correlation spectroscopy data and compared with the standard model that incorporates anomalous diffusion. The maximum entropy method revealed evidence of different diffusion mechanisms that had not been revealed using the anomalous diffusion model. These mechanisms likely correspond to nanostructuring in crowded environments and to the relative dimensions of the crowding agent with respect to the tracer molecule. Analysis with the maximum entropy method also revealed information about the degree of heterogeneity in the environment as reported by the behavior of diffusive molecules.     

A natural and readily available crowding agent: NMR studies of proteins in hen egg white

In vitro studies of biological macromolecules are usually performed in dilute, buffered solutions containing one or just a few different biological macromolecules. Under these conditions, the interactions among molecules are diffusion limited. On the contrary, in living systems, macromolecules of a given type are surrounded by many others, at very high total concentrations. In the last few years, there has been an increasing effort to study biological macromolecules directly in natural crowded environments, as in intact bacterial cells or by mimicking natural crowding by adding proteins, polysaccharides, or even synthetic polymers. Here, we propose the use of hen egg white (HEW) as a simple natural medium, with all features of the media of crowded cells, that could be used by any researcher without difficulty and inexpensively. We present a study of the stability and dynamics behavior of model proteins in HEW, chosen as a prototypical, readily accessible natural medium that can mimic cytosol. We show that two typical globular proteins, dissolved in HEW, give NMR spectra very similar to those obtained in dilute buffers, although dynamic parameters are clearly affected by the crowded medium. The thermal stability of one of these proteins, measured in a range comprising both heat and cold denaturation, is also similar to that in buffer. Our data open new possibilities to the study of proteins in natural crowded media.     

Molecular crowding reduces to a similar extent the diffusion of small solutes and macromolecules: measurement by fluorescence correlation spectroscopy

Aqueous environments in living cells are crowded, with up to >50 wt% small and macromolecule-size solutes. We investigated quantitatively one important consequence of molecular crowding--reduced diffusion of biologically important solutes. Fluorescence correlation spectroscopy (FCS) was used to measure the diffusion of a series of fluorescent small solutes and macromolecules. In water, diffusion coefficients (D(o)w) were (in cm2/s x 10(-8)): rhodamine green (270), albumin (52), dextrans (75, 10 kDa; 10, 500 kDa), double-stranded DNAs (96, 20 bp; 10, 1 kb; 3.4, 4.5 kb) and polystyrene nanospheres (5.4, 20 nm diameter; 2.3, 100 nm). Aqueous-phase diffusion (Dw) in solutions crowded with Ficoll-70 (0-60 wt%) was reduced by up to 650-fold in an exponential manner: Dw = D(o)w exp (-[C]/[C]exp), where [C]exp is the concentration (in wt%) of crowding agent reducing D(o)w by 63%. FCS data for all solutes and Ficoll-70 concentrations fitted well to a model of single-component, simple (non-anomalous) diffusion. Interestingly [C]exp were nearly identical (11+/-2 wt%, SD) for diffusion of the very different types of macromolecules in Ficoll-70 solutions. However, [C]exp was dependent on the nature of the crowding agent: for example, [C]exp for diffusion of rhodamine green was 30 wt% for glycerol and 16 wt% for 500 kDa dextran. Our results indicate that molecular crowding can greatly reduce aqueous-phase diffusion of biologically important macromolecules, and demonstrate a previously unrecognized insensitivity of crowding effects on the size and characteristics of the diffusing species.     

Accounting for diffusion in agent based models of reaction-diffusion systems with application to cytoskeletal diffusion

Diffusion plays a key role in many biochemical reaction systems seen in nature. Scenarios where diffusion behavior is critical can be seen in the cell and subcellular compartments where molecular crowding limits the interaction between particles. We investigate the application of a computational method for modeling the diffusion of molecules and macromolecules in three-dimensional solutions using agent based modeling. This method allows for realistic modeling of a system of particles with different properties such as size, diffusion coefficients, and affinity as well as the environment properties such as viscosity and geometry. Simulations using these movement probabilities yield behavior that mimics natural diffusion. Using this modeling framework, we simulate the effects of molecular crowding on effective diffusion and have validated the results of our model using Langevin dynamics simulations and note that they are in good agreement with previous experimental data. Furthermore, we investigate an extension of this framework where single discrete cells can contain multiple particles of varying size in an effort to highlight errors that can arise from discretization that lead to the unnatural behavior of particles undergoing diffusion. Subsequently, we explore various algorithms that differ in how they handle the movement of multiple particles per cell and suggest an algorithm that properly accommodates multiple particles of various sizes per cell that can replicate the natural behavior of these particles diffusing. Finally, we use the present modeling framework to investigate the effect of structural geometry on the directionality of diffusion in the cell cytoskeleton with the observation that parallel orientation in the structural geometry of actin filaments of filopodia and the branched structure of lamellipodia can give directionality to diffusion at the filopodia-lamellipodia interface.     

Diffusional anisotropy in collagenous tissues: fluorescence imaging of continuous point photobleaching

Molecular transport in avascular collagenous tissues such as articular cartilage occurs primarily via diffusion. The presence of ordered structures in the extracellular matrix may influence the local transport of macromolecules, leading to anisotropic diffusion depending on the relative size of the molecule and that of extracellular matrix structures. Here we present what we believe is a novel photobleaching technique for measuring the anisotropic diffusivity of macromolecules in collagenous tissues. We hypothesized that macromolecular diffusion is anisotropic in collagenous tissues, depending on molecular size and the local organization of the collagen structure. A theoretical model and experimental protocol for fluorescence imaging of continuous point photobleaching was developed to measure diffusional anisotropy. Significant anisotropy was observed in highly ordered collagenous tissues such as ligament, with diffusivity ratios >2 along the fiber direction compared to the perpendicular direction. In less-ordered tissues such as articular cartilage, diffusional anisotropy was dependent on site in the tissue and size of the diffusing molecule. Anisotropic diffusion was also dependent on the size of the diffusing molecule, with greatest anisotropy observed for larger molecules. These findings suggest that diffusional transport of macromolecules is anisotropic in collagenous tissues, with higher rates of diffusion along primary orientation of collagen fibers.     

Applying Molecular Crowding Models to Simulations of Virus Capsid Assembly In Vitro

Virus capsid assembly has been widely studied as a biophysical system, both for its biological and medical significance and as an important model for complex self-assembly processes. No current technology can monitor assembly in detail and what information we have on assembly kinetics comes exclusively from in vitro studies. There are many differences between the intracellular environment and that of an in vitro assembly assay, however, that might be expected to alter assembly pathways. Here, we explore one specific feature characteristic of the intracellular environment and known to have large effects on macromolecular assembly processes: molecular crowding. We combine prior particle simulation methods for estimating crowding effects with coarse-grained stochastic models of capsid assembly, using the crowding models to adjust kinetics of capsid simulations to examine possible effects of crowding on assembly pathways. Simulations suggest a striking difference depending on whether or not a system uses nucleation-limited assembly, with crowding tending to promote off-pathway growth in a nonnucleation-limited model but often enhancing assembly efficiency at high crowding levels even while impeding it at lower crowding levels in a nucleation-limited model. These models may help us understand how complicated assembly systems may have evolved to function with high efficiency and fidelity in the densely crowded environment of the cell.     

Optical tweezers reveal force plateau and internal friction in PEG-induced DNA condensation

The simplified artificial environments in which highly complex biological systems are studied do not represent the crowded, dense, salty, and dynamic environment inside the living cell. Consequently, it is important to investigate the effect of crowding agents on DNA. We used a dual-trap optical tweezers instrument to perform force spectroscopy experiments at pull speeds ranging from 0.3 to 270 μm/s on single dsDNA molecules in the presence of poly(ethylene glycol) (PEG) and monovalent salt. PEG of sizes 1,500 and 4,000 Da condensed DNA, and force–extension data contained a force plateau at approximately 1 pN. The level of the force plateau increased with increasing pull speed. During slow pulling the dissipated work increased linearly with pull speed. The calculated friction coefficient did not depend on amount of DNA incorporated in the condensate, indicating internal friction is independent of the condensate size. PEG300 had no effect on the dsDNA force–extension curve. The force plateau implies that condensation induced by crowding agents resembles condensation induced by multivalent cations.     

Applying Molecular Crowding Models to Simulations of Virus Capsid Assembly In?Vitro

Virus capsid assembly has been widely studied as a biophysical system, both for its biological and medical significance and as an important model for complex self-assembly processes. No current technology can monitor assembly in detail and what information we have on assembly kinetics comes exclusively from in vitro studies. There are many differences between the intracellular environment and that of an in vitro assembly assay, however, that might be expected to alter assembly pathways. Here, we explore one specific feature characteristic of the intracellular environment and known to have large effects on macromolecular assembly processes: molecular crowding. We combine prior particle simulation methods for estimating crowding effects with coarse-grained stochastic models of capsid assembly, using the crowding models to adjust kinetics of capsid simulations to examine possible effects of crowding on assembly pathways. Simulations suggest a striking difference depending on whether or not a system uses nucleation-limited assembly, with crowding tending to promote off-pathway growth in a nonnucleation-limited model but often enhancing assembly efficiency at high crowding levels even while impeding it at lower crowding levels in a nucleation-limited model. These models may help us understand how complicated assembly systems may have evolved to function with high efficiency and fidelity in the densely crowded environment of the cell.     

Cross-Strain Quorum Sensing Inhibition by Staphylococcus Aureus. Part 2: A Spatially Inhomogeneous Model

Staphylococcus aureus uses quorum sensing (QS) to enhance its pathogenicity. An intriguing aspect of this is that different strains are capable of inactivating the QS systems of opposing strains. In Part 1 of this study, we presented a model of this phenomenon in a well-mixed environment; here, we incorporate spatial structure. Two competitive strains occupying adjacent habitats with freely diffusing QS signal molecules (QSSMs) are considered. We investigate the effect of the QSSM diffusion coefficient and the relative size of the two populations on the ability of one population to dominate the other. Regarding population size, a larger population is generally at an advantage (initial conditions permitting), while the implications of different diffusivities are more complex and depend upon the sizes of the populations.     

Protein-protein association in polymer solutions: from dilute to semidilute to concentrated

In a typical cell, proteins function in the crowded cytoplasmic environment where 30% of the space is occupied by macromolecules of varying size and nature. This environment may be simulated in vitro using synthetic polymers. Here, we followed the association and diffusion rates of TEM1-beta-lactamase (TEM) and the beta-lactamase inhibitor protein (BLIP) in the presence of crowding agents of varying molecular mass, from monomers (ethylene glycol, glycerol, or sucrose) to polymeric agents such as different polyethylene glycols (PEGs, 0.2-8 kDa) and Ficoll. An inverse linear relation was found between translational diffusion of the proteins and viscosity in all solutions tested, in accordance with the Stokes-Einstein (SE) relation. Conversely, no simple relation was found between either rotational diffusion rates or association rates (k(on)) and viscosity. To assess the translational diffusion-independent steps along the association pathway, we introduced a new factor, alpha, which corrects the relative change in k(on) by the relative change in solution viscosity, thus measuring the deviations of the association rates from SE behavior. We found that these deviations were related to the three regimes of polymer solutions: dilute, semidilute, and concentrated. In the dilute regime PEGs interfere with TEM-BLIP association by introducing a repulsive force due to solvophobic preferential hydration, which results in slower association than predicted by the SE relation. Crossing over from the dilute to the semidilute regime results in positive deviations from SE behavior, i.e., relatively faster association rates. These can be attributed to the depletion interaction, which results in an effective attraction between the two proteins, winning over the repulsive force. In the concentrated regime, PEGs again dramatically slow down the association between TEM and BLIP, an effect that does not depend on the physical dimensions of PEGs, but rather on their mass concentration. This is probably a manifestation of the monomer-like repulsive depletion effect known to occur in concentrated polymer solutions. As a transition from moderate to high crowding agent concentration can occur in the cellular milieu, this behavior may modulate protein association in vivo, thereby modulating biological function.     

A Method for Molecular Dynamics on Curved Surfaces

Dynamics simulations of constrained particles can greatly aid in understanding the temporal and spatial evolution of biological processes such as lateral transport along membranes and self-assembly of viruses. Most theoretical efforts in the field of diffusive transport have focused on solving the diffusion equation on curved surfaces, for which it is not tractable to incorporate particle interactions even though these play a crucial role in crowded systems. We show here that it is possible to take such interactions into account by combining standard constraint algorithms with the classical velocity Verlet scheme to perform molecular dynamics simulations of particles constrained to an arbitrarily curved surface. Furthermore, unlike Brownian dynamics schemes in local coordinates, our method is based on Cartesian coordinates, allowing for the reuse of many other standard tools without modifications, including parallelization through domain decomposition. We show that by applying the schemes to the Langevin equation for various surfaces, we obtain confined Brownian motion, which has direct applications to many biological and physical problems. Finally we present two practical examples that highlight the applicability of the method: 1) the influence of crowding and shape on the lateral diffusion of proteins in curved membranes; and 2) the self-assembly of a coarse-grained virus capsid protein model.     

Anomalous subdiffusion is a measure for cytoplasmic crowding in living cells

Macromolecular crowding dramatically affects cellular processes such as protein folding and assembly, regulation of metabolic pathways, and condensation of DNA. Despite increased attention, we still lack a definition for how crowded a heterogeneous environment is at the molecular scale and how this manifests in basic physical phenomena like diffusion. Here, we show by means of fluorescence correlation spectroscopy and computer simulations that crowding manifests itself through the emergence of anomalous subdiffusion of cytoplasmic macromolecules. In other words, the mean square displacement of a protein will grow less than linear in time and the degree of this anomality depends on the size and conformation of the traced particle and on the total protein concentration of the solution. We therefore propose that the anomality of the diffusion can be used as a quantifiable measure for the crowdedness of the cytoplasm at the molecular scale.