Crowding effects on association reactions at membranes

Nonmonotonic Modulation of the Protein-Ligand Recognition Event by Inert Crowders

The ubiquitous event of a protein recognizing small molecules or ligands at its native binding site is crucial for initiating major biological processes. However, how a crowded environment, as is typically represented by a cellular interior, would modulate the protein-ligand search process is largely debated. Excluded volume-based theory suggests that the presence of an inert crowder would reinforce a steady stabilization and enhancement of the protein-ligand recognition process. Here, we counter this long-held perspective via the molecular dynamics simulation and Markov state model of the protein-ligand recognition event in the presence of inert crowders. Specifically, we demonstrate that, depending on concentration, even purely inert crowders can exert a nonmonotonic effect via either stabilizing or destabilizing the protein-ligand binding event. Analysis of the kinetic network of binding pathways reveals that the crowders would either modulate precedent non-native on-pathway intermediates or would devise additional ones in a multistate recognition event across a wide range of concentrations. As an important insight, crowders gradually shift the relative transitional preference of these intermediates toward a native-bound state, with ligand residence time at the binding pocket dictating the trend of nonmonotonic concentration dependence by simple inert crowders.

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

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

Macromolecular crowding in solution alters huntingtin interaction and aggregation at interfaces

Huntington's disease (HD) is a fatal neurodegenerative disease caused by an extended polyglutamine (polyQ) domain within the first exon of the huntingtin protein (htt). PolyQ expansion directly invokes the formation of a heterogenous mixture of toxic htt aggregates, including fibrils and oligomers. While htt is a cytosolic protein, it also associates with numerous membranous surfaces within the cell, leading to altered organelle morphology and dysfunction. Here, the impact of macromolecular crowding on htt aggregation in bulk solution and at solid/liquid or membrane/liquid interfaces was investigated. Dextran, Ficoll, and polyethylene glycol (PEG) were used as crowding agents. In bulk solution, crowding enhanced the heterogeneity of non-fibrillar aggregate species formed in a crowder dependent manner. However, crowding agents interfered with the deposition of htt fibrils on mica, suggesting that a crowded aqueous phase influences the interaction of htt with interfaces. By use of in situ atomic force microcopy (AFM), the aggregation of htt directly at mica and bilayer interfaces was tracked. The predominate aggregates type observed to form at the mica interface was fibrillar, but oligomeric aggregates of various stabilities were also observed. Crowding in the aqueous phase suppressed deposition and formation of htt aggregates on mica. In contrast, the addition of crowders enhanced deposition of htt aggregates onto supported total brain lipid extract (TBLE) bilayers. Different crowding agents led to distinct htt aggregates on supported bilayers with unique morphological impact on bilayer integrity. Collectively, these observations point to the complexity of htt aggregation at interfaces and that crowding in the aqueous phase profoundly influences this process.

Macromolecular Crowding Effect on the Activity of Liposome-Bound Alkaline Phosphatase: A Paradoxical Inhibitory Action

The cytoplasm of a cell is extremely crowded, with 20-30% being large biomolecules. This crowding enforces a significant amount of the physical and chemical barrier around biomolecules, so understanding any biomolecular event within the cellular system is challenging. Unsurprisingly, enzymes show a diverse kind of catalytic behavior inside a crowded environment and thus have remained an area of active interest in the last few decades. The situation can become even more complex and exciting in the case of understanding the behavior of a membrane-bound enzyme (almost 25-30% of enzymes are membrane-bound) in such a crowded environment that until now has remained unexplored. Herein, we have particularly investigated how a membrane-bound enzyme (using liposome-bound alkaline phosphatase) can behave in a crowded environment comprising polymer molecule-like poly(ethylene glycol) (PEG) of different weights (PEG400, PEG4000, and PEG9000) and Ficoll 400. We have compared the activity using a polymer microbead conjugated enzyme and have found that liposome-bound alkaline phosphatase had much higher activity in crowded environments, showing the importance and superiority of soft-deformable particles (i.e., vesicles) over hard spheres in macro-molecularly crowded media. Interstingly, we have found a paradoxical behavior of inhibitors in terms of both their extent and pathway of inhibitory action. For instance, phosphates, known as competitive inhibitors in buffer, behave as uncompetitive inhibitors in liposome-bound enzymes in crowded media with an ∼5-fold less inhibitory effect, whereas phenyl alanine (an uncompetitive inhibitor in buffer) did not show any inhibitory potential when the enzyme was membrane-bound and in crowded media containing PEG9000 (30 wt %). Overall, this demonstration elucidates aspects of membrane-bound enzymes in crowded media in terms of both catalytic behavior and inhibitory actions and can lead to further studies of the understanding of enzymatic behavior in such complex crowded environments having a dampening effect in regular diffusive transport.

Interplay of reactive interference and crowding effects in the diffusion-influenced reaction kinetics

We investigate the interplay of reactive interference and crowding effects in the irreversible diffusion-influenced bimolecular reactions of the type A+B→P+B by using the Brownian dynamics simulation method. It is known that the presence of nonreactive crowding agents retards the reaction rate when the volume fraction of the crowding agents is large enough. On the other hand, a high concentration of B is known to increase the reaction rate more than expected from the mass action law, although the B's may also act as crowders. Therefore, it would be interesting to see which effect dominates when the number density of B as well as the number density of the crowders increases. We will present an approximate theory that provides a reasonable account for the Brownian dynamics simulation results.

Diffusion-influenced reaction rates in the presence of pair interactions

The kinetics of bimolecular reactions in solution depends, among other factors, on intermolecular forces such as steric repulsion or electrostatic interaction. Microscopically, a pair of molecules first has to meet by diffusion before the reaction can take place. In this work, we establish an extension of Doi's volume reaction model to molecules interacting via pair potentials, which is a key ingredient for interacting-particle-based reaction-diffusion (iPRD) simulations. As a central result, we relate model parameters and macroscopic reaction rate constants in this situation. We solve the corresponding reaction-diffusion equation in the steady state and derive semi-analytical expressions for the reaction rate constant and the local concentration profiles. Our results apply to the full spectrum from well-mixed to diffusion-limited kinetics. For limiting cases, we give explicit formulas, and we provide a computationally inexpensive numerical scheme for the general case, including the intermediate, diffusion-influenced regime. The obtained rate constants decompose uniquely into encounter and formation rates, and we discuss the effect of the potential on both subprocesses, exemplified for a soft harmonic repulsion and a Lennard-Jones potential. The analysis is complemented by extensive stochastic iPRD simulations, and we find excellent agreement with the theoretical predictions.

Fibrillar fibronectin plays a key role as nucleator of collagen I polymerization during macromolecular crowding-enhanced matrix assembly

Macromolecular crowding is used by tissue engineers to accelerate extracellular matrix assembly in vitro, however, most mechanistic studies focus on the impact of crowding on collagen fiber assembly and largely ignore the highly abundant provisional matrix protein fibronectin. We show that the accelerated collagen I assembly as induced by the neutral crowding molecule Ficoll is regulated by cell access to fibronectin. Ficoll treatment leads to significant increases in the amount of surface adherent fibronectin, which can readily be harvested by cells to speed up fibrillogenesis. FRET studies reveal that Ficoll crowding also upregulates the total amount of fibronectin fibers in a low-tension state through upregulating fibronectin assembly. Since un-stretched fibronectin fibers have more collagen binding sites to nucleate the onset of collagen fibrillogenesis, our data suggest that the Ficoll-induced upregulation of low-tension fibronectin fibers contributes to enhanced collagen assembly in crowded conditions. In contrast, chemical cross-linking of fibronectin to the glass substrate prior to cell seeding prevents early force mediated fibronectin harvesting from the substrate and suppresses upregulation of collagen I assembly in the presence of Ficoll, even though the crowded environment is known to drive enzymatic cleavage of procollagen and collagen fiber formation. To show that our findings can be exploited for tissue engineering applications, we demonstrate that the addition of supplemental fibronectin in the form of an adsorbed coating markedly improves the speed of tissue formation under crowding conditions.

Anomalous transport in the soft-sphere Lorentz model

The sensitivity of anomalous transport in crowded media to the form of the inter-particle interactions is investigated through computer simulations. We extend the highly simplified Lorentz model towards realistic natural systems by modeling the interactions between the tracer and the obstacles with a smooth potential. We find that the anomalous transport at the critical point happens to be governed by the same universal exponent as for hard exclusion interactions, although the mechanism of how narrow channels are probed is rather different. The scaling behavior of simulations close to the critical point confirm this exponent. Our result indicates that the simple Lorentz model may be applicable to describing the fundamental properties of long-range transport in real crowded environments.

A Membraneless Organelle Associated with the Endoplasmic Reticulum Enables 3'UTR-Mediated Protein-Protein Interactions

Approximately half of human genes generate mRNAs with alternative 3' untranslated regions (3'UTRs). Through 3'UTR-mediated protein-protein interactions, alternative 3'UTRs enable multi-functionality of proteins with identical amino acid sequence. While studying how information on protein features is transferred from 3'UTRs to proteins, we discovered that the broadly expressed RNA-binding protein TIS11B forms a membraneless organelle, called TIS granule, that enriches membrane protein-encoding mRNAs with multiple AU-rich elements. TIS granules form a reticular meshwork intertwined with the endoplasmic reticulum (ER). The association between TIS granules and the ER creates a subcellular compartment-the TIGER domain-with a biophysically and biochemically distinct environment from the cytoplasm. This compartment promotes 3'UTR-mediated interaction of SET with membrane proteins, thus allowing increased surface expression and functional diversity of proteins, including CD47 and PD-L1. The TIGER domain is a subcellular compartment that enables formation of specific and functionally relevant protein-protein interactions that cannot be established outside.

Effect of the Organization of Rhodopsin on the Association between Transducin and a Photoactivated Receptor

After photoactivation, rhodopsin (R), a G-protein-coupled receptor, rapidly activates multiple transducin G-proteins (G) in an initial amplification step of phototransduction. G-protein activation requires diffusion-mediated association with an active rhodopsin (R*) at the rod disk membrane. Different organizations of R within the membrane have been revealded by several microscopy studies, including static and freely diffusing situations. However, it is unclear how such different scenarios influence the activation rate of G proteins. Through Monte Carlo simulations, we study the association reaction between a photoactivated rhodopsin and transducin under different reported receptor organizations including (a) R monomers diffusing freely, (b) R forming static dispersed crystalline domains made of rows of dimers, and (c) R arranged in static tracks formed by two adjacent rows of dimers. A key parameter in our simulations is the probability of binding following a collision ( p). For high p, the association rate between R* and G is higher in the freely diffusive system than in the static organizations, but for low collision efficiencies, the static organizations can result in faster association rates than the mobile system. We also observe that with low p, increasing the concentration of R increases the association rate significantly in the dispersed crystals configuration and just slightly in the free diffusive system. In summary, the lateral organization of rhodopsin influences the association rate between R* and G in a manner strongly dependent on the collision efficiency.

Reactions, diffusion, and volume exclusion in a conserved system of interacting particles

Complex biological and physical transport processes are often described through systems of interacting particles. The effect of excluded volume on these transport processes has been well studied; however, the interplay between volume exclusion and reactions between heterogenous particles is less well studied. In this paper we develop a framework for modeling reaction-diffusion processes which directly incorporates volume exclusion. We consider simple reactions (unimolecular and bimolecular) that conserve the total number of particles. From an off-lattice microscopic individual-based model we use the Fokker-Planck equation and the method of matched asymptotic expansions to derive a low-dimensional macroscopic system of nonlinear partial differential equations describing the evolution of the particles. A biologically motivated, hybrid model of chemotaxis with volume exclusion is explored, where reactions occur at rates dependent upon the chemotactic environment. Further, we show that for reactions that require particle contact the appropriate reaction term in the macroscopic model is of lower order in the asymptotic expansion than the nonlinear diffusion term. However, we find that the next reaction term in the expansion is needed to ensure good agreement with simulations of the microscopic model. Our macroscopic model allows for more direct parametrization to experimental data than existing models.

Product interactions and feedback in diffusion-controlled reactions

Steric or attractive interactions among reactants or between reactants and inert crowders can substantially influence the total rate of a diffusion-influenced reaction in the liquid phase. However, the role of the product species, which has typically different physical properties than the reactant species, has been disregarded so far. Here we study the effects of reactant-product and product-product interactions as well as asymmetric diffusion properties on the rate of diffusion-controlled reactions in the classical Smoluchowski-setup for chemical transformations at a perfect catalytic sphere. For this, we solve the diffusion equation with appropriate boundary conditions coupled by a mean-field approach on the second virial level to account for the particle interactions. We find that all particle spatial distributions and the total rate can change significantly, depending on the diffusion and interaction properties of the accumulated products. Complex competing and self-regulating (homeostatic) or self-amplifying effects are observed for the system, leading to both decrease and increase in the rates, as the presence of interacting products feeds back to the reactant flux and thus the rate with which the products are generated.

An electrostatic and probabilistic simulation model to describe neurosecretion at the synaptic scale

A hybrid simulation model (macro-molecular dynamics and Monte Carlo method) is proposed to reproduce neurosecretion and exocytosis. A theory has been developed for vesicular dynamics based on quasi-static electric interactions and a simple transition-state model for the vesicular fusion. Under the non-equilibrium electric conditions in an electrolytic fluid, it is considered that the motion of each synaptic vesicle is influenced by electrostatic forces exerted by the membranes of the synaptic bouton, other vesicles, the intracellular and intravesicular fluids, and external elements to the neuron. In addition, friction between each vesicle and its surrounding intracellular fluid is included in the theory, resulting in a drift type movement. To validate the vesicle equations of motion, a molecular dynamics method has been implemented, where the synaptic pool was replaced by a straight angle parallelepiped, the vesicles were represented by spheres and the fusion between each vesicle and the presynaptic membrane was simulated by a Monte Carlo type probabilistic change of state. Density profiles showing clusters of preferential activity as well as fusion distributions similar to the Poisson distributions associated with miniature end-plate potentials were obtained in the simulations.

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.

Hyperbranched polyglycerols: recent advances in synthesis, biocompatibility and biomedical applications

Hyperbranched polyglycerol is one of the most widely studied biocompatible dendritic polymer and showed promising applications. Here, we summarized the recent advancements in the field.

Population dynamics, information transfer, and spatial organization in a chemical reaction network under spatial confinement and crowding conditions

We investigate, via Brownian dynamics simulations, the reaction dynamics of a generic, nonlinear chemical network under spatial confinement and crowding conditions. In detail, the Willamowski-Rossler chemical reaction system has been "extended" and considered as a prototype reaction-diffusion system. Our results are potentially relevant to a number of open problems in biophysics and biochemistry, such as the synthesis of primitive cellular units (protocells) and the definition of their role in the chemical origin of life and the characterization of vesicle-mediated drug delivery processes. More generally, the computational approach presented in this work makes the case for the use of spatial stochastic simulation methods for the study of biochemical networks in vivo where the "well-mixed" approximation is invalid and both thermal and intrinsic fluctuations linked to the possible presence of molecular species in low number copies cannot be averaged out.

Abnormal polymer transport in crowded attractive micropost arrays

We investigate polymer diffusion in a quasi-two-dimensional environment decorated with attractive cylindrical posts using Langevin dynamics simulation. We find that the polymer diffusivity has non-monotonic dependence on the post array density. This diffusive behavior strongly depends on the adsorption-desorption transition and the critical adsorption strength ε_c. For ε < ε_c, the polymer undergoes normal diffusion and the diffusivity decreases as the post density increases due to the reduction of the void volume. For ε > ε_c, polymer dynamics is strongly mediated by post adsorption, and we observe a regime where the polymer diffusivity increases as the post density increases. The polymer diffusivity reaches a maximum, which can be attributed to cross-post translation enabled by large polymer conformation fluctuations. We find both cross-post transport and polymer conformation fluctuations strongly depend on the post absorption strength and the chain length.

What macromolecular crowding can do to a protein

The intracellular environment represents an extremely crowded milieu, with a limited amount of free water and an almost complete lack of unoccupied space. Obviously, slightly salted aqueous solutions containing low concentrations of a biomolecule of interest are too simplistic to mimic the “real life” situation, where the biomolecule of interest scrambles and wades through the tightly packed crowd. In laboratory practice, such macromolecular crowding is typically mimicked by concentrated solutions of various polymers that serve as model “crowding agents”. Studies under these conditions revealed that macromolecular crowding might affect protein structure, folding, shape, conformational stability, binding of small molecules, enzymatic activity, protein-protein interactions, protein-nucleic acid interactions, and pathological aggregation. The goal of this review is to systematically analyze currently available experimental data on the variety of effects of macromolecular crowding on a protein molecule. The review covers more than 320 papers and therefore represents one of the most comprehensive compendia of the current knowledge in this exciting area.

Enhancement of biological reactions on cell surfaces via macromolecular crowding

The reaction of macromolecules such as enzymes and antibodies with cell surfaces is often an inefficient process, requiring large amounts of expensive reagent. Here we report a general method based on macromolecular crowding with a range of neutral polymers to enhance such reactions, using red blood cells (RBCs) as a model system. Rates of conversion of type A and B red blood cells to universal O type by removal of antigenic carbohydrates with selective glycosidases are increased up to 400-fold in the presence of crowders. Similar enhancements are seen for antibody binding. We further explore the factors underlying these enhancements using confocal microscopy and fluorescent recovery after bleaching (FRAP) techniques with various fluorescent protein fusion partners. Increased cell-surface concentration due to volume exclusion, along with two-dimensionally confined diffusion of enzymes close to the cell surface, appear to be the major contributing factors.

Nonideal diffusion effects and short-range ordering lead to higher aggregation rates in concentrated hard-sphere dispersions

Brownian aggregation in concentrated hard-sphere dispersions is studied using models and Brownian dynamics (BD) simulations. Two new theoretical models are presented and compared to several existing approaches and BD simulation results, which serve as benchmarks. The first new model is an improvement over an existing local density approximation (LDA)-based model. The other is based on the more rigorous Fundamental measure theory (FMT) applied to the "liquid-state" dynamic density-functional theory (DDFT). Both models provide significant improvements over the classical Smoluchowski model. The predictions of the new FM-DDFT-based model for aggregation kinetics are in excellent agreement with BD simulation results for dispersions with initial particle volume fractions, ϕ, up to 0.35 (close to the hard-sphere freezing transition at ϕ = 0.494). In contrast to previous approaches, the nonideal particle diffusion effects and the initial and time-dependent short-range ordering in concentrated dispersions due to entropic packing effects are explicitly considered here, in addition to the unsteady-state effects. The greater accuracy of the FM-DDFT-based model compared to that of the LDA-based models indicates that nonlocal contributions to particle diffusion (only accounted for in the former) play important roles in aggregation. At high concentrations, the FM-DDFT-based model predicts aggregation half-times and gelation times that are up to 2 orders of magnitude shorter than those of the Smoluchowski model. Moreover, the FM-DDFT-based model predicts asymmetric cluster-cluster aggregation rate constants, at least for short times. Overall, a rigorous mechanistic understanding of the enhancement of aggregation kinetics in concentrated dispersions is provided.

Crowding-induced formation and structural alteration of nuclear compartments: insights from computer simulations

Our understanding of the structural and dynamical characteristics of nuclear structures such as chromosomes and nuclear bodies (NBs) has increased significantly in recent days owing to advances in biophysical and biochemical techniques. These techniques include the use of computer simulations, which have provided further physical insights complementary to findings from experiments. In this chapter, we review recent computer simulation studies on the structural alteration of chromosome subcompartments and the formation and maintenance of NBs in the highly crowded cell nucleus. It is found that because of macromolecular crowding, the degree of chromosome compaction changes significantly and the formation of NBs is facilitated. We further discuss the physical consequences of these phenomena, which may be of critical importance in understanding genome processes.

Anomalous transport in the crowded world of biological cells

A ubiquitous observation in cell biology is that the diffusive motion of macromolecules and organelles is anomalous, and a description simply based on the conventional diffusion equation with diffusion constants measured in dilute solution fails. This is commonly attributed to macromolecular crowding in the interior of cells and in cellular membranes, summarizing their densely packed and heterogeneous structures. The most familiar phenomenon is a sublinear, power-law increase of the mean-square displacement (MSD) as a function of the lag time, but there are other manifestations like strongly reduced and time-dependent diffusion coefficients, persistent correlations in time, non-Gaussian distributions of spatial displacements, heterogeneous diffusion and a fraction of immobile particles. After a general introduction to the statistical description of slow, anomalous transport, we summarize some widely used theoretical models: Gaussian models like fractional Brownian motion and Langevin equations for visco-elastic media, the continuous-time random walk model, and the Lorentz model describing obstructed transport in a heterogeneous environment. Particular emphasis is put on the spatio-temporal properties of the transport in terms of two-point correlation functions, dynamic scaling behaviour, and how the models are distinguished by their propagators even if the MSDs are identical. Then, we review the theory underlying commonly applied experimental techniques in the presence of anomalous transport like single-particle tracking, fluorescence correlation spectroscopy (FCS) and fluorescence recovery after photobleaching (FRAP). We report on the large body of recent experimental evidence for anomalous transport in crowded biological media: in cyto- and nucleoplasm as well as in cellular membranes, complemented by in vitro experiments where a variety of model systems mimic physiological crowding conditions. Finally, computer simulations are discussed which play an important role in testing the theoretical models and corroborating the experimental findings. The review is completed by a synthesis of the theoretical and experimental progress identifying open questions for future investigation.

Crowding effects on the formation and maintenance of nuclear bodies: insights from molecular-dynamics simulations of simple spherical model particles

The physics of structure formation and maintenance of nuclear bodies (NBs), such as nucleoli, Cajal bodies, promyelocytic leukemia bodies, and speckles, in a crowded nuclear environment remains largely unknown. We investigate the role of macromolecular crowding in the formation and maintenance of NBs using computer simulations of a simple spherical model, called Lennard-Jones (LJ) particles. LJ particles form a one-phase, dilute fluid when the intermolecular interaction is weaker than a critical value, above which they phase separate and form a condensed domain. We find that when volume-exclusive crowders exist in significant concentrations, domain formation is induced even for weaker intermolecular interactions, and the effect is more pronounced with increasing crowder concentration. Simulation results show that a previous experimental finding that promyelocytic leukemia bodies disappear in the less-crowded condition and reassemble in the normal crowded condition can be interpreted as a consequence of the increased intermolecular interactions between NB proteins due to crowding. Based on further analysis of the simulation results, we discuss the acceleration of macromolecular associations that occur within NBs, and the delay of diffusive transport of macromolecules within and out of NBs when the crowder concentration increases. This study suggests that in a polydisperse nuclear environment that is enriched with a variety of macromolecules, macromolecular crowding not only plays an important role in the formation and maintenance of NBs, but also may perform some regulatory functions in response to alterations in the crowding conditions.

A computational approach to increase time scales in Brownian dynamics-based reaction-diffusion modeling

Particle-based Brownian dynamics simulations offer the opportunity to not only simulate diffusion of particles but also the reactions between them. They therefore provide an opportunity to integrate varied biological data into spatially explicit models of biological processes, such as signal transduction or mitosis. However, particle based reaction-diffusion methods often are hampered by the relatively small time step needed for accurate description of the reaction-diffusion framework. Such small time steps often prevent simulation times that are relevant for biological processes. It is therefore of great importance to develop reaction-diffusion methods that tolerate larger time steps while maintaining relatively high accuracy. Here, we provide an algorithm, which detects potential particle collisions prior to a BD-based particle displacement and at the same time rigorously obeys the detailed balance rule of equilibrium reactions. We can show that for reaction-diffusion processes of particles mimicking proteins, the method can increase the typical BD time step by an order of magnitude while maintaining similar accuracy in the reaction diffusion modelling.

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.

Rule-based modeling of signal transduction: a primer

Biological cells accomplish their physiological functions using interconnected networks of genes, proteins, and other biomolecules. Most interactions in biological signaling networks, such as bimolecular association or covalent modification, can be modeled in a physically realistic manner using elementary reaction kinetics. However, the size and combinatorial complexity of such reaction networks have hindered such a mechanistic approach, leading many to conclude that it is premature and to adopt alternative statistical or phenomenological approaches. The recent development of rule-based modeling languages, such as BioNetGen (BNG) and Kappa, enables the precise and succinct encoding of large reaction networks. Coupled with complementary advances in simulation methods, these languages circumvent the combinatorial barrier and allow mechanistic modeling on a much larger scale than previously possible. These languages are also intuitive to the biologist and accessible to the novice modeler. In this chapter, we provide a self-contained tutorial on modeling signal transduction networks using the BNG Language and related software tools. We review the basic syntax of the language and show how biochemical knowledge can be articulated using reaction rules, which can be used to capture a broad range of biochemical and biophysical phenomena in a concise and modular way. A model of ligand-activated receptor dimerization is examined, with a detailed treatment of each step of the modeling process. Sections discussing modeling theory, implicit and explicit model assumptions, and model parameterization are included, with special focus on retaining biophysical realism and avoiding common pitfalls. We also discuss the more advanced case of compartmental modeling using the compartmental extension to BioNetGen. In addition, we provide a comprehensive set of example reaction rules that cover the various aspects of signal transduction, from signaling at the membrane to gene regulation. The reader can modify these reaction rules to model their own systems of interest.

Excluded-volume effects in the diffusion of hard spheres

Excluded-volume effects can play an important role in determining transport properties in diffusion of particles. Here, the diffusion of finite-sized hard-core interacting particles in two or three dimensions is considered systematically using the method of matched asymptotic expansions. The result is a nonlinear diffusion equation for the one-particle distribution function, with excluded-volume effects enhancing the overall collective diffusion rate. An expression for the effective (collective) diffusion coefficient is obtained. Stochastic simulations of the full particle system are shown to compare well with the solution of this equation for two examples.

Unified regression model of binding equilibria in crowded environments

Molecular crowding is a critical feature distinguishing intracellular environments from idealized solution-based environments and is essential to understanding numerous biochemical reactions, from protein folding to signal transduction. Many biochemical reactions are dramatically altered by crowding, yet it is extremely difficult to predict how crowding will quantitatively affect any particular reaction systems. We previously developed a novel stochastic off-lattice model to efficiently simulate binding reactions across wide parameter ranges in various crowded conditions. We now show that a polynomial regression model can incorporate several interrelated parameters influencing chemistry under crowded conditions. The unified model of binding equilibria accurately reproduces the results of particle simulations over a broad range of variation of six physical parameters that collectively yield a complicated, non-linear crowding effect. The work represents an important step toward the long-term goal of computationally tractable predictive models of reaction chemistry in the cellular environment.

Diffusion in crowded biological environments: applications of Brownian dynamics

Biochemical reactions in living systems occur in complex, heterogeneous media with total concentrations of macromolecules in the range of 50 - 400 mgml. Molecular species occupy a significant fraction of the immersing medium, up to 40% of volume. Such complex and volume-occupied environments are generally termed 'crowded' and/or 'confined'. In crowded conditions non-specific interactions between macromolecules may hinder diffusion - a major process determining metabolism, transport, and signaling. Also, the crowded media can alter, both qualitatively and quantitatively, the reactions in vivo in comparison with their in vitro counterparts. This review focuses on recent developments in particle-based Brownian dynamics algorithms, their applications to model diffusive transport in crowded systems, and their abilities to reproduce and predict the behavior of macromolecules under in vivo conditions.

Effects of macromolecular crowding on the inhibition of virus assembly and virus-cell receptor recognition

Biological fluids contain a very high total concentration of macromolecules that leads to volume exclusion by one molecule to another. Theory and experiment have shown that this condition, termed macromolecular crowding, can have significant effects on molecular recognition. However, the influence of molecular crowding on recognition events involving virus particles, and their inhibition by antiviral compounds, is virtually unexplored. Among these processes, capsid self-assembly during viral morphogenesis and capsid-cell receptor recognition during virus entry into cells are receiving increasing attention as targets for the development of new antiviral drugs. In this study, we have analyzed the effect of macromolecular crowding on the inhibition of these two processes by peptides. Macromolecular crowding led to a significant reduction in the inhibitory activity of: 1), a capsid-binding peptide and a small capsid protein domain that interfere with assembly of the human immunodeficiency virus capsid, and 2), a RGD-containing peptide able to block the interaction between foot-and-mouth disease virus and receptor molecules on the host cell membrane (in this case, the effect was dependent on the conditions used). The results, discussed in the light of macromolecular crowding theory, are relevant for a quantitative understanding of molecular recognition processes during virus infection and its inhibition.

Crowding effects on protein association: effect of interactions between crowding agents

The cell cytoplasm is a dense environment where the presence of inert cosolutes can significantly alter the rates of protein folding and protein association reactions. Most theoretical studies focus on hard sphere crowding agents and quantify the effect of excluded volume on reaction rates. In this work the effect of interactions between the crowding agents on the thermodynamics of protein association is studied using computer simulation. Three cases are considered, where the crowding agents are (i) hard spheres, (ii) hard spheres with additional attractive or repulsive interactions, and (iii) chains of hard spheres. Reactants and products of the protein association are modeled as hard spheres. Although crowding effects are sensitive to the shape of the reaction product, in most cases the excess free energy difference between the product and reactants (nonideality factor) is insensitive to the interactions between crowding agents, due to a cancellation of effects. The simulations therefore suggest that the hard sphere model of crowding agents has a surprisingly large regime of validity and should be sufficient for a qualitative understanding of the thermodynamics of crowding effects when the interactions of associating proteins with crowding agents other than excluded volume interactions are not significant.

Diffusion-limited reactions in crowded environments

Diffusion-limited reactions are usually described within the Smoluchowski theory, which neglects interparticle interactions. We propose a simple way to incorporate excluded-volume effects building on simulations of hard sphere in the presence of a sink. For large values of the sink-to-particle size ratio R(s), the measured encounter rate is in good agreement with a simple generalization of the Smoluchowski equation at high densities. Reducing R(s), the encounter rate is substantially depressed and becomes even nonmonotonic for R(s)<<1. Concurrently with the saturation of the rate, stationary density waves set in close to the sink. A mean-field analysis helps to shed light on the subtle link between such ordering and the slowing down of the encounter dynamics. Finally, we show how an infinitesimal amount of nonreacting impurities can equally slow down dramatically the reaction.

Monte Carlo investigation of diffusion of receptors and ligands that bind across opposing surfaces

Studies of receptor diffusion on a cell surface show a variety of behaviors, such as diffusive, sub-diffusive, or super-diffusive motion. However, most studies to date focus on receptor molecules diffusing on a single cell surface. We have previously studied receptor diffusion to probe the molecular mechanism of receptor clustering at the cell–cell junction between two opposing cell surfaces. Here, we characterize the diffusion of receptors and ligands that bind to each other across two opposing cell surfaces, as in cell–cell and cell–bilayer interactions. We use a Monte Carlo method, where receptors and ligands are simulated as independent agents that bind and diffuse probabilistically. We vary receptor–ligand binding affinity and plot the molecule-averaged mean square displacement (MSD) of ligand molecules as a function of time. Our results show that MSD plots are qualitatively different for flat and curved interfaces, as well as between the cases of presence and absence of directed transport of receptor–ligand complexes toward a specific location on the interface. Receptor–ligand binding across two opposing surfaces leads to transient sub-diffusive motion at early times provided the interface is flat. This effect is entirely absent if the interface is curved, however, in this instance we observe sub-diffusive motion. In addition, a decrease in the equilibrium value of the MSD occurs as affinity increases, something which is absent for a flat interface. In the presence of directed transport of receptor–ligand complexes, we observe super-diffusive motion at early times for a flat interface. Super-diffusive motion is absent for a curved interface, however, in this case we observe a transient decrease in MSD with time prior to equilibration for high-affinity values.