Effect of macromolecular crowding on reaction rates: a computational and theoretical study

Effect of crowding and confinement on first-passage times: A model study

We study the "color dynamics" of a hard-disk fluid confined in an annulus, as well as the corresponding hard-sphere system in three dimensions, using event-driven simulation in order to explore the effect of confinement and self-crowding on the search for targets. We compute the mean first-passage times (MFPTs) of red particles transiting from the outer to the inner boundary as well as those of blue particles passing from the inner to the outer boundary for different packing fractions and geometries. In the steady state the reaction rate, defined as the rate of collision of red particles with the inner boundary, is inversely proportional to the sum of the MFPTs. The reaction rate is wall mediated (ballistic) at low densities and diffusion controlled at higher densities and displays a maximum at intermediate densities. At moderate to high densities, the presence of layering has a strong influence on the search process. The numerical results for the reaction rate and MFPTs are compared with a ballistic model at low densities and a Smoluchowski approach with uniform diffusivities at higher densities. We discuss the reasons for the limited validity of the theoretical approaches. The maximum in the reaction rate is qualitatively well rendered by a Bosanquet-like approach that interpolates between the two regimes. Finally, we compute the position-dependent diffusivity from the MFPTs and observe that it is out of phase with the radial density.

Theory of diffusion-influenced reactions in complex geometries

Chemical transformations involving the diffusion of reactants and subsequent chemical fixation steps are generally termed "diffusion-influenced reactions" (DIR). Virtually all biochemical processes in living media can be counted among them, together with those occurring in an ever-growing number of emerging nano-technologies. The role of the environment's geometry (obstacles, compartmentalization) and distributed reactivity (competitive reactants, traps) is key in modulating the rate constants of DIRs, and is therefore a prime design parameter. Yet, it is a formidable challenge to build a comprehensive theory that is able to describe the environment's "reactive geometry". Here we show that such a theory can be built by unfolding this many-body problem through addition theorems for special functions. Our method is powerful and general and allows one to study a given DIR reaction occurring in arbitrary "reactive landscapes", made of multiple spherical boundaries of given size and reactivity. Importantly, ready-to-use analytical formulas can be derived easily in most cases.

The predicted role of steric specificity in crowding-mediated effects on reversible biomolecular association

A fundamental question in biology is whether the presence of non-reacting macromolecules in the cytoplasm affects the rates and extents of reversible association reactions, a phenomenon often referred to as 'macromolecular crowding.' Under certain conditions, crowding has been proposed to dramatically alter the kinetics and thermodynamics of chemical reactions, making it difficult to quantitatively relate rates and extents of reactions measured in vitro to those occurring in vivo. In this work, we use Brownian dynamics simulation and Monte Carlo methods to (1) quantify the overall thermodynamic and kinetic effects of crowding by independently investigating each step of reversible bimolecular association (i.e. translational diffusion, steric specific binding, and dissociation), and (2) provide an explicit, quantitative investigation of how the degree of steric specificity of protein dimerization influences crowding-mediated effects on association and dissociation. We find that k on decreases by ∼2-fold for non-steric specific reactions, and increases by ∼3-fold for highly steric specific reactions. In addition, k off decreases by only ∼30%-60% in the presence of crowders, depending on the strength of the bond between the reactant pair, so that the equilibrium constant is increased by ∼4-fold, at most. These results suggest that crowding-mediated effects on globular protein dimerization reactions in the cytoplasm are modulated by the steric specificity of the reactants, and that reversible protein-protein association is relatively insensitive to the physical presence of crowders (i.e. steric repulsion effects in the cytoplasm) for crowders of similar size and shape to reactants over a range of volume fractions (0-0.3).

Effective reaction rates in diffusion-limited phosphorylation-dephosphorylation cycles

We investigate the kinetics of the ubiquitous phosphorylation-dephosphorylation cycle on biological membranes by means of kinetic Monte Carlo simulations on the triangular lattice. We establish the dependence of effective macroscopic reaction rate coefficients as well as the steady-state phosphorylated substrate fraction on the diffusion coefficient and concentrations of opposing enzymes: kinases and phosphatases. In the limits of zero and infinite diffusion, the numerical results agree with analytical predictions; these two limits give the lower and the upper bound for the macroscopic rate coefficients, respectively. In the zero-diffusion limit, which is important in the analysis of dense systems, phosphorylation and dephosphorylation reactions can convert only these substrates which remain in contact with opposing enzymes. In the most studied regime of nonzero but small diffusion, a contribution linearly proportional to the diffusion coefficient appears in the reaction rate. In this regime, the presence of opposing enzymes creates inhomogeneities in the (de)phosphorylated substrate distributions: The spatial correlation function shows that enzymes are surrounded by clouds of converted substrates. This effect becomes important at low enzyme concentrations, substantially lowering effective reaction rates. Effective reaction rates decrease with decreasing diffusion and this dependence is more pronounced for the less-abundant enzyme. Consequently, the steady-state fraction of phosphorylated substrates can increase or decrease with diffusion, depending on relative concentrations of both enzymes. Additionally, steady states are controlled by molecular crowders which, mostly by lowering the effective diffusion of reactants, favor the more abundant enzyme.

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.

Macromolecular crowding as a regulator of gene transcription

Studies of macromolecular crowding have shown its important effects on molecular transport and interactions in living cells. Less clear is the effect of crowding when its influence is incorporated into a complex network of interactions. Here, we explore the effects of crowding in the cell nucleus on a model of gene transcription as a network of reactions involving transcription factors, RNA polymerases, and DNA binding sites for these proteins. The novelty of our approach is that we determine the effects of crowding on the rates of these reactions using Brownian dynamics and Monte Carlo simulations, allowing us to integrate molecular-scale information, such as the shapes and sizes of each molecular species, into the rate equations of the model. The steady-state cytoplasmic mRNA concentration shows several regimes with qualitatively different dependences on the volume fraction, ϕ, of crowding agents in the nucleus, including a broad range of parameter values where it depends nonmonotonically on ϕ, with maximum mRNA production occurring at a physiologically relevant value. The extent of this crowding dependence can be modulated by a variety of means, suggesting that the transcriptional output of a gene can be regulated jointly by the local level of macromolecular crowding in the nucleus, together with the local concentrations of polymerases and DNA-binding proteins, as well as other properties of the gene's physical environment.

Nonmonotonic diffusion of particles among larger attractive crowding spheres

We study the diffusive motion of particles among fixed spherical crowders. The diffusers interact with the crowders through a combination of a hard-core repulsion and a short-range attraction. The long-time effective diffusion coefficient of the diffusers is found to depend nonmonotonically on the strength of their attraction to the crowders. That is, for a given concentration of crowders, a weak attraction to the crowders enhances diffusion. We show that this counterintuitive fact can be understood in terms of the mesoscopic excess chemical potential landscape experienced by the diffuser. The roughness of this excess chemical potential landscape quantitatively captures the nonmonotonic dependence of the diffusion rate on the strength of crowder-diffuser attraction; thus, it is a purely static predictor of dynamic behavior. The mesoscopic view given here provides a unified explanation for enhanced diffusion effects that have been found in various systems of technological and biological interest.

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.

Calculated rates of diffusion-limited reactions in a three-dimensional network of connected compartments: application to porous catalysts and biological systems

We describe the diffusion limit for reaction rates in a three-dimensional system of connected compartments. This model exhibits the length-scale dependent diffusion that can be observed in many heterogeneous environments, such as porous catalysts and biological environments. We obtain a simple analytical expression for the diffusion limit applicable to any scale of the compartment confinement. This diffusion limit exceeds the classic Smoluchowski diffusion limit that was derived for homogeneous environments but is often applied to biological reactions in heterogeneous environments. We expect our new diffusion limit to provide a more appropriate upper bound on reaction rates in biological systems, porous structures, and other heterogeneous environments where obstacles create local confinement.

The effect of macromolecular crowding on the electrostatic component of barnase-barstar binding: a computational, implicit solvent-based study

Macromolecular crowding within the cell can impact both protein folding and binding. Earlier models of cellular crowding focused on the excluded volume, entropic effect of crowding agents, which generally favors compact protein states. Recently, other effects of crowding have been explored, including enthalpically-related crowder–protein interactions and changes in solvation properties. In this work, we explore the effects of macromolecular crowding on the electrostatic desolvation and solvent-screened interaction components of protein–protein binding. Our simple model enables us to focus exclusively on the electrostatic effects of water depletion on protein binding due to crowding, providing us with the ability to systematically analyze and quantify these potentially intuitive effects. We use the barnase–barstar complex as a model system and randomly placed, uncharged spheres within implicit solvent to model crowding in an aqueous environment. On average, we find that the desolvation free energy penalties incurred by partners upon binding are lowered in a crowded environment and solvent-screened interactions are amplified. At a constant crowder density (fraction of total available volume occupied by crowders), this effect generally increases as the radius of model crowders decreases, but the strength and nature of this trend can depend on the water probe radius used to generate the molecular surface in the continuum model. In general, there is huge variation in desolvation penalties as a function of the random crowder positions. Results with explicit model crowders can be qualitatively similar to those using a lowered “effective” solvent dielectric to account for crowding, although the “best” effective dielectric constant will likely depend on multiple system properties. Taken together, this work systematically demonstrates, quantifies, and analyzes qualitative intuition-based insights into the effects of water depletion due to crowding on the electrostatic component of protein binding, and it provides an initial framework for future analyses.

Anomalous versus slowed-down Brownian diffusion in the ligand-binding equilibrium

Measurements of protein motion in living cells and membranes consistently report transient anomalous diffusion (subdiffusion) that converges back to a Brownian motion with reduced diffusion coefficient at long times after the anomalous diffusion regime. Therefore, slowed-down Brownian motion could be considered the macroscopic limit of transient anomalous diffusion. On the other hand, membranes are also heterogeneous media in which Brownian motion may be locally slowed down due to variations in lipid composition. Here, we investigate whether both situations lead to a similar behavior for the reversible ligand-binding reaction in two dimensions. We compare the (long-time) equilibrium properties obtained with transient anomalous diffusion due to obstacle hindrance or power-law-distributed residence times (continuous-time random walks) to those obtained with space-dependent slowed-down Brownian motion. Using theoretical arguments and Monte Carlo simulations, we show that these three scenarios have distinctive effects on the apparent affinity of the reaction. Whereas continuous-time random walks decrease the apparent affinity of the reaction, locally slowed-down Brownian motion and local hindrance by obstacles both improve it. However, only in the case of slowed-down Brownian motion is the affinity maximal when the slowdown is restricted to a subregion of the available space. Hence, even at long times (equilibrium), these processes are different and exhibit irreconcilable behaviors when the area fraction of reduced mobility changes.

FOUNDATIONS FOR MODELING THE DYNAMICS OF GENE REGULATORY NETWORKS: A MULTILEVEL-PERSPECTIVE REVIEW

A promising alternative for unraveling the principles under which the dynamic interactions among genes lead to cellular phenotypes relies on mathematical and computational models at different levels of abstraction, from the molecular level of protein-DNA interactions to the system level of functional relationships among genes. This review article presents, under a bottom–up perspective, a hierarchy of approaches to modeling gene regulatory network dynamics, from microscopic descriptions at the single-molecule level in the spatial context of an individual cell to macroscopic models providing phenomenological descriptions at the population-average level. The reviewed modeling approaches include Molecular Dynamics, Particle-Based Brownian Dynamics, the Master Equation approach, Ordinary Differential Equations, and the Boolean logic abstraction. Each of these frameworks is motivated by a particular biological context and the nature of the insight being pursued. The setting of gene network dynamic models from such frameworks involves assumptions and mathematical artifacts often ignored by the non-specialist. This article aims at providing an entry point for biologists new to the field and computer scientists not acquainted with some recent biophysically-inspired models of gene regulation. The connections promoting intuition between different abstraction levels and the role that approximations play in the modeling process are highlighted throughout the paper.

Importance of crowding in signaling, genetic, and metabolic networks

It is now well established that the cell is a highly crowded environment. Yet, the effects of crowding on the dynamics of signaling pathways, gene regulation networks, and metabolic networks are still largely unknown. Crowding can alter both molecular diffusion and the equilibria of biomolecular reactions. In this chapter, we first discuss how diffusion can affect biochemical networks. Diffusion of transcription factors can increase noise in gene expression, while diffusion of proteins between intracellular compartments or between cells can reduce concentration fluctuations. In push-pull networks diffusion can impede information transmission, while in multisite protein modification networks diffusion can qualitatively change the macroscopic response of the system, such as the loss or emergence of bistability. Moreover, diffusion can directly change the metabolic flux. We describe how crowding affects diffusion, and thus how all these phenomena are influenced by crowding. Yet, a potentially more important effect of crowding on biochemical networks is mediated via the shift in the equilibria of bimolecular reactions, and we provide computational evidence that supports this idea. Finally, we discuss how the effects of crowding can be incorporated in models of biochemical networks.

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.

Effects of carbohydrates on the oNPG converting activity of β-galactosidases

The effects of high concentrations of carbohydrates on the o-nitrophenyl β-d-galactopyranoside (oNPG) converting activity of β-galactosidase from Bacillus circulans are studied to get a better understanding of the enzyme behavior in concentrated and complicated systems in which enzymatic synthesis of galacto-oligosaccharides is usually performed. The components that were tested were glucose, galactose, lactose, sucrose, trehalose, raffinose, Vivinal GOS, dextran-6000, dextran-70,000, and sarcosine. Small carbohydrates act as acceptors in the reaction. This speeds up the limiting step, which is binding of the galactose residue with the acceptor and release of the product. Simultaneously, both inert and reacting additives seem to cause some molecular crowding, which results in a higher enzyme affinity for the substrate. The effect of molecular crowding on the enzyme activity is small compared to the effect of carbohydrates acting in the reactions as acceptors. The effects of reactants on β-galactosidases from B. circulans, A. oryzae, and K. lactis are compared.

Irreversible bimolecular reactions with inertia: from the trapping to the target setting at finite densities

We investigate numerically pseudo-first-order irreversible bimolecular reactions of the type A + B → B between hard spheres undergoing event-driven Brownian dynamics. We study the encounter rate and the survival probability of A particles as functions of the packing fraction ϕ in the trapping (a single particle diffusing among static non-overlapping traps) and target (many traps diffusing in the presence of a single static target particle) settings, as well as in the case of diffusing traps and particles (full mobility). We show that, since inertial effects are accounted for in our simulation protocol, the standard Smoluchowski theory of coagulation of non-interacting colloids is recovered only at times greater than a characteristic time Δt, marking the transition from the under-damped to the over-damped regime. We show that the survival probability S(t) decays exponentially during this first stage, with a rate 1/τ0 is proportional to φ. Furthermore, we work out a simple analytical expression that is able to capture to an excellent extent the numerical results for t < Δt at low and intermediate densities. Moreover, we demonstrate that the time constant of the asymptotic exponential decay of S(t) for diffusing traps and particles is k(S)(-1), where kS = 4π(DA + DB)Rρ is the Smoluchowski rate. Detailed analyses of the effective decay exponent β = d [log(-logS(t))]/d (logt) and of the steady-state encounter rate reveal that the full mobility and trapping problem are characterized by very similar kinetics, rather different from the target problem. Our results do not allow one to ascertain whether the prediction S(t) is proportional to exp(-at(3/2)) (a = const.) as t → ∞ for the trapping problem in 3D is indeed recovered. In fact, at high density, S(t) is dominated by short encounter times, which makes it exceedingly hard to record the events corresponding to the exploration of large, trap-free regions. As a consequence, at high densities the steady-state rate simply tends to 1/τ0. Finally, we work out an analytical formula for the rate that shows a remarkable agreement with the numerics up φ = 0.4.

Protein-protein interactions in a crowded environment

Protein-protein interactions are important in many essential biological functions, such as transcription, translation, and signal transduction. Much progress has been made in understanding protein-protein association in dilute solution via experimentation and simulation. Cells, however, contain various macromolecules, such as DNA, RNA, proteins, among many others, and a myriad of non-specific interactions (usually weak) are present between these cellular constituents. In this review article, we describe the important developments in recent years that have furthered our understanding and even allowed prediction of the consequences of macromolecular crowding on protein-protein interactions. We outline the development of our crowding theory that can predict the change in binding free energy due to crowding quantitatively for both repulsive and attractive protein-crowder interactions. One of the most important findings from our recent work is that weak attractive interactions between crowders and proteins can actually destabilize protein complex formation as opposed to the commonly assumed stabilizing effect predicted based on traditional crowding theories that only account for the entropic-excluded volume effects. We also discuss the implications of macromolecular crowding on the population of encounter versus specific native complex.

Crowding induced entropy-enthalpy compensation in protein association equilibria

A statistical mechanical theory is presented to predict the effects of macromolecular crowding on protein association equilibria, accounting for both excluded volume and attractive interactions between proteins and crowding molecules. Predicted binding free energies are in excellent agreement with simulation data over a wide range of crowder sizes and packing fractions. It is shown that attractive interactions between proteins and crowding agents counteract the stabilizing effects of excluded volume interactions. A critical attraction strength, for which there is no net effect of crowding, is approximately independent of the crowder packing fraction.

Influences of excluded volume of molecules on signaling processes on the biomembrane

We investigate the influences of the excluded volume of molecules on biochemical reaction processes on 2-dimensional surfaces using a model of signal transduction processes on biomembranes. We perform simulations of the 2-dimensional cell-based model, which describes the reactions and diffusion of the receptors, signaling proteins, target proteins, and crowders on the cell membrane. The signaling proteins are activated by receptors, and these activated signaling proteins activate target proteins that bind autonomously from the cytoplasm to the membrane, and unbind from the membrane if activated. If the target proteins bind frequently, the volume fraction of molecules on the membrane becomes so large that the excluded volume of the molecules for the reaction and diffusion dynamics cannot be negligible. We find that such excluded volume effects of the molecules induce non-trivial variations of the signal flow, defined as the activation frequency of target proteins, as follows. With an increase in the binding rate of target proteins, the signal flow varies by monotonically increasing; increasing then decreasing in a bell-shaped curve; or increasing, decreasing, then increasing in an S-shaped curve. We further demonstrate that the excluded volume of molecules influences the hierarchical molecular distributions throughout the reaction processes. In particular, when the system exhibits a large signal flow, the signaling proteins tend to surround the receptors to form receptor-signaling protein clusters, and the target proteins tend to become distributed around such clusters. To explain these phenomena, we analyze the stochastic model of the local motions of molecules around the receptor.

Computer Simulations of the Bacterial Cytoplasm

Ever since the pioneering work of Minton, it has been recognized that the highly crowded interior of biological cells has the potential to cause dramatic changes to both the kinetics and thermodynamics of protein folding and association events relative to behavior that might be observed in dilute solution conditions. One very productive way to explore the effects of crowding on protein behavior has been to use macromolecular crowding agents that exclude volume without otherwise strongly interacting with the protein under study. An alternative, complementary approach to understanding the potential differences between behavior in vivo and in vitro is to develop simulation models that explicitly attempt to model intracellular environments at the molecular scale, and that thereby can be used to directly monitor biophysical behavior in conditions that accurately mimic those encountered in vivo. It is with studies of this type that the present review will be concerned. We review in detail four published studies that have attempted to simulate the structure and dynamics of the bacterial cytoplasm and that have each explored different biophysical aspects of the cellular interior. While each of these studies has yielded important new insights, there are important questions that remain to be resolved in terms of determining the relative contributions made by energetic and hydrodynamic interactions to the diffusive behavior of macromolecules and to the thermodynamics of protein folding and associations in vivo. Some possible new directions for future generation simulation models of the cytoplasm are outlined.

Contrasting factors on the kinetic path to protein complex formation diminish the effects of crowding agents

The crowded environment of cells poses a challenge for rapid protein-protein association. Yet, it has been established that the rates of association are similar in crowded and in dilute solutions. Here we probe the pathway leading to fast association between TEM1 β-lactamase and its inhibitor protein BLIP in crowded solutions. We show that the affinity of the encounter complex, the rate of final complex formation, and the structure of the transition state are similar in crowded solutions and in buffer. The experimental results were reproduced by calculations based on the transient-complex theory for protein association. Both experiments and calculations suggest that while crowding agents decrease the diffusion constant of the associating proteins, they also induce an effective excluded-volume attraction between them. The combination of the two opposing effects thus results in nearly identical overall association rates in diluted and crowded solutions.

Formation of protein complexes in crowded environments--from in vitro to in vivo

Traditionally, biochemical studies are performed in dilute homogenous solutions, which are very different from the dense mixture of molecules found in cells. Thus, the physiological relevance of these studies is in question. This recognition motivated scientists to formulate the effect of crowded solutions in general, and excluded volume in particular, on biochemical processes. Using polymers or proteins as crowders, it was shown that while crowding tends to significantly enhance the formation of complexes containing many subunits, dimerizations are only mildly affected. Computer simulations, together with experimental evidence, indicate soft interactions and diffusion as critical factors that operate in a concerted manner with excluded volume to modulate protein binding. Yet, these approaches do not truly mimic the cellular environment. In vivo studies may overcome this shortfall. The few studies conducted thus far suggest that in cells, binding and folding occur at rates close to those determined in dilute solutions. Obtaining quantitative biochemical information on reactions inside living cells is currently a main challenge of the field, as the complexity of the intracellular milieu was what motivated crowding research to begin with.

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 method for computing association rate constants of atomistically represented proteins under macromolecular crowding

In cellular environments, two protein molecules on their way to form a specific complex encounter many bystander macromolecules. The latter molecules, or crowders, affect both the energetics of the interaction between the test molecules and the dynamics of their relative motion. In earlier work (Zhou and Szabo 1991 J. Chem. Phys. 95 5948-52), it has been shown that, in modeling the association kinetics of the test molecules, the presence of crowders can be accounted for by their energetic and dynamic effects. The recent development of the transient-complex theory for protein association in dilute solutions makes it possible to easily incorporate the energetic and dynamic effects of crowders. The transient complex refers to a late on-pathway intermediate, in which the two protein molecules have near-native relative separation and orientation, but have yet to form the many short-range specific interactions of the native complex. The transient-complex theory predicts the association rate constant as k(a) = k(a0)exp(-ΔG*(el)/k(B)T), where k(a0) is the 'basal' rate constant for reaching the transient complex by unbiased diffusion, and the Boltzmann factors captures the influence of long-range electrostatic interactions between the protein molecules. Crowders slow down the diffusion, therefore reducing the basal rate constant (to k(ac0)), and induce an effective interaction energy ΔG(c). We show that the latter interaction energy for atomistic proteins in the presence of spherical crowders is 'long'-ranged, allowing the association rate constant under crowding to be computed as k(ac) = k(ac0)exp[-(ΔG*(el) + ΔG*(c))/k(B)T]. Applications demonstrate that this computational method allows for realistic modeling of protein association kinetics under crowding.

Protein-binding dynamics imaged in a living cell

Historically, rate constants were determined in vitro and it was unknown whether they were valid for in vivo biological processes. Here, we bridge this gap by measuring binding dynamics between a pair of proteins in living HeLa cells. Binding of a β-lactamase to its protein inhibitor was initiated by microinjection and monitored by Förster resonance energy transfer. Association rate constants for the wild-type and an electrostatically optimized mutant were only 25% and 50% lower than in vitro values, whereas no change in the rate constant was observed for a slower binding mutant. These changes are much smaller than might be anticipated considering the high macromolecular crowding within the cell. Single-cell analyses of association rate constants and fluorescence recovery after photobleaching reveals a naturally occurring variation in cell density, which is translated to an up to a twofold effect on binding rate constants. The data show that for this model protein interaction the intracellular environment had only a small effect on the association kinetics, justifying the extrapolation of in vitro data to processes in the cell.

Diffusion-mediated geminate reactions under excluded volume interactions

In this paper, influence of crowding by inert particles on the geminate reaction kinetics is theoretically investigated. Time evolution equations for the survival probability of a geminate pair are derived from the master equation taking into account the correlation among all diffusing particles, and the results are compared with those obtained by Monte Carlo simulations. In general, excluded volume interactions by the inert particles slow down the diffusive motion of reactants. However, when the initial concentration of the inert particles is uniform and high, we show that additional influence of interference between reaction and correlated diffusion accelerates the transient decay of the survival probability in the diffusion-controlled limit. We also study the escape probability for a nonuniform initial distribution of the inert particles by taking the continuous limit in space. We show that reaction yield is increased when the reaction proceeds in the presence of a positive density gradient of the inert particles which inhibits the escape of reactants. The effect can be interpreted as a cage effect.

Studying IDP stability and dynamics by fast relaxation imaging in living cells

Fast relaxation imaging (FReI) temperature-tunes living cells and applies small temperature jumps to them, to monitor biomolecular stability and kinetics in vivo. The folding or aggregation state of a target protein is monitored by Förster resonance energy transfer (FRET). Intrinsically disordered proteins near the structured-unstructured boundary are particularly sensitive to their environment. We describe, using the IDP α-synuclein as an example, how FReI can be used to measure IDP stability and folding inside the cell.

Effects of macromolecular crowding on genetic networks

The intracellular environment is crowded with proteins, DNA, and other macromolecules. Under physiological conditions, macromolecular crowding can alter both molecular diffusion and the equilibria of bimolecular reactions and therefore is likely to have a significant effect on the function of biochemical networks. We propose a simple way to model the effects of macromolecular crowding on biochemical networks via an appropriate scaling of bimolecular association and dissociation rates. We use this approach, in combination with kinetic Monte Carlo simulations, to analyze the effects of crowding on a constitutively expressed gene, a repressed gene, and a model for the bacteriophage λ genetic switch, in the presence and absence of nonspecific binding of transcription factors to genomic DNA. Our results show that the effects of crowding are mainly caused by the shift of association-dissociation equilibria rather than the slowing down of protein diffusion, and that macromolecular crowding can have relevant and counterintuitive effects on biochemical network performance.

Cell-free protein expression under macromolecular crowding conditions

Background

Cell-free protein expression (CFPE) comprised of in vitro transcription and translation is currently manipulated in relatively dilute solutions, in which the macromolecular crowding effects present in living cells are largely ignored. This may not only affect the efficiency of protein synthesis in vitro, but also limit our understanding of the functions and interactions of biomolecules involved in this fundamental biological process.

Methodology/Principal Findings

Using cell-free synthesis of Renilla luciferase in wheat germ extract as a model system, we investigated the CFPE under macromolecular crowding environments emulated with three different crowding agents: PEG-8000, Ficoll-70 and Ficoll-400, which vary in chemical properties and molecular size. We found that transcription was substantially enhanced in the macromolecular crowding solutions; up to 4-fold increase in the mRNA production was detected in the presence of 20% (w/v) of Ficoll-70. In contrast, translation was generally inhibited by the addition of each of the three crowding agents. This might be due to PEG-induced protein precipitation and non-specific binding of translation factors to Ficoll molecules. We further explored a two-stage CFPE in which transcription and translation was carried out under high then low macromolecular crowding conditions, respectively. It produced 2.2-fold higher protein yield than the coupled CFPE control. The macromolecular crowding effects on CFPE were subsequently confirmed by cell-free synthesis of an approximately two-fold larger protein, Firefly luciferase, under macromolecular crowding environments.

Conclusions/Significance

Three macromolecular crowding agents used in this research had opposite effects on transcription and translation. The results of this study should aid researchers in their choice of macromolecular crowding agents and shows that two-stage CFPE is more efficient than coupled CFPE.

Brownian dynamics simulation of protein solutions: structural and dynamical properties

The study of solutions of biomacromolecules provides an important basis for understanding the behavior of many fundamental cellular processes, such as protein folding, self-assembly, biochemical reactions, and signal transduction. Here, we describe a Brownian dynamics simulation procedure and its validation for the study of the dynamic and structural properties of protein solutions. In the model used, the proteins are treated as atomically detailed rigid bodies moving in a continuum solvent. The protein-protein interaction forces are described by the sum of electrostatic interaction, electrostatic desolvation, nonpolar desolvation, and soft-core repulsion terms. The linearized Poisson-Boltzmann equation is solved to compute electrostatic terms. Simulations of homogeneous solutions of three different proteins with varying concentrations, pH, and ionic strength were performed. The results were compared to experimental data and theoretical values in terms of long-time self-diffusion coefficients, second virial coefficients, and structure factors. The results agree with the experimental trends and, in many cases, experimental values are reproduced quantitatively. There are no parameters specific to certain protein types in the interaction model, and hence the model should be applicable to the simulation of the behavior of mixtures of macromolecules in cell-like crowded environments.

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.

Exploring the spatial and temporal organization of a cell's proteome

To increase our current understanding of cellular processes, such as cell signaling and division, knowledge is needed about the spatial and temporal organization of the proteome at different organizational levels. These levels cover a wide range of length and time scales: from the atomic structures of macromolecules for inferring their molecular function, to the quantitative description of their abundance, and spatial distribution in the cell. Emerging new experimental technologies are greatly increasing the availability of such spatial information on the molecular organization in living cells. This review addresses three fields that have significantly contributed to our understanding of the proteome's spatial and temporal organization: first, methods for the structure determination of individual macromolecular assemblies, specifically the fitting of atomic structures into density maps generated from electron microscopy techniques; second, research that visualizes the spatial distributions of these complexes within the cellular context using cryo electron tomography techniques combined with computational image processing; and third, methods for the spatial modeling of the dynamic organization of the proteome, specifically those methods for simulating reaction and diffusion of proteins and complexes in crowded intracellular fluids. The long-term goal is to integrate the varied data about a proteome's organization into a spatially explicit, predictive model of cellular processes.

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.

The influence of hydrodynamic interactions on protein dynamics in confined and crowded spaces-assessment in simple models

We consider several systems that are confined within a softly repulsive sphere. The first one is a model protein, crambin, which is described by a structure-based coarse grained model. We demonstrate that the folding process is accelerated by the hydrodynamic interactions (HI) in a way that depends on the radius of the sphere. The tighter the encompassing sphere, the smaller the effect, independent of the nature of the starting conformations. The second system is a protein surrounded by protein-like softly repulsive spheres that make the confined space crowded. In this case, the HI shorten the folding times in a way which depends on the degree of crowdedness only weakly. The third system is a collection of spheres that are meant to represent molecules. We show that confinement increases association times. We also observe that the HI either facilitate or obstruct association of two spheres depending on the crowding conditions. The dependence of the association time on crowdedness in the confining sphere is qualitatively distinct from that derived by Wieczorek and Zielenkiewicz for a cube with the periodic boundary conditions.

Crowding effects on association reactions at membranes

The effect of macromolecular crowding on the binding of ligands to a receptor near membranes is studied using Brownian dynamics simulations. The receptor is modeled as a reactive patch on a hard surface and the ligands and crowding agents are modeled as spheres that interact via a steep repulsive interaction potential. When a ligand collides with the patch, it reacts with probability p(rxn). The association rate constant (k(infinity)) can be decomposed into contributions from diffusion-limited (k(D)) and reaction-limited (k(R)) rates, i.e., 1/k(infinity) = 1/k(D) + 1/k(R). The simulations show that k(D) is a nonmonotonic function of the volume fraction of crowding agents for receptors of small sizes. k(R) is always an increasing function of the volume fraction of crowding agents, and the association rate constant k(infinity) determined from both contributions has a qualitatively different dependence on the macromolecular crowding for high and low values of the reaction probability p(rxn). The simulation results are used to predict the velocity of the membrane protrusion driven by actin filament elongation. Based on the simple model where the protrusive force on the membrane is generated by the intercalation of actin monomers between the membrane and actin filament ends, we predict that crowding increases the local concentration of actin monomers near the filament ends and hence accelerates the membrane protrusion.

Rate theories for biologists

Some of the rate theories that are most useful for modeling biological processes are reviewed. By delving into some of the details and subtleties in the development of the theories, the review will hopefully help the reader gain a more than superficial perspective. Examples are presented to illustrate how rate theories can be used to generate insight at the microscopic level into biomolecular behaviors. An attempt is made to clear up a number of misconceptions in the literature regarding popular rate theories, including the appearance of Planck's constant in the transition-state theory and the Smoluchowski result as an upper limit for protein-protein and protein-DNA association rate constants. Future work in combining the implementation of rate theories through computer simulations with experimental probes of rate processes, and in modeling effects of intracellular environments so that theories can be used for generating rate constants for systems biology studies is particularly exciting.

Crowding-induced anisotropic transport modulates reaction kinetics in nanoscale porous media

We quantify the emergence of persistent anisotropy in the diffusion of spherical tracer particles through a nanoscale porous medium composed of a uniform distribution of purely symmetric crowding particles. We focus on the interior of a biological cell as an example of such a medium and find that diffusion is highly directional for distances comparable to the size of some organelles. We use a geometrical procedure that avoids the standard orientational averaging to quantify the anisotropy of diffusive paths and show that the point source distributions are predominantly of prolate ellipsoidal shape as a result of local volume exclusion. This geometrical symmetry breaking strongly skews the distribution of kinetic rates of diffusion-limited reactions toward small values, leading to the result that, for short to intermediate times, almost 80% of the rates measured in an ensemble of heterogeneous media are smaller than the expected rate in an ideal homogeneous medium of similar excluded volume fraction. This crowding-induced modulation may have implications for our understanding and measurement of diffusion-controlled intracellular reaction kinetics and for experimental nanotechnology applications, such as nanoparticle-based bioimaging and drug delivery, where diffusion plays an important role.

Macromolecular crowding induces polypeptide compaction and decreases folding cooperativity

A cell's interior is comprised of macromolecules that can occupy up to 40% of its available volume. Such crowded environments can influence the stability of proteins and their rates of reaction. Using discrete molecular dynamics simulations, we investigate how both the size and number of neighboring crowding reagents affect the thermodynamic and folding properties of structurally diverse proteins. We find that crowding induces higher compaction of proteins. We also find that folding becomes less cooperative with the introduction of crowders into the system. The crowders may induce alternative non-native protein conformations, thus creating barriers for protein folding in highly crowded media.

Models of macromolecular crowding effects and the need for quantitative comparisons with experiment

In recent years significant effort has been devoted to exploring the potential effects of macromolecular crowding on protein folding and association phenomena. Theoretical calculations and molecular simulations have, in particular, been exploited to describe aspects of protein behavior in crowded and confined conditions and many aspects of the simulated behavior have reflected, at least at a qualitative level, the behavior observed in experiments. One major and immediate challenge for the theorists is to now produce models capable of making quantitatively accurate predictions of in vitro behavior. A second challenge is to derive models that explain results obtained from experiments performed in vivo, the results of which appear to call into question the assumed dominance of excluded-volume effects in vivo.

Common crowding agents have only a small effect on protein-protein interactions

Studies of protein-protein interactions, carried out in polymer solutions, are designed to mimic the crowded environment inside living cells. It was shown that crowding enhances oligomerization and polymerization of macromolecules. Conversely, we have shown that crowding has only a small effect on the rate of association of protein complexes. Here, we investigated the equilibrium effects of crowding on protein heterodimerization of TEM1-beta-lactamase with beta-lactamase inhibitor protein (BLIP) and barnase with barstar. We also contrasted these with the effect of crowding on the weak binding pair CyPet-YPet. We measured the association and dissociation rates as well as the affinities and thermodynamic parameters of these interactions in polyethylene glycol and dextran solutions. For TEM1-BLIP and for barnase-barstar, only a minor reduction in association rate constants compared to that expected based on solution viscosity was found. Dissociation rate constants showed similar levels of reduction. Overall, this resulted in a binding affinity that is quite similar to that in aqueous solutions. On the other hand, for the CyPet-YPet pair, aggregation, and not enhanced dimerization, was detected in polyethylene glycol solutions. The results suggest that typical crowding agents have only a small effect on specific protein-protein dimerization reactions. Although crowding in the cell results from proteins and other macromolecules, one may still speculate that binding in vivo is not very different from that measured in dilute solutions.