Substrate concentration dependence of the diffusion-controlled steady-state rate constant

Target searches of interacting Brownian particles in dilute systems

We study the target searches of interacting Brownian particles in a finite domain, focusing on the effect of interparticle interactions on the search time. We derive the integral equation for the mean first-passage time and acquire its solution as a series expansion in the orders of the Mayer function. We analytically obtain the leading order correction to the search time for dilute systems, which are most relevant to target search problems and prove a universal relation given by the particle density and the second virial coefficient. Finally, we validate our theoretical prediction by Langevin dynamics simulations for the various types of the interaction potential.

Electrostatic Reaction Inhibition in Nanoparticle Catalysis

Electrostatic reaction inhibition in heterogeneous catalysis emerges if charged reactants and products with similar charges are adsorbed on the catalyst and thus repel the approaching reactants. In this work, we study the effects of electrostatic inhibition on the reaction rate of unimolecular reactions catalyzed on the surface of a spherical model nanoparticle using particle-based reaction-diffusion simulations. Moreover, we derive closed rate equations based on an approximate Debye-Smoluchowski rate theory, valid for diffusion-controlled reactions, and a modified Langmuir adsorption isotherm, relevant for reaction-controlled reactions, to account for electrostatic inhibition in the Debye-Hückel limit. We study the kinetics of reactions ranging from low to high adsorptions on the nanoparticle surface and from the surface- to diffusion-controlled limits for charge valencies 1 and 2. In the diffusion-controlled limit, electrostatic inhibition drastically slows down the reactions for strong adsorption and low ionic concentration, which is well described by our theory. In particular, the rate decreases with adsorption affinity because, in this case, the inhibiting products are generated at a high rate. In the (slow) reaction-controlled limit, the effect of electrostatic inhibition is much weaker, as semiquantitatively reproduced by our electrostatic-modified Langmuir theory. We finally propose and verify a simple interpolation formula that describes electrostatic inhibition for all reaction speeds ("diffusion-influenced" reactions) in general.

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.

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.

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 Crowding Effects on Bimolecular Reaction Rates

An analytical expression for the rate constant of a diffusion-influenced bimolecular reaction in a crowded environment is derived in the framework of a microscopic model that accounts for: (1) the slowdown of diffusion due to crowding and the dependence of the diffusivity on the distance between the reactants, (2) a crowding-induced attractive short-range potential of mean force, and (3) nonspecific reversible binding to the crowders. This expression spans the range from reaction to diffusion control. Crowding can increase the reaction-controlled rate by inducing an effective attraction between reactants but decrease the diffusion-controlled rate by reducing their relative diffusivity.

Concentration effects on the rates of irreversible diffusion-influenced reactions

We formulate a new theory of the effects of like-particle interactions on the irreversible diffusion-influenced bimolecular reactions of the type A + B → P + B by considering the evolution equation of the triplet ABB number density field explicitly. The solution to the evolution equation is aided by a recently proposed method for solving the Fredholm integral equation of the second kind. We evaluate the theory by comparing its predictions with the results of extensive computer simulations. The present theory provides a reasonable explanation of the simulation results.

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.

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.

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.

Theory of molecular crowding in Brownian hard-sphere liquids

We derive an analytical pair potential of mean force for Brownian molecules in the liquid state. Our approach accounts for many-particle correlations of crowding particles of the liquid and for diffusive transport across the spatially modulated local density of crowders in the dense environment. Focusing on the limit of equal-size particles, we show that this diffusive transport leads to additional density- and structure-dependent terms in the interaction potential and to a much stronger attraction (by a factor of ≈4 at average volume fraction of crowders φ{0}=0.25) than in the standard depletion interaction where the diffusive effects are neglected. As an illustration of the theory, we use it to study the size of a polymer chain in a solution of inert crowders. Even in the case of an athermal background solvent, when a classical chain should be fully swollen, we find a sharp coil-globule transition of the ideal chain collapsing at a critical value of the crowder volume fraction φ{c}≈0.145.

Theory of thermally activated ionization and dissociation of bound states

Calculating the microscopic dissociation rate of a bound state, such as a classical diatomic molecule, has been difficult so far. The problem was that standard theories require an energy barrier over which the bound particle (or state) escapes into the preferred low-energy state. This is not the case when the long-range repulsion responsible for the barrier is either absent or screened (as in Cooper pairs, plasmas, or biomolecular complexes). We solve this classical problem by accounting for entropic driving forces at the microscopic level. The theory predicts dissociation rates for arbitrary potentials and is successfully tested on the example of plasma, where it yields an estimate of ionization in the core of the Sun in excellent agreement with experiments. In biology, the new theory accounts for crowding in receptor-ligand kinetics and protein aggregation.

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.

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.

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.

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.

Kinetics of diffusion-controlled enzymatic reactions with charged substrates

The Debye-Hückel limiting law (DHL) has often been used to estimate rate constants of diffusion-controlled reactions under different ionic strengths. Two main approximations are adopted in DHL: one is that the solution of the linearized Poisson-Boltzmann equation for a spherical cavity is used to estimate the excess electrostatic free energy of a solution; the other is that details of electrostatic interactions of the solutes are neglected. This makes DHL applicable only at low ionic strengths and dilute solutions (very low substrate/solute concentrations). We show in this work that through numerical solution of the Poisson-Nernst-Planck equations, diffusion-reaction processes can be studied at a variety of conditions including realistically concentrated solutions, high ionic strength, and certainly with non-equilibrium charge distributions. Reaction rate coefficients for the acetylcholine-acetylcholinesterase system are predicted to strongly depend on both ionic strength and substrate concentration. In particular, they increase considerably with increase of substrate concentrations at a fixed ionic strength, which is open to experimental testing. This phenomenon is also verified on a simple model, and is expected to be general for electrostatically attracting enzyme-substrate systems.PACS Codes: 82.45.Tv, 87.15.VvMSC Codes: 92C30.

Simulation of coagulation, flocculation, and sedimentation

The computer program ZB2 was used to study simulated coagulation, flocculation, and sedimentation. The effect of the initial velocity of soil particles on the values of reaction-rate constants was investigated. In this respect, the results obtained with the program ZB2 corresponded to the theory and realities of coagulation carried out under practical conditions. The effect of coagulant excess/deficiency on the formation of the first floc and on the rate of coagulation of 50% of a soil was also estimated. An increase in simulated coagulant excess caused a decrease in the simulated rate of soil coagulation. In this respect, the results obtained with the program ZB2 did not correspond to the realities of coagulation carried out under practical conditions. An attempt was made to explain this inconsistency referring to the coagulation-flocculation theory. The simulation program ZB2 may provide a basis for developing "local" programs simulating coagulation-flocculation, which can be successfully applied at wastewater treatment plants.

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

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

Toward realistic modeling of dynamic processes in cell signaling: quantification of macromolecular crowding effects

One of the major factors distinguishing molecular processes in vivo from biochemical experiments in vitro is the effect of the environment produced by macromolecular crowding in the cell. To achieve a realistic modeling of processes in the living cell based on biochemical data, it becomes necessary, therefore, to consider such effects. We describe a protocol based on Brownian dynamics simulation to characterize and quantify the effect of various forms of crowding on diffusion and bimolecular association in a simple model of interacting hard spheres. We show that by combining the elastic collision method for hard spheres and the mean field approach for hydrodynamic interaction (HI), our simulations capture the correct dynamics of a monodisperse system. The contributions from excluded volume effect and HI to the crowding effect are thus quantified. The dependence of the results on size distribution of each component in the system is illustrated, and the approach is applied as well to the crowding effect on electrostatic-driven association in both neutral and charged environments; values for effective diffusion constants and association rates are obtained for the specific conditions. The results from our simulation approach can be used to improve the modeling of cell signaling processes without additional computational burdens.

Continuum simulations of acetylcholine consumption by acetylcholinesterase: a Poisson-Nernst-Planck approach

The Poisson-Nernst-Planck (PNP) equation provides a continuum description of electrostatic-driven diffusion and is used here to model the diffusion and reaction of acetylcholine (ACh) with acetylcholinesterase (AChE) enzymes. This study focuses on the effects of ion and substrate concentrations on the reaction rate and rate coefficient. To this end, the PNP equations are numerically solved with a hybrid finite element and boundary element method at a wide range of ion and substrate concentrations, and the results are compared with the partially coupled Smoluchowski-Poisson-Boltzmann model. The reaction rate is found to depend strongly on the concentrations of both the substrate and ions; this is explained by the competition between the intersubstrate repulsion and the ionic screening effects. The reaction rate coefficient is independent of the substrate concentration only at very high ion concentrations, whereas at low ion concentrations the behavior of the rate depends strongly on the substrate concentration. Moreover, at physiological ion concentrations, variations in substrate concentration significantly affect the transient behavior of the reaction. Our results offer a reliable estimate of reaction rates at various conditions and imply that the concentrations of charged substrates must be coupled with the electrostatic computation to provide a more realistic description of neurotransmission and other electrodiffusion and reaction processes.