Kinetics of reversible diffusion influenced reactions: The self-consistent relaxation time approximation

Kinetics of diffusion-influenced multisite phosphorylation with enzyme reactivation

The simplest way to account for the influence of diffusion on the kinetics of multisite phosphorylation is to modify the rate constants in the conventional rate equations of chemical kinetics. We have previously shown that this is not enough and new transitions between the reactants must also be introduced. Here we extend our results by considering enzymes that are inactive after modifying the substrate and need time to become active again. This generalization leads to a surprising result. The introduction of enzyme reactivation results in a diffusion-modified kinetic scheme with a new transition that has a negative rate constant. The reason for this is that mapping non-Markovian rate equations onto Markovian ones with time-independent rate constants is not a good approximation at short times. We then developed a non-Markovian theory that involves memory kernels instead of rate constants. This theory is now valid at short times, but is more challenging to use. As an example, the diffusion-modified kinetic scheme with new connections was used to calculate kinetics of double phosphorylation and steady-state response in a phosphorylation-dephosphorylation cycle. We have reproduced the loss of bistability in the phosphorylation-dephosphorylation cycle when the enzyme reactivation time decreases, which was obtained by particle-based computer simulations.

Multiscale molecular kinetics by coupling Markov state models and reaction-diffusion dynamics

A novel approach to simulate simple protein-ligand systems at large time and length scales is to couple Markov state models (MSMs) of molecular kinetics with particle-based reaction-diffusion (RD) simulations, MSM/RD. Currently, MSM/RD lacks a mathematical framework to derive coupling schemes, is limited to isotropic ligands in a single conformational state, and lacks multiparticle extensions. In this work, we address these needs by developing a general MSM/RD framework by coarse-graining molecular dynamics into hybrid switching diffusion processes. Given enough data to parameterize the model, it is capable of modeling protein-protein interactions over large time and length scales, and it can be extended to handle multiple molecules. We derive the MSM/RD framework, and we implement and verify it for two protein-protein benchmark systems and one multiparticle implementation to model the formation of pentameric ring molecules. To enable reproducibility, we have published our code in the MSM/RD software package.

Cluster Channeling in Cascade Reactions

Enzymatic cascade reactions, where a substrate is converted into a product in several steps, play a critical role in many biological systems. The enzymes in such reactions are often clustered inside intracellular compartments. To understand the effect of localization, we develop a theory for cascade reactions converting substrates into intermediates and then into products when the enzymes are localized in clusters. The theory shows that the kinetic scheme that describes the reaction with dispersed enzymes changes as a result of clustering. A new reaction channel, in which the substrate is directly converted into product, appears with a diffusion-influenced rate that is expressed in terms of enzyme catalytic efficiencies, diffusion coefficient, and cluster size. This rate is proportional to the cluster channeling probability, which is the probability that an intermediate is converted into product within the cluster in which the intermediate was formed. Simple analytic formulas allow one to quantify how enzyme clustering can affect product formation and regulate the direction of metabolic reaction flux in biological and synthetic systems. The rate of the substrate conversion decreases whereas the cluster channeling probability increases as the number of enzyme molecules in a cluster increases. The interplay between these factors leads to an optimal number of enzyme molecules that maximizes the clustering efficiency.

Quantifying the roles of space and stochasticity in computer simulations for cell biology and cellular biochemistry

Most of the fascinating phenomena studied in cell biology emerge from interactions among highly organized multimolecular structures embedded into complex and frequently dynamic cellular morphologies. For the exploration of such systems, computer simulation has proved to be an invaluable tool, and many researchers in this field have developed sophisticated computational models for application to specific cell biological questions. However, it is often difficult to reconcile conflicting computational results that use different approaches to describe the same phenomenon. To address this issue systematically, we have defined a series of computational test cases ranging from very simple to moderately complex, varying key features of dimensionality, reaction type, reaction speed, crowding, and cell size. We then quantified how explicit spatial and/or stochastic implementations alter outcomes, even when all methods use the same reaction network, rates, and concentrations. For simple cases, we generally find minor differences in solutions of the same problem. However, we observe increasing discordance as the effects of localization, dimensionality reduction, and irreversible enzymatic reactions are combined. We discuss the strengths and limitations of commonly used computational approaches for exploring cell biological questions and provide a framework for decision making by researchers developing new models. As computational power and speed continue to increase at a remarkable rate, the dream of a fully comprehensive computational model of a living cell may be drawing closer to reality, but our analysis demonstrates that it will be crucial to evaluate the accuracy of such models critically and systematically.

Multisite reversible association in membranes and solutions: From non-Markovian to Markovian kinetics

The role of diffusion on the kinetics of reversible association to a macromolecule with two inequivalent sites is studied. Previously, we found that, in the simplest possible description, it is not sufficient to just renormalize the rate constants of chemical kinetics, but one must introduce direct transitions between the bound states in the kinetic scheme. The physical reason for this is that a molecule that just dissociated from one site can directly rebind to the other rather than diffuse away into the bulk. Such a simple description is not valid in two dimensions because reactants can never diffuse away into the bulk. In this work, we consider a variety of more sophisticated implementations of our recent general theory that are valid in both two and three dimensions. We compare the predicted time dependence of the concentrations for a wide range of parameters and establish the range of validity of various levels of the general theory.

Diffusion-induced competitive two-site binding

The influence of diffusion on the kinetics of ligand binding to a macromolecule with two sites is considered for a simple model where, in the reaction-controlled limit, there is no cooperativity and hence the sites are independent. By applying our recently developed formalism to describe a network of coupled diffusion-influenced reactions, we show that the rate constants of chemical kinetics cannot just be renormalized. Rather a new reaction channel, which connects the two singly occupied states, must be introduced. The rate constants of this new channel depend on the committor or capture probability that a ligand that just dissociated from one site rebinds to the other. This result is rederived in an elementary way using the encounter complex model. Illustrative calculations are presented where the kinetics of the fractional saturation of one site is compared with that of a macromolecule that has only this site. If all sites are initially empty, then the second site slows down binding to the first due to competition between the sites. On the other hand, if the second site is initially occupied, the binding of the first site speeds up because of the direct diffusion-induced transitions between the two singly bound states.

Theory of Diffusion-Influenced Reaction Networks

A formalism is developed to describe how diffusion alters the kinetics of coupled reversible association-dissociation reactions in the presence of conformational changes that can modify the reactivity. The major difficulty in constructing a general theory is that, even to the lowest order, diffusion can change the structure of the rate equations of chemical kinetics by introducing new reaction channels (i.e., modifies the kinetic scheme). Therefore, the right formalism must be found that allows the influence of diffusion to be described in a concise and elegant way for networks of arbitrary complexity. Our key result is a set of non-Markovian rate equations involving stoichiometric matrices and net reaction rates (fluxes), in which these rates are coupled by a time-dependent pair association flux matrix, whose elements have a simple physical interpretation. Specifically, each element is the probability density that an isolated pair of reactants irreversibly associates at time t via one reaction channel on the condition that it started out with the dissociation products of another (or the same) channel. In the Markovian limit, the coupling of the chemical rates is described by committors (or splitting/capture probabilities). The committor is the probability that an isolated pair of reactants formed by dissociation at one site will irreversibly associate at another site rather than diffuse apart. We illustrate the use of our formalism by considering three reversible reaction schemes: (1) binding to a single site, (2) binding to two inequivalent sites, and (3) binding to a site whose reactivity fluctuates. In the first example, we recover the results published earlier, while in the second one we show that a new reaction channel appears, which directly connects the two bound states. The third example is particularly interesting because all species become coupled and an exchange-type bimolecular reaction appears. In the Markovian limit, some of the diffusion-modified rate constants that describe new transitions become negative, indicating that memory effects cannot be ignored.

Grand canonical diffusion-influenced reactions: A stochastic theory with applications to multiscale reaction-diffusion simulations

Smoluchowski-type models for diffusion-influenced reactions (A + B → C) can be formulated within two frameworks: the probabilistic-based approach for a pair A, B of reacting particles and the concentration-based approach for systems in contact with a bath that generates a concentration gradient of B particles that interact with A. Although these two approaches are mathematically similar, it is not straightforward to establish a precise mathematical relationship between them. Determining this relationship is essential to derive particle-based numerical methods that are quantitatively consistent with bulk concentration dynamics. In this work, we determine the relationship between the two approaches by introducing the grand canonical Smoluchowski master equation (GC-SME), which consists of a continuous-time Markov chain that models an arbitrary number of B particles, each one of them following Smoluchowski's probabilistic dynamics. We show that the GC-SME recovers the concentration-based approach by taking either the hydrodynamic or the large copy number limit. In addition, we show that the GC-SME provides a clear statistical mechanical interpretation of the concentration-based approach and yields an emergent chemical potential for nonequilibrium spatially inhomogeneous reaction processes. We further exploit the GC-SME robust framework to accurately derive multiscale/hybrid numerical methods that couple particle-based reaction-diffusion simulations with bulk concentration descriptions, as described in detail through two computational implementations.

Efficient reactive Brownian dynamics

We develop a Split Reactive Brownian Dynamics (SRBD) algorithm for particle simulations of reaction-diffusion systems based on the Doi or volume reactivity model, in which pairs of particles react with a specified Poisson rate if they are closer than a chosen reactive distance. In our Doi model, we ensure that the microscopic reaction rules for various association and dissociation reactions are consistent with detailed balance (time reversibility) at thermodynamic equilibrium. The SRBD algorithm uses Strang splitting in time to separate reaction and diffusion and solves both the diffusion-only and reaction-only subproblems exactly, even at high packing densities. To efficiently process reactions without uncontrolled approximations, SRBD employs an event-driven algorithm that processes reactions in a time-ordered sequence over the duration of the time step. A grid of cells with size larger than all of the reactive distances is used to schedule and process the reactions, but unlike traditional grid-based methods such as reaction-diffusion master equation algorithms, the results of SRBD are statistically independent of the size of the grid used to accelerate the processing of reactions. We use the SRBD algorithm to compute the effective macroscopic reaction rate for both reaction-limited and diffusion-limited irreversible association in three dimensions and compare to existing theoretical predictions at low and moderate densities. We also study long-time tails in the time correlation functions for reversible association at thermodynamic equilibrium and compare to recent theoretical predictions. Finally, we compare different particle and continuum methods on a model exhibiting a Turing-like instability and pattern formation. Our studies reinforce the common finding that microscopic mechanisms and correlations matter for diffusion-limited systems, making continuum and even mesoscopic modeling of such systems difficult or impossible. We also find that for models in which particles diffuse off lattice, such as the Doi model, reactions lead to a spurious enhancement of the effective diffusion coefficients.

New exact solutions of the reaction–diffusion equation with variable coefficients via the mathematical computation

In this paper, we construct new exact solutions of the reaction–diffusion equation with time dependent variable coefficients by employing the mathematical computation via the Painlevé test. We describe the behaviors and their interactions of the obtained solutions under certain constraints and various variable coefficients.

On the generality of Michaelian kinetics

The reversible Michaelis-Menten equation is shown to follow from a very broad class of steady-state kinetic models involving enzymes that adopt a unique free (i.e., not complexed to substrate/product) state in solution. In the case of enzymes with multiple free states/conformations (e.g., fluctuating, hysteretic, or co-operative monomeric enzymes), Michaelian behavior is still assured if the relative steady-state populations of free enzyme states are independent of substrate and product concentration. Prior models for Michaelian behavior in multiple conformer enzymes are shown to be special cases of this single condition.

Effective reaction rates for diffusion-limited reaction cycles

Biological signals in cells are transmitted with the use of reaction cycles, such as the phosphorylation-dephosphorylation cycle, in which substrate is modified by antagonistic enzymes. An appreciable share of such reactions takes place in crowded environments of two-dimensional structures, such as plasma membrane or intracellular membranes, and is expected to be diffusion-controlled. In this work, starting from the microscopic bimolecular reaction rate constants and using estimates of the mean first-passage time for an enzyme-substrate encounter, we derive diffusion-dependent effective macroscopic reaction rate coefficients (EMRRC) for a generic reaction cycle. Each EMRRC was found to be half of the harmonic average of the microscopic rate constant (phosphorylation c or dephosphorylation d), and the effective (crowding-dependent) motility divided by a slowly decreasing logarithmic function of the sum of the enzyme concentrations. This implies that when c and d differ, the two EMRRCs scale differently with the motility, rendering the steady-state fraction of phosphorylated substrate molecules diffusion-dependent. Analytical predictions are verified using kinetic Monte Carlo simulations on the two-dimensional triangular lattice at the single-molecule resolution. It is demonstrated that the proposed formulas estimate the steady-state concentrations and effective reaction rates for different sets of microscopic reaction rates and concentrations of reactants, including a non-trivial example where with increasing diffusivity the fraction of phosphorylated substrate molecules changes from 10% to 90%.

Reversible Stochastically Gated Diffusion-Influenced Reactions

An approximate but accurate theory is developed for the kinetics of reversible binding of a ligand to a macromolecule when either can stochastically fluctuate between reactive and unreactive conformations. The theory is based on a set of reaction-diffusion equations for the deviations of the pair distributions from their bulk values. The concentrations are shown to satisfy non-Markovian rate equations with memory kernels that are obtained by solving an irreversible geminate (i.e., two-particle) problem. The relaxation to equilibrium is not exponential but rather a power law. In the Markovian limit, the theory reduces to a set of ordinary rate equations with renormalized rate constants.

Theory of bi-molecular association dynamics in 2D for accurate model and experimental parameterization of binding rates

The dynamics of association between diffusing and reacting molecular species are routinely quantified using simple rate-equation kinetics that assume both well-mixed concentrations of species and a single rate constant for parameterizing the binding rate. In two-dimensions (2D), however, even when systems are well-mixed, the assumption of a single characteristic rate constant for describing association is not generally accurate, due to the properties of diffusional searching in dimensions d ≤ 2. Establishing rigorous bounds for discriminating between 2D reactive systems that will be accurately described by rate equations with a single rate constant, and those that will not, is critical for both modeling and experimentally parameterizing binding reactions restricted to surfaces such as cellular membranes. We show here that in regimes of intrinsic reaction rate (ka) and diffusion (D) parameters ka/D > 0.05, a single rate constant cannot be fit to the dynamics of concentrations of associating species independently of the initial conditions. Instead, a more sophisticated multi-parametric description than rate-equations is necessary to robustly characterize bimolecular reactions from experiment. Our quantitative bounds derive from our new analysis of 2D rate-behavior predicted from Smoluchowski theory. Using a recently developed single particle reaction-diffusion algorithm we extend here to 2D, we are able to test and validate the predictions of Smoluchowski theory and several other theories of reversible reaction dynamics in 2D for the first time. Finally, our results also mean that simulations of reactive systems in 2D using rate equations must be undertaken with caution when reactions have ka/D > 0.05, regardless of the simulation volume. We introduce here a simple formula for an adaptive concentration dependent rate constant for these chemical kinetics simulations which improves on existing formulas to better capture non-equilibrium reaction dynamics from dilute to dense systems.

The influence of the "cage effect" on the mechanism of reversible bimolecular multistage chemical reactions in solutions

Manifestations of the "cage effect" at the encounters of reactants are theoretically treated by the example of multistage reactions in liquid solutions including bimolecular exchange reactions as elementary stages. It is shown that consistent consideration of quasi-stationary kinetics of multistage reactions (possible only in the framework of the encounter theory) for reactions proceeding near reactants contact can be made on the basis of the concepts of a "cage complex." Though mathematically such a consideration is more complicated, it is more clear from the standpoint of chemical notions. It is established that the presence of the "cage effect" leads to some important effects not inherent in reactions in gases or those in solutions proceeding in the kinetic regime, such as the appearance of new transition channels of reactant transformation that cannot be caused by elementary event of chemical conversion for the given mechanism of reaction. This results in that, for example, rate constant values of multistage reaction defined by standard kinetic equations of formal chemical kinetics from experimentally measured kinetics can differ essentially from real values of these constants.

Influence of diffusion on the kinetics of multisite phosphorylation

When an enzyme modifies multiple sites on a substrate, the influence of the relative diffusive motion of the reactants cannot be described by simply altering the rate constants in the rate equations of chemical kinetics. We have recently shown that, even as a first approximation, new transitions between the appropriate species must also be introduced. The physical reason for this is that a kinase, after phosphorylating one site, can rebind and modify another site instead of diffusing away. The corresponding new rate constants depend on the capture or rebinding probabilities that an enzyme-substrate pair, which is formed after dissociation from one site, reacts at the other site rather than diffusing apart. Here we generalize our previous work to describe both random and sequential phosphorylation by considering inequivalent modification sites. In addition, anisotropic reactive sites (instead of uniformly reactive spheres) are explicitly treated by using localized sink and source terms in the reaction-diffusion equations for the enzyme-substrate pair distribution function. Finally, we show that our results can be rederived using a phenomenological approach based on introducing transient encounter complexes into the standard kinetic scheme and then eliminating them using the steady-state approximation.

The Berg-Purcell limit revisited

Biological systems often have to measure extremely low concentrations of chemicals with high precision. When dealing with such small numbers of molecules, the inevitable randomness of physical transport processes and binding reactions will limit the precision with which measurements can be made. An important question is what the lower bound on the noise would be in such measurements. Using the theory of diffusion-influenced reactions, we derive an analytical expression for the precision of concentration estimates that are obtained by monitoring the state of a receptor to which a diffusing ligand can bind. The variance in the estimate consists of two terms, one resulting from the intrinsic binding kinetics and the other from the diffusive arrival of ligand at the receptor. The latter term is identical to the fundamental limit derived by Berg and Purcell (Biophys. J., 1977), but disagrees with a more recent expression by Bialek and Setayeshgar. Comparing the theoretical predictions against results from particle-based simulations confirms the accuracy of the resulting expression and reaffirms the fundamental limit established by Berg and Purcell.

Free-Propagator Reweighting Integrator for Single-Particle Dynamics in Reaction-Diffusion Models of Heterogeneous Protein-Protein Interaction Systems

We present a new algorithm for simulating reaction-diffusion equations at single-particle resolution. Our algorithm is designed to be both accurate and simple to implement, and to be applicable to large and heterogeneous systems, including those arising in systems biology applications. We combine the use of the exact Green's function for a pair of reacting particles with the approximate free-diffusion propagator for position updates to particles. Trajectory reweighting in our free-propagator reweighting (FPR) method recovers the exact association rates for a pair of interacting particles at all times. FPR simulations of many-body systems accurately reproduce the theoretically known dynamic behavior for a variety of different reaction types. FPR does not suffer from the loss of efficiency common to other path-reweighting schemes, first, because corrections apply only in the immediate vicinity of reacting particles and, second, because by construction the average weight factor equals one upon leaving this reaction zone. FPR applications include the modeling of pathways and networks of protein-driven processes where reaction rates can vary widely and thousands of proteins may participate in the formation of large assemblies. With a limited amount of bookkeeping necessary to ensure proper association rates for each reactant pair, FPR can account for changes to reaction rates or diffusion constants as a result of reaction events. Importantly, FPR can also be extended to physical descriptions of protein interactions with long-range forces, as we demonstrate here for Coulombic interactions.

Effect of ligand diffusion on occupancy fluctuations of cell-surface receptors

The role of diffusion in the kinetics of reversible ligand binding to receptors on a cell surface or to a macromolecule with multiple binding sites is considered. A formalism is developed that is based on a Markovian master equation for the distribution function of the number of occupied receptors containing rate constants that depend on the ligand diffusivity. The formalism is used to derive (1) a nonlinear rate equation for the mean number of occupied receptors and (2) an analytical expression for the relaxation time that characterizes the decay of equilibrium fluctuations of the occupancy of the receptors. The relaxation time is shown to depend on the ligand diffusivity and concentration, the number of receptors, the cell radius, and intrinsic association/dissociation rate constants. This result is then used to estimate the accuracy of the ligand concentration measurements by the cell, which, according to the Berg-Purcell model, is related to fluctuations in the receptor occupancy, averaged over a finite interval of time. Specifically, a simple expression (which is exact in the framework of our formalism) is derived for the variance in the measured ligand concentration in the limit of long averaging times.

Diffusion modifies the connectivity of kinetic schemes for multisite binding and catalysis

The simplest way to describe the influence of the relative diffusion of the reactants on the time course of bimolecular reactions is to modify or renormalize the phenomenological rate constants that enter into the rate equations of conventional chemical kinetics. However, for macromolecules with multiple inequivalent reactive sites, this is no longer sufficient, even in the low concentration limit. The physical reason is that an enzyme (or a ligand) that has just modified (or dissociated from) one site can bind to a neighboring site rather than diffuse away. This process is not described by the conventional chemical kinetics, which is only valid in the limit that diffusion is fast compared with reaction. Using an exactly solvable many-particle reaction-diffusion model, we show that the influence of diffusion on the kinetics of multisite binding and catalysis can be accounted for by not only scaling the rates, but also by introducing new connections into the kinetic scheme. The rate constants that describe these new transitions or reaction channels turn out to have a transparent physical interpretation: The chemical rates are scaled by the appropriate probabilities that a pair of reactants, which are initially in contact, bind rather than diffuse apart. The theory is illustrated by application to phosphorylation of a multisite substrate.

Theoretical modeling of masking DNA application in aptamer-facilitated biomarker discovery

In aptamer-facilitated biomarker discovery (AptaBiD), aptamers are selected from a library of random DNA (or RNA) sequences for their ability to specifically bind cell-surface biomarkers. The library is incubated with intact cells, and cell-bound DNA molecules are separated from those unbound and amplified by the polymerase chain reaction (PCR). The partitioning/amplification cycle is repeated multiple times while alternating target cells and control cells. Efficient aptamer selection in AptaBiD relies on the inclusion of masking DNA within the cell and library mixture. Masking DNA lacks primer regions for PCR amplification and is typically taken in excess to the library. The role of masking DNA within the selection mixture is to outcompete any nonspecific binding sequences within the initial library, thus allowing specific DNA sequences (i.e., aptamers) to be selected more efficiently. Efficient AptaBiD requires an optimum ratio of masking DNA to library DNA, at which aptamers still bind specific binding sites but nonaptamers within the library do not bind nonspecific binding sites. Here, we have developed a mathematical model that describes the binding processes taking place within the equilibrium mixture of masking DNA, library DNA, and target cells. An obtained mathematical solution allows one to estimate the concentration of masking DNA that is required to outcompete the library DNA at a desirable ratio of bound masking DNA to bound library DNA. The required concentration depends on concentrations of the library and cells as well as on unknown cell characteristics. These characteristics include the concentration of total binding sites on the cell surface, N, and equilibrium dissociation constants, K(nsL) and K(nsM), for nonspecific binding of the library DNA and masking DNA, respectively. We developed a theory that allows the determination of N, K(nsL), and K(nsM) based on measurements of EC50 values for cells mixed separately with the library and masking DNA (EC50 is the concentration of fluorescently labeled DNA at which half of the maximum fluorescence signal from DNA-bound cells is reached). We also obtained expressions for signals from bound DNA (measured by flow cytometry) in terms of N, K(nsL), and K(nsM). These expressions can be used for the verification of N, K(nsL), and K(nsM) values found from EC50 measurements. The developed procedure was applied to MCF-7 breast cancer cells, and corresponding values of N, K(nsL), and K(nsM) were established for the first time. The concentration of masking DNA required for AptaBiD with MCF-7 breast cancer cells was also estimated.

Spatial simulations in systems biology: from molecules to cells

Cells are highly organized objects containing millions of molecules. Each biomolecule has a specific shape in order to interact with others in the complex machinery. Spatial dynamics emerge in this system on length and time scales which can not yet be modeled with full atomic detail. This review gives an overview of methods which can be used to simulate the complete cell at least with molecular detail, especially Brownian dynamics simulations. Such simulations require correct implementation of the diffusion-controlled reaction scheme occurring on this level. Implementations and applications of spatial simulations are presented, and finally it is discussed how the atomic level can be included for instance in multi-scale simulation methods.

Role of diffusion in the kinetics of reversible enzyme-catalyzed reactions

The accurate expression for the steady-state velocity of an irreversible enzyme-catalyzed reaction obtained by Shin and co-workers is generalized to allow for the rebinding of the product. The amplitude of the power-law (t(-1/2)) relaxation of the free- and bound-enzyme concentrations to steady-state values is expressed in terms of the steady-state velocity and the intrinsic (chemical) rate constants. This result is conjectured to be exact, even though our expression for the steady-state velocity in terms of microscopic parameters is only approximate.

Diffusion-influenced ligand binding to buried sites in macromolecules and transmembrane channels

We consider diffusion-influenced binding to a buried binding site that is connected to the surface by a narrow tunnel. Under the single assumption of an equilibrium distribution of ligands over the tunnel cross section, we reduce the calculation of the time-dependent rate coefficient to the solution of a one-dimensional diffusion equation with appropriate boundary conditions. We obtain a simple analytical expression for the steady-state rate that depends on the potential of mean force in the tunnel and the diffusion-controlled rate of binding to the tunnel entrance. Potential applications of our theory include substrate binding to a buried active site of an enzyme and permeant ion binding to an internal site in a transmembrane channel.

A solvable model for the diffusion and reaction of neurotransmitters in a synaptic junction

BACKGROUND

The diffusion and reaction of the transmitter acetylcholine in neuromuscular junctions and the diffusion and binding of Ca2+ in the dyadic clefts of ventricular myocytes have been extensively modeled by Monte Carlo simulations and by finite-difference and finite-element solutions. However, an analytical solution that can serve as a benchmark for testing these numerical methods has been lacking.

RESULT

Here we present an analytical solution to a model for the diffusion and reaction of acetylcholine in a neuromuscular junction and for the diffusion and binding of Ca2+ in a dyadic cleft. Our model is similar to those previously solved numerically and our results are also qualitatively similar.

CONCLUSION

The analytical solution provides a unique benchmark for testing numerical methods and potentially provides a new avenue for modeling biochemical transport.

Manifestation of macroscopic correlations in elementary reaction kinetics. II. Irreversible reaction A+B→C

The applicability of the Encounter Theory (ET) (the prototype of the Collision Theory) concepts for widely occurring diffusion assisted irreversible bulk reaction A+B→C (for example, radical reaction) in dilute solutions with arbitrary ratio of initial concentrations of reactants has been treated theoretically with modern many-particle method for the derivation of non-Markovian binary kinetic equations. The method shows that, just as in the reaction A+A→C considered earlier, the agreement with the Encounter Theory is observed when the familiar Integral Encounter Theory is used which is just a step in the derivation of kinetic equations in the framework of the method employed. It allows for two-particle correlations only, and fails to consider the correlation of reactant simultaneously with a partner and with a reactant in the bulk. However, the next step leading to the Modified Encounter Theory under reduction of equations to a regular form both extends the time applicability interval of ET homogeneous rate equation (as for reactions proceeding in excess of one of the reactants), and yields the inhomogeneous equation of the Generalized Encounter Theory (GET) that reveals macroscopic correlations induced by the encounters in a reservoir of free walks in full agreement with physical considerations. This means that the encounters of reactants in solution are correlated at rather large time interval of the reaction course. However, unlike the reaction A+A→C of identical reactants, the reaction A+B→C accumulation of the above macroscopic correlations (even with the initial concentrations of reactants being equal) proceeds much slower. Another distinction is that for the reaction A+A→C the long-term behavior of ET and GET kinetics is the same, while in the reaction A+B→C these kinetics behave differently. It is of interest that just taking account of the above macroscopic correlations in the reaction A+B→C (in GET) results in the universal character of the long-term behavior of the kinetics for the case of equal initial concentrations of reactants and that where one of the reactants is in excess. This is more natural from the point of view of the reaction course on the encounters of reactants in solutions.

Manifestation of macroscopic correlations in elementary reaction kinetics. I. Irreversible reaction A+A-->product

Using an modern many-particle method for the derivation of non-Markovian binary kinetic equations, we have treated theoretically the applicability of the encounter theory (ET) (the prototype of the collision theory) concepts to the widely known diffusion assisted irreversible bulk reaction A+A-->product (for example, radical reaction) in dilute solutions. The method shows that the agreement with the ET is observed when the familiar integral ET is employed which in this method is just a step in the derivation of kinetic equations. It allows for two-particle correlations only, but fails to take account of correlation of reactant simultaneously with the partner of the encounter and the reactant in the bulk. However, the next step leading to the modified ET under transformation of equations to the regular form both extends the time range of the applicability of ET rate equation (as it was for reactions proceeding with one of the reactants in excess), and gives the equation of the generalized ET. In full agreement with physical considerations, this theory reveals macroscopic correlations induced by the encounters in the reservoir of free walks. This means that the encounters of reactants in solution are correlated on a rather large time interval of the reaction. Though any nonstationary (non-Markovian) effects manifest themselves rather weakly in the kinetics of the bimolecular reaction in question, just the existence of the revealed macroscopic correlations in the binary theory is of primary importance. In particular, it means that the well-known phenomena which are generally considered to be associated solely with correlation of particles on the encounter (for example, chemically induced dynamic nuclear polarization) may be induced by correlation in the reservoir of free random walks of radicals in solution.

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.

A diffusional bimolecular propensity function

We derive an explicit formula for the propensity function (stochastic reaction rate) of a generic bimolecular chemical reaction in which the reactant molecules move about by diffusion, as solute molecules in a bath of much smaller and more numerous solvent molecules. Our derivation assumes that the solution is macroscopically well stirred and dilute in the solute molecules. It effectively extends the physical rationale for the chemical master equation and the stochastic simulation algorithm from well-stirred dilute gases to well-stirred dilute solutions, with the former becoming a limiting case of the latter. This extension is important for cellular systems, where the solvent molecules are typically water and the solute (reactant) molecules are much larger organic structures, whose relatively low populations often require a discrete-stochastic formalism. In the course of our derivation, we illuminate some limitations on the ability of the classical diffusion equation to accurately describe how a diffusing molecule moves on spatial and temporal scales that are relevant to collision-induced chemical reactions.

Contact and Distant Luminescence Quenching in Solutions

The limitations and advantages of modern encounter theories of remote transfer are discussed, as well as their application to particular transfer reactions assisted by encounter diffusion. Comparison is made with contact multiparticle theories, Brownian dynamic simulations, and the actual experimental data requiring a distant description of energy and/or electron transfer.

Excited-state reversible geminate recombination in two dimensions

Excited-state reversible geminate recombination with two different lifetimes and quenching is investigated in two dimensions. From the exact Green function in the Laplace domain, analytic expressions of two-dimensional survival and binding probabilities are obtained at short and long times. We find that a new pattern of kinetic transition occurs in two dimensions. The long-time effective survival probabilities show a pattern of (ln t)(-1)-->constant-->e(t) depending on the rate constants while the effective binding probabilities show t(-1)(ln t)(-2)-->t(-1)-->e(t).

Reversible geminate recombination of hydrogen-bonded water molecule pair

The (history independent) autocorrelation function for a hydrogen-bonded water molecule pair, calculated from classical molecular dynamics trajectories of liquid water, exhibits a t(-3/2) asymptotic tail. Its whole time dependence agrees quantitatively with the solution for reversible diffusion-influenced geminate recombination derived by Agmon and Weiss [J. Chem. Phys. 91, 6937 (1989)]. Agreement with diffusion theory is independent of the precise definition of the bound state. Given the water self-diffusion constant, this theory enables us to determine the dissociation and bimolecular recombination rate parameters for a water dimer. (The theory is indispensable for obtaining the bimolecular rate coefficient.) Interestingly, the activation energies obtained from the temperature dependence of these rate coefficients are similar, rather than differing by the hydrogen-bond (HB) strength. This suggests that recombination requires displacing another water molecule, which meanwhile occupied the binding site. Because these activation energies are about twice the HB strength, cleavage of two HBs may be required to allow pair separation. The autocorrelation function without the HB angular restriction yields a recombination rate coefficient that is larger than that for rebinding to all four tetrahedral water sites (with angular restrictions), suggesting the additional participation of interstitial sites. Following dissociation, the probability of the pair to be unbound but within the reaction sphere rises more slowly than expected, possibly because binding to the interstitial sites delays pair separation. An extended diffusion model, which includes an additional binding site, can account for this behavior.