Three-dimensional simulations of reversible bimolecular reactions: The simple target problem

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.

Surface reaction-diffusion kinetics on lattice at the microscopic scale

Microscopic models of reaction-diffusion processes on the cell membrane can link local spatiotemporal effects to macroscopic self-organized patterns often observed on the membrane. Simulation schemes based on the microscopic lattice method (MLM) can model these processes at the microscopic scale by tracking individual molecules, represented as hard spheres, on fine lattice voxels. Although MLM is simple to implement and is generally less computationally demanding than off-lattice approaches, its accuracy and consistency in modeling surface reactions have not been fully verified. Using the Spatiocyte scheme, we study the accuracy of MLM in diffusion-influenced surface reactions. We derive the lattice-based bimolecular association rates for two-dimensional (2D) surface-surface reaction and one-dimensional (1D) volume-surface adsorption according to the Smoluchowski-Collins-Kimball model and random walk theory. We match the time-dependent rates on lattice with off-lattice counterparts to obtain the correct expressions for MLM parameters in terms of physical constants. The expressions indicate that the voxel size needs to be at least 0.6% larger than the molecule to accurately simulate surface reactions on triangular lattice. On square lattice, the minimum voxel size should be even larger, at 5%. We also demonstrate the ability of MLM-based schemes such as Spatiocyte to simulate a reaction-diffusion model that involves all dimensions: three-dimensional (3D) diffusion in the cytoplasm, 2D diffusion on the cell membrane, and 1D cytoplasm-membrane adsorption. With the model, we examine the contribution of the 2D reaction pathway to the overall reaction rate at different reactant diffusivity, reactivity, and concentrations.

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.

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.

Simulation of the radiolysis of water using Green's functions of the diffusion equation

Radiation chemistry is of fundamental importance in the understanding of the effects of ionising radiation, notably with regard to DNA damage by indirect effect (e.g. damage by ·OH radicals created by the radiolysis of water). In the recent years, Green's functions of the diffusion equation (GFDEs) have been used extensively in biochemistry, notably to simulate biochemical networks in time and space. In the present work, an approach based on the GFDE will be used to refine existing models on the indirect effect of ionising radiation on DNA. As a starting point, the code RITRACKS (relativistic ion tracks) will be used to simulate the radiation track structure and calculate the position of all radiolytic species formed during irradiation. The chemical reactions between these radiolytic species and with DNA will be done by using an efficient Monte Carlo sampling algorithm for the GFDE of reversible reactions with an intermediate state that has been developed recently. These simulations should help the understanding of the contribution of the indirect effect in the formation of DNA damage, particularly with regards to the formation of double-strand breaks.

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.

A computational kinetic model of diffusion for molecular systems

Regulation of biomolecular transport in cells involves intra-protein steps like gating and passage through channels, but these steps are preceded by extra-protein steps, namely, diffusive approach and admittance of solutes. The extra-protein steps develop over a 10-100 nm length scale typically in a highly particular environment, characterized through the protein's geometry, surrounding electrostatic field, and location. In order to account for solute energetics and mobility of solutes in this environment at a relevant resolution, we propose a particle-based kinetic model of diffusion based on a Markov State Model framework. Prerequisite input data consist of diffusion coefficient and potential of mean force maps generated from extensive molecular dynamics simulations of proteins and their environment that sample multi-nanosecond durations. The suggested diffusion model can describe transport processes beyond microsecond duration, relevant for biological function and beyond the realm of molecular dynamics simulation. For this purpose the systems are represented by a discrete set of states specified by the positions, volumes, and surface elements of Voronoi grid cells distributed according to a density function resolving the often intricate relevant diffusion space. Validation tests carried out for generic diffusion spaces show that the model and the associated Brownian motion algorithm are viable over a large range of parameter values such as time step, diffusion coefficient, and grid density. A concrete application of the method is demonstrated for ion diffusion around and through the Eschericia coli mechanosensitive channel of small conductance ecMscS.

Dynamic transition states of ErbB1 phosphorylation predicted by spatial stochastic modeling

ErbB1 overexpression is strongly linked to carcinogenesis, motivating better understanding of erbB1 dimerization and activation. Recent single-particle-tracking data have provided improved measures of dimer lifetimes and strong evidence that transient receptor coconfinement promotes repeated interactions between erbB1 monomers. Here, spatial stochastic simulations explore the potential impact of these parameters on erbB1 phosphorylation kinetics. This rule-based mathematical model incorporates structural evidence for conformational flux of the erbB1 extracellular domains, as well as asymmetrical orientation of erbB1 cytoplasmic kinase domains during dimerization. The asymmetric dimer model considers the theoretical consequences of restricted transactivation of erbB1 receptors within a dimer, where the N-lobe of one monomer docks with the C-lobe of the second monomer and triggers its catalytic activity. The dynamic nature of the erbB1 phosphorylation state is shown by monitoring activation states of individual monomers as they diffuse, bind, and rebind after ligand addition. The model reveals the complex interplay between interacting liganded and nonliganded species and the influence of their distribution and abundance within features of the membrane landscape.

Diffusional effects on the reversible excited-state proton transfer. From experiments to Brownian dynamics simulations

We have studied an excited state proton transfer (ESPT) from the cationic "super" photoacid N-methyl 6-hydroxyquinolinium perfluorobutane sulfonate to non-aqueous solvents using picosecond and nanosecond time-resolved fluorescence spectroscopy. Upon the photoinduced adiabatic deprotonation from the hydroxyl moiety, a quinolinium zwitterion with a highly anisotropic charge distribution is formed. Due to the complexity of the resultant photodissociated system, the typical description of the reversible ESPT within the framework of the Spherically Symmetric Diffusion Problem (SSDP) is not possible. Additional complications are caused by the presence of a counteranion particle which affects the proton mobility. To better understand the ESPT process, we have performed extensive Brownian dynamics (BD) simulations of this three-body system as a tool to reveal the nature of the nonstationary interaction potentials and to elucidate the role of a counterion in the diffusion and reactive properties of the proton. Moreover, our results demonstrated that the anisotropy of the potential force can be taken into account after adapting this force for use in the SSDP. The results of both BD simulations and SSDP calculation with the adapted force field were used to fit the experimental kinetics of this three-body problem adequately.

A new multicompartmental reaction-diffusion modeling method links transient membrane attachment of E. coli MinE to E-ring formation

Many important cellular processes are regulated by reaction-diffusion (RD) of molecules that takes place both in the cytoplasm and on the membrane. To model and analyze such multicompartmental processes, we developed a lattice-based Monte Carlo method, Spatiocyte that supports RD in volume and surface compartments at single molecule resolution. Stochasticity in RD and the excluded volume effect brought by intracellular molecular crowding, both of which can significantly affect RD and thus, cellular processes, are also supported. We verified the method by comparing simulation results of diffusion, irreversible and reversible reactions with the predicted analytical and best available numerical solutions. Moreover, to directly compare the localization patterns of molecules in fluorescence microscopy images with simulation, we devised a visualization method that mimics the microphotography process by showing the trajectory of simulated molecules averaged according to the camera exposure time. In the rod-shaped bacterium Escherichia coli, the division site is suppressed at the cell poles by periodic pole-to-pole oscillations of the Min proteins (MinC, MinD and MinE) arising from carefully orchestrated RD in both cytoplasm and membrane compartments. Using Spatiocyte we could model and reproduce the in vivo MinDE localization dynamics by accounting for the previously reported properties of MinE. Our results suggest that the MinE ring, which is essential in preventing polar septation, is largely composed of MinE that is transiently attached to the membrane independently after recruited by MinD. Overall, Spatiocyte allows simulation and visualization of complex spatial and reaction-diffusion mediated cellular processes in volumes and surfaces. As we showed, it can potentially provide mechanistic insights otherwise difficult to obtain experimentally.

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.

Stochastic simulations of ErbB homo and heterodimerisation: potential impacts of receptor conformational state and spatial segregation

ErbB overexpression is linked to carcinogenesis. It is hypothesised that this is due to increased receptor density and receptor clustering, leading to increased receptor dimerisation and activation. Herein, spatial stochastic simulations have been performed to shed light receptor dimerisation processes. First, ligand-independent homodimerisation, is considered, based upon constitutive oligomerisation estimates (14%) in A431 cells that overexpress epidermal growth factor receptor (EGFR). When autocrine stimulation is blocked, ligand-independent EGFR activation is demonstrated by persistent, low levels of phosphorylation. The possibility that ligand-independent signalling is due to the fluctuation of EGFR conformation is considered. The agent-based model predicts the frequency (expressed as a probability) that uniformly distributed receptors would need to flux to the open conformation to reach 14% EGFR dimers at high receptor density. Simulations suggest that ligand-independent EGFR homodimerisation is highly density dependent, since collisions between 'open', dimerisation-competent receptors are a rare event at low receptor levels. Simulations that incorporate receptor clustering lower the threshold for homodimerisation of unoccupied receptors as well as the estimate of the probability for fluxing to the dimer-competent conformation. The impact of ErbB receptor clustering patterns on hetero and homodimerisation rates is also considered, using immunoelectron microscopy data derived from SKBR3 breast cancer cells that express ErbB2>>EGFR>ErbB3. Partial spatial segregation of ErbB receptors has a profound effect on simulated heterodimerisation rates. Despite the general assumption that ErbB2 is a preferred heterodimerising partner for other ErbBs, it is predicted that most ErbB2 will form homodimers. Overall, it is proposed that both receptor density and membrane spatial organisation contribute to the carcinogenesis process.

Diffusion-influenced excited-state reversible geminate ABCD reaction in the presence of an external field

We obtained the exact Green functions, in the Laplace domain, for a diffusion-influenced excited-state reversible geminate ABCD reaction with two different lifetimes and quenching processes under a constant external field in one dimension. Analytic expressions for the survival probabilities of the initial and final states are obtained in the time domain at short and long times, respectively. The short-time approximations obtained in this work are valid for t</K/(-1), where K depends on several parameters of the system. The analysis of the long-time asymptotic behaviors reveals rather complex kinetic transitions dependent upon the field and lifetimes. We also find a destructive interplay leading to the reduction in the number of kinetic transitions similar to that found for the excited-state geminate ABC reaction with an external field in one dimension.

Extraction of elementary rate constants from global network analysis of E. coli central metabolism

Background

As computational performance steadily increases, so does interest in extending one-particle-per-molecule models to larger physiological problems. Such models however require elementary rate constants to calculate time-dependent rate coefficients under physiological conditions. Unfortunately, even when in vivo kinetic data is available, it is often in the form of aggregated rate laws (ARL) that do not specify the required elementary rate constants corresponding to mass-action rate laws (MRL). There is therefore a need to develop a method which is capable of automatically transforming ARL kinetic information into more detailed MRL rate constants.

Results

By incorporating proteomic data related to enzyme abundance into an MRL modelling framework, here we present an efficient method operating at a global network level for extracting elementary rate constants from experiment-based aggregated rate law (ARL) models. The method combines two techniques that can be used to overcome the difficult properties in parameterization. The first, a hybrid MRL/ARL modelling technique, is used to divide the parameter estimation problem into sub-problems, so that the parameters of the mass action rate laws for each enzyme are estimated in separate steps. This reduces the number of parameters that have to be optimized simultaneously. The second, a hybrid algebraic-numerical simulation and optimization approach, is used to render some rate constants identifiable, as well as to greatly narrow the bounds of the other rate constants that remain unidentifiable. This is done by incorporating equality constraints derived from the King-Altman and Cleland method into the simulated annealing algorithm. We apply these two techniques to estimate the rate constants of a model of E. coli glycolytic pathways. The simulation and statistical results show that our innovative method performs well in dealing with the issues of high computation cost, stiffness, local minima and uncertainty inherent with large-scale non-convex nonlinear MRL models.

Conclusion

In short, this new hybrid method can ensure the proper solution of a challenging parameter estimation problem of nonlinear dynamic MRL systems, while keeping the computational effort reasonable. Moreover, the work provides us with some optimism that physiological models at the particle scale can be rooted on a firm foundation of parameters generated in the macroscopic regime on an experimental basis. Thus, the proposed method should have applications to multi-scale modelling of the real biological systems allowing for enzyme intermediates, stochastic and spatial effects inside a cell.

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.

Mobile trap algorithm for zinc detection using protein sensors

We present a mobile trap algorithm to sense zinc ions using protein-based sensors such as carbonic anhydrase (CA). Zinc is an essential biometal required for mammalian cellular functions although its intracellular concentration is reported to be very low. Protein-based sensors like CA molecules are employed to sense rare species like zinc ions. In this study, the zinc ions are mobile targets, which are sought by the mobile traps in the form of sensors. Particle motions are modeled using random walk along with the first passage technique for efficient simulations. The association reaction between sensors and ions is incorporated using a probability (p1) upon an ion-sensor collision. The dissociation reaction of an ion-bound CA molecule is modeled using a second, independent probability (p2). The results of the algorithm are verified against the traditional simulation techniques (e.g., Gillespie's algorithm). This study demonstrates that individual sensor molecules can be characterized using the probability pair (p1,p2), which, in turn, is linked to the system level chemical kinetic constants, kon and koff. Further investigations of CA-Zn reaction using the mobile trap algorithm show that when the diffusivity of zinc ions approaches that of sensor molecules, the reaction data obtained using the static trap assumption differ from the reaction data obtained using the mobile trap formulation. This study also reveals similar behavior when the sensor molecule has higher dissociation constant. In both the cases, the reaction data obtained using the static trap formulation reach equilibrium at a higher number of complex molecules (ion-bound sensor molecules) compared to the reaction data from the mobile trap formulation. With practical limitations on the number sensors that can be inserted/expressed in a cell and stochastic nature of the intracellular ionic concentrations, fluorescence from the number of complex sensor molecules at equilibrium will be the measure of the intracellular ion concentration. For reliable detection of zinc ions, it is desirable that the sensors must not bind all the zinc ions tightly, but should rather bind and unbind. Thus for a given fluorescence and with association-dissociation reactions between ions and sensors, the static trap approach will underestimate the number of zinc ions present in the system.

Green's-function reaction dynamics: a particle-based approach for simulating biochemical networks in time and space

We have developed a new numerical technique, called Green's-function reaction dynamics (GFRD), that makes it possible to simulate biochemical networks at the particle level and in both time and space. In this scheme, a maximum time step is chosen such that only single particles or pairs of particles have to be considered. For these particles, the Smoluchowski equation can be solved analytically using Green's functions. The main idea of GFRD is to exploit the exact solution of the Smoluchoswki equation to set up an event-driven algorithm, which combines in one step the propagation of the particles in space with the reactions between them. The event-driven nature allows GFRD to make large jumps in time and space when the particles are far apart from each other. Here, we apply the technique to a simple model of gene expression. The simulations reveal that spatial fluctuations can be a major source of noise in biochemical networks. The calculations also show that GFRD is highly efficient. Under biologically relevant conditions, GFRD is up to five orders of magnitude faster than conventional particle-based techniques for simulating biochemical networks in time and space. GFRD is not limited to biochemical networks. It can also be applied to a large number of other reaction-diffusion problems.

Serial rebinding of ligands to clustered receptors as exemplified by bacterial chemotaxis

Serial ligation is the repeated reversible binding of a ligand to one receptor after another. It is a widespread phenomenon throughout biochemical systems, occurring anytime receptors are clustered together and ligand binding is reversible. Computer simulations are used in this work to investigate a representative example, which is the serial ligation of an extracellular aspartate molecule to the membrane-bound chemotaxis receptors of an Escherichia coli bacterium. It is found that the initial binding site of a ligand to a cluster of receptors is more likely to be near the edge of the cluster than near the middle, although there is no overall bias when all rebindings are considered. Serial ligation does not lead directly to signal amplification or attenuation but instead causes binding events to be correlated in both space and time: a ligand is likely to bind many times in rapid succession in a small region of the receptor cluster, but there can also be long intervals between bindings. This leads to an increased level of noise in the received signal but may allow a single ligand to be sensed above a uniform level of background noise. The focus of this paper is on the interpretation of simulation results so they can be generalized to a wide variety of other systems and to allow the identification of systems in which serial ligation is likely to be important. In the process, several characteristic times are identified, as are scaling laws for the spatial and temporal dynamics.

Activated N-formyl peptide receptor and high-affinity IgE receptor occupy common domains for signaling and internalization

Immune cells display multiple cell surface receptors that integrate signals for survival, proliferation, migration, and degranulation. Here, immunogold labeling is used to map the plasma membrane distributions of two separate receptors, the N-formyl peptide receptor (FPR) and the high-affinity IgE receptor (FepsilonRI). We show that the FPR forms signaling clusters in response to monovalent ligand. These domains recruit Gi, followed by the negative regulatory molecule arrestin2. There are low levels of colocalization of FPR with FcepsilonRI in unstimulated cells, shown by computer simulation to be a consequence of receptor density. Remarkably, there is a large increase in receptor coclustering when cells are simultaneously treated with N-formyl-methionyl-leucyl-phenylalanine and IgE plus polyvalent antigen. The proximity of two active receptors may promote localized cross-talk, leading to enhanced inositol-(3,4,5)-trisphosphate production and secretion. Some cointernalization and trafficking of the two receptors can be detected by live cell imaging, but the bulk of FPR and FcepsilonRI segregates over time. This segregation is associated with more efficient internalization of cross-linked FcepsilonRI than of arrestin-desensitized FPR. The observation of receptors in lightly coated membrane invaginations suggests that, despite the lack of caveolin, hematopoietic cells harbor caveolae-like structures that are candidates for nonclathrin-mediated endocytosis.

Stochastic simulation of chemical reactions with spatial resolution and single molecule detail

Methods are presented for simulating chemical reaction networks with a spatial resolution that is accurate to nearly the size scale of individual molecules. Using an intuitive picture of chemical reaction systems, each molecule is treated as a point-like particle that diffuses freely in three-dimensional space. When a pair of reactive molecules collide, such as an enzyme and its substrate, a reaction occurs and the simulated reactants are replaced by products. Achieving accurate bimolecular reaction kinetics is surprisingly difficult, requiring a careful consideration of reaction processes that are often overlooked. This includes whether the rate of a reaction is at steady-state and the probability that multiple reaction products collide with each other to yield a back reaction. Inputs to the simulation are experimental reaction rates, diffusion coefficients and the simulation time step. From these are calculated the simulation parameters, including the 'binding radius' and the 'unbinding radius', where the former defines the separation for a molecular collision and the latter is the initial separation between a pair of reaction products. Analytic solutions are presented for some simulation parameters while others are calculated using look-up tables. Capabilities of these methods are demonstrated with simulations of a simple bimolecular reaction and the Lotka-Volterra system.

Molecular-dynamics simulations for nonclassical kinetics of diffusion-controlled bimolecular reactions

Molecular-dynamics simulations are presented for the diffusion-controlled bimolecular reaction A+B<==>C in two and three dimensions. The reactants and solvent molecules are modeled as spheres interacting via continuous potential-energy functions. The interaction potential between two reactants contains a deep well that results in a reaction. When the solvent concentration is low and the reactant dynamics is essentially ballistic, the system reaches equilibrium rapidly, and the reaction follows classical kinetics with exponential decay to the equilibrium. When the solvent concentration is high the particles enter the normal diffusion regime quickly and nonclassical behavior is observed, i.e., the reactant concentrations approach equilibrium as t(-d/2) where d is the dimensionality of space. When the reaction well depth is large, however, the reaction becomes irreversible within the simulation time. In this case the reactant concentrations decay as t(-d/4). Interestingly this behavior is also observed at intermediate times for reversible reactions.

Diffusion-influenced excited-state reversible transfer reactions, A*+B⇌C*+D, with two different lifetimes: Theories and simulations

We report accurate Brownian simulation results for the kinetics of the pseudo-first-order diffusion-influenced excited-state reversible transfer reaction A(*) + Bright harpoon over left harpoonC(*) + D with two different lifetimes using two different propagation algorithms. The results are used to test approximate solutions for this many-particle problem. Available theories fail when one of the two reactions or (decay) rate constants is large. To remedy this situation, we develop two uniform approximations, which are based on introducing a generalized Smoluchowski term into the relaxation-time approximation. The best of these is the extended unified theory of reversible target reactions, which reduces correctly in all limits and exhibits superior agreement with simulations.

Simulating biochemical networks at the particle level and in time and space: Green's function reaction dynamics

We present a technique, called Green's function reaction dynamics (GFRD), for particle-based simulations of reaction-diffusion systems. GFRD uses a maximum time step such that only single particles or pairs of particles have to be considered. For these particles, the Smoluchowski equations are solved analytically using Green's functions, which are used to set up an event-driven algorithm. We apply the technique to a model of gene expression. Under biologically relevant conditions, GFRD is up to 5 orders of magnitude faster than conventional particle-based schemes.

Elementary Steps in Excited-State Proton Transfer

The absorption of a photon by a hydroxy-aromatic photoacid triggers a cascade of events contributing to the overall phenomenon of intermolecular excited-state proton transfer. The fundamental steps involved were studied over the last 20 years using a combination of theoretical and experimental techniques. They are surveyed in this sequel in sequential order, from fast to slow. The excitation triggers an intramolecular charge transfer to the ring system, which is more prominent for the anionic base than the acid. The charge redistribution, in turn, triggers changes in hydrogen-bond strengths that set the stage for the proton-transfer step itself. This step is strongly influenced by the solvent, resulting in unusual dependence of the dissociation rate coefficient on water content, temperature, and isotopic substitution. The photolyzed proton can diffuse in the aqueous solution in a mechanism that involves collective changes in hydrogen-bonding. On longer times, it may recombine adiabatically with the excited base or quench it. The theory for these diffusion-influenced geminate reactions has been developed, showing nice agreement with experiment. Finally, the effect of inert salts, bases, and acids on these reactions is analyzed.

Fluorescence quenching by reversible excimer formation: Kinetics and yield predictions for a classical potential association–dissociation model

Fluorescence quenching by reversible excimer formation is studied on the assumption that excimer formation and dissociation can be modelled as entering and leaving the attractive region of an monomer excited-monomer interaction potential by diffusion. To get some general insight in the kinetic consequences of such a type of modelling, the simple case of an attractive square-well potential is investigated. It is shown that three different kinetic regimes have to be distinguished: Two "reversible" ones in case of slow excimer radiative decay, in which the quenching kinetics can be formulated by Markovian or non-Markovian rate equations with both excimer formation and excimer dissociation terms, and an effectively "irreversible" regime if the excimer radiative decay is too rapid to allow the excimer equilibration. In the latter case a dissociation coefficient can no longer be defined and the quenching kinetics can only be predicted on the basis of generalized rate equations of a net-excimer-formation type. It is shown how the quenching constant formula must be generalized to be applicable in all kinetic situations.

Diffusion-influenced reversible geminate recombination in one dimension. III. Field effect on the excited-state reaction

We obtain exact analytic solutions of the diffusion-influenced excited-state reversible geminate recombination reaction, A* + B<-->(AB)*, with two different lifetimes and quenching under the influence of a constant external field in one dimension. These fundamental solutions generalize two previous results [Kim et al., J. Chem. Phys. 111, 3791 (1999); 114, 3905 (2001)] and provide us with the insight necessary to analyze their specific relations and asymptotic kinetic transition behaviors. We find that the number of kinetic transitions can be changed due to interplay between the field strength and lifetimes. Unlike the previous works, the number of lifetime dependent transitions is found to be one or zero. On the other hand, the number of the field dependent transitions becomes two, one, or zero. We find a new pattern of kinetic transition e(t)-->t(-1/2)-->e(t) when there is only one field dependent transition.

Mesoscopic model for diffusion-influenced reaction dynamics

A hybrid mesoscopic multiparticle collision model is used to study diffusion-influenced reaction kinetics. The mesoscopic particle dynamics conserves mass, momentum, and energy so that hydrodynamic effects are fully taken into account. Reactive and nonreactive interactions with catalytic solute particles are described by full molecular dynamics. Results are presented for large-scale, three-dimensional simulations to study the influence of diffusion on the rate constants of the A + C <==> B + C reaction. In the limit of a dilute solution of catalytic C particles, the simulation results are compared with diffusion equation approaches for both the irreversible and reversible reaction cases. Simulation results for systems where the volume fraction phi of catalytic spheres is high are also presented, and collective interactions among reactions on catalytic spheres that introduce volume fraction dependence in the rate constants are studied.

Influence of diffusion on the kinetics of excited-state association–dissociation reactions: Comparison of theory and simulation

Several recent theories of the kinetics of diffusion influenced excited-state association--dissociation reactions are tested against accurate Brownian dynamics simulation results for a wide range of parameters. The theories include the relaxation time approximation (RTA), multiparticle kernel decoupling approximations and the so-called kinetic theory. In the irreversible limit, none of these theories reduce to the Smoluchowski result. For the pseudo-first-order target problem, we show how the RTA can be modified so that the resulting formalism does reduce correctly in the irreversible limit. We call this the unified Smoluchowski approximation, because it unites modern theories of reversible reactions with Smoluchowski's theory of irreversible reactions.