First-passage time to clear the way for receptor-ligand binding in a crowded environment

Certain biological reactions, such as receptor-ligand binding at cell-cell interfaces and macromolecules binding to biopolymers, require many smaller molecules crowding a reaction site to be cleared. Examples include the T cell interface, a key player in immunological information processing. Diffusion sets a limit for such cavitation to occur spontaneously, thereby defining a timescale below which active mechanisms must take over. We consider $N$ independent diffusing particles in a closed domain, containing a sub-region with $N_{0}$ particles, on average. We investigate the time until the sub-region is empty, allowing a subsequent reaction to proceed. The first passage time is computed using an efficient exact simulation algorithm and an asymptotic approximation in the limit that cavitation is rare. In this limit, we find that the mean first passage time is sub-exponential, $T \propto e^{N_{0}}/N_{0}^2$. For the case of T cell receptors, we find that stochastic cavitation is exceedingly slow, $10^9$ seconds at physiological densities, however can be accelerated to occur within 5 second with only a four-fold dilution

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 disassociation 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 (RDME) 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- and diffusion-limited irreversible association in three dimensions. We also study long-time tails in the time correlation functions for reversible association at thermodynamic equilibrium. Finally, we compare different particle and continuum methods on a model exhibiting a Turing-like instability and pattern formation. We 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.Comment: To appear in J. Chem. Phy

Multiscale stochastic reaction-diffusion modelling: application to actin dynamics in filopodia

Two multiscale (hybrid) stochastic reaction-diffusion models of actin dynamics in a filopodium are investigated. Both hybrid algorithms combine compartment-based and molecular-based stochastic reaction-diffusion models. The first hybrid model is based on the models previously\ud developed in the literature. The second hybrid model is based on the application of recently developed two-regime method (TRM) to a fully molecular-based model which is also developed in this paper. The results of hybrid models are compared with the results of the molecular-based model. It is shown that both approaches give comparable results, although the TRM model better agrees quantitatively with the molecular-based model

Convergence of methods for coupling of microscopic and mesoscopic reaction-diffusion simulations

In this paper, three multiscale methods for coupling of mesoscopic (compartment-based) and microscopic (molecular-based) stochastic reaction-diffusion simulations are investigated. Two of the three methods that will be discussed in detail have been previously reported in the literature; the two-regime method (TRM) and the compartment-placement method (CPM). The third method that is introduced and analysed in this paper is the ghost cell method (GCM). Presented is a comparison of sources of error. The convergent properties of this error are studied as the time step $\Delta t$ (for updating the molecular-based part of the model) approaches zero. It is found that the error behaviour depends on another fundamental computational parameter $h$, the compartment size in the mesoscopic part of the model. Two important limiting cases, which appear in applications, are considered: (i) \Delta t approaches 0 and h is fixed; and (ii) \Delta t approaches 0 and h approaches 0 such that \Delta t/h^2 is fixed. The error for previously developed approaches (the TRM and CPM) converges to zero only in the limiting case (ii), but not in case (i). It is shown that the error of the GCM converges in the limiting case (i). Thus the GCM is superior to previous coupling techniques if the mesoscopic description is much coarser than the microscopic part of the model

Reaction-diffusion kinetics on lattice at the microscopic scale

Lattice-based stochastic simulators are commonly used to study biological reaction-diffusion processes. Some of these schemes that are based on the reaction-diffusion master equation (RDME), can simulate for extended spatial and temporal scales but cannot directly account for the microscopic effects in the cell such as volume exclusion and diffusion-influenced reactions. Nonetheless, schemes based on the high-resolution microscopic lattice method (MLM) can directly simulate these effects by representing each finite-sized molecule explicitly as a random walker on fine lattice voxels. The theory and consistency of MLM in simulating diffusion-influenced reactions have not been clarified in detail. Here, we examine MLM in solving diffusion-influenced reactions in 3D space by employing the Spatiocyte simulation scheme. Applying the random walk theory, we construct the general theoretical framework underlying the method and obtain analytical expressions for the total rebinding probability and the effective reaction rate. By matching Collins-Kimball and lattice-based rate constants, we obtained the exact expressions to determine the reaction acceptance probability and voxel size. We found that the size of voxel should be about 2% larger than the molecule. MLM is validated by numerical simulations, showing good agreement with the off-lattice particle-based method, eGFRD. MLM run time is more than an order of magnitude faster than eGFRD when diffusing macromolecules with typical concentrations in the cell. MLM also showed good agreements with eGFRD and mean-field models in case studies of two basic motifs of intracellular signaling, the protein production-degradation process and the dual phosphorylation cycle. Moreover, when a reaction compartment is populated with volume-excluding obstacles, MLM captures the non-classical reaction kinetics caused by anomalous diffusion of reacting molecules