Reaction-diffusion kinetics on lattice at the microscopic scale

Abstract

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. The theoretical framework of 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 observed 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-dephosphorylation cycle. In addition, when a reaction compartment is populated with volume-excluding obstacles, MLM captures the non-classical reaction kinetics caused by anomalous diffusion of reacting molecules.

pSpatiocyte: a high-performance simulator for intracellular reaction-diffusion systems

“…This walk interval is fixed and identical across all processes throughout the simulation procedure. With the above bimolecular reaction scheme, we have shown that the rebinding-time probability distribution of a reactive molecule pair on HCP lattice agrees well with continuum theory (see Section III.A of [43]). We have also verified the accuracy of the reaction rate coefficient and its time-dependent behavior by comparison to SCK theory.…”

“…W ij captures the intrinsic reaction rate k ij of the pair according to the Smoluchowski-Collins-Kimball (SCK) model [54,55]. The accuracy and consistency of the bimolecular reaction on lattice in both activation-limited and diffusion-limited regimes have been verified [43,45]. To represent volume occupying molecules, a voxel can be occupied by a single molecule at any given time.…”

“…is inversely proportional to the diffusion coefficients of reactants, highly diffusion-limited reactions can cause W ij > 1 [42,43]. To address this inaccurate condition, we first determine the maximum W ij of all bimolecular reactions involving A B Figure 2: Schemes for large-scale parallel simulation of particles.…”

“…In Spatiocyte, fine and fast-diffusing molecules such as messengers, metabolites and ions are simulated at the compartment scale using the Next-Reaction method [42,44]. The accuracy and consistency of Spatiocyte have been validated in detail recently in both volume [43] and surface [45] compartments. Spatiocyte achieves better execution time scaling behavior compared to other methods [43] because it resolves molecular collisions by looking only at the target voxel for occupancy.…”

“…On the other hand, when the molecules have physical dimensions, the runtime of Spatiocyte is at least two times faster than Smoldyn. These recent performance benchmarks [41,43] imply that at present Spatiocyte is one of the fastest methods for simulating individual molecules in crowded systems.…”