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.

During cell division, cross-linking motors determine the architecture of the spindle, a dynamic microtubule network that segregates the chromosomes in eukaryotes. It is unclear how motors with opposite directionality coordinate to drive both contractile and extensile behaviors in the spindle. Particularly, the impact of different cross-linker designs on network self-organization is not understood, limiting our understanding of self-organizing structures in cells but also our ability to engineer new active materials. Here, we use experiment and theory to examine active microtubule networks driven by mixtures of motors with opposite directionality and different cross-linker design. We find that although the kinesin-14 HSET causes network contraction when dominant, it can also assist the opposing kinesin-5 KIF11 to generate extensile networks. This bifunctionality results from HSET’s asymmetric design, distinct from symmetric KIF11. These findings expand the set of rules underlying patterning of active microtubule assemblies and allow a better understanding of motor cooperation in the spindle.

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 verifed. 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 surface-surface reaction and one-dimensional 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 diffusion in the cytoplasm, two-dimensional diffusion on the cell membrane and one-dimensional 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.1