The convergent reaction-diffusion master equation (CRDME) was recently developed to provide a lattice particle-based stochastic reaction-diffusion model that is a convergent approximation in the lattice spacing to an underlying spatially-continuous particle dynamics model. The CRDME was designed to be identical to the popular lattice reaction-diffusion master equation (RDME) model for systems with only linear reactions, while overcoming the RDME's loss of bimolecular reaction effects as the lattice spacing is taken to zero. In our original work we developed the CRDME to handle bimolecular association reactions on Cartesian grids. In this work we develop several extensions to the CRDME to facilitate the modeling of cellular processes within realistic biological domains. Foremost, we extend the CRDME to handle reversible bimolecular reactions on unstructured grids. Here we develop a generalized CRDME through discretization of the spatially continuous volume reactivity model, extending the CRDME to encompass a larger variety of particle-particle interactions. Finally, we conclude by examining several numerical examples to demonstrate the convergence and accuracy of the CRDME in approximating the volume reactivity model.

Signalling by surface receptors often relies on tethered reactions whereby an enzyme bound to the cytoplasmic tail of a receptor catalyses reactions on substrates within reach. The overall length and stiffness of the receptor tail, the enzyme, and the substrate determine a biophysical parameter termed the molecular reach of the reaction. This parameter determines the probability that the receptor-tethered-enzyme will contact the substrate, in the volume proximal to the membrane, when separated by different distances within the membrane plane. In this work we develop particlebased stochastic reaction-diffusion models to study the interplay between molecular reach and diffusion. We find that increasing the molecular reach can increase reaction efficacy for slowly diffusing receptors, while for rapidly diffusing receptors increasing molecular reach reduces reaction efficacy. In contrast, if reactions are forced to take place within the 2D plasma membrane instead of the 3D volume proximal to it, or if molecules diffuse in 3D, increasing molecular reach increases reaction efficacy for all diffusivities. We show results in the context of immune checkpoint receptors (PD-1 dephosphorylating CD28), a standard opposing kinase-phosphatase reaction, and a minimal two-particle model. The work highlights the importance of the 3D nature of many 2D membraneconfined interactions, illustrating a role for molecular reach in controlling biochemical reactions.O.D. and S.A.I. contributed equally to this work.

In particle-based stochastic reaction-diffusion models, reaction rate and placement kernels are used to decide the probability per time a reaction can occur between reactant particles, and to decide where product particles should be placed. When choosing kernels to use in reversible reactions, a key constraint is to ensure that detailed balance of spatial reaction-fluxes holds at all points at equilibrium. In this work we formulate a general partial-integral differential equation model that encompasses several of the commonly used contact reactivity (e.g. Smoluchowski-Collins-Kimball) and volume reactivity (e.g. Doi) particle models. From these equations we derive a detailed balance condition for the reversible $\textrm{A} + \textrm{B} \leftrightarrows \textrm{C}$ reaction. In bounded domains with no-flux boundary conditions, when choosing unbinding kernels consistent with several commonly used binding kernels, we show that preserving detailed balance of spatial reaction-fluxes at all points requires spatially varying unbinding rate functions near the domain boundary. Brownian Dynamics simulation algorithms can realize such varying rates through ignoring domain boundaries during unbinding and rejecting unbinding events that result in product particles being placed outside the domain.

The geometry of the blood vessel wall plays a regulatory role on the motion of red blood cells (RBCs). The overall topography of the vessel wall depends on many features, among which the endothelial lining of the endothelial surface layer (ESL) is an important one. The endothelial lining of vessel walls presents a large surface area for exchanging materials between blood and tissues. The ESL plays a critical role in regulating vascular permeability, hindering leukocyte adhesion as well as inhibiting coagulation during inflammation. Changes in the ESL structure are believed to cause vascular hyperpermeability and entrap immune cells during sepsis, which could significantly alter the vessel wall geometry and disturb interactions between RBCs and the vessel wall, including the wall-induced migration of RBCs and the thickening of a cell-free layer. To investigate the influence of the vessel wall geometry particularly changed by the ESL under various pathological conditions, such as sepsis, on the motion of RBCs, we developed two models to represent the ESL using the immersed boundary method in two dimensions. In particular, we used simulations to study how the lift force and drag force on a RBC near the vessel wall vary with different wall thickness, spatial variation, and permeability associated with changes in the vessel wall geometry. We find that the spatial variation of the wall has a significant effect on the wall-induced migration of the RBC for a high permeability, and that the wall-induced migration is significantly inhibited as the vessel diameter is increased.