Spatial modeling of dimerization reaction dynamics in the plasma membrane: Monte Carlo vs. continuum differential equations

Bimolecular reactions in the plasma membrane, such as receptor dimerization, are a key signaling step for many signaling systems. For receptors to dimerize, they must first diffuse until a collision happens, upon which a dimerization reaction may occur. Therefore, study of the dynamics of cell signaling on the membrane may require the use of a spatial modeling framework. Despite the availability of spatial simulation methods, e.g., stochastic spatial Monte Carlo (MC) simulation and partial differential equation (PDE) based approaches, many biological models invoke well-mixed assumptions without completely evaluating the importance of spatial organization. Whether one is to utilize a spatial or non-spatial simulation framework is therefore an important decision. In order to evaluate the importance of spatial effects a priori, i.e., without performing simulations, we have assessed the applicability of a dimensionless number, known as second Damköhler number (Da), defined here as the ratio of time scales of collision and reaction, for 2-dimensional bimolecular reactions. Our study shows that dimerization reactions in the plasma membrane with Da ∼> 0.1 (tested in the receptor density range of 102-105/μm2) require spatial modeling. We also evaluated the effective reaction rate constants of MC and simple deterministic PDEs. Our simulations show that the effective reaction rate constant decreases with time due to time dependent changes in the spatial distribution of receptors. As a result, the effective reaction rate constant of simple PDEs can differ from that of MC by up to two orders of magnitude. Furthermore, we show that the fluctuations in the number of copies of signaling proteins (noise) may also depend on the diffusion properties of the system. Finally, we used the spatial MC model to explore the effect of plasma membrane heterogeneities, such as receptor localization and reduced diffusivity, on the dimerization rate. Interestingly, our simulations show that localization of epidermal growth factor receptor (EGFR) can cause the diffusion limited dimerization rate to be up to two orders of magnitude higher at higher average receptor densities reported for cancer cells, as compared to a normal cell.