Continuum and discrete modeling of binding-site distribution-mediated reactions on lipid surfaces

Cell surface-dependent biochemical reactions play a critical role in many biological processes. These include interactions between macromolecules in a 3D bulk solution, macromolecules confined to a 2D membrane surface, and/or lipids that comprise the membrane. In blood coagulation, for instance, biochemical reactions that generate the key enzyme thrombin occur predominantly on the membrane surfaces of activated platelets. However, how the spatial distribution of the membrane binding sites affects enzymatic activity and reaction efficiency remains poorly understood. To bridge this gap, we employed both a partial differential equation (PDE) model and a particle-based model to analyze a simplified biochemical reaction system in the presence of a lipid surface, inspired by coagulation reactions characterized by surface binding, inhibition, and positive feedback. Our analyses show that when binding sites are localized in a single patch, the optimal patch size reflects a trade-off between surface-bound reactant density and transport time to the surface. When binding sites are distributed among multiple patches, the PDE model predicts that increasing the number of patches monotonically enhances efficiency. The particle-based model, which captures molecular-scale effects, reveals a non-monotonic trend: efficiency increases with patch number initially, then declines as patches become overly fragmented. This discrepancy arises because continuum models allow for fractions of molecules to bind and interact, whereas biological reactions can only occur with whole molecules. Our results suggest that, for a fixed number of binding sites, moderately-sized patches with high binding-site density are more efficient for enzyme generation compared to many small patches or few large patches of the same density. Our findings highlight the limitations of continuum models at molecular spatial scales, underscore the importance of discrete modeling in such regimes, and provide mechanistic insights regarding optimization of surface-dependent biomolecular reactions such as thrombin generation.