Lattice theory of reaction efficiency in compartmentalized systems

Abstract

We consider the problem of reaction efficiency in compartmentalized systems. In our model, we position a target molecule (active site) at a centrosymmetric location in a reaction space defined by N-1 locations; the coordinates of these sites are specified by their positions on a d-dimensional lattice of valency ν. We then consider the factors that may influence the efficiency of an (eventual) irreversible reaction at the reaction center between a diffusing coreactant A and the target molecule B, i.e., we consider the bimolecular reaction A+B → C. We suppose there exists a biasing potential of the form v(r)=r−s centered at the active site and then explore the interplay between this longer-range biasing potential and other short-range (nonspecific) chemical or cage effects in influencing the efficiency of the underlying reaction–diffusion process. These effects are studied as a function of the geometry of the compartmentalized system (i.e., its spatial extent, dimensionality d, and valency ν), the strength of the biasing potential (values of the potential exponent s ranging from s=1 to s=6 are studied), the temperature and finally the role of the initial conditions. Our calculations are carried out using an approach developed recently wherein group theoretic arguments are coupled with theorems from the theory of finite Markov processes to produce exact resuls on the efficiency of reaction–diffusion processes taking place on finite, d-dimensional networks of valency ν. That is, all results presented in this paper represent the exact solution of the underlying lattice statistical problem, and may be taken as an accurate documentation of the effects studied. Our calcuations also have relevance to the related problem of reaction–diffusion processes in systems where a set of reaction centers is distributed uniformly throughout an extended system, and this relationship and its implications will be brought out in the text.