Diffusion-controlled reactions in small and structured spaces as a tool for describing living cell biochemistry

A simplified model of the living cell is studied. The reaction space is divided into compartments and the structured (non-compact) geometry is described in terms of a network consisting of containers connected by tubes. By assumption, reactions in the containers ( tubes) are allowed ( forbidden). It is assumed that the number of reactants is low, leading to stochastic (noisy) dynamics. By varying the transport rate among various containers D relative to the reaction rate within each container lambda, using either D >> lambda or D << lambda, a transition from a reaction- controlled ( reactants mix well) towards a diffusion-controlled ( large spatial fluctuations) regime can be studied. The focus is on a study of the timing of chemical reactions. For a single set of chemical reactions, the reaction times t = (t(1), t(2),...) are defined as the time intervals needed to synthesize a given amount of molecules ( of various types and in different regions of the system). The components of t are stochastic (non-independent) variables described in terms of two moments: average t = ( t1, t2,...) and standard deviation sigma = (sigma(1), sigma(2),...). In such a way it is possible to have a measure of the reaction speed ( t) and noise content ( s). A large number of chemical reactions were classified by monitoring how norms t and s vary as the geometry of the system changes from compact (tau(0), sigma(0)) towards structured (tau(n), sigma(n)). It is found that there are reactions that draw benefits in terms of both increased reaction speed (tau(n) < tau(0)) and noise reduction (sigma(n) < sigma(0)). Such reactions become faster and synchronize better in structured space. There are reactions that exhibit an increase in speed (tau(n) < tau(0)) but become more noisy and harder to synchronize (sigma(n) > sigma(0)). Opposite cases are possible where reactions become slower (tau(n) > tau(0)) but more accurate (sigma(n) < sigma(0)).