Lattice theory of reaction efficiency in compartmentalized systems. II. Reduction of dimensionality

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The timing and efficiency of a diffusion-controlled kinetic process in a compartmentalized system can be enhanced by reducing the dimensionality of the reaction space of the system. This idea, introduced by Adam and Delbrück and referred to as ‘‘reduction of dimensionality,’’ is explored quantitatively in this paper using a lattice theory of reaction efficiency developed in our earlier work. In particular, we study the interplay between system geometry and reaction efficiency using an approach in which group theoretic arguments are used within the framework of the theory of finite Markov processes to determine the average number 〈n〉 of steps required for a diffusing coreactant A to undergo an irreversible reaction with a stationary target molecule B. We study in detail three classes of problems in this paper. First, we study as a function of the position of the reaction center how the efficiency of the underlying, irreversible, reaction-diffusion process A+B → C changes with increase in system size for symmetrical geometries. We show how reducing the dimensionality of the flow of the diffusing co-reactant leads to a crossover in reaction efficiency with increase in the size of the system, and document this effect as a function of N (the total number of sites characterizing the reaction space of the system), d (the dimensionality of the system), and ν (the valency or connectivity between adjacent sites in the reaction space). Secondly, we study how the calculated value of 〈n〉, and hence the efficiency of the process, changes when the compartmentalized system is characterized by tubular or platelet geometries, and show how the process of reduction of dimensionality is dependent on the further geometrical characteristics of eccentricity ε and the surface-to-volume ratio S/V. Finally, we study the consequences of reduction of dimensionality for (two) consecutive (say, enzymatic) reactions taking place in a compartmentalized system and demonstrate the advantages of a further collapse in the dimensionality of the flow from d=3 → 2 to d=2 →1 for tubular geometries. The (exact) results presented in these studies provide a detailed and comprehensive picture of the role of system morphology in influencing the efficiency of diffusion-controlled irreversible processes involving a single reaction center in organized molecular assemblies (micelles, vesicles, microemulsions). Our results also cast light on the geometrical factors which may govern the optimal configuration of (two) reaction centers when consecutive enzymatic reactions take place in evolving cellular systems.