Degrees of Rate Control in Interconnected Reaction Networks

Overall reactions in interconnected networks exhibit net, forward, and reverse rates that are governed by both constitutive elementary steps in the pathway of interest and branching elementary steps that lead to alternative products. Accordingly, steps in branching pathways exhibit negative net, forward, and reverse degrees of rate control, as they reduce reaction flux to the desired product. We here contextualize the forward and reverse degrees of rate control in terms of kinetic resistances (inverse of rates) and leverage the additive nature of kinetic resistance to decouple kinetic driving forces contributed by constitutive elementary steps and branching points (nodal species) in interconnected networks. Regardless of the network connectivity, forward and reverse degrees of rate control are shown to converge at equilibrium. Away from equilibrium, we identify two critical features of interconnected networks: stoichiometric regularity─condition where all stoichiometric numbers are unity─and pathway symmetry around nodal species─condition where branching pathways share the same rate constants, stoichiometry, and species concentrations/activities─that result in (i) equal forward, reverse, and consequently net degrees of rate control and (ii) forward and reverse degrees of rate control that exhibit constant offsets, respectively, across all extents of reaction. Our discourse further provides a mathematical description for the influence of stoichiometric irregularity and pathway asymmetry on forward and reverse degrees of rate control. Altogether, the presented work details the effects of network (inter)connectivity and stoichiometry on reaction kinetics and, in doing so, establishes general protocols for capturing these effects as additive terms in the formulation of forward and reverse degrees of rate control.