Flux-concentration duality in dynamic nonequilibrium biological networks

The structure of dynamic states in biological networks is of fundamental importance in understanding their function. Considering the elementary reaction structure of reconstructed metabolic networks, we show how appreciation of a gradient matrix, G =dv/dx (where v is the vector of fluxes and x is the vector of concentrations), enables the formulation of dual Jacobian matrices. One is for concentrations, J(x) =S x G, and the other is for fluxes, J(v) =G x S. The fundamental properties of these two Jacobians and the underlying duality that relates them are delineated. We describe a generalized approach to decomposing reaction networks in terms of the thermodynamic and kinetic components in the context of the network structure. The thermodynamic and kinetic influences can be viewed in terms of direction-driver relationships in the network.