Universal properties of cellular input-output functions
Cellular systems – enzymes, motors, allosteric proteins, genes, ion channels, receptors, etc – are often described by the functional dependence of some output property on the concentrations of input factors, with the system itself described as a Markov process at steady state. Such input-output functions are calculated by a variety of methods with few common features. In fact, this typical biological heterogeneity conceals a remarkable underlying mathematical unity, which emerges from the graph-theoretic “linear framework” developed in our lab over many years. All such input-output functions are rational functions of their inputs, with coefficients which are themselves rational functions of the Markov transition rates. There is a uniform procedure for calculating each function in terms of the linear-framework graph associated to the Markov process. Furthermore, input-output functions exhibit a universal Hopfield barrier: if the Markov process can reach thermodynamic equilibrium, then the degree of the rational function depends only on the numbers of input binding sites and is otherwise entirely model independent. This universality vanishes away from thermodynamic equilibrium: the degree of the input-output function depends sensitively on the structure of the graph and simple families of models exist whose input-output functions have unbounded degree.