enzymatic cycles

This paper introduces a notion of non-oscillation, proposes a constructive method for its robust verification, and studies its application to biological interaction networks. The paper starts by revisiting Muldowney's result on non-existence of periodic solutions based on the study of the variational system of the second additive compound of the Jacobian of a nonlinear system. It then shows that exponential stability of the latter rules out limit cycles, quasi-periodic solutions, and broad classes of oscillatory behavior. The focus then turns ton nonlinear equations arising in biological interaction networks with general kinetics, the paper shows that the dynamics of the variational system can be embedded in a linear differential inclusion. This leads to algorithms for constructing piecewise linear Lyapunov functions to certify global robust non-oscillatory behavior. Finally, the paper applies the new techniques to study several regulated enzymatic cycles where available methods are not able to provide any information about their qualitative global behavior.

Complex networks of biochemical reactions, such as intracellular protein signaling pathways and genetic networks, are often conceptualized in terms of ``modules,'' semi-independent collections of components that perform a well-defined function and which may be incorporated in multiple pathways. However, due to sequestration of molecular messengers during interactions and other effects, collectively referred to as retroactivity, real biochemical systems do not exhibit perfect modularity. Biochemical signaling pathways can be insulated from impedance and competition effects, which inhibit modularity, through enzymatic ``futile cycles'' which consume energy, typically in the form of ATP. We hypothesize that better insulation necessarily requires higher energy consumption. We test this hypothesis through a combined theoretical and computational analysis of a simplified physical model of covalent cycles, using two innovative measures of insulation, as well as a new way to characterize optimal insulation through the balancing of these two measures in a Pareto sense. Our results indicate that indeed better insulation requires more energy. While insulation may facilitate evolution by enabling a modular ``plug and play'' interconnection architecture, allowing for the creation of new behaviors by adding targets to existing pathways, our work suggests that this potential benefit must be balanced against the metabolic costs of insulation necessarily incurred in not affecting the behavior of existing processes.