We introduce the notion of non-oscillation, propose a constructive method for its robust verification, and study its application to biological interaction networks (also known as, chemical reaction networks). We begin by revisiting Muldowneys 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. We show that exponential stability of the latter rules out limit cycles, quasi-periodic solutions, and broad classes of oscillatory behavior. We focus then on nonlinear equations arising in biological interaction networks with general kinetics, and we show that the dynamics of the aforementioned variational system can be embedded in a linear differential inclusion. We then propose algorithms for constructing piecewise linear Lyapunov functions to certify global robust non-oscillatory behavior. Finally, we apply our techniques to study several regulated enzymatic cycles where available methods are not able to provide any information about their qualitative global behavior.
A dynamical system entrains to a periodic input if its state converges globally to an attractor with the same period. In particular, for a constant input the state converges to a unique equilibrium point for any initial condition. We consider the pro
blem of maximizing a weighted average of the systems output along the periodic attractor. The gain of entrainment is the benefit achieved by using a non-constant periodic input relative to a constant input with the same time average. Such a problem amounts to optimal allocation of resources in a periodic manner. We formulate this problem as a periodic optimal control problem which can be analyzed by means of the Pontryagin maximum principle or solved numerically via powerful software packages. We then apply our framework to a class of occupancy models that appear frequently in biological synthesis systems and other applications. We show that, perhaps surprisingly, constant inputs are optimal for various architectures. This suggests that the presence of non-constant periodic signals, which frequently appear in biological occupancy systems, is a signature of an underlying time-varying objective functional being optimized.
We study in this paper certain properties of the responses of dynamical systems to external inputs. The motivation arises from molecular systems biology. and, in particular, the recent discovery of an important transient property, related to Webers l
aw in psychophysics: fold-change detection in adapting systems, the property that scale uncertainty does not affect responses. FCD appears to play an important role in key signaling transduction mechanisms in eukaryotes, including the ERK and Wnt pathways, as well as in E.coli and possibly other prokaryotic chemotaxis pathways. In this paper, we provide further theoretical results regarding this property. Far more generally, we develop a necessary and sufficient characterization of adapting systems whose transient behaviors are invariant under the action of a set (often, a group) of symmetries in their sensory field. A particular instance is FCD, which amounts to invariance under the action of the multiplicative group of positive real numbers. Our main result is framed in terms of a notion which extends equivariant actions of compact Lie groups. Its proof relies upon control theoretic tools, and in particular the uniqueness theorem for minimal realizations.
