Homeostasis in Input-Output Networks: Structure, Classification and Applications
arxiv(2024)
Abstract
Homeostasis is concerned with regulatory mechanisms, present in biological
systems, where some specific variable is kept close to a set value as some
external disturbance affects the system. Mathematically, the notion of
homeostasis can be formalized in terms of an input-output function that maps
the parameter representing the external disturbance to the output variable that
must be kept within a fairly narrow range. This observation inspired the
introduction of the notion of infinitesimal homeostasis, namely, the derivative
of the input-output function is zero at an isolated point. This point of view
allows for the application of methods from singularity theory to characterize
infinitesimal homeostasis points (i.e. critical points of the input-output
function). In this paper we review the infinitesimal approach to the study of
homeostasis in input-output networks. An input-output network is a network with
two distinguished nodes `input' and `output', and the dynamics of the network
determines the corresponding input-output function of the system. This class of
dynamical systems provides an appropriate framework to study homeostasis and
several important biological systems can be formulated in this context.
Moreover, this approach, coupled to graph-theoretic ideas from combinatorial
matrix theory, provides a systematic way for classifying different types of
homeostasis (homeostatic mechanisms) in input-output networks, in terms of the
network topology. In turn, this leads to new mathematical concepts, such as,
homeostasis subnetworks, homeostasis patterns, homeostasis mode interaction. We
illustrate the usefulness of this theory with several biological examples:
biochemical networks, chemical reaction networks (CRN), gene regulatory
networks (GRN), Intracellular metal ion regulation and so on.
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