Stability in generic mitochondrial models

Abstract

In this paper, we use a variety of mathematical techniques to explore existence, local stability, and global stability of equilibria in abstract models of mitochondrial metabolism. The class of models constructed is defined by the biological description of the system, with minimal mathematical assumptions. The key features are an electron transport chain coupled to a process of charge translocation across a membrane. In the absence of charge translocation these models have previously been shown to behave in a very simple manner with a single, globally stable equilibrium. We show that with charge translocation the conclusion about a unique equilibrium remains true, but local and global stability do not necessarily follow. In sufficiently low dimensions -i.e. for short electron transport chains -it is possible to make claims about local and global stability of the equilibrium. On the other hand, for longer chains, these general claims are no longer valid. Some particular conditions which ensure stability of the equilibrium for chains of arbitrary length are presented.

P-matrices and signed digraphs

“…A well-known result of Gale and Nikaido [2], whose extensions and corollaries are discussed in [3], states that a differentiable function on a rectangular domain in R n with P -matrix Jacobian is injective. This result has a number of practical applications -see for example [4,5].…”

Independent sets on the Towers of Hanoi graphs

“…On one hand, all these graph invariants reflect the structure of a graph in some way, and therefore, some of them are even of interest in theoretical chemistry for the study of molecular graphs (see [32,38]). For example, the number of independent sets is called Merrifield-Simmons index, the number of matchings is known as Hosoya index in chemistry.…”