Multi-valued characteristics and Morse decompositions

A rapid growth of molecular and systems biology in recent years challenges mathematicians to develop robust modeling and analytical tools for this area. We combine a theory of monotone input-output systems with a classical theory of Morse decompositions in the context of ordinary differential equations models of biochemical reactions. We show that a multi-valued input-output characteristic can be used to define non-trivial Morse decompositions which provide information about a global structure of the attractor. The previous work on input-output characteristics is shown to apply locally to individual Morse sets and is seamlessly incorporated into our global theory. We apply our tools to a model of cell cycle maintenance. We show that changing the strength of the negative feedback loop can lead to cessation of cell cycle in two different ways: it can either lead to globally attracting equilibrium or to a pair of equilibria that attract almost all solutions. (C) 2009 Elsevier Inc. All rights reserved.