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Unformatted text preview: Chemical reaction network theory for in-silico biologists Jeremy Gunawardena Bauer Center for Genomics Research Harvard University 7 Divinity Avenue, Cambridge, MA 02138, USA email@example.com June 20, 2003 Contents 1 Introduction 1 2 Chemical reaction networks 2 3 Linearity in chemical reaction networks 6 4 The kernel of A 9 5 The deficiency formula and some biological examples 14 6 Fixed points for which A ( c ) = 0 16 7 Existence of fixed points 23 1 1 Introduction Mathematical models of intra-cellular biological systems are often based on systems of nonlinear ordinary differential equations (ODEs). Recent studies of some signalling cascades using both com- putational analysis of an ODE model and experimental manipulation of the signalling pathway have revealed the presence of multiple fixed points, [3, 7, 21]. While biological systems evidently exhibit more complex behaviours, such as oscillations, , the significance of fixed points in biolog- ical dynamics has been a recurring theme in the literature at least as far back as Max Delbrucks observations on cytoplasmic inheritance, [9, 24, 33]. It would be helpful if there were mathematical theorems that tell us whether a relevant system of ODEs has any fixed points. Most mathematicians would regard this as hopeless, given the nonlinearity of the equations and the complexity of the systems. The biological studies referred to above were all based on computational simulation. However, there are at least two contexts in which potentially relevant mathematical results have been established. The first is the recent proof by Christophe Soul e of Ren e Thomas longstanding conjecture about multiple fixed points, [31, 32]. This result gives an elegant necessary condition for multiple nondegenerate fixed points of a system of nonlinear ODEs. The second is Chemical Reaction Network Theory (CRNT), which establishes several theorems about systems of nonlinear ODEs which describe the behaviour of reactor vessels used in chemical engineering. These systems are derived from mass-action kinetics, which is probably a more realistic assumption in chemical engineering than in biology (more about this below). In these notes we discuss CRNT and leave the work of Soul e and Thomas for later consideration. CRNT has been developed over the last 30 years, initially through the work of Horn and Jackson and subsequently by Martin Feinberg and his students. The theory introduces new concepts, such as the deficiency of a reaction network, and gives conditions on such networks for the existence, uniqueness, multiplicity and stability of fixed points. These conclusions sometimes hold irrespective of the values of the parameters in the system, which should be of interest to those who associate such behaviour with biological robustness, [5, 34]. The deficiency 0 theorem is proved in the first four lectures of  and draws on earlier work of Horn and Jackson, , and Feinberg and Horn, .lectures of  and draws on earlier work of Horn and Jackson, , and Feinberg and Horn, ....
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