By: When is a dynamical system of mass action type? « Hydrobates

When is a dynamical system of mass action type? « Hydrobates — Fri, 01 Jul 2011 06:10:23 +0000

[...] On Tuesday I went to a session on biochemical reaction networks. This included a talk by Gheorghe Craciun with a large expository component which I found enlightening. He raised the question of when a system of ODE with polynomial coefficients can be interpreted as coming from a system of chemical reactions with mass action kinetics. He mentioned a theorem about this and after asking him for details I was able to find a corresponding paper by Hars and Toth. This is in the Colloquia Mathematica Societatis Janos Bolyai, which is a priori not easily accessible. The paper is, however, available as a PDF file on the web page of Janos Toth. A chemical reaction network gives rise to a system of equations of the form where the and are polynomials with positive coefficients. They represent the contributions from reactions where the species with concentration is on the right and left side respectively. The result of Hars and Toth is that any system of this algebraic form can be obtained from a reaction network. It was pointed out by Craciun in his talk that this means that arbitrarily complicated dynamics can be incorporated into systems coming from reaction networks. If we have a system of the form we can replace it by . This changes the system but does not change the orbits of solutions. If, for instance, we start with the Lorenz system with unknowns , and we can simply translate the coordinates so as to move the interesting dynamics into the region where all coordinates are positive and then multiply the result by . This preserves the strange attractor structure. This result may be compared with the Perelson-Wallwork theorem discussed in a previous post. [...]

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By: When is a dynamical system of mass action type? « Hydrobates