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Systems biology uses large networks of biochemical reactions to model the functioning of biological cells from the molecular to the cellular scale. The dynamics of dissipative reaction networks with many well separated time scales can be described as a sequence of successive equilibrations of different subsets of variables of the system. Polynomial systems with separation are equilibrated when at least two monomials, of opposite signs, have the same order of magnitude and dominate the others. These equilibrations and the corresponding truncated dynamics, obtained by eliminating the dominated terms, find a natural formulation in tropical analysis and can be used for model reduction.

We propose a new mechanism for robust biological patterning. The mechanism bears analogy to interface dynamics in condensed media. We apply this method to study how gene networks control segmentation of Drosophila. The proposed model is minimal involving only 4 genes and a morphogen gradient. We discuss experimental data for which developmental genes are expressed within domains spatially limited by kinks (interfaces) and the gene interaction scheme contains both weak and strong repulsion. We show how kink-kink interactions can be calculated from the gene interactions and how the gene interaction scheme ensures the control of proportions (size regulation).

We discuss a method of approximate model reduction for networks of biochemical reactions. This method can be applied to networks with polynomial or rational reaction rates and whose parameters are given by their orders of magnitude. In order to obtain reduced models we solve the problem of tropical equilibration that is a system of equations in max-plus algebra. In the case of networks with fast nonlinear cycles we have to compute the tropical equilibrations at least twice, once for the initial system and a second time for an extended system obtained by adding to the full system the differential equations satisfied by the conservation laws of the fast subsystem. Our method can be used for formal model reduction in computational systems biology.

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