In computational systems biology, often biochemical reaction networks are modelled using systems of ordinary differential equations derived from mass action kinetics [1]. In such a scenario, the rate of change of concentration of every chemical species (x), denoted by ẋ, can be represented in a compact form as ẋ =Nv where N is the stoichiometric matrix and v denotes the vector of flux through the reaction which constitutes the monomials in the equation system. This gives rise to an equation system which is non-linear with respect to the concentration of species (i.e. the variables of the system). In steady state, ẋ=0 and the system reduces to a polynomial equation system. Solving such a system for zeroes has applications in model reduction, multistationarity analysis and computing steady state concentrations. Unfortunately, the complexity of algorithms to solve such systems is generally exponential in the square of the number of variables, thereby limiting the applicability of computer algebraic methods. In this context, we present an algorithm to solve for the zeros of polynomial equation systems in tropical semiring [2]. Tropical semiring is the set of real numbers where addition is replaced by taking the maximum, and multiplication is replaced by the usual sum. The relationship between the solutions in this semiring to classical polynomial systems solving is through Puiseux series solutions. Our algorithm computes the lowest term of such a series solution using the Newton polygon approach which is also called as tropical equilibrations [3]. The complexity of our algorithm is exponential in the number of variables because of the combinatorial step involved in the algorithm. Nonetheless, in practice our algorithm can scale up to medium sized systems (e.g. number of variables ~ 50) by using a tree pruning step which reduces the number of combinations to explore. It has been shown in recent studies that our calculated tropical equilibrations can facilitate automatic model reduction by identifying the dominating terms in the equation system [3].

References

1. Edelstein-Keshet, Leah. Mathematical models in biology. Vol. 46. Siam, 1988.

2. Sturmfels, Bernd. Solving systems of polynomial equations. No. 97. American Mathematical Soc., 2002.

3. Noel, Vincent, et al. "Tropicalization and tropical equilibration of chemical reactions." arXiv preprint arXiv:1303.3963 (2013).