AlgSignSen · The Algebraic Geometry of Chemical Reaction Networks: Structural conditions for uniquely determined Sign-sensitivities.

The Algebraic Geometry of Chemical Reaction Networks: Structural conditions for uniquely determined Sign-sensitivities.

Many biological processes of communication between cells, activation and deactivation of cellular processes and some regulatory processes can be modelled by means of dynamical systems, determined by the interaction of the species involved in each process. These species can be, for instance, different molecules or proteins. The interactions between them can be represented by directed graphs which are known in the biochemical context as Reaction Networks (RNs) [F]. The project AlgSignSen was designed to introduce some new ideas from Algebraic Geometry to the study of RNs, and in particular to the sensitivity analysis at steady state. As an example, the following RN represents a possible way of interaction between two species, A and B: A+B→2B B→A Equations that describe the evolution of such a process can be given, in terms of the concentrations of the species involved and some constants called reaction rate parameters. These rates carry information about the speed at which the system evolves. Interesting information about this evolution can be obtained by looking at the states (concentrations) in which the system is in equilibrium, the so-called steady states. Under certain assumptions, these states are the common solutions of a finite collection of polynomial equations, where the variables encode the concentrations of species and the coefficients depend on the reaction rates. For the former network, we can obtain the polynomial equations -k_1 ab+k_2 b=0, k_1 ab-k_2 b=0. The set of common solutions to these equations can be seen as a geometrical object, and geometrical objects which are defined by polynomial equations are the subject of study of Algebraic Geometry. The subset of those solutions which correspond to positive concentrations is called the positive steady state variety. The steady state (or states) that can be reached by a system depends on the initial concentrations of the involved species: the initial amounts of the species evolve according to the specific dynamics of the system, and eventually reach a steady state. Different initial concentrations lead, under the same environmental conditions, to different steady states. An interesting question is: how do small changes in the initial concentrations affect the steady state reached in each case? This is what we refer to as the sensitivity of the system to (small) perturbations of the initial concentrations. A species X can see its concentration at steady state increased, decreased or untouched by the effect of a perturbation. This is the kind of sensitivities that we have investigated in the AlgSignSen project. When X is not affected (at steady state) by any perturbation of initial concentrations, we say that the system has zero sensitivity for X. [F] M. Feinberg. Foundations of Chemical Reaction Network Theory, Applied Mathematical Sciences, 202. Springer International Publishing, 2019

Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз