Mathematical Analysis of Chemical Reaction Systems

Abstract

The use of mathematical methods for the analysis of chemical reaction systems has a very long history, and involves many types of models: deterministic versus stochastic, continuous versus discrete, and homogeneous versus spatially distributed. Here we focus on mathematical models based on deterministic mass‐action kinetics. These models are systems of coupled nonlinear differential equations on the positive orthant. We explain how mathematical properties of the solutions of mass‐action systems are strongly related to key properties of the networks of chemical reactions that generate them, such as specific versions of reversibility and feedback interactions.

Gain and Loss of Function mutations in biological chemical reaction networks: a mathematical model with application to colorectal cancer cells

“…, x n ) T ∈ R n + , where the upper T denotes transposition, R + is the set of non-negative real number, and the generic component x i is the molar concentration (nM) of the species A i . According to this chemical interpretation, x is also called concentration vector (Yu and Craciun, 2018). We assume that the law of mass action holds: when two or more reactants are involved in a reaction, the reaction rate is proportional to the product of their concentrations.…”

“…This shows that x(t) − x 0 is a linear combination of the reaction vectors, with time-dependent coefficients. Therefore x(t) − x 0 belongs to a vector space of dimension equal to rank(S) defined by the image of the stoichiometric matrix (Feinberg, 1987(Feinberg, , 1995Yu and Craciun, 2018). Accordingly, we provide the following definition of stoichiometric compatibility class (SCC).…”

“…Similarly to what happens for many networks of biological interest, a signaling network may comprise hundreds of reacting chemical species and reactions. Any CRN can be mapped into a system of ordinary differential equations (ODEs) by following standard procedures (Feinberg, 1987;Yu and Craciun, 2018;Chellaboina et al, 2009) thus allowing for simulations of the kinetics of the signaling process in the biologic case (see, e.g., Anderson et al (2019); Roy and Finley (2017)). In principle, the resulting mathematical model is capable of describing healthy physiologic states, and can be adapted to simulate individual pathological conditions associated to mutations.…”

Metabolite-mediated modelling of microbial community dynamics captures emergent behaviour more effectively than species–species modelling

“…The generalized Lotka-Volterra model faithfully models direct pairwise interactions between agents (e.g. physically interacting organisms or reactants in an industrial reactor) under the assumption of mass action kinetics [45,25], but does not include any possible environmental variation in interaction. It is therefore a fair representation of species-species interaction in a controlled environment.…”

“…Focusing on species-species interactions is notably the strategy of the popular Lotka-Volterra (LV) model and its generalizations [16,17,18,19]. The Lotka-Volterra model reproduces the dynamics of interacting species according the law of mass action [24,25]. Therefore, it is an appropriate model of direct interaction between species in a well mixed and stable environment.…”