We consider solutions to parametrized systems of generalized polynomial equations (with real exponents) in $n$ positive variables, involving $m$ monomials with positive parameters; that is, $x \in \mathbb R^n_>$ such that ${A \, (c \circ x^B)=0}$ with coefficient matrix $A \in \mathbb R^{l \times m}$, exponent matrix $B \in \mathbb R^{n \times m}$, and parameter vector $c \in \mathbb R^m_>$...
Glycolysis, the process by which energy is extracted from sugar, is known to exhibit oscillations and it has been shown experimentally that these arise in a reaction catalysed by the enzyme phosphofructokinase. Selkov introduced a simple model for this phenomenon which is a system of two ODE with mass action kinetics (in what follows the MA system). It is derived by a limiting process from a...
Reaction-diffusion equations provide a framework for understanding how spatial patterns emerge from biochemical networks. In quantitative biology, however, parameter values are frequently accompanied by large confidence intervals due to measurement uncertainty and limited experimental repetitions. This necessitates the study of entire families of parametrized PDEs to determine their...
Time delays are often present in natural and technological processes, and can be essential for the precise understanding and description of important phenomena. On the other hand, chemical reaction networks (CRNs) provide a general framework for describing general nonnegative nonlinear dynamics. It is known that complex balance is a property of fundamental importance, which guarantees a strong...
Fitting functions to data with nonlinear parameter dependence is challenging, particularly for differential equations common in chemical kinetics, due to the need for good initial estimates. We propose a method to automatically generate robust initial estimates for mass-action kinetic ODEs (where the right-hand side is linear in parameters). Illustrative examples show these estimates often...
Conditions for Turing instability in reaction networks involving two interacting species are well understood and typically require self-activation or autocatalysis. However, general criteria for the emergence of Turing instabilities in large-scale reaction networks are less studied. We consider a reaction network with an arbitrary number of species, in which only a single species diffuses. We...
In the bifurcation analysis of biochemical reaction networks, the determinant of the Jacobian of the corresponding ODE system plays a crucial role. When network parameters are treated symbolically, computing this determinant becomes challenging, even for moderately sized systems. This computation can be simplified if one decomposes the network into subnetworks such that the Jacobi determinant...
In the practical analysis of chemical reaction networks, it is often assumed that no explicit catalysts exist, i.e., that species appear both as a reactant and a product in the same reaction.
In this case, the stoichiometric matrix uniquely identifies the corresponding reaction network (RN), and methods from matrix theory naturally apply. However, catalysis plays an important role, in...
Mathematical aspects in the analysis of reaction networks range from graph theory and dynamical systems (ODEs and PDEs) to positive algebraic geometry and inverse problems. In recent years, substantial progress has been made in the mathematics of reaction networks, and the proposed minisymposium will bring together leading experts to present both theoretical advances and analyses of concrete...
