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.

Mass-action kinetics and its generalizations appear in mathematical models of (bio-)chemical reaction networks, population dynamics, and epidemiology. The dynamical systems arising from directed graphs are generally non-linear and difficult to analyze. One approach to studying them is to find conditions on the network which either imply or preclude certain dynamical properties. For example, a vertex-balanced steady state for a generalized mass-action system is a state where the net flux through every vertex of the graph is zero. In particular, such steady states admit a monomial parametrization. The problem of existence and uniqueness of vertex-balanced steady states can be reformulated in two different ways, one of which is related to Birch's theorem in statistics, and the other one to the bijectivity of generalized polynomial maps, similar to maps appearing in geometric modelling. We present a generalization of Birch's theorem, by providing a sufficient condition for the existence and uniqueness of vertex-balanced steady states.

Periodic gene expression dynamics are key to cell and organism physiology. Studies of oscillatory expression have focused on networks with intuitive regulatory negative feedback loops, leaving unknown whether other common biochemical reactions can produce oscillations. Oscillation and noise have been proposed to support mammalian progenitor cells’ capacity to restore heterogenous, multimodal expression from extreme subpopulations, but underlying networks and specific roles of noise remained elusive. We use mass-action-based models to show that regulated RNA degradation involving as few as two RNA species—applicable to nearly half of human protein-coding genes—can generate sustained oscillations without explicit feedback. Diverging oscillation periods synergize with noise to robustly restore cell populations’ bimodal expression on timescales of days. The global bifurcation organizing this divergence relies on an oscillator and bistable switch which cannot be decomposed into two structural modules. Our work reveals surprisingly rich dynamics of post-transcriptional reactions and a potentially widespread mechanism underlying development, tissue regeneration, and cancer cell heterogeneity.

The induced kinetic differential equations of a reaction network endowed with mass action type kinetics is a system of polynomial differential equations. The problem studied here is: Given a system of polynomial differential equations, is it possible to find a network which induces these equations; in other words: is it possible to find a kinetic realization of this system of differential equations? If yes, can we find a network with some chemically relevant properties (implying also important dynamic consequences), such as reversibility, weak reversibility, zero deficiency, detailed balancing, complex balancing, mass conservation, etc.? The constructive answers presented to a series of questions of the above type are useful when fitting differential equations to datasets, or when trying to find out the dynamic behavior of the solutions of differential equations. It turns out that some of these results can be applied when trying to solve seemingly unrelated mathematical problems, like the existence of positive solutions to algebraic equations.2. There is an internal requirement within this branch of science: One should like to know as much as possible about the structure of differential equations that arise from modelling a chemical system.3. Given a system of polynomial differential equations in any field of pure or applied mathematics, one may wish to have statements on stability or oscillations, similar to those offered by the Horn-Jackson Theorem [26], Zero Deficiency Theorem [21], Volpert's theorem [50], or the Global Attractor Conjecture, where several cases have been proven [3,12,23,35] 1 . Then it comes in handy to see that the system of differential equations of interest belongs to a well behaving class.4. Lastly, results of formal reaction kinetics (to use an expression introduced in [5, 6]), e.g. on the existence of stationary points, may offer alternative methods for solving problems in algebraic geometry [17,32]. For instance, one might be able to show the existence of positive roots of a polynomial if the system of polynomial equations is known to be the right hand side of the induced kinetic differential equation of a reversible or weakly reversible reaction network [7].The structure of our paper is as follows. Section 2 introduces the essential concepts of reaction networks and mass action systems. Section 3 formulates the problem we are interested in, that of realizability of kinetic differential equations. Section 4 treats two special cases: finding realizations for compartmental models (defined later) and weakly reversible networks. Section 5 focuses on the general problem of realizability. We first review existing algorithms available. Then we outline several procedures that modify a reaction network while preserving the system of differential equations, including adding and removing vertices from the reaction graph. Section 6 explores the relation between weakly reversible and complex balanced realizations. Here we work with families of symbolic kinetic differential equations. In Section 7, w...

Very often, models in biology, chemistry, physics and engineering are systems of polynomial or power-law ordinary differential equations, arising from a reaction network. Such dynamical systems can be generated by many different reaction networks. On the other hand, networks with special properties (such as reversibility or weak reversibility) are known or conjectured to give rise to dynamical systems that have special properties: existence of positive steady states, persistence, permanence, and (for well-chosen parameters) complex balancing or detailed balancing. These last two are related to thermodynamic equilibrium, and therefore the positive steady states are unique and stable. We describe a computationally efficient characterization of polynomial or powerlaw dynamical systems that can be obtained as complex-balanced, detailed-balanced, weakly reversible, and reversible mass-action systems. redirect the reaction z → y * . Instead of the reaction z → y * with flux J z→y * , we have M reactions z → y j with fluxes J ′ z→y j = J y * →y j . Let (G ′ , J ′ ) denote this newest flux system. Recall that flux equivalence means Equation (11) holds at each vertex of G and G ′ . Here we only need to look at the vertex z to show that (G ′ , J ′ ) ∼ (G, J ). Note that y j − z = w j − w 0 . From P (G,J) (y * ) = 0, we also have M j=1 J y * →y j = J z→y * . Thus, the weighted sum of vectors coming out of z isFinally, we prove that the potentials are unchanged. Trivially P (G,J ) (y * ) = P (G ′ ,J ′ ) (y * ) = 0. Also P (G,J) (y j ) = J y * →y j = J ′ z→y j = P (G ′ ,J ′ ) (y j ) for j = 1, 2, . . . , M . Last but not least,J y * →y j = J z→y * = −P (G,J) (z).We have shown that the resulting flux system (G ′ , J ′ ) is flux equivalent to the original flux system (G, J ), and the potential at each vertex is preserved.Remark. In Lemma 4.3, the source vertex z may not be distinct from y j .