János Tóth

Global controllability of chemical reactions

Controllability of chemical reactions is an important problem in chemical engineering science. In control theory, analysis of the controllability of linear systems is well-founded, however the dynamics of chemical reactions is usually nonlinear. Global controllability properties of chemical reactions are analyzed here based on the Lie-algebra of the vector fields associated to the reaction steps. A chemical reaction is controllable almost everywhere if all the reaction rate coefficients can be used as control inputs. The problem where one can not control all the reaction rate coefficients is also analyzed. The reaction steps whose reaction rate coefficients need to be control inputs are identified. A general definition of consecutive reactions is given, and it turns out that they are controllable almost everywhere by using the reaction rate coefficient of only one reaction step as control input.

Microscopic reversibility or detailed balance in ion channel models

Mass action type deterministic kinetic models of ion channels are usually constructed in such a way as to obey the principle of detailed balance (or, microscopic reversibility) for two reasons: first, the authors aspire to have models harmonizing with thermodynamics, second, the conditions to ensure detailed balance reduce the number of reaction rate coefficients to be measured. We investigate a series of ion channel models which are asserted to obey detailed balance, however, these models violate mass conservation and in their case only the necessary conditions (the so-called circuit conditions) are taken into account. We show that ion channel models have a very specific structure which makes the consequences true in spite of the imprecise arguments. First, we transform the models into mass conserving ones, second, we show that the full set of conditions ensuring detailed balance (formulated by Feinberg) leads to the same relations for the reaction rate constants in these special cases, both for the original models and the transformed ones.

Structural analysis of combustion mechanisms

Thirty-nine detailed mechanisms for combustion of hydrogen, carbon monoxide and methanol are investigated using ReactionKinetics, a Mathematica based package published earlier. Our methods involved mainly structural and graph theoretical approaches as well as techniques which are related to the time evolution of the considered mechanisms. Our investigations support the view that the hydrogen mechanisms tend to take on a final form in these days. CO combustion mechanisms, however, showed a larger variety both in species and in reaction steps. There exist only a few mechanisms directly developed to describe methanol combustion (mechanisms developed for other purposes may contain a submechanism for methanol combustion); the big differences between them shows that the modeling community is only at the very beginning of exploring this process. Most of our results do not depend on the choice of reaction rate coefficients, the methods only use the underlying sets of reaction steps, hence they are robust and general. These investigations can be used before or in parallel with usual numerical investigations, such as pathway analysis, sensitivity analysis, parameter estimation or simulation. The package and the methods may be useful for automatic mechanism generations, testing, comparing and reduction of mechanisms as well, especially in the case of large systems.

The Induced Kinetic Differential Equation

This chapter focuses on the formal analysis of the most relevant model describing the time evolution of concentrations of reactions. This model is deterministic; both the time and the state space are continuous. The corresponding mathematical object is a system of ordinary differential equations, called the induced kinetic differential equation of the reaction in question. This model can be constructed in different ways and has different forms; we are going to study these forms in the present chapter. Special emphasis is laid on those reactions which are endowed with mass action type kinetics: In these cases the induced kinetic differential equation is a special polynomial differential equation. The right-hand side of the induced kinetic differential equation of reactions can be factorized in many different ways, and these factorizations facilitate the analyses of the stationary states (see Chap. 7) and the dynamic behavior (see Chap. 8), as well; furthermore, they help solve some of the inverse problems of reaction kinetics (see Chap. 11). A remark with far-reaching consequences is that although the induced kinetic differential equation is a nonlinear differential equation, it has many linear features. At the end of the chapter, we introduce and discuss models with diffusion, i.e. space inhomogeneities, and those in which the change and interplay with chemistry of a further physical parameter, temperature, is also taken into consideration.