Reduktion und asymptotische Reduktion von Reaktionsgleichungen
- Reduction and asymptotic reduction of reaction equations
Goeke, Alexandra; Walcher, Sebastian (Thesis advisor)
Dissertation / PhD Thesis
Aachen, Techn. Hochsch., Diss., 2013
This thesis deals with ordinary differential equations which model reacting systems obeying mass-action kinetics. We focus on the mathematical characterization and analysis of quasi-steady state. Frequently, singular perturbation methods are used to model this biochemical phenomenon. Essentially, the reduction of dimension goes back to Tikhonov and Fenichel and their fundamental results for singularly perturbed systems. In general, an explicit application of Tikhonov's theorem is impossible. But there is a way to directly compute a reduced system, which does not require a so-called Tikhonov standard form. Substantially this is a result of the geometrical interpretation of Fenichel. To determine a reduced system, Nöthen and Walcher give a straightforward procedure based on a projection, but this method is not feasible for practical purposes. We present a different approach in Chapter 2 for differential equations with rational right hand side, especially for reaction equations. Fundamentally, the method is based on a suitable local representation of the Slow Manifold. We construct a product representation for the slow system, which leads to a simple formula for the Tikhonov-Fenichel reduction. This formula only requires elementary algebraic operations. In particular, for slow and fast reactions the stoichiometry of the fast system leads to a canonical product representation. Moreover, we specify locally unique initial values for the reduced system on the slow manifold to complete the reduction. In general, these cannot be computed exactly. Therefore we develop an iterative procedure to generate approximate values. One area of application concerns enzymatic reactions with quasi-steady state assumptions for intermediates whose reaction equations are reduced to a real affine subspace. As described in Chapter 4, the special structure of the slow manifold yields a simplification of the reduction formula. Moreover, we are able to determine a domain of validity for an ad-hoc method that is commonly used in chemical literature. The reduction procedure in Chapter 2 is neither based on an explicit transformation of a system in Tikhonov standard form, nor does it make use of a parametrization of the slow manifold. In this respect, the new approach extends well-known reduction formulas in the literature. Frequently, a reduced equation is computable only via this extension. Given that a Tikhonov-Fenichel reduction exists, we are able to calculate a reduced equation explicitly for every autonomous differential equation with rational right hand side. Applications in Chapters 5 and 6 include generalizations and corrections of quasi-steady state reductions for several important examples. In particular, we are able to reduce reversible model extensions and higher dimensional systems. Moreover, Chapter 7 examines reaction-diffusion systems and the determination of a Tikhonov-Fenichel reduction for spatial discretization schemes. As a consequence, we obtain a heuristic to find candidates for reduced partial differential equations. Since there exists no conterpart to the Tikhonov-Fenichel reduction for systems of partial differential equations in the literature, even this step provides an approach to the solution of a nontrivial problem. A systematic method to classify all possible Tikhonov-Fenichel reductions for a parameter-dependent system is the central result of Part II. Thus, one is interested in points of phase-parameter space such that the conditions for Tikhonov-Fenichel hold for small perturbations along any fixed direction in the parameter space. We take this as a defining property of a so called Tikhonov parameter value. Moreover, for a strong Tikhonov parameter value there is an attractive slow manifold. The mathematical conditions for the existence of a reduction yield exact criteria for Tikhonov parameter values. Consequently, methods of algorithmic algebra are applicable. Among other results, all Tikhonov parameter values for the classical irreversible and reversible Michaelis-Menten dynamics are determined. In particular, in the irreversible Micha-elis--Menten-scheme a reduction with a small parameter is possible only for the well-known ones in the literature. Concerning the structure we show that the strong Tikhonov parameter values of a system form a semi-algebraic set. Furthermore, it is easy to apply necessary conditions for more complicated systems. Our computations for the applications in Chapter 9 show that only a small number of reductions is possible (apart from a solution preserving transformation).