Local bifurcation theory typically deals with the response of a degenerate but isolated equilibrium state or periodic orbit of a dynamical system to perturbations controlled by one or more independent parameters, and characteristically uses tools fro
m singularity theory. There are many situations, however, in which the equilibrium state or periodic orbit is not isolated but belongs to a manifold $S$ of such states, typically as a result of continuous symmetries in the problem. In this case the bifurcation analysis requires a combination of local and global methods, and is most tractable in the case of normal nondegeneracy, that is when the degeneracy is only along $S$ itself and the system is nondegenerate in directions normal to $S$. In this paper we consider the consequences of relaxing normal nondegeneracy, which can generically occur within 1-parameter families of such systems. We pay particular attention to the simplest but important case where $dim S=1$ and where the normal degeneracy occurs with corank 1. Our main focus is on uniform degeneracy along $S$, although we also consider aspects of the branching structure for solutions when the degeneracy varies at different places on $S$. The tools are those of singularity theory adapted to global topology of $S$, which allow us to explain the bifurcation geometry in natural way. In particular, we extend and give a clear geometric setting for earlier analytical results of Hale and Taboas.
In most natural sciences there is currently the insight that it is necessary to bridge gaps between different processes which can be observed on different scales. This is especially true in the field of chemical reactions where the abilities to form
bonds between different types of atoms and molecules create much of the properties we experience in our everyday life, especially in all biological activity. There are essentially two types of processes related to biochemical reaction networks, the interactions among molecules and interactions involving their conformational changes, so in a sense, their internal state. The first type of processes can be conveniently approximated by the so-called mass-action kinetics, but this is not necessarily so for the second kind where molecular states do not define any kind of density or concentration. In this paper we demonstrate the necessity to study reaction networks in a stochastic formulation for which we can construct a coherent approximation in terms of specific space-time scales and the number of particles. The continuum limit procedure naturally creates equations of Fokker-Planck type where the evolution of the concentration occurs on a slower time scale when compared to the evolution of the conformational changes, for example triggered by binding or unbinding events with other (typically smaller) molecules. We apply the asymptotic theory to derive the effective, i.e. macroscopic dynamics of the biochemical reaction system. The theory can also be applied to other processes where entities can be described by finitely many internal states, with changes of states occuring by arrival of other entities described by a birth-death process.
The paper analyses stochastic systems describing reacting molecular systems with a combination of two types of state spaces, a finite-dimensional, and an infinite dimenional part. As a typical situation consider the interaction of larger macro-molecu
les, finite and small in numbers per cell (like protein complexes), with smaller, very abundant molecules, for example metabolites. We study the construction of the continuum approximation of the associated Master Equation (ME) by using the Trotter approximation [27]. The continuum limit shows regimes where the finite degrees of freedom evolve faster than the infinite ones. Then we develop a rigourous asymptotic adiabatic theory upon the condition that the jump process arising from the finite degrees of freedom of the Markov Chain (MC, typically describing conformational changes of the macro-molecules) occurs with large frequency. In a second part of this work, the theory is applied to derive typical enzyme kinetics in an alternative way and interpretation within this framework.
In this paper we consider deterministic limits of molecular stochastic systems with finite and infinite degrees of freedom. The method to obtain the deterministic vector field is based on the continuum limit of such microscopic systems which has been
derived in [11]. With the aid of the theory we finally develop a new approach for molecular systems that describe typical enzyme kinetics or other interactions between molecular machines like genetic elements and smaller communicating molecules. In contrast to the literature on enzyme kinetics the resulting deterministic functional responses are not derived by time-scale arguments on the macroscopic level, but are a result of time scaling transition rates on the discrete microscopic level. We present several examples of common functional responses found in the literature, like Michaelis-Menten and Hill equation. We finally give examples of more complex but typical macro-molecular machinery.
