Onset of birhythmicity in a regulated biochemical system

These notes are devoted to showing that the topological degree is a useful tool in the study of the properties of stability of periodic solutions of a scalar, time-dependent differential equation of Newtons type. Two different situations are considered depending on whether the equation has damping or not. When there is linear friction the asymptotic stability of a periodic solution can be characterized in terms of degree. When there is no friction the equation has a hamiltonian structure and some connections between Lyapunov stability and degree are discussed. These results are applied in two different directions: to prove that some classical methods in the theory of existence lead to instability (minimization of the action functional, upper and lower solutions) and to study the stability of the solutions of a concrete class of equations (equations of pendulum-type).

Some applications of the topological degree to stability theory

The glycolytic oscillator, mainly studied in yeast, is described with respect to its overall dynamic and biochemical properties and the kinetics of its master enzyme phosphofructokinase. Biochemical and kinetic analyses are complemented by analysis of mathematical models. In addition to the discussion of structure and function of the glycolytic oscillator under homogeneous spatial conditions, recent theoretical and biochemical experiments demonstrating spatial pattern formation are also discussed, followed by a presentation of physiological viewpoints.

The glycolytic oscillator.

In this paper, we prove that a stable isolated fixed point of an orientation preserving local homeomorphism onR2 has fixed point index 1. We also give a number of applications to differential equations. In particular, we deduce that a number of existence methods for producing periodic solutions of differential equations in the plane always produce unstable solutions.

The index of lyapunov stable fixed points in two dimensions

Oscillatory isozymes as the simplest model for coupled biochemical oscillators

Many models of open enzyme regulated systems are given by two first order differential equations of a certain type. We present a set of biochemically reasonable conditions for the rate laws which imply the existence of observable oscillations. A simple positive feedback mechanism is discussed as an example. Results on the allosteric model of glycolytic oscillations, which were formerly obtained by linear stability analysis and computer simulation, are extended and verified by mathematical proofs.

The existence of stable limit cycles for enzyme catalyzed reactions with positive feedback

Nonuniqueness of stable limit cycles in a class of enzyme catalyzed reactions

Stable equilibria and vector field index

For a class of dynamical systems of the form 
$$\dot x = q{\text{(}}x{\text{)}} - u{\text{(}}x,y{\text{)}},\;\;\dot y = \varepsilon {\text{(}}v{\text{(}}x,y) - r(y))$$

 , we prove boundedness of all solutions in the positive time direction. We discuss the existence of stable limit cycles for the simplest autocatalytic reaction involving two internal and two external reactants, as well as for a number of other models arising in applications.

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Dieter Erle

Papers

The existence of stable limit cycles for enzyme catalyzed reactions with positive feedback

Nonuniqueness of stable limit cycles in a class of enzyme catalyzed reactions

Stable equilibria and vector field index

A boundedness theorem with application to oscillation of autocatalytic chemical reactions

Some non-linear equivariant sphere bundles