Spatial patterns in coupled biochemical oscillators

SummaryThe effects of diffusion on the dynamics of biochemical oscillators are investigated for general kinetic mechanisms and for a simplified model of glycolysis. When diffusion is sufficiently rapid a population of oscillators relaxes to a globally-synchronized oscillation, but when diffusion of one or more species is slow enough, the synchronized oscillation can be unstable and a nonuniform steady state or an asynchronous oscillation can arise. The significance of these results vis-a-vis models of contact inhibition and zonation patterns is discussed.

[1]  A. Winfree.  Polymorphic pattern formation in the fungus Nectria. , 1973, Journal of theoretical biology.

[2]  Bernard J. Matkowsky,et al.  A simple nonlinear dynamic stability problem , 1970 .

[3]  S. Kauffman,et al.  Control of mitosis by a continuous biochemical oscillation: Synchronization; spatially inhomogeneous oscillations , 1975 .

[4]  L. E. Scriven,et al.  Interactions of reaction and diffusion in open systems , 1969 .

[5]  G. Sell,et al.  The Hopf Bifurcation and Its Applications , 1976 .

[6]  H. Othmer,et al.  The effects of cell density and metabolite flux on cellular dynamics , 1978, Journal of mathematical biology.

[7]  A. Goldbeter,et al.  An allosteric enzyme model with positive feedback applied to glycolytic oscillations , 1976 .

[8]  A. C. Burton,et al.  The behaviour of coupled biochemical oscillators as a model of contact inhibition of cellular division. , 1973, Journal of theoretical biology.

[9]  D. Hoff,et al.  LARGE TIME BEHAVIOR OF SOLUTIONS OF SYSTEMS OF NONLINEAR REACTION-DIFFUSION EQUATIONS* , 1978 .

[10]  P. Morse,et al.  Methods of theoretical physics , 1955 .

[11]  Avner Friedman,et al.  Partial differential equations , 1969 .

[12]  Bifurcation analysis of nonlinear reaction-diffusion equations—I. Evolution equations and the steady state solutions , 1975 .

[13]  Theodosios Pavlidis,et al.  Biological Oscillators: Their Mathematical Analysis , 1973 .

[14]  W. Walter.  Differential and Integral Inequalities , 1970 .

[15]  L E Scriven,et al.  Instability and dynamic pattern in cellular networks. , 1971, Journal of theoretical biology.

[16]  B. Hess,et al.  Oscillatory phenomena in biochemistry. , 1971, Annual review of biochemistry.

[17]  J. Calvin Giddings,et al.  AN EXPANDING ROLE FOR GAS CHROMATOGRAPHY: MEASUREMENT OF PHYSICAL AND CHEMICAL PARAMETERS , 1967 .

[18]  J. Auchmuty.  Positivity for elliptic and parabolic systems , 1978, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[19]  S. Smale,et al.  A Mathematical Model of Two Cells Via Turing’s Equation , 1976 .

[20]  G. Franck,et al.  Self-synchronization in yeast and other fungi. , 1973, International journal of chronobiology.

[21]  R. Sani,et al.  On the stability of the steady state in systems of coupled diffusion and reaction , 1973 .

[22]  J. Kevorkian,et al.  The two variable expansion procedure for the approximate solution of certain nonlinear differential equations , 1966 .

[23]  E. Sel'kov,et al.  Self-oscillations in glycolysis. 1. A simple kinetic model. , 1968, European journal of biochemistry.

[24]  J. Keener.  SECONDARY BIFURCATION IN NONLINEAR DIFFUSION REACTION EQUATIONS. , 1976 .

[25]  Aubrey B. Poore,et al.  On the theory and application of the Hopf-Friedrichs bifurcation theory , 1976 .

[26]  E. D. Conway,et al.  A comparison technique for systems of reaction-diffusion equations , 1977 .

[27]  H. Meinhardt,et al.  A model of pattern formation in insect embryogenesis. , 1977, Journal of cell science.

[28]  Hans G. Othmer,et al.  On the significance of finite propagation speeds in multicomponent reacting systems , 1976 .