Summary
The 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.
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References
Aldridge, J.: Short-range intercellular communication, biochemical oscillations, and circadian rhythms. In: Hndbk. of Eng. in Med. and Biol. (R. Fleming, ed.) Cleveland: Chemical Rubber Co. 1976
Auchmuty, J. F. G., Nicholis, G.: Bifurcation analysis of nonlinear reaction-diffusion equations. Bull. Math. Biology 37, 323–365 (1975)
Auchmuty, J. F. G.: Positivity for elliptic and parabolic systems. Proc. Roy. Soc. Edin. (To appear, 1977)
Burton, A. C., Canham, P. B.: The behavior of coupled biochemical oscillators as a model of contact inhibition of cellular division. J. Theor. Biol. 39, 555–580 (1973)
Chueh, K. N., Conley, C. C., Smoller, J. A.: Positively invariant regions for systems of nonlinear diffusion equations. Ind. Univ. Math. Jour. 26, 373–392 (1977)
Conway, E., Hoff, D., Smoller, J.: Large time behavior of solutions of systems of nonlinear reaction diffusion equations, to appear in SIAM J. Appl. Math. (1977)
Conway, E. D., Smoller, J. A.: A comparison theorem for systems of reaction-diffusion equations (preprint, 1977)
Eidel'man, S. D.: Parabolic Systems. Amsterdam: North-Holland 1969
Friedman, A.: Partial Differential Equations. New York: Holt, Rinehart and Winston 1969
Georgakis, C., Sani, R. L.: On the stability of the steady state in systems of coupled diffusion and reaction. Arch. Rat. Mech. and Anal. 52, 266–296 (1973)
Gibbs, R. G., Murray, J. D.: On models of oscillations in the glycolytic pathways (preprint, 1977).
Gierer, A., Meinhardt, H.: A theory of biological pattern formation. Kybernetik 12, 30–39 (1972)
Goldbeter, A., Nicolis, G.: An allosteric enzyme model with positive feedback applied to glycolytic oscillations, in Prog. Theor. Biol. 4 (R. Rosen and F. Snell, eds.) New York: Academic Press 1976
Hartman, P.: Ordinary Differential Equations. Baltimore: P. Hartman 1973
Hess, B., Boiteux, A.: Oscillatory phenomena in biochemistry. Ann. Rev. Biochem. 40, 237–258 (1971)
Higgins, J.: Oscillatory reactions. Ind. and Eng. Chem. 59, 19–62 (1967)
Hopf, E.: Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differential Systems. Ber. Math.-Phys. Klasse Sächs. Akad. Wiss. Leipzig. 94, 3–22 (1942)
Keener, J. P.: Secondary bifurcation in nonlinear diffusion reaction equations. Studies Appl. Math. 55, 187–211 (1976)
Kevorkian, J.: The two variable expansion procedure for the approximate solution of certain nonlinear differential equations, in Space Mathematics, part 3: Lectures in Applied Math. 7, Providence; Amer. Math. Soc. 1966
Kraeplein, G., Franck, G.: Self-synchronization in yeast and other fungi. Int. J. Chronobiology 1, 163–172 (1973)
Marsden, J. E., McCracken, M.: The Hopf Bifurcation and its Applications. New York: Springer 1976
Matkowsky, B. J.: A simple nonlinear dynamic stability problem. Bull. Amer. Math. Soc., 76, 620–625 (1970)
Meinhardt, H.: A model of pattern formation in insect embryogenesis. J. Cell. Sci. 23, 117–139 (1977)
Morse, P. M., Feshbach, H.: Methods of Theoretical Physics. New York: McGraw-Hill 1953
Othmer, H. G., Scriven, L. E.: Interactions of reaction and diffusion in open systems. Ind. & Eng. Chem. Fund. 8, 302–313 (1969)
Othmer, H. G., Scriven, L. E.: Instability and dynamic pattern in cellular networks. J. Theor. Biol. 32, 507–537 (1971)
Othmer, H. G.: Temporal oscillations in chemically-reacting systems, in: Proc. IEEE Int. Conf. on Cybernetics and Society. Washington, D.C.: IEEE 1976
Othmer, H. G.: On the significance of finite propagation speeds in multicomponent reacting systems. J. Chem. Phys. 64, 460–470 (1976)b
Othmer, H. G.: Current problems in pattern formation, in: Lectures on Mathematics in the Life Sciences, (S. A. Levin, ed.) Providence: Amer. Math. Soc. 1977
Othmer, H. G.; Aldridge, J.: The effects of cell density and metabolite flux on cellular dynamics, to appear in J. Math. Biology (1978)
Pavlidis, T.: Biological Oscillators: Their Mathematical Analysis. New York: Academic Press 1973
Poore, A.: On the theory and application of Hopf-Friedrichs bifurcation theory. Arch. Rat. Mech. Anal. 60, 371–393 (1976)
Pye, E. K.: Periodicities in intermediary metabolism, in: Symp. on Biochronometry. Washington, D.C.: Nat. Acad. Sci. Press 1971
Sel'kov, E. E.: Self-oscillations in glycolysis. A simple kinetic model. Eur. J. Biochem. 4, 79–86 (1968)
Smale, S.: A mathematical model of two cells via Turing's equations, in: Lectures on Mathematics in the Life Sciences 7 (J. Cowan, ed.) Providence: Amer. Math. Soc. 1975
Torre, V.: Synchronization of non-linear biochemical oscillators coupled by diffusion. Biol. Cybern. 17, 137–144 (1975)
Tyson, J., Kauffman, S.: Control of mitosis by a continuous biochemical oscillation. J. Math. Biology 1, 289 (1975)
Walter, W.: Differential and Integral Inequalities. New York: Springer 1970
Weinberger, H. F.: Invariant sets for weakly coupled parabolic and elliptic systems. Rend. di Mat. 8, 295–310 (1975)
Winfree, A. T.: Polymorphic pattern formation in the fungus nectria. J. Theor. Biol. 38, 363–382 (1973)
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Ashkenazi, M., Othmer, H.G. Spatial patterns in coupled biochemical oscillators. J. Math. Biology 5, 305–350 (1977). https://doi.org/10.1007/BF00276105
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DOI: https://doi.org/10.1007/BF00276105