Abstract
A class of models based on the Jacob and Monod theory of genetic repression for control of biosynthetic pathways in cells is considered. Both spatial diffusion and time delays are taken into account. A method is developed for representing the effects of spatial diffusion as distributed delay terms. This method is applied to two specific models and the interaction between the diffusion and the delays is treated in detail. The destabilization of the steadystate and the bifurcation of oscillatory solutions are studied as functions of the diffusivities and the delays. The limits of very small and very large diffusivities are analyzed and comparisons with well-mixed compartment models are made.
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Allwright, D. J.: A global stability criterion for simple loops. J. Math. Biol.4, 363–373 (1977)
an der Heiden, U.: Periodic solutions of a nonlinear second order differential equation with delay. J. Math. Anal. Appl.70, 599–609 (1979)
Banks, H.T, Mahaffy, J. M.: Global asymptotic stability of certain models for protein synthesis and repression. Quart. Appl. Math.36, 209–221 (1978)
Bliss, R. D., Painter, P. R., Marr, A. G.: Role of feedback inhibition in stabilizing the classical operon. J. Theor. Biol.97, 177–193 (1982)
Busenberg, S., Mahaffy, J.: Interaction of spatial diffusion and delays in models of genetic control by repression. Harvey Mudd College Technical Report, June 1984
Goodwin, B. C.: Oscillatory behavior of enzymatic control processes. Adv. Enzyme Reg.3, 425–439 (1965)
Goodwin, B. C.: Temporal organization in cells. New York: Academic Press 1963
Hadeler, K. P., Tomiuk, J.: Periodic solutions of difference-differential equations. Arch. Rational Mech. Anal.65, 87–95 (1977)
Jacob, F., Monod, J.: On the regulation of gene activity. Cold Spring Harbor Symp. Quant. Biol.26, 193–211, 389–401 (1961)
Kernevez, J-P.: Enzyme Mathematics. New York: North-Holland Publishing Co. 1980
MacDonald, N. Time lag in biological models. Lect. Notes Biomath., Vol. 27, Berlin Heidelberg, New York: Springer 1978
MacDonald, N.: Time lag in a model of a biochemical reaction sequence with end-product inhibition. J. Theor. Biol.67, 549–556 (1977)
Mahaffy, J. M.: Periodic solutions for certain protein synthesis models. J. Math. Anal. Appl.74, 72–105
Mahaffy, J. M., Pao, C. V.: Models of genetic control by repression with time delays and spatial effects. J. Math. Biol.20, 39–57 (1984)
Pao, C. V., Mahaffy, J. M.: Qualitative analysis of a coupled reaction-diffusion model in biology with time delays. J. Math. Anal. Appl., to appear
Pazy, A.: Semigroups of linear operators and applications to partial differential equations, Berlin Heidelberg, New York: Springer 1983
Shymko, R. M., Glass, L.: Spatial switching in chemical reactions with heterogeneous catalysis. J. Chem. Phys.60, 835–841 (1974)
Thames, H. D., Aronson, D. G.: Oscillation in a nonlinear parabolic model of separated, cooperatively coupled enzymes. Nonlinear Systems and Applications V. Lakshmikantham, (ed) New York: Academic Press 1977
Thames, H. D., Elster, A. D.: Equilibrium states and oscillations for localized two enzyme kinetics: Models for circadian rhythms. J. Theor. Biol.59, 415–427 (1976)
Tyson, J. J.: Periodic enzyme synthesis and oscillatory repression: Why is the period of oscillation close to the cell cycle time? J. Theor. Biol.103, 313–328 (1983)
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On leave from North Carolina State University
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Busenberg, S., Mahaffy, J. Interaction of spatial diffusion and delays in models of genetic control by repression. J. Math. Biology 22, 313–333 (1985). https://doi.org/10.1007/BF00276489
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DOI: https://doi.org/10.1007/BF00276489