Stability analysis for a mathematical model of the lac operon
Authors:
Joseph M. Mahaffy and Emil Simeonov Savev
Journal:
Quart. Appl. Math. 57 (1999), 37-53
MSC:
Primary 92C40; Secondary 92D10
DOI:
https://doi.org/10.1090/qam/1672171
MathSciNet review:
MR1672171
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Abstract: A mathematical model for induction of the lac operon is derived using biochemical kinetics and includes delays for transcription and translation. Local analysis of the unique equilibrium of this nonlinear model provides conditions for stability. Techniques are developed to determine Hopf bifurcations, and stability switching is found for the delayed system. Near a double bifurcation point a hysteresis of solutions to two stable periodic orbits is studied. Global analysis provides conditions on the model for asymptotic stability. The biological significance of our results is discussed.
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J. M. Mahaffy, Cellular control models with linked positive and negative feedback and delays, I. The models, J. Theor. Biol. 106, 89–102 (1984)
J. M. Mahaffy, Cellular control models with linked positive and negative feedback and delays, II. Linear analysis and local stability, J. Theor. Biol. 106, 103–118 (1984)
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G. Yagil and F. Yagil, On the relation between effector concentration and the rate of induced enzyme synthesis, Biophysical J. 11, 11–27 (1971)
H. T. Banks and J. M. Mahaffy, Global asymptotic stability of certain models for protein synthesis and repression, Quart. Appl. Math. 36, 209–211 (1978)
H. T. Banks and J. M. Mahaffy, Stability of cyclic gene models for systems involving repression, J. Theor. Biol. 74, 323–334 (1978)
J. R. Beckwith, The lactose operon. In F. Neidhardt, editor, Escherichia coli and Salmonella typhimurium cellular and molecular biology, volume 2, ASM, Washington, D.C., 1987, pp. 1444–1452
J. R. Beckwith and D. Zipser, The Lactose Operon, Cold Spring Harbor Laboratory, Cold Spring Harbor, N.Y., 1970
J. Bélair and S. A. Campbell, Stability and bifurcations of equilibria in a multiple-delayed differential equation, SIAM J. Appl. Math. 54, 1402–1424 (1994)
K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl. 86, 592–627 (1982)
F. R. Gantmacher, The Theory of Matrices, volume II, Chelsea, New York, 1959
B. C. Goodwin, Temporal Organization in Cells, Academic Press, New York, 1963
J. S. Griffith, Mathematics of cellular control processes, I; II, J. Theor. Biol. 20, 202–208 (1968)
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, revised edition, 1983
J. H. Miller and W. S. Reznikoff, editors, The Operon, Cold Spring Harbor Laboratory, Cold Spring Harbor, N.Y., 1978
J. Ji-Fa, A Liapunov function for four-dimensional positive-feedback systems, Quart. Appl. Math. 52, 601–614 (1994)
W. A. Knorre, Oscillations of the rate of synthesis of β-galactosidase in Escherichia coli ML30 and ML308, Biochem. Biophys. Res. Com. 31, 812–817 (1968)
J. M. Mahaffy, Periodic solutions for certain protein synthesis models, J. Math. Anal. Appl. 74, 72–105 (1980)
J. M. Mahaffy, A test for stability of linear differential delay equations, Quart. Appl. Math. 40, 193–202 (1982)
J. M. Mahaffy, Cellular control models with linked positive and negative feedback and delays, I. The models, J. Theor. Biol. 106, 89–102 (1984)
J. M. Mahaffy, Cellular control models with linked positive and negative feedback and delays, II. Linear analysis and local stability, J. Theor. Biol. 106, 103–118 (1984)
J. M. Mahaffy, Stability of periodic solutions for a model of genetic repression with delays, J. Math. Biol. 22, 137–144 (1985)
H. G. Othmer, The qualitative dynamics of a class of biochemical control circuits, J. Math. Biol. 3, 53–78 (1976)
N. P. Pih and P. Dhurjati, Oscillatory behavior of β-galactosidase enzyme activity in Escherichia coli during perturbed batch experiments, Biotech. Bioeng. 29, 292–296 (1987)
J. F. Selgrade, Mathematical analysis of a cellular control process with positive feedback, SIAM J. Appl. Math. 36, 219–229 (1979)
J. F. Selgrade, A Hopf bifurcation in single-loop positive-feedback systems, Quart. Appl. Math. 40 347–351 (1982)
G. Yagil and F. Yagil, On the relation between effector concentration and the rate of induced enzyme synthesis, Biophysical J. 11, 11–27 (1971)
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© Copyright 1999
American Mathematical Society