Global asymptotic stability of certain models for protein synthesis and repression
Authors:
H. T. Banks and J. M. Mahaffy
Journal:
Quart. Appl. Math. 36 (1978), 209-221
MSC:
Primary 92A05
DOI:
https://doi.org/10.1090/qam/508768
MathSciNet review:
508768
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Abstract: We consider a class of mathematical models involving nonlinear differential equations with hereditary terms. Included as special cases are a number of models that have been proposed as qualitative models for protein synthesis in eukaryotic cells. We establish global stability for these models and discuss the implications of our results.
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B. C. Goodwin, Oscillatory behavior in enzymatic control processes, Adv. Enzyme Reg. 3, 425–439 (1965)
J. S. Griffith, Mathematics of cellular control processes, I, II, J. Theor. Biol. 20, 202–208, 209–216 (1968)
Z. Grossman and I. Gumowski, Self-sustained oscillations in the Jacob-Monod mode of gene regulation, 7th IFIP Conf. on Optimization Techniques, September, 1975, Nice, France
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F. Jacob and J. Monod, On the regulation of gene activity, Cold Spring Harbor Symp. Quant. Biol. 26, 193–211, 389–401 (1961)
A. Johnsson and H. G. Karlsson, A feedback model for biological rhythms: I. Mathematical description and basic properties of the model, J. Theor. Biol. 36, 153–174 (1972)
W. A. Knorre, Oscillations of the rate of synthesis of $\beta$-galactosidase in E. Coli ML 30 and ML 308, Biochem. Biophys. Res. Comm. 31, 812–817 (1968)
W. A. Knorre, Oscillations in the epigenetic system: biophysical model of the $\beta$-galactosidase control system, in Biological and biochemical oscillators (B. Chance et al., eds.), Academic Press, New York, 1973, 449–457
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J. Tiwari, A. Fraser, and R. Beckman, Genetical feedback repression: I. Single locus models, J. Theor. Biol. 45, 311–326 (1976)
A. Travers, Bacterial transcription, in MTP international review of science: biochemistry of nucleic acids (K. Burton, ed.), Butterworth and Co., London, Vol. 6, 1974, 191–218
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H. E. Umbarger, Regulation of amino acid biosynthesis in microorganisms, in MTP international review of science: synthesis of amino acids and proteins (H.R.V. Arnstein, ed.), Butterworth and Company, London, Vol. 7, 1975, 1–56
C. F. Walter, Kinetic and thermodynamic aspects of biological and biochemical control mechanisms, in Biochemical regulatory mechanisms in eukaryotic cells (E. Kun and S. Grisolia, eds.) Wiley, New York, 1972, 355–489
G. Yagil and E. Yagil, On the relation between effector concentration and the rate of induced enzyme synthesis, Biophysical J. 11, 11–27 (1971).
D. G. Aronson and H. D. Thames, Stability and oscillation in biochemical systems with localized enzymes, J. Theor. Biol., to appear.
H. T. Banks and J. M. Mahaffy, Mathematical models for protein biosynthesis, LCDS Technical Report 77-2, Division of Applied Mathematics, Brown University, December, 1977
J. Caperon, Time lag in population growth response of Isochrysis Galbana to a variable nitrate environment, Ecology 50, 188–192 (1969)
L. E. El’sgol’ts, Introduction to the theory of differential equations with deviating arguments, Holden-Day, Inc., San Francisco, 1966
A. Fraser and J. Tiwari, Genetical feedback repression: II. cyclic genetic systems, J. Theor. Biol. 47, 397–412 (1974)
B. C. Goodwin, Temporal organization in cells, Academic Press, New York, 1963
B. C. Goodwin, Oscillatory behavior in enzymatic control processes, Adv. Enzyme Reg. 3, 425–439 (1965)
J. S. Griffith, Mathematics of cellular control processes, I, II, J. Theor. Biol. 20, 202–208, 209–216 (1968)
Z. Grossman and I. Gumowski, Self-sustained oscillations in the Jacob-Monod mode of gene regulation, 7th IFIP Conf. on Optimization Techniques, September, 1975, Nice, France
B. Hess and A. Boiteux, Oscillatory phenomena in biochemistry, Ann. Rev. Biochemistry 40, 237–258 (1971)
F. Jacob and J. Monod, On the regulation of gene activity, Cold Spring Harbor Symp. Quant. Biol. 26, 193–211, 389–401 (1961)
A. Johnsson and H. G. Karlsson, A feedback model for biological rhythms: I. Mathematical description and basic properties of the model, J. Theor. Biol. 36, 153–174 (1972)
W. A. Knorre, Oscillations of the rate of synthesis of $\beta$-galactosidase in E. Coli ML 30 and ML 308, Biochem. Biophys. Res. Comm. 31, 812–817 (1968)
W. A. Knorre, Oscillations in the epigenetic system: biophysical model of the $\beta$-galactosidase control system, in Biological and biochemical oscillators (B. Chance et al., eds.), Academic Press, New York, 1973, 449–457
A. L. Lehninger, Biochemistry, 2nd ed., Worth Publishers, New York, 1975
N. MacDonald, Bifurcation theory applied to a simple model of a biochemical oscillator, J. Theor. Biol. 65, 727–734.
N. MacDonald, Time lag in a model of a biochemical reaction sequence with end-product inhibition, J. Theor. Biol. 67, 549–556 (1977)
M. Morales and D. McKay, Biochemical oscillations in “controlled” systems, Biophys. J. 7, 621–625 (1967)
H. G. Othmer, The qualitative dynamics of a class of biochemical control circuits, J. Math. Biol. 3, 53–78 (1976)
T. Pavlidis, Populations of biochemical oscillators as circadian clocks, J. Theor. Biol. 33, 319–338 (1971)
P. E. Rapp, Biochemical oscillators—a search procedure, Math. Biosci. 23, 289–303 (1975)
P. E. Rapp, A theoretical investigation of a large class of biochemical oscillators, Math. Biosci. 25, 165–188 (1975)
B. Sikyta and J. Slezák, A periodic phenomenon in regulation of pyruvate biosynthesis in E. Coli B, Biochem. Biophys. Acta 100, 311–313 (1965)
J. M. Smith, Mathematical ideas in biology, Cambridge University Press, 1968
B. M. Sweeney, A physiological model for circadian rhythms derived from the Acetabularia rhythm paradoxes, Intl. J. Chronobiology 2, 25–33 (1974)
H. D. Thames and D. Aronson, Oscillation in a non-linear parabolic model of separated, cooperatively coupled enzymes, in Nonlinear systems and applications (V. Lakshmikantham, ed.), Academic Press, New York, 1977, 687–693
H. D. Thames and A. D. Elster, Equilibrium states and oscillations for localized two-enzyme kinetics: a model for circadian rhythms, J. Theor. Biol. 59, 415–427 (1976)
J. Tiwari and A. Fraser, Genetic regulation by feedback repression, J. Theor. Biol. 39, 679–681 (1973)
J. Tiwari, A. Fraser, and R. Beckman, Genetical feedback repression: I. Single locus models, J. Theor. Biol. 45, 311–326 (1976)
A. Travers, Bacterial transcription, in MTP international review of science: biochemistry of nucleic acids (K. Burton, ed.), Butterworth and Co., London, Vol. 6, 1974, 191–218
J. J. Tyson, On the existence of oscillatory solutions in negative feedback cellular control processes, J. Math. Biol. 1, 311–315 (1975)
J. J. Tyson and H. G. Othmer, The dynamics of feedback control circuits in biochemical pathways, 1977, submitted to Prog, in Theor. Biol.
H. E. Umbarger, Regulation of amino acid biosynthesis in microorganisms, in MTP international review of science: synthesis of amino acids and proteins (H.R.V. Arnstein, ed.), Butterworth and Company, London, Vol. 7, 1975, 1–56
C. F. Walter, Kinetic and thermodynamic aspects of biological and biochemical control mechanisms, in Biochemical regulatory mechanisms in eukaryotic cells (E. Kun and S. Grisolia, eds.) Wiley, New York, 1972, 355–489
G. Yagil and E. 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 1978
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