Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

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 | References | Similar Articles | Additional Information

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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  • [1] D. G. Aronson and H. D. Thames, Stability and oscillation in biochemical systems with localized enzymes, J. Theor. Biol., to appear.
  • [2] 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
  • [3] J. Caperon, Time lag in population growth response of Isochrysis Galbana to a variable nitrate environment, Ecology 50, 188-192 (1969)
  • [4] L. È. Èl′sgol′c, Introduction to the theory of differential equations with deviating arguments, Translated from the Russian by Robert J. McLaughlin, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1966. MR 0192154
  • [5] A. Fraser and J. Tiwari, Genetical feedback repression: II. cyclic genetic systems, J. Theor. Biol. 47, 397-412 (1974)
  • [6] B. C. Goodwin, Temporal organization in cells, Academic Press, New York, 1963
  • [7] B. C. Goodwin, Oscillatory behavior in enzymatic control processes, Adv. Enzyme Reg. 3, 425-439 (1965)
  • [8] J. S. Griffith, Mathematics of cellular control processes, I, II, J. Theor. Biol. 20, 202-208, 209-216 (1968)
  • [9] 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
  • [10] B. Hess and A. Boiteux, Oscillatory phenomena in biochemistry, Ann. Rev. Biochemistry 40, 237-258 (1971)
  • [11] F. Jacob and J. Monod, On the regulation of gene activity, Cold Spring Harbor Symp. Quant. Biol. 26, 193-211, 389-401 (1961)
  • [12] 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)
  • [13] 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)
  • [14] 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
  • [15] A. L. Lehninger, Biochemistry, 2nd ed., Worth Publishers, New York, 1975
  • [16] N. MacDonald, Bifurcation theory applied to a simple model of a biochemical oscillator, J. Theor. Biol. 65, 727-734.
  • [17] N. MacDonald, Time lag in a model of a biochemical reaction sequence with end product inhibition, J. Theoret. Biol. 67 (1977), no. 3, 549–556. MR 0490019, https://doi.org/10.1016/0022-5193(77)90056-X
  • [18] M. Morales and D. McKay, Biochemical oscillations in ``controlled'' systems, Biophys. J. 7, 621-625 (1967)
  • [19] H. G. Othmer, The qualitative dynamics of a class of biochemical control circuits, J. Math. Biol. 3 (1976), no. 1, 53–78. MR 0406568, https://doi.org/10.1007/BF00307858
  • [20] T. Pavlidis, Populations of biochemical oscillators as circadian clocks, J. Theor. Biol. 33, 319-338 (1971)
  • [21] P. E. Rapp, A theoretical investigation of a large class of biochemical oscillators, Math. Biosci. 25 (1975), no. 1/2, 165–188. MR 0429169, https://doi.org/10.1016/0025-5564(75)90059-0
  • [22] P. E. Rapp, A theoretical investigation of a large class of biochemical oscillators, Math. Biosci. 25 (1975), no. 1/2, 165–188. MR 0429169, https://doi.org/10.1016/0025-5564(75)90059-0
  • [23] B. Sikyta and J. Slezák, A periodic phenomenon in regulation of pyruvate biosynthesis in E. Coli B, Biochem. Biophys. Acta 100, 311-313 (1965)
  • [24] J. M. Smith, Mathematical ideas in biology, Cambridge University Press, 1968
  • [25] B. M. Sweeney, A physiological model for circadian rhythms derived from the Acetabularia rhythm paradoxes, Intl. J. Chronobiology 2, 25-33 (1974)
  • [26] H. D. Thames Jr. and D. G. Aronson, Oscillation in a non-linear parabolic model of separated, cooperatively coupled enzymes, Nonlinear systems and applications (Proc. Internat. Conf., Univ. Texas, Arlington, Tex., 1976) Academic Press, New York, 1977, pp. 687–693. MR 0472115
  • [27] Howard D. Thames Jr. and Allen D. Elster, Equilibrium states and oscillations for localized two-enzyme kinetics: a model for circadian rhythms, J. Theoret. Biol. 59 (1976), no. 2, 415–427. MR 0437126, https://doi.org/10.1016/0022-5193(76)90180-6
  • [28] J. Tiwari and A. Fraser, Genetic regulation by feedback repression, J. Theor. Biol. 39, 679-681 (1973)
  • [29] J. Tiwari, A. Fraser, and R. Beckman, Genetical feedback repression: I. Single locus models, J. Theor. Biol. 45, 311-326 (1976)
  • [30] 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
  • [31] J. J. Tyson, On the existence of oscillatory solutions in negative feedback cellular control processes, J. Math. Biol. 1 (1974/75), no. 4, 311–315. MR 0390383, https://doi.org/10.1007/BF00279849
  • [32] J. J. Tyson and H. G. Othmer, The dynamics of feedback control circuits in biochemical pathways, 1977, submitted to Prog, in Theor. Biol.
  • [33] 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
  • [34] 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
  • [35] 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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DOI: https://doi.org/10.1090/qam/508768
Article copyright: © Copyright 1978 American Mathematical Society

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