×

Models of genetic control by repression with time delays and spatial effects. (English) Zbl 0577.92010

For cellular control by repression two models are considered by means of standard theory from compartmental analysis and biochemical kinetics. They include time delays to account for the processes of transcription and translation and diffusion to account for spatial effects in the cell.
The first model considers repression of the gene by an endproduct produced at a distance from the gene. The diffusion equations then involve delays and are formulated by a system of difference-differential equations for the concentrations of mRNA and repressors in three interacting compartments with appropriate boundary conditions.
The second model accounts for translation occurring uniformly throughout the cytoplasm and contains two compartments. This leads also to a similar system of equations.
Analysis of the steady-state problems is given. Some results on the existence and uniqueness of a global solution and the stability of the steady-state problems are summarized. A Hopf bifurcation result and a theorem on asymptotic stability are given for the limiting case of the models without diffusion. A proof for this theorem is stated in an appendix.
Reviewer: Y.Komatu

MSC:

92D10 Genetics and epigenetics
92Cxx Physiological, cellular and medical topics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Allwright, D. J.: A global stability criterion for simple loops. J. Math. Biol. 4, 363-373 (1977) · Zbl 0372.93042 · doi:10.1007/BF00275084
[2] an der Heiden, U.: Periodic solutions of a nonlinear second order differential equation with delays. J. Math. Anal. Appl 70, 599-609 (1979) · Zbl 0426.34059 · doi:10.1016/0022-247X(79)90068-4
[3] Banks, H. T., Mahaffy, J. M.: Mathematical models for protein synthesis. Technical Report, Division of Applied Mathematics, Lefschetz Center for Dynamical Systems, Providence, R. I., 1979 · Zbl 0406.92011
[4] Banks, H. T., Mahafiy, J. M.: Global asymptotic stability of certain models for protein synthesis and repression. Quart. Appl. Math 36, 209-221 (1978) · Zbl 0406.92011
[5] Britten, R. J., Davidson, E. H.: Gene regulation for higher cell: A theory. Science 165, 349-357 (1969) · doi:10.1126/science.165.3891.349
[6] Goodwin, B. C.: Oscillatory behavior in enzymatic control processes. Adv. Enzyme Reg. 3, 425-439 (1965) · doi:10.1016/0065-2571(65)90067-1
[7] Goodwin, B. C.: Temporal Organization in Cells. New York: Academic Press 1963
[8] Griffith, J. S.: Mathematics of cellular control processes, I. J. Theoret. Biol. 20, 202-208 (1968) · doi:10.1016/0022-5193(68)90189-6
[9] Hadeler, K. P., Tomiuk, J.: Periodic solutions of difference-differential equations. Arch. Rational Mech. Anal. 65, 87-95 (1977) · Zbl 0426.34058 · doi:10.1007/BF00289359
[10] Hastings, S. P., Tyson, J. J., Webster, D.: Existence of periodic solutions for negative feedback cellular control systems. J. Differential Equations 25, 39-64 (1977) · Zbl 0361.34038 · doi:10.1016/0022-0396(77)90179-6
[11] Jacob, F., Monod, J.: On the regulation of gene activity. Cold Spring Harbor Symp. Quant. Biol. 26, 193-211, 389-401 (1961)
[12] Kernevez, J-P.: Enzyme Mathematics. New York: North-Holland Publishing Co., 1980 · Zbl 0446.92007
[13] Lehninger, A. L.: Biochemistry, 2nd ed., New York: Worth Publishers 1975
[14] MacDonald, N.: Time Lag in Biological Models. Lectures Notes in Biomathematics, vol. 27. Berlin-Heidelberg-New York: Springer 1978 · Zbl 0403.92020
[15] MacDonald, N.: Time lag in a model of a biochemical reaction sequence with end-product inhibition. J. Theoret. Biol. 67, (1977)
[16] Mahaffy, J. M.: A test for stability of linear delay differential equations. Quart. Appl. Math. 40, 193-202 (1982) · Zbl 0499.34049
[17] Mahaffy, J. M.: Periodic solutions for certain protein synthesis models. J. Math. Anal. Appl. 74, 72-105 (1980) · Zbl 0441.92006 · doi:10.1016/0022-247X(80)90115-8
[18] Morales, M., McKay, D.: Biochemical oscillations in ?controlled? systems. Biophys. J. 7, 621-625 (1967) · doi:10.1016/S0006-3495(67)86611-6
[19] Othmer, H. G.: The qualitative dynamics of a class of biochemical control circuits. J. Math. Biol 3, 53-70 (1976) · Zbl 0334.92001 · doi:10.1007/BF00307858
[20] Pao, C. V.: Mathematical analysis of enzyme-substrate reaction diffusion in some biochemical systems. J. Nonlinear Analysis: TMA 4, 369-392 (1980). · Zbl 0431.92014 · doi:10.1016/0362-546X(80)90061-9
[21] Pao, C. V.: Reaction-diffusion equations with nonlinear boundary conditions. J. Nonlinear Anal. TMA 5, 1077-1094 (1981) · Zbl 0519.35038 · doi:10.1016/0362-546X(81)90004-3
[22] Pao, C. V., Mahaffy, J. M.: Qualitative analysis of a coupled reaction-diffusion model in biology with time delays. J. Math. Anal. Appl. in press (1984) · Zbl 0583.92012
[23] Rapp, P. E.: Mathematical techniques for the study of oscillations in biochemical control loops. Bull. Inst. Math. Appl. 12, 11-20 (1976)
[24] Shymko, R. M., Glass, L.: Spatial switching in chemical reactions with heterogeneous catalysis. J. Chem. Phys. 60, 835-841 (1974) · doi:10.1063/1.1681157
[25] Thames, H. D., Aronson, D. G.: Oscillation in a nonlinear parabolic model of separated, cooperatively coupled enzymes. Nonlinear Systems and Applications Lakshmikantham, V., (ed.). New York: Academic Press 1977
[26] Thames, H. D., Elster, A. D.: Equilibrium states and oscillations for localized two enzyme kinetics: Model for circadian rhythms. J. Theoret. Biol. 59, 415-427 (1976) · doi:10.1016/0022-5193(76)90180-6
[27] Tyson, J. J., Othmer, H. G.: The dynamics of feedback control circuits in biochemical pathways. Prog in Theor. Biol. Rosen, R., Snell, F. M. (eds.). New York: Academic Press 1978 · Zbl 0448.92010
[28] Walter, C. F.: Kinetic and thermodynamic aspects of biological and biochemical control mechanisms, in: Biochemical Regulatory Mechanisms in Eukaryotic Cells, Kun, E., Grisolia, S., (eds.) New York: Wiley 1972, pp 355-480
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.