Stability results for a model of repression with external control
Authors:
Joseph M. Mahaffy, David A. Jorgensen and Robert L. Vanderheyden
Journal:
Quart. Appl. Math. 50 (1992), 415-435
MSC:
Primary 92D25; Secondary 92C30
DOI:
https://doi.org/10.1090/qam/1178425
MathSciNet review:
MR1178425
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Abstract: A stability analysis is performed for a mathematical model with negative feedback, diffusion, and time delays. The model of a cell includes three biochemical species that interact to control the transport of an extracellular nutrient. This study examines the effects of diffusion, cell size, and extracellular nutrient concentration on the model. With certain assumptions the symmetry, a linearized version of the model is studied in detail. The characteristic equation is shown to have no solutions with positive real part when extracellular nutrient concentration or the diffusivities are sufficiently small or the radius of the cell is sufficiently large. These results are compared to an earlier study that showed that biochemical oscillations could occur for certain parameter values. A discussion is provided for how the bifurcations from regions of stability to regions of instability could affect the biological response of the cell.
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J. B. Neilands, Microbial iron compounds, Ann. Rev. Biochem. 50, 715–731 (1981)
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P. E. Rapp, Mathematical techniques for study of oscillations in biochemical control loops, Bull. Inst. Math. Appl. 12, 11–20 (1976)
J. J. Tyson and H. G. Othmer, The dynamics of feedback control circuits in biochemical pathways, Progress in Theoretical Biology, R. Rosen and F. M. Snell, eds., Academic Press, New York, 1978.
S. N. Busenberg and J. M. Mahaffy, Interaction of spatial diffusion and delays in models of genetic control by repression, J. Math. Biol. 22, 313–333 (1985)
S. N. Busenberg and J. M. Mahaffy, The effects of dimension and size for a compartmental model of repression, SIAM J. Appl. Math. 48, 882–903 (1988)
S. N. Busenberg and J. M. Mahaffy, A compartmental reaction-diffusion cell cycle model, Computers Math. Appl. 18, 883–892 (1989)
B. C. Goodwin, Oscillatory behavior of enzymatic control processes, Adv. Enzyme Reg. 3, 425–439 (1965)
M. D. Lundrigan and R.J. Kadner, Nucleotide sequence of the gene for the ferrienterochelin receptor FepA in Escherichia coli, J. Biol. Chem. 261, 10797–10801 (1980)
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, D. A. Jorgensen, and R. L. Vanderheyden, Oscillations in a model of repression with external control, to appear in J. Theoret. Biol.
J. M. Mahaffy and C. V. Pao, Models of genetic control by repression with time delays and spatial effects, J. Math. Biol. 20, 39–58 (1984)
J. B. Neilands, Microbial iron compounds, Ann. Rev. Biochem. 50, 715–731 (1981)
J. B. Neilands, Microbial envelope proteins related to iron, Ann. Rev. Microbiol. 36, 285–309 (1982)
H. G. Othmer, The qualitative dynamics of a class of biochemical control circuits, J. Math. Biol. 3, 53–78 (1976)
P. E. Rapp, Mathematical techniques for study of oscillations in biochemical control loops, Bull. Inst. Math. Appl. 12, 11–20 (1976)
J. J. Tyson and H. G. Othmer, The dynamics of feedback control circuits in biochemical pathways, Progress in Theoretical Biology, R. Rosen and F. M. Snell, eds., Academic Press, New York, 1978.
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© Copyright 1992
American Mathematical Society