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Temporal variability in a system of coupled mitotic timers

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Abstract

Cell proliferation is considered a periodic process governed by a relaxation timer. The collective behavior of a system composed of three identical relaxation oscillators in numerically studied under the condition that diffusion of the slow mode dominates. We demonstrate: (1) the existence of three periodic regimes with different periods and phase relations and an unsymmetrical, stable steady-state (USSS); (2) the coexistence of in-phase oscillations and USSS; (3) the coexistence of periodic attractors; and (4) the emergence of a two-loop limit cycle coexisting with both in-phase oscillations and a stable steady-state. The qualitative reasons for such a diversitiy and its possible role in the generation of cell cycle variability are discussed.

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References

  • Aronson DG, Doedel EJ, Othmer HG (1987) An analytical and numerical study of the bifurcations in a system of linearly-coupled oscillators. Physica 25D:20–104

    Google Scholar 

  • Ashwin P, King GP, Swift JW (1990) Three identical oscillators with symmetrical coupling. Nonlinearity 3:585–601

    Google Scholar 

  • Baesens C, Guckenheimer J, Kim S, MacKey RS (1991) Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos. Physica 49D:387–475

    Google Scholar 

  • Bar-Eli KJ (1984a) Coupling of chemical oscillators. J Phys Chem 88:3616–3622

    Google Scholar 

  • Bar-Eli KJ (1984b) The dynamics of coupled chemical oscillators. J Phys Chem 88:6174–6177

    Google Scholar 

  • Bar-Eli K (1990) Coupling of identical chemical oscillators. J Phys Chem 94:2368–2374

    Google Scholar 

  • Bar-Eli K, Reuveni SY (1985) Stable stationary states of coupled chemical oscillators. J Phys Chem 89:1329–1330

    Google Scholar 

  • Brooks RF, Riddle PN (1988) Differences in growth factor sensitivity between individual 3T3 cells arise at high frequency:possible relevance to cell senescence. Exp Cell Res 174:378–387

    Google Scholar 

  • Chernavskii DS, Palamarchuk EK, Polezhaev AA, Solyanik GI, Burlakova EB (1977) A mathematical model of periodic processes in membranes (with application to cell cycle regulation). BioSystems 9:187–193

    Google Scholar 

  • Crowley MF, Epstein IR (1989) Experimental and theoretical studies of coupled chemical oscillators:phase death, multistability, and in-phase and out-of-phase entrainment. J Phys Chem 93:2496–2502

    Google Scholar 

  • Doedel EJ (1981) AUTO:a program for the automatic bifurcation analysis of autonomous systems. Cong Num 30:265–284

    Google Scholar 

  • Dolnik M, Marek MY (1988) Extinction of oscillators in forced and coupled reaction cells. J Phys Chem 92:2452–2455

    Google Scholar 

  • Epstein IR (1990) Coupled oscillators in chemistry and biology. Comment Mol Cell Biophys 6:299–327

    Google Scholar 

  • Grasman J, Jansen MJW (1979) Mutually synhronized relaxation oscillators as prototypes of oscillating systems in biology. J Math Biol 7:171–197

    Google Scholar 

  • Hairer E, Wanner G (1990) Solving ordinary differential equations. II. Stiff and differential-algebraic problems. (Springer Series in Computational Mathematics) Springer, Berlin Heidelberg New York

    Google Scholar 

  • Helfand E (1979) Numerical integration of stochastic differential equations. Bell Sys Tech J 58:2289–2299

    Google Scholar 

  • Mustafin AT, Volkov EI (1984) The role of lipid and antioxidant exchanges in cell division synchronization (mathematical model). Biol Cybern 49:149–154

    Google Scholar 

  • Petrovic AC, Oudet CL, Stutzmann JJ (1984) Temporal organization of rat and human skeletal cells. In:Edmunds LNJ (ed) Cell cycle clocks. Marcel Dekker, New York

    Google Scholar 

  • Pol van der, Mark van der (1928) The heartbeat considered as a relaxation oscillation, and an electrical model of the heart. Philos Mag 6:763–775

    Google Scholar 

  • Prigogine I, Lefever R (1968) Symmetry breaking instabilities in dissipative systems. J Chem Phys 48:1665–1700

    Google Scholar 

  • Schimtz A, Hildebrand E (1992) Non-random structures in the locomotor behaviour of Halobacterium: a bifurcation route to chaos. Proc Natl Acad Sci USA 89:457–460

    Google Scholar 

  • Tyson J, Kauffman SI (1975) Control of mitosis by continuous biochemical oscillation:synchronization; spatially inhomogenous oscillations. J Math Biol 1:289–310

    Google Scholar 

  • Volkov EI, Mustafin AT (1985) Mathematical model of the lipid peroxidation in membranes. Proc Acad Sci USSR [Biol] 6:805–821

    Google Scholar 

  • Volkov EI, Pertsova TB (1987) Formation of a frequency trigger in a system of interacting relaxation oscillators. Soviet Physics Lebedev Institute Reports 11:48–51

    Google Scholar 

  • Volkov EI, Stolyarov MN (1991) Birhythmicity in a system of two coupled identical oscillators. Phys Lett 159A:61–66

    Google Scholar 

  • Volkov EI, Stolyarov MN, Brooks RF (1991) The modelling of heterogeneity in proliferative capacity during clonal growth. In: Volkov EI (ed) Biophysical approach to complex biological phenomena. Nova Science Publishers, New York

    Google Scholar 

  • Winfree AT (1967) Biological rhythms and the behavior of populations of coupled oscillators. J Theor Biol 16:15–42

    Google Scholar 

  • Yoshimoto M, Yoshikawa K, Mori Y (1993) Coupling among three chemical oscillators:synchronization, phase death, and frustration. Phys Rev 47E:864–874

    Google Scholar 

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Volkov, E.I., Stolyarov, M.N. Temporal variability in a system of coupled mitotic timers. Biol. Cybern. 71, 451–459 (1994). https://doi.org/10.1007/BF00198921

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  • DOI: https://doi.org/10.1007/BF00198921

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