Summary
It often happens that scalar-valued observables of biological interest are points not on the real line, but on the circle… for example phases of periodic events, and colors. Sometimes one such quantity depends upon two others in a symmetric way, for example, in determining a compromise phase after slime mold plasmodia are fused at different phases of the cell cycle, or in determining the color of a mixture of two colors. In such cases the experimental result cannot depend on the two inputs in an unreservedly continuous way: there must be a point of ambiguity or discontinuity. Experiments involving the cell cycle and glycolysis are examined, in which the discontinuity appears to take two different forms.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aschoff, J.: Phasing of diurnal rhythms as a function of season and latitude (in German). Oecologia3, 125–165 (1969).
Beck, S. D.: Insect Photoperiodism. New York: Academic Press 1968.
Braemer, W.: A critical review of the sun-azimuth hypothesis. Cold Spring Harbor Symposia on Quantitative Biology25, 413–427 (1960).
Brinkmann, K.: Temperatureinflüsse auf die Circadiane Rhythmik vonEuglena gracilis bei Mixotrophie und Auxotrophie. Planta70, 344–389 (1966).
Brinkmann, K.: Der Einfluß von Populationseffekten auf die Circadiane Rhythmik vonEuglena gracilis. Biologische Rhythmen10, 1380–1400 (1967).
Cohen, M. H.: Models for the control of development. Symposia of the society of experimental biology,25: Control mechanism of growth and differentiation, p. 445-476. Cambridge: 1971.
Danilevsky, A. S., Goryshin, N. I., Tyschenko, V. P.: Biological rhythms in Arthropods. Ann. Rev. Entomol.51, 201–244 (1970).
Dewey, W. C., Miller, H. H., Nagaswa, H.: Interactions betweenS andG 1 cells: effects on decay of synchrony. Exp. Cell Res.77, 73–78 (1973).
Edmunds, L. N., Chuang, L., Jarrett, R. M., Terry, O. W.: Long-term persistence of free-running circadian rhythms of cell division inEuglena and the implication of autosynchrony. J. Interdiscipl. Cycle Res.2, 121–132 (1971).
Fitzhugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J.1, 445–566 (1961).
Ghosh, A, K., Chance, B., Pye, E. K.: Metabolic coupling and synchronization of NADH oscillations in yeast cell populations. Arch. Biochem. Biophys.145, 319–331 (1971).
Guttes, E., Devi, V. R., Guttes, S.: Synchronization of mitosis inPhysarum polycephalum by coalescence of postmitotic and premitotic plasmodial fragments. Experientia25, 615–616 (1969).
Hale, J. K.: Ordinary differential equations. New York: Wiley-Interscience 1969.
Hanusse, P.: De l'existence d'un cycle limite dans l'evolution des systèmes chimiques ouverts. C. R. Acad. Sci. Ser.C 274, 1245–1247 (1972).
Hardeland, R.: Circadian rhythmicity of tyrosine aminotransferase activity in suspensions of isolated rat liver cells. J. Interdiscipl. Cycle Res.3, 109–114 (1972).
Hastings, J. W., Sweeney, B. M.: A persistent diurnal rhythm of luminescence in Gonyaulax polyedra. Biol. Bull.115, 440–458 (1958).
Irwin, M. C.: A classification of elementary cycles. Topology9, 35–47 (1970).
Kauffman, S.: Measuring the mitotic oscillator. Bull. Math. Biophys. (In press, 1974.)
Kauffman, S., Wille, J. J.: The mitotic oscillation inPhysarum polycephalum. (In preparation.)
Pavlidis, T.: Biological oscillators: Their mathematical analysis. New York: Academic Press 1973.
Rao, P. N., Johnson, R. T.: Mammalian cell fusion: Studies on the regulation of DNA synthesis and mitosis. Nature225, 159–164 (1970).
Rusch, H. P., Sachsenmaier, W., Behrens, K., Gruter, V.: Synchronization of mitosis by the fusion of the plasmodia ofPhysarum polycephalum. J. Cell Biol.31, 204–209 (1966).
Sachsenmaier, W., Remy, U., Plattner-Schobel, R.: Initiation of synchronous mitosis inPhysarum polycephalum. Exp. Cell Res.73, 41–48 (1972).
Sheppard, J. J.: Human Color Perception. New York: Elseview 1968.
Tyson, J. J., Light, J. C.: Properties of two-component bimolecular and trimolecular chemical reaction systems. (In press.)
Tyson, J. J., and Kauffman, S.: Continuous biochemical control of mitosis: phase synchronization and inhomogeneous oscillations. (In preparation, 1974.)
Winfree, A. T.: Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol.16, 15–42 (1967).
Winfree, A. T.: The temporal morphology of a biological clock, in: Lectures on Mathematics in the Life Sciences,2 (Gerstenhaber, M., ed.), pp. 109–150. Providence, R. I.: Am. Math Soc. 1970.
Winfree, A. T.: Oscillatory glycolysis in yeast: The pattern of phase resetting by oxygen. Arch Biochem. Biophys.149, 388–401 (1972).
Winfree, A. T.: Time and timelessness in biological clocks, in: Temporal Aspects of Therapeutics (Urquardt, J., Yates, F. E., eds.), pp. 35–57. New York: Plenum 1973.
Zeeman, E. C.: Differential equations for the heartbeat and nerve impulse, in: Towards a Theoretical Biology,4 (Waddington, C. H., ed.), pp. 8–67. Chicago: Aldine 1972.
Zeuthen, E., Williams, N. E.: Division-limiting morphogenetic processes inTetrahymena, in: Nucleic Acid Metabolism, Cell Differentiation and Cancer Growth (Cowdry, E. V., Seno, S., eds.), pp. 203–216. Oxford: Pergamon Press 1969.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Winfree, A.T. Patterns of phase compromise in biological cycles. J. Math. Biology 1, 73–93 (1974). https://doi.org/10.1007/BF02339491
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02339491