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Physico-chemical model of a protocell. (English) Zbl 0571.92003
A physico-chemical model of a self-maintaining unity or protocell is constructed on the basis of reaction and diffusion processes. The surface motion of the protocell is taken into account explicitly by a so-called Stefan condition, which leads to a nonlinear feedback to the reaction and diffusion processes.
The spatio-temporal dynamics in the neighbourhood of the steady states is investigated in the framework of linear stability analysis with the use of an expansion in terms of spherical harmonics $$Y_ l^ m$$. It is shown that modes with $$l\geq 2$$ become successively unstable with increasing nutrient supply. The leading instability with $$l=2$$ initiates a process of the nonlinear dynamics which is interpreted as the onset of division. A stabilizing effect of surface tension is also discussed.

##### MSC:
 92B05 General biology and biomathematics 92Cxx Physiological, cellular and medical topics 35A99 General topics in partial differential equations 35B35 Stability in context of PDEs
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##### References:
 [1] An der Heiden, U., Roth, G., Schwegler, H.: System-theoretic characterization of living systems. In: Systemanalyse biologischer Prozesse, 29-34. Möller, D. P. F. (Hrsg.) Berlin-Heidelberg-New York-Tokyo: Springer 1984 [2] An der Heiden, U., Roth, G., Schwegler, H.: Principles of self-generation and self-maintenance. Acta Biotheor. (1985), in press [3] Fife, P.: Mathematical approach of reacting and diffusing systems. Berlin-Heidelberg-New York: Springer 1979 · Zbl 0403.92004 [4] Free boundary problems: theory and applications. Fasano, A., Primicero, M. (eds.) Boston: Pitman 1983 [5] Fox, S. W.: A theory of macromolecular and cellular origins. Nature205, 328-340 (1965) [6] Fox, S. W.: Self-ordered polymers and propagative cell-like systems. Naturwissenschaften56, 1-9 (1969) [7] Fox, S. W., Nakashima, T.: The assembly and properties of protobiological structures. Biosystems12, 155-166 (1980) [8] Gierer, A., Meinhardt, H.: A theory of biological pattern formation. Kybernetik12, 30-39 (1972) · Zbl 0297.92007 [9] Greenspan, H. P.: On the self-inhibited growth of cell cultures. Growth38, 81-95 (1974) [10] Haken, H.: Synergetics. Berlin-Heidelberg-New York: Springer 1978 · Zbl 0396.93001 [11] Kessler, D. A., Koplik, J., Levine, H.: Numerical simulation of two-dimensional snowflake growth. Schlumberger-Doll Research Note, March 1984 [12] Landau, H. G.: Heat conduction in a melting solid. Quart. Appl. Math.8, 81-94 (1950) · Zbl 0036.13902 [13] Langer, J. S.: Instabilities and pattern formation in crystal growth. Rev. Mod. Phys.52, 1-28 (1980) [14] Langer, J. S.: Dynamics of dentritic pattern formation. Mater. Sci. Eng.65, 37-44 (1984) [15] Meinhardt, H.: Models of biological pattern formation. London-New York: Academic Press 1982 [16] Nicolis, G., Prigogine, I.: Self-organization in non-equilibrium systems. New York: Wiley 1977 · Zbl 0363.93005 [17] Oparin, A. J.: Genesis and evolutionary development of life. New York: Academic Press 1968 [18] Rothe, F.: Global solutions of reaction-diffusion systems. Berlin-Heidelberg-New York-Tokyo: Springer 1984 · Zbl 0546.35003 [19] Rubinstein, L. I.: The Stefan problem. Providence: Am. Math. Soc. 1971 · Zbl 0208.09804 [20] Schwegler, H.: Structure and organization of biological systems. In: Self-organizing systems. Roth, G. Schwegler, H. (Eds.) pp. 24-38. Frankfurt-New York: Campus 1981 [21] Schwegler, H., Tarumi, K.: Stability analysis of some models of tissue growth control. Funkt. Biol. Med.2, 226-233 (1983) [22] Schwegler, H., Tarumi, K.: The onset of division in a protocell mode. Springer Series in Synergetics, vol. 29. Rensing, L., Jaeger, I. N. (eds.) pp. 321, 322. Berlin-Heidelberg-New York-Tokyo: Springer 1985 · Zbl 0571.92003 [23] Smoller, J.: Shock waves and reaction-diffusion equations. Berlin-Heidelberg-New York-Tokyo: Springer 1983 · Zbl 0508.35002 [24] Tarumi, K., Schwegler, H.: Oscillating spatial structures of a tissue. J. Theor. Biol.101, 373-386 (1983)
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