Abstract
Many developing systems obey the principle of continuity: a morphogenetic field, when perturbed, tends to restore the normal local pattern of structures in its organ district. We have investigated physical field theories for a morphogenetic field, seeking constraints which would make a field theory produce the principle of continuity. We assume that during embryonic (ontogenetic) development a leg develops a pattern of positional values and a length which extremize a time-independent functional—the integral, over the length of the leg, of a function of positional values and position. For a single state variable which represents positional value, if a unique extremizing solution for the ontogenetically generated pattern and the length exists, and if no position-dependent functions other than the state variable appear in the integrand, then the principle of continuity is valid: in any regenerated leg the state variable is continuous and each region is locally identical to a region of the ontogenetically generated leg. This proposition is applied to three simple examples. For an exponential gradient and a Jacobi elliptic function there is a set of parameter values and boundary values for which a functional is minimized and the ontogenetically generated leg has an optimal length. Thus a leg which meets these constraints will obey the principle of continuity. However, a functional which when extremized gives a sinusoidal pattern does not in general provide a unique extremal length. Mathematical conditions are discussed under which an ontogenetically generated limb or a regenerated limb represents an asymptotically stable steady state. For a specific model of the transient dynamics in the exponential gradient case, the steady state gradient is asymptotically stable.
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Literature
Abraham, R. H. and C. D. Shaw.Dynamics—The Geometry of Behavior. Part I: Periodic Behavior, Santa Cruz: Aerial Press.
Barrett, A. N. and D. Summerbell. 1984. “Mathematical Modelling of the Growth Processes in the Developing Chick Wing Bud.”Comput. biol. Med.,14, 411–418.
Ben-Jacob, E., H. Brand, G. Dee, L. Kramer and J. S. Langer. 1985. “Pattern Propagation in Nonlinear Dissipative Systems.”Physica 14D, 348–364.
Bolza, O. 1904.Lectures on the Calculus of Variations. Chicago: University of Chicago Press.
Bowman, F. 1961.Introduction to Elliptic Functions with Applications. Dover, New York.
Bulliere, D. and F. Bulliere. 1985. “Regeneration.” InComprehensive Insect Physiology, Biochemistry, and Pharmacology, Vol. 2: Postembryonic Development, G. A. Kerkut and L. I. Gilbert (Eds), pp. 387–440. New York: Pergamon Press.
Byrd, P. F. and M. D. Friedman. 1956.Handbook of Elliptic Integrals for Engineers and Physicists. Berlin: Springer.
Chafee, N. and E. Infante. 1974. A Bifurcation Problem for a Nonlinear Parabolic Equation.J. Applic. Anal. 4, 17–37.
Courant, R. and D. Hilbert. 1965.Methods of Mathematical Physics, Vol. 1. New York: Interscience.
Cummings, F. W. 1985. A Pattern-Surface Interactive Model of Morphogenesis.J. theor. Biol. 116, 243–273.
Do Carmo, M. P. 1976.Differential Geometry of Curves and Surfaces. New Jersey: Prentice-Hall.
Ede, D. A. and J. T. Law. 1969. “Computer Simulation of Vertebrate Limb Morphogenesis.”Nature 221, 244–248.
Elsgolts, L. 1970.Differential Equations and the Calculus of Variations. Moscow: Mir.
French, V., P. J. Bryant and S. V. Bryant. 1976. Pattern Regulation in Epimorphic Fields.Science 193, 969–981.
Goodwin, B. C. 1963.Temporal Organization in Cells. Academic Press, New York.
— and L. E. H. Trainor. 1983. “The Ontogeny and Phylogeny of the Pentadactyl Limb.” InDevelopment and Evolution, B. C. Goodwin, N. Holder and C. C. Wylie (Eds), pp. 75–98. New York: Cambridge University Press.
Henry, D.Geometric Theory of Semilinear Parabolic Equations. 1981. Berlin: Springer.
Hinchliffe, J. R. and D. R. Johnson.The Development of the Vertebrate Limb. 1980. Oxford: Clarendon Press.
Jordan, D. W. and P. Smith. 1977.Nonlinear Ordinary Differential Equations. p. 39. Oxford: Clarendon Press.
Lipschutz, M. M. 1969.Differential Geometry. New York: McGraw-Hill.
Malvern, L. E. 1969.Introduction to the Mechanics of a Continuous Medium. New Jersey: Prentice-Hall.
Meinhardt, H. 1983. “A Boundary Model for Pattern Formation in Vertebrate Limbs.J. Embryol. Exp. Morph. 76, 115–137.
Mittenthal, J. E. 1980. “On the Form and Size of Crayfish Legs Regenerated after Grafting.”Biol. Bull. 159, 700–713.
— 1985. “Morphogenetic Fields and the Control of Form in the Limbs of Decapods.” InCrustacean growth: Factors in adult growth, A. M. Wenner (Ed.), pp. 47–71. Rotterdam: Balkema.
— and R. M. Mazo. 1983. “A Model for Shape Generation by Strain and Cell-Cell Adhesion in the Epithelium of an Arthropod Leg Segment.”J. theor. Biol. 100, 443–483.
Mitolo, V. 1971. “Un Programma in Fortran per la Simulazione dell'Accrescimento e della Morfogenesi.Boll. Soc. ital. Biol. sper. 47, 18–20.
Morgan, T. H. 1901.Regeneration. New York: Macmillan.
Muneoka, K. and S. V. Bryant. 1982. “Evidence that Patterning Mechanisms in Developing and Regenerating Limbs are the Same.Nature 298, 369–371.
Newman, S. and J. Frisch. 1979. “Dynamics of Skeletal Pattern Formation in Developing chick Limb.Science 205, 662–668.
Oster, G. F., J. D. Murray and A. K. Harris. 1983. “Mechanical Aspects of Mesenchymal Morphogenesis.”J. Embryol. Exp. Morph. 78, 83–125.
—— and P. K. Maini. 1985. “A Model for Chondrogenic Condensations in the Developing Limb: The Role of Extracellular Matrix and Cell Tractions.”J. Embryol. Exp. Morph. 89, 93–112.
Papageorgiou, S. 1984. “A Hierarchical Polar Co-ordinate Model for Epimorphic Regeneration.J. theor. Biol. 109, 533–554.
Pritchard, A. J. 1968. “A Study of Two of the Classical Problems of Hydrodynamic Stability by the Liapunov Method.”J. Inst. Math. Applic. 4, 78–93.
Rosen, R. 1967.Optimality Principles in Biology, p. 68. London: Butterworths.
Ross, S. L. 1964.Differential Equations. New York: Blaisdell.
Shames, I. H. and C. L. Dym. 1985.Energy and Finite Element Methods in Structural Mechanics. New York: McGraw-Hill.
Spivak, M. 1975.A Comprehensive Introduction to Differential Geometry, Vols 3 and 4. Boston: Publish or Perish.
Stocum, D. L. 1984. “The Urodele Limb Regeneration Blastema. Determination and Organization of the Morphogenetic Field.”Differentiation 27, 13–28.
Tevlin, P. and L. E. H. Trainor. 1985. “A Two Vector Field Model of Limb Regeneration and Transplant Phenomena.”J. theor. Biol. 115, 495–513.
Todd, P. H. 1985a. “Gaussian Curvature as a Parameter of Biological Surface Growth.”J. theor. Biol. 113, 63–68.
—, 1985b. “Estimating Surface Growth Rates from Changes in Curvature.”math. Biosci. 74, 157–176.
Totafurno, J. 1985. “A Non-Linear Vector Field Model with Application to Supernumerary Production in Amphibian Limb Regeneration. Ph.D Thesis, Department of Physics, University of Toronto.
— and L. E. H. Trainor. 1987. “A Non-Linear Vector Field Model of Supernumerary Limb Production in Salamanders.J. theor. Biol. 124, 415–454.
Waddington, C. H. 1966. “Fields and Gradients.” InMajor Problems in Developmental Biology (25th Symposium of the Society for Developmental Biology), M. Locke (Ed.), pp. 105–124. New York: Academic Press.
Wilby, O. K. and D. A. Ede. 1975. “A Model Generating the Pattern of Cartilage Skeletal Elements in the Embryonic Chick Limb.”J. theor. Biol. 52, 199–217.
Winfree, A. T. 1984. “A Continuity Principle for Regeneration.” InPattern Formation, G. M. Malacinski, S. V. Bryant (Eds), pp. 103–124. New York: Macmillan.
Wolpert, L. 1971. “Positional Information and Pattern Formation.Curr. Topics Dev. Biol. 6, 183–224.
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Clarke, B.S., Mittenthal, J.E. & Arcuri, P.A. An extremal criterion for epimorphic regeneration. Bltn Mathcal Biology 50, 595–634 (1988). https://doi.org/10.1007/BF02460093
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DOI: https://doi.org/10.1007/BF02460093