An extremal criterion for epimorphic regeneration

Bertrand S. Clarke, Jay E. Mittenthal, Phillip A. Arcuri

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)595-634
Number of pages40
JournalBulletin of Mathematical Biology
Volume50
Issue number6
DOIs
StatePublished - Nov 1 1988

Fingerprint

Regeneration
limb
Leg
legs
regeneration
embryonic development
Gradient
Asymptotically Stable
Field Theory
limbs (animal)
Extremal Length
Extremities
Jacobi Elliptic Function
Transient Dynamics
Integrand
Boundary Value
Proposition
Unique Solution
Embryonic Development
Tend

ASJC Scopus subject areas

  • Neuroscience(all)
  • Immunology
  • Mathematics(all)
  • Biochemistry, Genetics and Molecular Biology(all)
  • Environmental Science(all)
  • Pharmacology
  • Agricultural and Biological Sciences(all)
  • Computational Theory and Mathematics

Cite this

Clarke, B. S., Mittenthal, J. E., & Arcuri, P. A. (1988). An extremal criterion for epimorphic regeneration. Bulletin of Mathematical Biology, 50(6), 595-634. https://doi.org/10.1007/BF02460093

An extremal criterion for epimorphic regeneration. / Clarke, Bertrand S.; Mittenthal, Jay E.; Arcuri, Phillip A.

In: Bulletin of Mathematical Biology, Vol. 50, No. 6, 01.11.1988, p. 595-634.

Research output: Contribution to journalArticle

Clarke, BS, Mittenthal, JE & Arcuri, PA 1988, 'An extremal criterion for epimorphic regeneration', Bulletin of Mathematical Biology, vol. 50, no. 6, pp. 595-634. https://doi.org/10.1007/BF02460093
Clarke, Bertrand S. ; Mittenthal, Jay E. ; Arcuri, Phillip A. / An extremal criterion for epimorphic regeneration. In: Bulletin of Mathematical Biology. 1988 ; Vol. 50, No. 6. pp. 595-634.
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