An extremal criterion for epimorphic regeneration

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.