Nowadays morphogenesis becomes commonly used term in some interdisciplinary fields. This term reflects general idea of structure formation governed (directly or indirectly) by sets of rules or algorithms. However, biological meaning of morphogenesis had inspiring by concepts of "Mechanics of Development" that have been working out in experimental embryology to the first half of our century. The biological morphogenesis conception includes, apart from self-organisation idea, so called "regulation" principles. By regulation biologists mean the ability of developing organisms to regulate their pattern to their size. Apparently it is one of the most striking properties of living organisms. Two sides of holistic idea of regulation preferably worked out in modern theoretical biology. It is modelling of regeneration (i) and size invariance of the morphogenesis models (ii).
Regulative development is development in which cells of different history are brought together by morphogenetic movements or surgical intervention, resulting in mutual inductive events. The idea of key significance is that inductive events are not tied to regulation by a single cell, but rather are mediated by cells in collectives interacting across borders in particular neighbourhoods.
A common example of an animal which always retains remarkable regulative ability is the fresh water coelenterate hydra. A piece excised from anywhere except the hypostome or tentacle region will reform itself into a perfectly formed miniature hydra, and this occurs down to excised pieces as small as a few percent [(Bode and Bode, 1984)]. Normal hydra are composed of about twenty different cell types, so this means that one, or at most a few, type of cells in a small excised piece will reform itself into the requisite twenty or so cell types as well as reform itself into the hydra shape, but on a miniature scale!
In modern theoretical biology literature we can find strict but elementary models of regulation which appropriate to use in ALife. Simplest tutorial bases on well-known idea of developmental gradients. Fred Cummings proposes "Laplace algorithm" for the purposes of formal description of regulating development (1985; 1994). According to Laplace algorithm, the value of gradient (chemical concentration in case of morphogen gradient) at one point in space is the average of the value at of that same quantity at all equidistant points. The Laplace algorithm size regulates since it contains no mention of size or scale.
Following Fred Cummings (1994), consider a "worm" of length L, and with fixed values 10 at one end and 1 at the other, as in (10-9-8-7-6-5-4-3-2-1). Next imagine cutting out a section of the worm, say, the values 7 through 4, and joining the 8 to the 3, to form a worm of about half the length of the original. Just after "surgical intervention" the worm contains the values of (10-9-8_3-2-1) in this order along the line, where 10 and 1 are fixed. Two possibilities ensure, depending on whether the worm can regenerate by adding new "cells" by cell division, or by adding no new elements, only re-valuing the existing cells. Clearly, the 8 and 3 do not satisfy the Laplace algorithm after joining them, and are not the average of their neighbours.
First consider the case when no new cells is added, but the existing cells take on new values in successive stages to satisfy the algorithm as best as possible at that moment. This will continue recursively until a steady state is reached where all values take on the average of their neighbours. Then the length of the finally regenerated worm will remain about half of its pre-excision length, but the values will go from 10 down to 1 as before. Then we have:
(10-9-8-3-2-1) (10-9-6-5-2-1) (10-8-7-6-3-1) (10-8-6-4-2-1) …
Finally we achieve regulation as size invariance with desired degree of accuracy all along the creature length.
Next consider the case of a Cummings' worm that regenerates via cell division. In this case the values 10, 9, 8, 3, 2, 1 are all fixed, but obviously the Laplace algorithm is again not satisfied with an 8 adjacent to a 3. This organism now uses the strategy of cell division at the juncture between the 8 and 3. As a result, the new cell is now assigned the value 11/2 = (8+3)/2, so that it now appears as (10-9-8-11/2-3-2-1). This is, of course, an improvement, but still the algorithm is not satisfied, for the 8 and 3 are still not the average of their neighbours. Cell division in the second time-step occurs between the 11/2 and 3, as well as between the 8 and 11/2. These two new cells are assigned the average of the neighbours, namely 17/4 and 27/4, respectively. This process of cell division continues until the worm has satisfied the Laplace algorithm and has regained its original length, as well as its original pattern of values. Hence, we achieve complete regeneration of the creature.
In frames of developmental gradients ideology, all upper levels of an
organisation are predetermined by local developmental gradient values.
If developmental gradient begins to rebuild, then overall structure undergoes
reorganisation. We sketched one-dimensional examples, but apparently this
is straightforwardly extendible to the situation in two dimensions. Word
description of the models allows elementary simulations in frame of cell
automates approach.
Get your own Free Homepage