A Probabilistic Model for Morphogenesis 



369 



increasing k (Table V). Estimates have been made for A; = 10 and A' ^ 16 from 

 the Monte Carlo samples. These estimates are certainly lower than the precise 

 values since an estimate of/?, was not available for every configuration and the 

 means for rather large groups of configurations were used instead. A functional 



Table V. Entropy oj \i-arrays 



relationship between liG^^j,) and k has so far not been found. One may con- 

 jecture that the relatedness increment /(C7y,fc) — I(Gj^j._i) approaches 0.5 as k 

 increases without limit. This may be interpreted to suggest that the rate of 

 information accumulation in an organism constructed according to such a plan 

 is half a bit per cell division. 



Other measures have been suggested as being useful to our purposes. 

 Following the terminology of McGill and Quastler (16), the relative uncertainty 



M(C \ 

 of the generating process is Dj^. = — ^ ^^ and the redundancy is Q^^ — \ — D 



H,} 



},k- 



The redundancy evaluated from the results presented in Table V increases 

 from a value of for A: = 2 to a value of 0.234. As with the measure HGj^j.) — 

 /(G;j.,_i), it seems plausible to expect that as k increases Q^j. will converge to 

 some value other than or 1, but no procedure has as yet been found to test 

 this conjecture and to determine the limit. 



In a qualitative way, this increase in /(C7,,fc) may be understood to mean that 

 the featureless generating function considered above determines the configura- 

 tions of large numbers of cells with a high degree of specificity. It is virtually 

 a certainty that large configurations will be essentially circular in outline; that 

 they will have a high density, i.e. they will contain very few 'holes' and short 

 'tentacles'. Thus if one considers the most probable outcomes of the generating 

 procedure in the large, then these configurations appear to resemble one another 

 very closely even though they exhibit no correspondence in detail. 



It is true that the results obtained with such a simple model are far removed 

 from the intricacy of development of living things. A few regularities in the 

 most probable forms may be introduced by small modifications of the initial 

 generating procedure. Objects that are ellipsoidal or cruciform or objects 

 characterized by large numbers of branches have been developed by such 

 modifications. However, it is unlikely that further complexity can be introduced 



