A Probabilistic Model for Morphogenesis 



367 



Since any direct extension of the model to larger values of ^ does not appear 

 feasible, another procedure was adopted. Starting as before from a single cell, 

 the edges were numbered, a random number table (14) was consulted to find 

 a number equal to or less than 4, and then a cell was adjoined to the appro- 

 priately numbered edge. The open edges were renumbered, another number 

 equal to or less than the number of open edges obtained from the table, and 

 a new cell adjoined. In this way samples of 1000 10-configurations and 

 16-configurations were constructed. A few larger configurations were prepared 

 by this procedure. One such containing 200 cells is shown in Fig. 5. 



In the case of the sample of the 10-array, one configuration was obtained 

 twenty-two times. Its probability was computed by the exact procedure des- 

 cribed above and found to be 2.06 per cent. All the configurations containing 

 four inner corners (77-= 16) (maximal for /: = 10) appeared more than ten 

 times each. With very few exceptions, in order of occurrence, there followed 

 the configurations with tt = 18, 20, 22. There were eighty-three occurrences 

 with 77 = 24 (no inner corners), but none of these was two-ended. Although it 

 was impossible to enumerate all the configurations, by judicious use of the 

 equality of probabihties found in configurations with the same graph, estimates 

 were made of the numbers of configurations of each kind up to rank 1150. 

 The data were plotted in Fig. 4. It can be seen that the portion of curve 

 obtainable is very close to the ordinate axis. 



A similar procedure was followed in the case of the sample of the 16-array, 

 Here, estimates v^ere considerably poorer, but the same general features were 

 revealed (Table IV). The thirty-two possible configurations with 77 = 18 



appeared twenty-eight times, or an expectation of occurrence of a particular 

 configuration of 8.75 x 10^*. (It is assumed that all configurations with 

 identical values of tt have approximately equal probabilities of occurrence.) 

 it was estimated that the expectation of occurrence of a configuration of 77 = 20 

 was 3 X 10-4; 77 = 22, 4 x 10"^; and 77 = 24, 1.6 X 10~^ In this sample of 

 1000 there were only five occurrences of configurations with 77 = 32, and no 

 occurrences of 77 = 34, although a low estimate of the number of distinct 



