A Probabilistic Model for Morphogenesis 



363 



digit in the identification; a cell above it, and a cell to the right. Thus, tiie 

 first cell Q in a configuration is identified by four binary digits. Next, identify 

 the adjacent cell which contributed the first T in the designation of the first 

 cell, as the second cell Cg, the second '1', as the third cell, C3, etc. Reorient 

 the lattice so that the first cell is beneath the second. We construct the desig- 

 nation number of the second cell as we did for the first. However, this time 

 there are only three binary digits required since the adjacency of C^ to Q is 

 already determined. Any of the cells adjacent to C^ that have not yet been 

 assigned a position in the order can be given one now in a perfectly well-defined 

 way. It is obvious that this procedure can be continued until designation 

 numbers have been obtained for each cell in the configuration. We thus have 

 a well-defined word in 3^ + 1 binary digits and a possible 2^''+^ such words. 



Since the initial cell and the orientation of the lattice were chosen arbitrarily, 

 each district configuration (as usual, excepting those exhibiting some internal 

 symmetry) will be given by Ak such words. Thus an upper bound for TV [A] is 

 23^-VA:. 



It is easily ascertained that a very large proportion of the 2^^^+^ words do 

 not represent A'-configurations. These forbidden words arise for essentially 

 the same reason that the unrestricted random walk on the square lattice fails to 

 serve as an estimate of two-ended configurations. No simple relations have 

 been found that will indicate which of the 2^'^"+^ words are permissible. However, 

 one can generate a random sample of these words by a Monte Carlo procedure 

 and arrive at a statistic that suggests that a satisfactory estimate of N[k'\ is 

 in the neighborhood of 2-^. 



Values of the bounds and the estimate mentioned above have been com- 

 puted for certain values of k (Table I). This serves to give some idea of the 



Table I. Estimate of Configurations for Large Arrays 



magnitudes one might expect for configurations of large numbers of cells. So 

 long as the number of cells is small, the distinct configurations can be exhibited 

 with relative ease. This has been done up to A: = 8 and the results are given 

 in Table II. 



In order to establish the assignment of a probability to each of these con- 

 figurations, a simple and nearly featureless generating function was adopted. 

 Starting with a single cell, equal probability is assigned to each of the four 

 possible two-celled configurations. These are all isomorphic. This two-celled 

 configuration has six open edges. Equal probabilities are assigned to each 



