A Probabilistic Model for Morphogenesis 361 



phenomena are not considered. The principal concern of this model is the 

 domain of forms an idealized organism can assume and the likelihood of an 

 egg developing into such a form. Tlie mathematics employed is elementary, 

 and as in so much of combinatorial analysis, it is ad hoc*. The attempt to 

 establish a relationship between the form of an organism and the information 

 content of the germ cell ancestor is treated here from a point of view that has 

 some resemblance to that of statistical mechanics. 



It is assumed here that there are only a finite number of different kinds 

 of cells in any organism and a finite number of cells of each kind. We neglect 

 the dynamic processes occurring continuously in an organism: changes within 

 cells, the migration and movements of cells, the death of certain cells and the 

 cleavage or maturation of others. If there is some well-defined way of des- 

 cribing the orientation of each cell in any organism relative to the other cells 

 in that organism or relative to some arbitrary system of coordinates, then it 

 is possible, in theory at least, to enumerate all the possible ways of arranging 

 cells into different configurations. Some of these arrangements would be 

 recognizable organisms, the overwhelming majority would not. In any case, 

 these objects, both the recognizable and otherwise, are elements of the set 

 of all possible configurations. This procedure might represent a means of 

 defining a given species by certain restrictions on the possible orientations 

 of cells and thus to identify the given species with a well defined subset of all 

 possible configurations. 



Most multicellular organisms can be said to arise from a single cell resulting 

 from the fusion of two germ cells. It is true that there are certain biological 

 objects, of which the slime-mold is a notable example, which take their form 

 from the migration, coalescence and specialization of a number of free-living 

 cells. However, such organisms are uncommon and will not be considered 

 further. 



This single germ cell divides into two cells and these cells will divide further 

 and so on until maturity. Throughout the course of this branching process 

 the growing organism will pass through a sequence of configurations, each 

 of which is an element in the set of all possible configurations. If there is a f: 



relationship between successive configurations which is recursive, then a 

 generating function can be constructed to describe the branching process. 

 Generating functions are useful because they may afford a means of assigning 

 a probability to each possible configuration. The actual model chosen for 

 investigation has been simplified to the extent that its relation to biological 

 reality is largely impressionistic. Its justification is heuristic, for the study of 

 relatively simple systems may suggest methods of approaching the real systems 

 which are so very much more complex. 



The element of the model is called a cell. All cells are considered to be 

 identical. We restrict ourselves to the consideration of arrangements of cells 

 in two dimensions. The shape of the individual cell is unspecified (they may 



* I should like to take this opportunity to express my debt to a number of mathematicians 

 both at Princeton and at the Institute for Advanced Study with whom I have discussed this 

 problem; and in particular to Professor Valentine Bargmann, Professor William Feller, ij 



Dr Hale Trotter and Dr Norman Shapiro for their stimulation and suggestions. Needless to ' 



say, the results and errors are my own. 



