H. WADDINGTON 



absolute values of P and Qj; and these will also be reduced if k 3 is fairly large, so that 

 P and CLare rapidly removed. 



Without going into further details, we can see that if two autocatalytic processes 

 compete for raw materials, we may under some conditions find that an initial change 

 in the supply of the materials produces an exaggerated effect on the steady-state 

 concentrations of the synthesized products, and thus on the rates at which these pro- 

 ducts can be made available outside the system. 



If we suppose that a, b and c are the raw materials out of which two genes manu- 

 facture their immediate products P and Q, we have now developed a picture by 

 means of which we can see how change in the concentrations of these raw materials 

 leads to exaggerated differences in the rate at which P and Q, are passed out of the 

 nucleus into the cytoplasm. I have previously suggested this model, without going 

 into such detail concerning it (Waddington, 1948). 



It is, however, by no means the only model which might be appropriate. As 

 Delbruck (1949) has suggested, there might be direct interactions between the two 

 synthetic processes. These are perhaps most simply formulated by supposing that P 

 is destroyed at some rate proportional to the concentration of Q (and vice versa). 



The equations for — and — will then contain terms in PQ. If we regard the system 



CtL (XV 



as closed, rather than open as was the system discussed above, and if the supplies of 

 raw materials are taken as constant, the equations which result are of the same type 

 as those which arise in the study of the growth of two populations of animals which 

 compete with one another for a limited food supply. Lotka (1934) had discussed the 

 relatively simple situation of two populations (or substances) for which the equations 

 take the form 



§ =m p P-k p P*-k pq Pd 



§'=m q (l-k Q (^-k qp P(l 



He shows that according as m p k q is greater or less than m q k pqy and m p k qp greater or 

 less than mjcp, so the final state of the system is either wholly P, or wholly (£, or a 

 certain fixed ratio between them, or finally the system is one which will finish up 

 either entirely P or entirely d according to the initial concentrations of these sub- 

 stances. 



Again, Kostitzin (1937) discusses a somewhat similar set of equations which he 

 takes to represent two species competing for a food supply which consists of another 

 species which multiplies in the normal way, but which could equally well represent 

 two autocatalytic substances which interfere with one another and also compete 

 for a raw material which is supplied at a more or less exponential rate. In this case 

 also he shows that, under some conditions, the system will be such that the initial 

 conditions will determine whether it goes wholly in the direction of one of the com- 

 peting elements or in that of the other. 



Kostitzin has also discussed shortly the more general case in which there are many 

 competing and interacting substances (or populations), so that we have a large series, 



116 



