THE DIFFERENTIATING SYSTEM 4O7 



Now in a system consisting of a mixture of raw materials for which 

 several synthetic processes are competing, situations can easily arise in 

 which a slight change in initial conditions will have a great effect on the 

 fmal state; in fact, such systems will often be such that there are only a 

 limited number of fmal states to which they can attain, the choice between 

 one end-state or another depending on the concentrations of substances 

 present at the beginning (Waddington I948l», 1954). To take a simple 

 example of what may happen, consider two substances P and Q, which 

 are being formed out of the raw materials A, B, and C, for the supplies 

 for which they compete. Suppose competition occurs because P is formed 

 from A and B, while Q is formed from B and C. Again for the sake of 

 simphcity, let the reaction constants be the same for the two syntheses, 

 as shown in Fig. 19. i; and let A, B and C diffuse into the system at 

 rates proportional to the difference in their concentration inside (A, B, C) 

 and outside {a, h, c), while P and Q are removed at rate ^3. Finally, let 

 us suppose that the coupling of ^ and B to form P, and of B and C to 

 form Q, are autocatalytic processes, i.e. are speeded up by the presence 

 of already-formed P and Q. This is a simple form of a 'feed-back' mech- 

 anism. The equations for the rates of change of the various components 

 will be 



dA 



— - = k{a -A) - kPAB + J^2P2 

 at 



-^ = k(h-B)- kPAB + kP^ - hQBC + k^Q^ 

 at 



Ac 



= k{c-C) - kQBC + k^Q" 

 at 



dp 



-— = kPAB - hP' - kP 



dt 



^ = kQBC - hQ' - kQ. 



At the steady state, we fmd a relation between P and Q of the form 



{khc + k\)P - {kha + k%)Q + kk{a -c) . . . (i) 



Now if ^3 is small compared with k (i.e. diffusion out of the system is 

 slower than diffusion in), then we can neglect its higher powers, and we 

 find 



P=^Q + f_^iZL_^ (,) 



c ko c 



