The cell physiology of early development 



The system will change progressively. The course of the change is complicated to 

 describe in detail, but we can discover something about the general characteristics 

 of the system if we consider only the final steady state, when no further change is 

 occurring, and the right-hand side of each equation is equal to 0. Under these con- 

 ditions it is easy to show that the ratio P/Q^will be equal to ajc. That is to say, if in one 

 region of the egg, the available supplies of A are increased in comparison with those 

 in some other region, then the steady-state concentration of P in the first region will 

 be increased in exact proportion. That is, of course, not very surprising. And it 

 hardly seems to provide much enlightenment as to the mechanisms of differentiation. 

 What we seem to meet in embryology are situations in which small initial differences 

 lead to large divergences in later development. To account for this, we need some- 

 thing more complicated than the very simple system we have just discussed. 



As a first step towards a more adequate picture, let us suppose that the coupling of 

 A 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' mechanism. Our equations will now be 



dA 

 It 



= k(a-A)-k 1 PAB+k 2 P 2 



d l=k(b-B)- k l PAB+kJ*-k 1 QBC+ h z Q? 



d £ =k(c-C)-k 1 QBC+k 2 Q? 



dP 



dt 



= k 1 PAB-k 2 P 2 -k 3 P 



^ = kl QBC - k 2 0J - k 3 Q 



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



{kk 2 c +k z *)P = (kk 2 a+k 3 *)(l+kk 3 {a-c) (i) 



(Note that although the dimensions in this look a bit odd at first sight, k and k 3 are 

 simple diffusion constants, while k x and k 2 are rate constants of third- and second- 

 order reactions.) 



Now if k 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 



_ a _ k 3 a— c , . 



p =-7^+v ~r (2) 



n, 2 



Thus if initially in a certain region the supply of A is increased relative to that of 

 C, we find that P will be increased relative to Q, by something more than a propor- 

 tionate amount, the excess being expressed by the last term on the right. And if the 

 rate of removal of P (that is k 3 ), is greater than the rate at which P breaks down again 

 into A and B, (that is k 2 ), this excess can be considerable. We can also see from 

 expression (2) that the exaggeration will be the more important the smaller the 



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