38 TEMPORAL ORGANIZATION IN CELLS 



occurs only when the jc/s and j/s satisfy the Euler-Lagrange equations 



d_dA_dA^ 

 dt dxj dXi 



dtdyi dyi 

 Substituting for the partial differentials in these equations, we get 



or 



dG 

 dXi 



Xi = 



dG 



dy, 



which are just the equations (20). Thus the equations derived for the dynamics 

 of a simple epigenetic control loop can be summarized in a "least-action" 

 type of principle. No fundamental use will actually be made of this principle in 

 the following, for a much more general principle will be introduced when a 

 statistical mechanics is constructed in the next chapter. 



More Complex Control Circuits 



Before deriving equations for control circuits which interact strongly, let 

 us see how far the simplest model can be extended to cover some of the more 

 obvious deficiencies in the theory. There are many structural proteins such as 

 keratin, collagen, lens protein, etc., which are metabolically inert or very 

 nearly so, and cannot generate a feed-back repression signal in the manner 

 that enzymes can. It is conceivable that a partial degradation product of such 

 metabolically inactive proteins could serve a feed-back repression function, 

 since examples are now known of small polypeptides acting as regulatory 

 substances, bradykinin and kallidin being perhaps the most famihar (cf. 

 Elliot, 1963). Such a control mechanism certainly cannot be ruled out; but 

 there is another control scheme which fits more readily into the ideas which 

 have been introduced regarding the role of metabolites in cellular regulatory 

 processes. This is represented in Fig. 5, where the metabolite M^ which is 

 controlled by enzyme Y^ acts to repress not only the activity of locus Li, but 

 also that of the locus L2, the structural gene for the metabolically inactive 

 protein Yi- On the basis of our earlier arguments leading to equations (14), 



