4. THE DYNAMICS OF THE EPIGENETIC SYSTEM 35 



rather different from those arising in mechanical systems, and it will be of 

 considerable interest to study the properties of these new oscillators. The 

 primary source of the non-linearity in the equations (14) is the feed-back 

 repression term which enters in the denominator, a consequence of the 

 assumption that repression is a surface adsorption phenomenon of the type 

 which is so characteristic of macromolecular activity. 



If we let the subscript / run over all values from 1 to n, then we have a 

 system consisting of /; pairs of variables, 2n variables in all. Each pair of 

 variables represents a closed control loop of the kind represented in Fig. 1. 

 Assume now that these n components are weakly coupled together by being 

 enclosed in a space in which there are common metabolic pools for the com- 

 ponents. That is to say, the precursors required by each component for the 

 synthesis of RNA and protein are drawn from common pools of activated 

 nucleotides and amino acids, and the metabolites, M,-, flow into common pools. 

 (These metabolic pools are of course biochemical entities without any strict 

 geographical location in the cell. They constitute a sort of metabolic "bath" 

 in which the epigenetic components are immersed.) Biochemically this means 

 that we have partitioned the cell into n components, each of which regulates its 

 own steady state level of mRNA, protein, and metabolite, and the only inter- 

 actions which occur between them are of the weak kind, due to competition for 

 precursors from the common metabolic pools. This model is still a far cry from 

 the biochemical organization of real cells, and it would seem that very little in 

 the way of interesting or significant macroscopic behaviour could be derived 

 from it. What we have here is a crude model which is probably not even as close 

 to the real system as the billiard-ball model of gases is to real gas structure. 

 However, some of the most obvious discrepancies between the idealized system 

 described above and what is known today about real cellular control patterns 

 can be removed without altering the fundamental structure of the model. And 

 a considerable amount of complexity can be added to the system in the manner 

 shown in Fig. 2, where a pattern of "strong" interactions between components 

 is built up so that a great richness of interconnections arises and the model 

 begins to take on some of the comphcated structure which is the most obvious 

 characteristic of cellular organization. It is of considerable interest, moreover, 

 to study the consequences of increasing complexity in the model, starting from 

 the simplest case, for in this procedure we can see how a more integrated and 

 highly organized time structure emerges as interactions are added to the 

 simplest or "ideal" control system. 



Analogies with Classical Mechanics 



To complete the dynamic description of this ideal system (ideal from the 

 point of view of the analyst, certainly not from the point of view of the cell), and 

 to obtain an integral which has a more convenient form from that of (15), we 

 transform the variables so that the steady state becomes the origin of co- 

 ordinates. Write first 



