4. THE DYNAMICS OF THE EPIGENETIC SYSTEM 29 



these metabolic pools. In the absence of relevant data we will assume that the 

 pools have the following characteristics. The pool for the /th metabolic species 

 is taken to have a storage capacity which is denoted by S/. So long as the total 

 amount of metabolite of the /th species in a cell is less than this value so that 

 Mi < Si, it will be assumed that the feed-back signal is zero and there is no 

 repression. But when A/",- > 5,-, we assume that the quantity of this excess 

 metabolite which serves a repressive function is directly proportional to some 

 power of the difference [M,-S,]. The feed-back signal is then a quantity 

 represented by 



a,[M,-5,r (9) 



where a,- is a constant and n an integer. For simplicity we will take « = 1 in 

 the following, thus assuming that the amount of metabolite feeding back from 

 the pool is linearly related to the quantity of metabolite in excess of the storage 

 capacity. However the analysis can be carried out for any integral value ofn. 

 The storage capacity. Si, will be assumed to be constant relative to the relaxa- 

 tion time of the epigenetic system, although this quantity is probably capable 

 of some variation with different cell states. The model thus obtained for the 

 kinetic behaviour of metabohc pools is an extremely simple one ; but once again 

 the hope is that the essential qualitative features of the processes involved have 

 been included, and that refinements can be added to the model as more detailed 

 knowledge of cellular organization is obtained. 



The other question with which we must deal in discussing repression is 

 whether the feed-back metabolite acts directly upon the genetic locus, or 

 whether it first combines with an aporepressor in the manner suggested by 

 Pardee, Jacob, and Monod (1959). If no aporepressor is involved, then the 

 repressor i?,- of equation (7) is simply a,[M,— 5,]. However, if the metabolite 

 must form a complex with a protein to produce an active repressor, then we 

 have a surface reaction of the form 



Mi+Hi 7^^ Mi Hi 



where //,■ is the aporepressor and the complex M^T/,- is R„ the active repressor. 

 Assuming that //,- is present in total constant amount [7/,]o, the by now 

 familiar argument leads to the following expression for the concentration 

 of active repressor: 



Wi Hi] - -Y.^rKf{M^ ' ^^''' ^' - ku 



In this equation (M,) is the concentration of metabolite available for reaction 

 with the aporepressor //,-, which we assumed to be (t,[M,-5',]. Thus we have 



Now the rate of synthesis of mRNA will be directly proportional to the 



