4. THE DYNAMICS OF THE EPIGENETIC SYSTEM 45 



closed circuit whose dynamics will be identical with those obtained for the 

 simple control loop described by equation (14). 



There are obviously still many discrepancies between the simplest model we 

 have constructed and cellular control, although the above discussion has shown 

 that the model can accommodate more situations than might appear at first 

 sight. Some of these limitations will be removed when we come to discuss 

 strong interactions between control units or epigenetic components, but a 

 more fundamental discrepancy relates to the fact that our model really applies 

 to a homogeneous system, not to a heterogeneous one. This question will be 

 discussed briefly towards the end of this chapter, after we have considered the 

 question of strong coupling between components. But as has been said 

 already, a strictly classical analysis is necessary to explore the theoretical and 

 experimental implications of the approach being used here before embarking 

 upon the much more complicated functional analysis which could greatly 

 improve the theory. 



Systems with Strong Coupling 



Turning now to the problem of strong coupling between autonomous 

 components, considering first the situation described by Fig. 2. Here the 

 metabolite Afi feeds back to repress not only the locus Lj, but also a second 

 locus, L2; while the metabolite M2 also has a double repressive effect, acting 

 on Li as well as L2. Again we make the assumption that repression obeys the 

 laws of surface adsorption isotherms; and we assume further that the two 

 repressors interact competitively for a single repressor site at each locus. Such 

 a repression site is then the "operon" which has been shown to have a genetic 

 existence by the Pasteur school in the case of E. coli, although we go well 

 beyond their evidence in assuming that diff'erent repressors compete for a 

 single operator site. By using an argument similar to that which led to expres- 

 sion (7), but extended now to include the effect of a second repressor, R2, on 

 the first locus, we get the following result for the amount of template-precursor 

 complex which is effective in mRNA synthesis: 



[TiAi] = 



l+LiMi] + /:n[^i] + ^i2[^2l 



Here the constants have the same meaning as those in (7), but we have written 

 A'li in place of K^ and K^ is now the equilibrium constant for the reaction 

 between R2 and Ti. The expression for the amount of template-precursor 

 complex for the synthesis of mRNA of the second species is analogous to 

 the above: 



^ ' 'J l+L2[A2] + K2dRl] + K22[R2] 



By using the further arguments concerning the relation between /?, and M,- 



