4. THE DYNAMICS OF THE EPIGENETIC SYSTEM 43 



The first equation reduces to 



Bi + nii' 



f^^--) 



"' -ft, 



Ai+k, y, 



where Ai = Bi — Wtl — 1-5, 



c '"i 



^ ill / 



k,= 



miCi 



c 



This is the same form as the equations previously derived, and so the 

 dynamics of the (^i, ¥{) pair is basically the same as that of the simple system. 

 However, what about the dynamics of all the other pairs (A',-, F,)in the sequence ? 

 First, any locus which can respond to the feed-back signal, M,„, but which is 

 not controlling its magnitude through a rate-limiting enzyme, will be driven 

 by the signal M,„ into an oscillation in the same way that the metabolically 

 inert components discussed previously are driven by a repression signal to 

 which they are sensitive. And again in this case it is not possible on the basis 

 of our assumptions to determine whether or not the quantities of protein 

 (enzyme) synthesized by these loci will be bounded above, for the equations 

 do not allow us to set any upper limit for these quantities. However, it is the 

 case that the enzymes of a biosynthetic sequence whose genetic loci respond 

 to the repression signal generated by the end product will never disappear 

 from the system; that is to say, there is always a positive lower bound for these 

 variables. This is because any enzyme, say Y„ which is decreasing in con- 

 centration will eventually reach a value at which it becomes rate limiting for 

 the whole sequence. At this point it will "take over" the dynamics of the 

 sequence, and the pair (X„ Y^) will begin to oscillate about some steady state 

 value in the same manner and for the same reasons that the pair (Xi, Y^) was 

 previously shown to undergo oscillatory motion. However, our assumptions 

 are not sufficient to settle the question of the stability of all members of the 

 biosynthetic sequence even if all loci respond to the feed-back repression signal 

 produced by M„,. All that can be concluded is that at least one pair of variables 

 in the set {Xj, Yj;i= l,...,m} will undergo stable oscillations about steady 

 state values, that all variables in the set will be bounded below by a positive 

 (non-zero) quantity, and that they will all undergo periodic variations but 

 may not be bounded above. (In real systems, of course, quantities are always 

 ultimately limited simply by the physical size and resources of cells. When we 

 speak of unbounded or unstable motion in a cell variable we mean that it 



