4. THE DYNAMICS OF THE EPIGENETIC SYSTEM 39 



the set of equations describing the control scheme of Fig. 5 would have the 



form 



dX, _ a, 



-b, 



(21) 



Here Yx occurs as a controlling variable in the expression for dX2Jdt. The 

 pair (Xi, ^i) still behaves as an autonomous non-linear oscillator, so that X^ 



Figure 5. 



and Fi are periodic functions of time, as has been shown. This means that 

 dX2Jdt is also a periodic function of time, with the same period as that of Y^. 

 However, this periodicity in the rate of mRNA synthesis at locus L2 does not 

 necessarily mean that the sizes of the variables X2 and Y2 oscillate about fixed 

 mean values. They could increase indefinitely, the oscillation being super- 

 imposed on a rising curve of synthesis which is unbounded ; or they could 

 decrease to zero. The actual behaviour of the variables {X2, Y2) would depend 

 upon the parameter values 02, A2, k2, and ^2- Only if ^,, the steady state value 

 of Yi, also makes dX2Jdt = Q can the second "driven" oscillator be stable 

 (bounded above and below by positive numbers). But in general, equations 

 (21) do not allow us to determine the stability of the pair {X2, Y^), and the most 

 that can be concluded is that the driven variables X2 and Y2 will have a 

 periodicity in their dynamics determined by that of the oscillator defined by 

 the first two equations. There are ways of altering the equations sHghtly so 

 that the stability of the coupled pair is assured, but this would require additional 

 assumptions about the kinetics of macromolecular synthesis. 



