4. THE DYNAMICS OF THE EPIGENETIC SYSTEM 33 



It is possible to combine these into a single equation of the form 



This can be integrated, and the result is 



«/^''-i8/ ^-, + 6, r,-^;iog(^, + A',- y,) ^ G^(X„ y,) = constant (15) 



We will show that this integral defines a closed trajectory in the space of the 

 variables (A',-, y,) so that what we have is an oscillating system. Now the 

 constant in the integral is determined by the initial values of the variables, i.e. 

 by the quantities (Xi)Q, ( y,)o both of which must be ^ 0. Therefore if these are 

 finite quantities, G(Xi, y,) must also be a finite quantity. This implies that both 

 the variables Xi and y, are bounded above (they are bounded below by zero), 

 since the integral G is a monotonic increasing function of Xj and Yj when these 

 quantities are greater than the steady state values (those values of Xj and y,- 

 which make dXiJdt and clYiJdt vanish). What this means is that the expression 

 G{Xi, y,) gets increasingly larger as Xi and/or y,- increase in size, as is readily 

 verified. It is thus clear that if Xi and y,- start off with finite values, then they 

 must remain finite. 



In fact, we can go further and show that neither variable can remain larger 

 than its steady state value. Let us call these quantities /?,• and ^/, so that they 

 are defined by the equations 



-^, ^-0 



Ai+kiqi 



o^iPi-^i = 0. 



(16) 



where we assume that/?/, ^/ > 0. Now it must be assumed that «,/y4/ > bi, for 

 OiJAi is the rate of mRNA synthesis when there is no repression occurring. 

 This rate must therefore be greater than the degradation rate of mRNA, 

 since otherwise the maximum rate of mRNA synthesis at the /th locus is 

 totally inadequate to supple the cell with messenger, and the gene is effectively 

 inactive. 



Returning to equations (14), it is readily verified that if A',- >/>,-, then 

 dViJdt > 0. Thus if Xi remains always larger than /?,•, then y,- will constantly 

 increase since its derivative is positive, and will approach oo as / ^ co. We have 

 shown that this is impossible. Therefore X,- cannot remain always greater than 



Pi- 



Observe further that for Xj > p,, y, increasing, y,- will at some point 

 become greater than^/,-. For y,- > ^/, dXi/dt < 0, as is easily seen from equations 

 (14) and (17). Therefore X,- must then begin to decrease. What we must show 

 now is that Xi cannot remain always smaller than /?,-. 



