7. STATISTICAL PROPERTIES OF THE EPIGENETIC SYSTEM 109 



The period of an oscillation is then (8v/27r) or about 20 h. If we increase 9 to 

 144, then the mean period of the oscillation is increased to about 24 h. These 

 very elevated ^-values are probably near the upper limit for the system we are 

 considering, the oscillatory amplitude of a 24 h oscillator being very large. 



The Dynamic Consequences of Strong Coupling 



Let us now see what information we can get about the dynamic properties 

 of components when they interact strongly by repressive coupling in the manner 

 described by equations (23). We may expect to find a more complicated type of 

 behaviour than in the case of weakly-interacting components, with some 

 qualitatively new features emerging. Unfortunately a complete investigation 

 of the new behaviour must await further mathematical and computational 

 analysis. However, it is possible to glimpse some of the richness of structure 

 which emerges with the introduction of strong interaction in the system, even 

 with the rather cursory treatment to which this study is confined. 



The differential equations defining the motion of the epigenetic system 

 with pair-wise interactions of components which we arbitrarily label 1 and 

 2 are 



^1 = ^Mr+ — / M -^^ ^ ai^2l(^liai^'l+^'l2a2^2) 



X2 = bi ( J— . 1 j >'2 = a2 ^'12(^21 «! Vi + k22 CC2 ^2) 



where yi = fii/^ziai, Yi = Q2^'i2a2 



gi = Ay + kuqi + ki2q2, Q2 = ^2 + ^21^1 + ^22^2 



The relations between the transformed variables and the original variables A', 

 and y,are 



A'l = Xi-pi yi = ocik2i[kn(Yi-gi) + ki2(Y2-q2)] 



X2 = X2-P2 J2 = a2^12[^2l(J^l-^l) + ^22(5^2 -^2)] 



We see that the A'/s give us direct information about the behaviour of RNA 

 populations, but the F/s are linear combinations of protein populations. 

 We will therefore find it more informative to investigate the behaviour of the 

 Xi's since the results are immediately interpretable in terms of biological 

 quantities. 



The integral of the above equations is 



x^ x^ 



G(.Xi,X2,yi,y2) = kn k2i oc^" ^-hk^zk 21 ccioc2XiX2 + k22ki2 0c^-;^ 



+^ibi-yilog(l+ji/yi)]+^2b2-y2log(l+;^2/y2)] 

 = constant (24) 



