4. THE DYNAMICS OF THE EPIGENETIC SYSTEM 



47 



In order to integrate this system of equations it is necessary first to make the 

 Hnear transformations 



•Vi = Xy-Pi 



-V2 = ^2-P2 



where yi and yi are auxihary parameters which will soon be defined in terms 

 of the original constants. With this transformation we can put the system of 

 equations into the form 



dx 



dt 



dx2 

 'dt 



1 _ ^1 / gi 



QiUi + knYi + ki 



ai I Qi 



2 ^2 



-1 



^2 I 



n 



y\+yi 



72 



.yi+yi 



Q2\A2 + k2lYi + k22Y2 



dy 



y? I , « -J" 1 I d Y2\ 72/7 , ; \ 



^2//- 1 . /. 21 _ -^ik2lXlX\ + k220C2X2) 



'dt-Q2r'~dT-^^''-it 



Qi 



This system is integrable if 



yiki2°^2 72^21 «! 



Qi fiz 



i.e. the "cross-coupling" coeflficients must be equal. Therefore take 



71 = Qik2lOCl 72 = S2^12a2 



This gives us the set of equations 



1 



dt \7i+>'i 



dt \y2+y2 



-ll = k2ioci{kncciXi + ki2a2X2) 



dy2 

 dt 



= ^12a2(^2iai^l + ^22a2^2) 



(23) 



