7. STATISTICAL PROPERTIES OF THE EPIGENETIC SYSTEM 107 



This value is about v = 15/100, giving us 



14 X 100 log M5 = 13-98 



Therefore for 6 = 1/240, the oscillations of ;^ will cross a line displaced a 

 distance of 3/20 above the steady state axis only about half as often as they cross 

 the line J,- = 0, This shows us that the "envelope" of the oscillating trajectories 

 is very close to the steady state axis for small values of 6. When v = ^, the 

 frequency of crossing of this line by the variable >', is already less than 10^^ 

 of the value on the steady state axis. 



Returning to equation (54), let us consider the approximation we get if 

 we let ^ approach zero. In this case we use the formula 



00 



/ 



e""e-"dt = 



- y^.-uVA 



This leads to the expression 



(.2^^,) 



1,2 



I2i8c„ 

 for the integral (54), so that we get 



Q-CiS'-i2? 



ji>M-^-''-~-:i*K-y(2;.)- 



y-'i^- 2^ 



In the limit of small /S, 

 so this reduces to 



CO 



Zp^ 



I 7T ' 



J „..,.-.-...,. Rl.^^J 



\ —g-CisV2P 



ds 



-Pi 



As before the integral on the r.h.s. is 



2V(2^') 



so we get 



00 



/ 



-Pi 



\yi\ e'^^^'dXi X -^ (j8 very small) 



Equation (53) then becomes, for this limit. 



aj(yi-v) 



^^ZpiZ,, 



