Chapter 7 

 STATISTICAL PROPERTIES OF THE EPIGENETIC SYSTEM 



Having obtained some idea of the time periods which are likely to be involved 

 in the dyanmics of epigenetic phenomena, we can now use the apparatus 

 introduced in Chapter 5 to obtain some results which are relevant to the ques- 

 tion of temporal organization in the epigenetic system of single cells. The 

 system of equations (18) with only weak interaction will be considered first, 

 and then a study will be made of the new features which arise in connection 

 with strong repressive couphng between components as described by equation 

 (23). 



We first introduce some functions which are useful in bringing out certain 

 oscillatory characteristics of our feed-back control system. In this we follow 

 the general procedures used by Kerner (1959) in his study of Volterra systems. 

 We saw in Chapter 5 that there is an asymmetry in the oscillations in the sense 

 that the variables make greater excursions above their steady state values than 

 below. Furthermore, this asymmetry becomes more exaggerated as 9, the 

 talandic temperature, increases. More information about this behaviour can 

 be obtained through the following function. Let T+/ThQ the fraction of a long 

 time interval T during which a variable, say x,-, is at values greater than its 

 steady state (which is 0). The time average of this is just the time average of 

 the function : 



h(xd = I, .v,> 

 /j(a-,) = 0, .v,< 



Using the canonical ensemble for evaluating the average, this becomes 



T 





-Pi -Pi 



00 00 





98 



