7. STATISTICAL PROPERTIES OF THE EPIGENETIC SYSTEM 101 



Since this has the indeterminate form 0/0, we differentiate numerator and 

 denominator, getting 



Hw%r '-•••■'■ wT---'- 







c,p2.e-^^i/'iV2 



= lim / ^ n / ^ \i/2 



. £^V-.c,.v. + (|-)' J ,-,v, 



Evaluating similarly the second term in the denominator, we find the expression 

 equal to 



Cip]e-^''P'''2 



^Z\ — e~^^''^^''2+^jPie-pc,p,v2 



= Pi 



Therefore in the limit d^<x> we have the result A-~^—pi which is precisely 

 what we expect since this is the lower bound for the variable x. 



It is possible to define similar expressions for the variables j,-, but the 

 evaluation of the quantities thus obtained is considerably complicated by the 

 properties and the asymptotic behaviour of the function r(v,z) which enters 

 into their definition. In the following analysis we will study sometimes the 

 behaviour of x, and sometimes that of j,, the aim being rather to extract 

 enough information about the behaviour of the theoretical model to suggest 

 experimental tests of its validity, and to indicate directions for further in- 

 vestigation. 



The Mean Frequency Function 



We want now to introduce a function which is of central importance for the 

 study of the behaviour of the epigenetic system in time, and for studying the 

 effects of strong interaction on this behaviour. Again the great utility of a 

 statistical mechanics for calculating mean values of various quantities in a 

 complex system will become evident, for without this mathematical apparatus 

 it would be extremely difficult to bring forth the results which we will now 

 obtain. Even in the most simplified situation, where one analyses the properties 

 of a single isolated feed-back control loop involving one species of RNA and 

 the homologous species of protein interacting in the manner described by 

 equation (18), there are some mathematical barriers to obtaining explicit 

 information about their oscillatory behaviour. But by going immediately to a 

 very complex, interacting system with many components amenable to a 

 statistical analysis, the calculations are greatly simplified. 



