5. THE STATISTICAL MECHANICS OF THE EPIGENETIC SYSTEM 63 



rest of the cell but will exchange talandic energy, G, with the other components, 

 n-v in number. Now when n is a large number, then in the canonical ensemble 

 a great preponderance of the components will have G/s (equation 18) in the 

 immediate vicinity of C,- (the canonical mean of (7,). It is true that the number 

 of degrees of freedom in the epigenetic system of a cell {2n) will seldom be as 

 large as the number usually encountered in physical systems, but they will 

 still be in the hundreds, as we will see in the next chapter. This is large enough 

 for statistical methods, but fluctuations from expectation values may be of 

 considerable importance in the biological case. Thus concentration changes 

 in single living cells would provide one set of observables for the present theory, 

 and an analysis of the data would depend upon the use of the canonical 

 ensemble. This theory shows that if j^, the number of components observed, is 

 large then the fluctuations of these v components about their mean G will be 

 small; while if r is small, the fluctuations may be quite substantial. 



We will now use the canonical ensemble to derive some results which 

 illustrate the way in which the statistical theory gives us information about the 

 general behaviour of the dynamic system. We may ask what is the probability 

 that the variable x,- will have a value in the range (x,-, X/+^x,). This is obtained 

 by integrating over all coordinates except x,-: 



\,.dxi = j'pdv'IJ pdv 



P 



where the dash means that x,- is left out of the integration. This gives us the 

 result 



P:,.dXi = — - — dXi 



which shows that the variable x,- has a normal or Gaussian probability distri- 

 bution. The most probable value of x,-, call it [x,], is obtained by solving the 

 expression for its maximum value. The result is easily obtained as [x,] = 0, the 

 value which makes P.^. a maximum. Therefore the most probable value of 

 Xi is the steady state value/?,. 



For the variable y, the analogous expression is 



g-(bilG)[yi-\o%(\+yi)] 



Py^dyt = dy 



z. 



Qi 



e-biyii\i +y.)^ 



^.. 



Solving for the most probable value, we find [yi] = 0. Therefore we again get 

 the result that the most probale value of 7,- is the steady state value, ^/. This 

 need not always be the case, however. We will see in Chapter 7 that when 

 strong coupling occurs between components, the most probable value of 



