148 TEMPORAL ORGANIZATION IN CELLS 



These can be integrated as we see from the combined expression 



We get 



Gr{xr,yr) = /SXe"'' - .v,) + 6,[j, - log (1 +)',)] = constant (94) 



as the new integral. The variable yr is unchanged from the previous system, 

 but Xr now varies between — oo and co as X^ varies between and yj. 



Assume now that a component whose kinetics are described by equations 

 (93) is part of an epigenetic system of « components, all drawing upon common 

 metabolic pools and so immersed in the same biochemical space in the manner 

 described in Chapter 5. Then statistical mechanical procedures can be applied 

 to the study of its behaviour, and we can proceed in the usual manner. In 

 particular the probability that .\v is to be found in the interval [av, x^ + dxr] is 

 given by 



P_,^dx, = i- e-(^^/^)f-''-^'V.Y, 



Zpr 



where now 



■^^ = j e-'^-'^'^^'-'^Ux, 



Z„. = 



If now we write 



Pr 



so that .Y;. = log I;, then in terms of this new variable the probability distribution 

 is 



= l-^f/^)-ie-^'^'/«^^, (95) 



The most probable value of ^^ call it [^J, is obtained by finding the maxi- 

 mum of this expression. It is therefore the root of the equation 



"2^ 



|I(/3r.e)-2] 



fi- 



')-'.■<■ 







Q 



hence [I.] =1-5 (96) 



Pr 



providing 9 < ^,. If ^ ^ jS^. then the exponent of ^r in (95) is negative or zero 

 and the maximum of P^^ is obtained at ^r = 0- That is to say 



[t] = if 6^ ^, 



