5. THE STATISTICAL MECHANICS OF THE EPIGENETIC SYSTEM 67 



Yi falls, and when Xi > pi, F, rises, the descending part of the 7, curve being 

 steeper than the ascending part since again A', takes larger excursions above/?, 

 than below, although the asymmetry is not so marked in the case of this 

 variable. (The curves in Fig. 4 are not drawn to scale for mRNA and protein 

 concentrations, and show qualitative properties only.) 



We can find the units of ^ from equations (33). The parameter A',, being an 

 equilibrium constant, has units 1 /concentration, which in our somewhat 

 unorthodox units is l/(molecules per cell). We write this as 1/C. Aj has no 

 units because it is in fact 1 +Lj[Ai], as we see from equation (7) in Chapter 4. 

 Since Qi = Ai+kigt, this has no units either. Now 



ociki 



and a, has the units of a rate constant, 



1 _ 1 



time T 

 Therefore 



CiXiiX^-pd = "^'X^iX^-pd = ^ 



_ C 

 ~ T 



C2 



The same result is obtained for the F, expression : 



b^iUY^-qj) ^ C 2_ 



_ C 



~ T 



because bj has units C/T. This fixes the absolute talandic temperature scale. 



In a statistical mechanics the parameter 6 is of importance in indicating the 

 direction of preferred flow of G from one component or set of components to 

 another weakly coupled to it. The one with higher 6 tends to lose G to the one 

 with lower 6, so that when the system has equilibrated they will both have the 

 same d. At equilibrium for the epigenetic system we will have over all sub- 

 scripts i, j the equalities 



c,X(X, p,)- g^ A,+k,Yr ' '^ ' '^'*~ a Aj+kjYj 



Thus the weak interactions which exist between synthetic units in cells, 

 by virtue of metabolic coupling through common metabolic pools, have a 

 very important consequence in the present theory, even though the interactions 

 are not sufficiently specific to allow for exact algebraic representation 

 in the dynamic equations. They are in effect represented by the statistical 

 hypothesis of weak interaction between components. A consequence of this 



