140 TEMPORAL ORGANIZATION IN CELLS 



one might derive an equation of state which relates T (physical temperature), 

 6 (talandic temperature), and some third, and as yet unidentified, variable in a 

 manner strictly analogous to the gas law :pV— nkT. Thus we want to see how 

 one might derive a relation of the form 



TFt = nXd 



where Ft is the thermodynamic variable conjugate to physical temperature in 

 the epigenetic system, n is the number of components, and A is some constant. 

 The derivation of this result should be looked upon as purely illustrative of 

 the way in which a general law of "thermodynamic" behaviour in cells might 

 be obtained. 



We restrict ourselves to the system without strong coupling, and our point 

 of view is to regard C, the talandic energy of the whole system, as a function of 

 6 and T, the latter variable entering G through the microscopic parameters 

 bi and c,. Now we have already suggested that bj will increase slightly with 

 temperature, so let us suppose that over some temperature range we can write 

 the functional dependence in the form 



b; = p^T'^^ 



where ct, is small. We must now consider how c, will vary with T. This is a 

 composite parameter which is made up of other parameters, 



Uiki 

 Since Q, = ^' 



aikjbi 



we get Ci = 



"/ 



The parameter a,- involves a rate constant for protein synthesis which would 

 increase with temperature, but it also includes an equilibrium constant for the 

 reaction between mRNA and activated amino acids which might increase or 

 decrease with T. a, is an even more complicated constant, being related to 

 mRNA synthesis but involving again a rate constant, equilibrium constants, 

 and mean nucleotide pool size. Let us assume that it increases with T to the 

 same extent as a,-, so that the ratio of these two quantities cancels out. We are 

 left with ki, which is essentially the equilibrium constant between the repressor 

 and the repressor site on the genome, although once again other parameters 

 enter into it. It seems likely that kj will decrease with increasing T, since one 

 would expect higher physical temperature would make the complex between 

 repressor and RNA less stable. This reaction might be quite sensitive to 

 temperature, and we will assume that, in fact, c, decreases appreciably with T, 

 the relation being 



over some temperature range. 



