36 TEMPORAL ORGANIZATION IN CELLS 



Since bi = ai/Qi and /3, = a,/?, (equations (16)), we can write equations (14) 

 in the form 



— = ai{Xi-pi) 



Now transform to new variables defined by 



Xi = Xi-pi 



Then the equations take the form 



^^^'=^y ^ 



(17) 



dt 'V+yi ) 

 dyi 



— — = CX- 



dt • • 



This transformation puts the integral in the new form 



Gi(xi,yd = ^ + ^,[j/-log(l +>',)] = constant (18) 



The new variables can take negative values, but they have lower limits -pi for Xj 

 and (Ai/Qi-\) for ;;,-. That [{Aj/Qi)-!] is a negative quantity is readily 

 verified by substituting for g,-: 



Ai , -ktgi 



1 = , , = -T„say. 



Ai+kiqi Ai+kiqt 



These lower limits for the new variables are obtained when we put Xi = 0, 

 y. = 0, which are the lower bounds for the original variables. We cannot set 

 upper limits to the variables. 



For the system made up of// pairs of equations (14) there will be an integral 

 of the above type for each pair. Thus regarding the // pairs as a single system 

 enclosed in a metabolic space in the manner previously described, there is a 

 general constant of the motion which can be written as 



n 



G(xu A'2, . . ., x„; yi,y2, . . .,>^„) = S C?,(-V/,7/) = constant (19) 



1=1 



where Gi(Xi,y,) is the integral of the /th component (1 8). The general integral G 

 will serve the same function in the study of cellujar activity that the energy 

 integral plays in classical mechanics. The construction of the statistical theory 

 forms the content of the next two chapters. Let us note here that for the simple 

 system of// weakly interacting components the structure of G is particularly 



