124 TEMPORAL ORGANIZATION IN CELLS 



Operate in the system. Such a principle would then be analogous to the condi- 

 tion of minimum potential energy in physics. The interaction between non- 

 hnear biochemical oscillators may be describable by some kind of field of force 

 which is defined by a potential function, and the system would then tend to 

 "move", i.e. the phase and frequency relations of the interacting oscillators 

 would charge, until the forces are at a minimum. The state of minimum 

 potential would thus correspond to some temporal ordering of constituent 

 rhythmic activities having a certain degree of stability, in analogy with the 

 way in which the condition of minimum potential energy corresponds to a 

 degree of structural order and stability in physical systems. The parameter 6 

 would enter into such a potential function because, as we have seen, it is a 

 measure of the non-linearity of the system, and hence of the intensity of the 

 interaction. Thus when 6 is very small, we should expect to find very little 

 interaction and the system would be only weakly ordered in a temporal sense, 

 this order increasing with d. This whole question is clearly of very consider- 

 able interest and it could lead to a general law of cellular organization which 

 would form the foundation of a true "thermodynamics" of cellular activity. 

 However, it is not yet clear how such a principle should be formulated, nor 

 are the "microscopic" interactions between non-linear oscillators sufficiently 

 well understood to allow of such a formulation. What we shall do now is to see 

 if there is any analytical evidence for the occurrence of entrainment in our 

 system and to see how this and other possible interactions between strongly- 

 coupled oscillators can be studied in the context of our statistical mechanics. 

 There is one condition on the microscopic parameters which immediately 

 presents itself as of possible significance in this relation. Looking at equations 

 (73) and (74) it is evident that if we take 



a2/:2i oc^kii 



^22 



ku 



(75) 



then both expressions are identical, independently of v. The ratio of the two 

 quantities under this condition is 



(^>x.)rel 

 (^$,)rel 



= 1 (76) 



Furthermore, we see from equations (68) and (69) that the condition (75) gives 

 (77), (A+Yj,, = (A+)%, independently of the size of ^. These identities mean that 

 for this particular constraint upon the microscopic parameters, the behaviour 

 of the two variables .Yi and a'2 represented by (a;v,)rei and (A+Yj,. is identical. 

 That is to say both these variables have the same mean frequency of zeros 

 relative to the line v = 0, and the same mean positive amplitudes. This simi- 

 larity of behaviour implies that the variables Aj and a'2 have a constant relation- 

 ship to one another when (75) is satisfied, but we cannot say exactly what this 

 relationship is on the basis of the limited information given by equations (76) 

 and (77). These equations are a necessary condition for entrainment, for 

 when the two oscillators are "locked" together the variables x^ and a-2 will 



