96 TEMPORAL ORGANIZATION IN CELLS 



upon the assumption of a particular type of steady state as its "equilibrium" 

 position. Clearly there are other types of steady state which could be used to 

 define the equilibrium condition of cellular or higher-order systems. This raises 

 the possibihty of a hierarchy of invariant theories which might be constructed 

 to treat biological processes at different organization levels. These theories 

 would have to be mutually consistent, but each level of behaviour could very 

 well have its own distinctive macroscopic laws and dynamic characteristics. 

 In this context it is of interest to see how consistency can be established between 

 the present statistical mechanics of cellular control mechanims and the laws of 

 thermodynamics, particularly the theorems of irreversible thermodynamics 

 which have been developed by the Belgian school. 



The process which constitutes the dynamic basis of the present study is the 

 biosynthesis of macromolecules in cells. This is a highly irreversible reaction, 

 but there is no reason to suspect that it is in any way inconsistent with the laws 

 of thermodynamics. The reaction will simply proceed with a liberation of free 

 energy and an overall increase of entropy. However, the occurrence of con- 

 tinuing oscillations in a biochemical system is not so obviously consistent with 

 the laws governing chemical processes. It has, in fact, often been argued that 

 periodic phenomena cannot occur in chemical systems because a chemical 

 reaction has no inertia. This argument, however, implies a comparison with 

 mechanical systems in which inertia is necessary to produce periodic displace- 

 ments around the position of equilibrium. For mechanical systems which are 

 not near equihbrium, inertia is not necessary for the occurrence of dynamic 

 periodicities. Now Prigogine and Balescu (1955) have actually shown that 

 in the neighbourhood of an equihbrium state, where the Onsager relations are 

 valid, a chemical reaction cannot in fact undergo a continuing oscillation about 

 an equilibrium state. Such a motion would violate thermodynamic laws. 

 However, they also showed that it is perfectly possible for a chemical system 

 to cycle indefinitely about a steady state providing only that the steady state is 

 sufficiently far from equilibrium — i.e. providing that the chemical reactions are 

 sufficiently irreversible. In this case the oscillatory motion of the system is 

 accompanied by a continuous production of entropy and is consistent with 

 thermodynamic laws. 



In a second paper Prigogine and Balescu (1956) pursued the study of 

 periodic chemical reactions further and showed that for the case of a pair 

 of coupled oscillating variables, the theorems of irreversible thermodynamics 

 require that the oscillation take place in a particular direction. No matter 

 what the initial conditions of the system, the direction of oscillation is 

 fixed by a fundamental inequality which holds for irreversible processes 

 (GlansdorfiF and Progogine, 1954). Furthermore, the steady state of the 

 system is unstable in the sense that any fluctuation will start the system oscil- 

 lating and it will not return to the steady state. These conclusions are directly 

 applicable to the biochemical oscillators which we have been considering in the 

 present study. In this case we have a pair of variables which interact in such a 

 manner as to produce continuing oscillations (except when both are at the 

 steady state, which is unstable in exactly the same sense as that defined above) 



