56 TEMPORAL ORGANIZATION IN CELLS 



single cell, so that even for the present theory it scarcely seems possible to 

 determine the initial microscopic conditions of the epigenetic system. The 

 situation thus becomes similar to that occurring in gas dynamics, so that one 

 is forced to adopt a statistical approach to the analysis of cell behaviour. 

 However, there is a more fundamental reason for introducing probabihstic 

 procedures into this study, and this is where the molecular theory of cellular 

 organization developed in this study differs markedly from the molecular 

 theory of gases. 



The biochemical oscillators introduced in the last chapter are strictly 

 deterministic. If the initial conditions and the parameters were given for any 

 particular oscillator, then it would behave in a perfectly predictable manner, 

 and the values of the variables could be obtained for any given time. However, 

 the biochemical space in which the oscillator is embedded and with which it 

 interacts weakly is not explicitly defined in our theory ; and as we mentioned in 

 Chapter 3 this dynamically undefined part of the cell must be regarded as 

 having the properties of random noise. This noise is transmitted to the 

 deterministic oscillators, and the result is that their trajectories are no longer 

 strictly predictable. It thus becomes necessary to talk only of mean or average 

 values of the trajectories, so that all the dynamic properties of the control 

 systems must be treated in a probabilistic manner. This is precisely what is 

 accomphshed by means of a statistical mechanics. 



It should be emphasized that it is not necessary to have a microscopic 

 representation of a macroscopic system in order that its phenomenological 

 properties be adequately described. Thus classical thermodynamics is a 

 perfectly self-contained theory which can be axiomatized on the basis of 

 Caratheodory's principles (cf. Margenau and Murphy, 1943), without any 

 recourse to a microscopic description of matter in terms of molecules. Simi- 

 larly it is by no means necessary to reduce cells to molecules in order to describe 

 phenomenologically such aspects of cell behaviour as cell division, differentia- 

 tion, or circadian periodicity in photosynthetic activity. However, there is a 

 very strong bias towards a molecular representation of phenomena in science, 

 and this is overwhelmingly evident in biology today. Having adopted this 

 attitude to cellular organization, it is necessary to attempt a resolution of 

 microscopic and macroscopic events, and in order to accomplish this it is 

 necessary to introduce procedures like those of statistical mechanics. We will 

 now consider in detail what conditions must be satisfied in the dynamic system 

 in order that these procedures be valid. 



Before we can construct a statistical mechanics of cellular control systems 

 we must show that our system of equations satisfies a theorem known as 

 Liouville's theorem. We have already guaranteed this result by showing that 

 the equations have a Hamiltonian representation, as shown in equations (20) 

 and (23), for the uncoupled and coupled components respectively. We will 

 now derive the theorem and explain its significance. 



Consider a large number of copies, or what is known as a Gibbs ensemble 

 of cells each organized dynamically according to the assumptions made in the 

 last chapter, so that their control systems obey equations (20) or (23), but 



