58 TEMPORAL ORGANIZATION IN CELLS 



value of any function/(;ci, ...,x„;yi,.. .,y„) of phase coordinates is then defined 

 to be 



\fpdxdy 



pdxdy 



where dx = dx^dxi, • • ., dx„, dy = dy^dyi, . . ., dy„ and the integration is taken 

 over all possible values of the variables x,-, j,- (/=],..., «). 



Now the above mean value is in fact an average over phase space, not in 

 time. A fundamental statistical hypothesis must now be introduced: for 

 purposes of finding expected values of variables of interest for a system about 

 which there is only limited knowledge, equal extensions in phase corresponding 

 equally well to this knowledge will be assigned equal a priori probabihties. 

 What this means is simply that for a statistical survey we consider all possible 

 copies of a system compatible with what information we have about it, and in 

 ignorance beyond this point we regard each copy as equally probable. Phase 

 space thus becomes the mathematical construction for carrying out the statis- 

 tical survey. 



The Canonical Ensemble 



The particular construct which we will use for the computation of expected 

 values of phase functions throughout the present study is known as the canon- 

 ical ensemble. Consider the behaviour of a part consisting of only v, say, of 

 the total system of ?j components (v < n). This part or subsystem does not 

 have its G constant in time but is assumed to exchange G with the rest of the 

 system, only the total G being conserved. There are two ways in which such an 

 exchange of " oscillatory energy " can occur in the epigenetic system. The first 

 is by weak interaction. This is analogous to the weak interactions which occur 

 in gases by means of collisions between molecules. In the context of cellular 

 control systems there is no such thing as collision between components (al- 

 though collisions between molecules still occur), but we do have competitive 

 interactions between components for the precursors required for macro- 

 molecular synthesis, as discussed in Chapter 4. A component or a group of 

 components with a large oscillation may be expected to deplete these pools 

 appreciably during their phases of synthesis, the pools filling up again to some 

 extent following their degradative phases. The size of the pools will therefore not 

 remain constant, and the motion of large oscillators (i.e. those having a large 

 amplitude) will tend to be transmitted, albeit weakly, to "smaller" oscillators 

 through the pools. In a very rough manner we may expect that during periods 

 of relative depletion of the pools, synthesis will be somewhat reduced in those 

 components which begin their rising phase of oscillation after a group of large 

 oscillators have commenced synthesis; and synthesis will be encouraged after 

 the degradative phase of these large oscillators, when the pools fill up. Large 

 pool size will tend to increase the amplitude of the smaller oscillators, and so we 

 see that some of the "oscillatory energy" of the large oscillators will be trans- 



