60 TEMPORAL ORGANIZATION IN CELLS 



have a much more complicated dynamic behaviour than will those consisting 

 of feed-back metabolites. Although the inputs to the pools of activated 

 molecules will generally be oscillatory because they come from oscillating 

 metabolite pools, the outputs from these pools, namely the incorporation of 

 the activated residues into proteins and nucleic acids, will have an extremely 

 irregular dynamic behaviour. To take a particular example, activated 

 threonine will be produced by enzymes from the threonine pool in a cyclic 

 manner which reflects the oscillatory behaviour of the metabolic pool, 

 according to our assumptions. The activated threonine molecules will then 

 be incorporated into a great variety of protein species, the dynamics of whose 

 synthesis will be extremely variable from species to species, many oscillating 

 but not generally in phase, and others possibly being synthesized at steady 

 rates. One would expect that the behaviour of the pool of activated threonine 

 would be highly irregular. This is what we mean by "noisy" behaviour in 

 these pools. 



The general tendency in a system made up of a large number of biochemical 

 oscillators which interact weakly is, then, an evening-out of the oscillations 

 both with respect to amplitude and with respect to phase. An oscillator of 

 large amphtude tends to get damped somewhat, one with small amphtude 

 tends to get excited ; oscillations in phase tend to get pushed out of phase (except 

 when entrainment between oscillators occurs as a result of strong coupling, 

 as will be discussed in Chapter 7). Weak coupling thus results in a distribution 

 of oscillatory motion more or less evenly over all oscillators, and a spreading 

 out of their phase relationships. 



The second way in which oscillatory motion can be transmitted between 

 components is the more obvious one of strong coupling. In this case the inter- 

 action is direct, occurring via repressors, and the motion of oscillators strongly 

 coupled in this way is in a rough sense the sum of the two individual oscillators. 

 For non-hnear oscillations it is not actually correct to speak of a sum of 

 oscillations, for such oscillators interact in such a manner as to produce very 

 complicated behaviour in the coupled pair. The nature of these interactions 

 and the phenomena which can occur in such coupled systems, such as entrain- 

 ment and subharmonic resonance (frequency demultiplication), are of great 

 importance in considerations of the temporal organization of cells, and will be 

 discussed in Chapter 7. For the present it is sufficient to note that a strongly- 

 coupled pair of oscillators will, in general, show an oscillatory pattern which 

 reflects or contains the characteristics of each individual oscillator. This 

 " sharing " of oscillatory energy is in fact represented by the mathematics of 

 strongly-coupled oscillators, since the invariant quantity G, for the oscillators 

 (equation (24)) is a function of the variables of both coupled components. 

 However, in a descriptive sense we can regard direct repressive coupling as a 

 strong form of G-exchange, whereas the previously discussed coupling through 

 common metabolic pools provides a weak form of G-exchange throughout 

 the system. 



Let us now return to a consideration of the part of the epigenetic system 

 consisting of i' components. This subsystem is not constrained to move on the 



