7, STATISTICAL PROPERTIES OF THE EPIGENETIC SYSTEM 1 3 I 



assumed to exist between components in virtue of their dependence upon 

 common metabolic pools, is not sufficiently strong for the establishment of 

 synchrony between oscillators. This is a result of our assumptions. It is quite 

 possible that under certain conditions the interactions occurring between 

 components in metabolic pools in cells might be strong enough to result in 

 dynamic interactions of some kind. For example, if two protein species are 

 both composed largely of a single amino acid and if the pool of this amino acid 

 is relatively small, then it is possible that a sufficiently strong interaction could 

 be established between these species to result in some kind of stable dynamic 

 ordering of their oscillatory motion, although this need not be entrainment. It 

 should be recalled at this point that the first recorded observation of entrain- 

 ment was by Huygens (1629-1695) who reported that two clocks which were 

 slightly "out of step" with each other when hung on a wall became synchro- 

 nized when fixed on a thin wooden board. This is not very strong coupling, 

 and might be comparable to that which may occur under particular conditions 

 between components through metabolic pools. A situation of this kind could 

 be investigated within the framework of the present theory by representing the 

 interactions explicitly in the diff"erential equation, and then studying its 

 dynamic consequences. Clearly, there are many other ways in which compo- 

 nents could interact strongly, and again these interactions would have dynamic 

 consequences for the time-ordering of oscillatory activities in the system. 

 The only conclusion that we can draw from the above result is that a weak 

 interaction in the sense employed in this study, i.e. an interaction sufficient to 

 result in a distribution of oscillatory motion throughout the whole epigenetic 

 system of a cell such that it is possible to speak of the estabUshment of an 

 equilibrium condition in the system, is insufficient to produce entrainment 

 between components. 



In concluding this rather cursory investigation of the interaction arising 

 between strongly-coupled components, we have moved somewhat closer to 

 defining the parametric constraints under which entrainment may occur. It is 

 of particular interest to note that entrainment definitely does not occur when d is 

 very small, whereas for 6 large it is possible to find values of the parameters 

 which satisfy some necessary conditions. These conditions are still not 

 sufficient, but it would seem that a complete analytical study of entrainment 

 should be possible by pursuing further the lines of the present investigation. 

 In comparison with the extremely difficult and laborious techniques that must 

 usually be used in the analytical study of synchrony in non-linear oscillators, 

 it is perhaps not an exaggeration to claim that considerable simplification is 

 afforded by the use of statistical methods. However, the limitation of our 

 approach is that it is restricted to integrable systems, a condition seldom 

 satisfied in non-linear mechanics. 



There seem to be other interactions in the strongly-coupled system which 

 could produce stable relations between the variables, besides the most familar 

 one of entrainment. We have mentioned Halberg's observation that syn- 

 chrony is not a sufficient basis for interpreting the mutual stable relationships 

 existing between various physiological activities in cells and organisms, so 



