7. STATISTICAL PROPERTIES OF THE EPIGENETIC SYSTEM 119 



resonance in Xi may have a more complicated wave-form than those shown in 

 Fig. 9. Furthermore, with mutual coupling of the type we are considering, it 

 is possible for subharmonics to appear in both oscillators, each one completing 

 a different whole number of cycles in a given time interval. This can result 

 from proper adjustment of the parameters, kn, kji, o^, (j2, as well as the 

 coupling parameters. However, we shall consider for the moment the condi- 

 tion of strong asymmetry, when one oscillator tends to drive the other without 

 itself being driven significantly. This we represent by the condition of large 9, 

 large ki2, and small A'21. We have seen that (/!+)$, then tends to be large, so 

 that Xi has a large positive amplitude of oscillation (we can restrict our atten- 

 tion to positive values of the variables, because for 6 large the asymmetry of 

 the oscillations about the steady state is such that the behaviour of the oscil- 

 lators can be investigated above the steady state axis). However, the variable 

 X2 would not be expected to show a marked subharmonic resonance because 

 it is not strongly driven by Oi, kii being small. The expression for the mean 

 positive amplitude of X2 is in fact easily shown to be (for large 6) 



U^Y ^ 1 // -d \ (knk22-knk2,)y^ 1 ™ 



^ ^^' 0C2A2ki2k22jWifcnk22) + Vikuk2i)]\^^_Jknk22_A'^^ ^ ^ 



which is decreased as ki2 increases. Therefore the large mean positive ampli- 

 tude of .Vi is not due to the transmission of a large oscillation from the "driving '* 

 oscillator O2 to the driven oscillator Oi under conditions of asymmetrical 

 coupling, but must arise from some more complicated interaction. 

 The ratio of these two quantities is 



(^+)$x _ oci /(ki2k22\ 



(A+)x, <X2N ykiikn/ 



This result shows us that when 6 is large, it is possible to control the relative 

 sizes of the oscillations in a strongly coupled pair by altering the ratio of the 

 coupling coefficients, ki2 and k2i. In this way an oscillation of arbitrarily 

 large amphtude (and hence arbitrarily large period) can be induced in one of 

 a pair of coupled oscillators when the talandic temperature of the system is 

 sufficiently large. This is just the kind of phenomenon which one would have 

 anticipated in view of the well-established consequences of non-linear inter- 

 action (cf. Hayashi, 1953; Minorsky, 1962), although we cannot yet conclude 

 that subharmonic resonance (frequency demultiplication) is an established 

 feature of our model. This will require a more thorough study of stability 

 relations in the system. Although we have been concentrating in our 

 analysis on the variables x,- and hence on the behaviour of mRNA popu- 

 lations, any subharmonic oscillation in these quantities would also occur in 

 the homologous species of protein. 



If one actually looks at the shape of well-plotted circadian rhythms, such 

 as those reported by Karakashian and Hastings (1962) for the case of the 

 luminescence rhythm in Gonyaulax polyedra, and by Mori (1960) for the 

 circadian rhythm of pH in the body fluid of the sea-pen, Cavemularia obesa 



5 



