120 TEMPORAL ORGANIZATION IN CELLS 



Valenciennes, there seems to be some evidence for the presence of an oscillation 

 of smaller frequency than 24 h. This shows up in the form of shoulders or 

 bumps in the circadian curve at roughly the same part of each cycle, in a 

 manner similar to the occurrence of smaller oscillations in the wave-form 

 shown in Fig. 9a. In the case of Gonyaulax it is of additional interest that the 

 pattern of luminescence appears to remain roughly periodic, even after the 

 strong diurnal rhythm has been suppressed by actinomycin at a concentration 

 which does not kill the cells (Karakashian and Hastings, 1962). The oscil- 

 lations are then small and somewhat irregular, but there seems to persist a 

 periodicity of 3-4 cycles per day. This is what one would expect if the result 

 of actinomycin treatment is a great reduction in 9, resulting from the inhibition 

 of mRNA synthesis. With d small the non-linearities of the oscillation would 

 be weak, so that a circadian rhythm dependent upon subharmonic resonance 

 would become unstable and die out, with only a small free-running oscillation 

 remaining. 



Alternatively actinomycin might reduce mRNA populations in the cells 

 to the point where they are too small to support regular oscillations, as we 

 argued in the case of bacteria. Then whatever variation is still observed in the 

 normally circadian observable is largely the result of biochemical noise. 



It should perhaps be remembered that the observables in luminescence 

 and pH are not in fact protein, much less mRNA, although there will certainly 

 be some fairly close correlations between protein levels and the observed 

 variables. Thus the luciferin-luciferase system responsible for luminescence 

 in Gonyaulax is controlled by the activity of luciferase, a protein, and by the 

 level of luciferin, controlled again by enzymes. 



A little more information about the behaviour of strongly interacting 

 oscillators can be obtained through a study of the mean frequency of zeros for 

 the variables x^ and .Y2. We have 



oj^xi -v) = I e-^^ \xi I S(jci - v) dvjj e-^^ dv 



Pi —pi —Oi 



z ^ 



\x\ 



00 



^P\Pi J 



-P2 



• ■—ah V' C 

 ^*_! ^ e-^[2\,vAr,+/,„x,«]^^^ (70) 



^p\p% J 



pipz 



p% 



where |ii | means the phase average of |a:i |. 



The integral can be reduced by the transformation 



= (^M-(.a + g^) 



