86 TEMPORAL ORGANIZATION IN CELLS 



the oscillation might take U-2 h, the whole period being 6-8 h. On the other 

 hand, if there are more than two DNA templates for synthesis of messenger, 

 the period could be less than 4 h; or if the amplitude of the oscillation in the 

 mRNA population is smaller than we have assumed, the period could again be 



smaller. 



Consider now the size of the oscillation which will occur in the protein 

 population for the case of the 4-h oscillator with a 50-molecule amplitude in the 

 mRNA population. We may say very roughly that the messenger population 

 in excess of the low value of 80 may have a mean value of about 40 during the 

 first two hours of the cycle because of the asymmetry in the oscillation. In this 

 two hours these 40 extra messengers can synthesize some 960 protein molecules, 

 so that as a crude estimate we might say that this is the amplitude of the oscilla- 

 ation in the protein population. With a mean protein population size of 

 24,000 this represents a 4% oscillation. If we assume a longer period for a 

 single cycle, then we get an increased amplitude for the protein oscillation, 

 approaching perhaps 6-8% of the mean population value. And if we were to 

 consider a protein species with a mean life-time of 10 h instead of 20, then the 

 population of protein molecules would be halved and the amplitude might 

 approach 15% of the mean population value. 



These estimates are extremely rough, but they do serve to give us some idea 

 of the amphtudes and the periods which could occur in the cells of higher 

 organisms if macromolecular populations do oscillate in the manner suggested. 

 The frequency of such oscillations would be quite small, but they would still 

 be in the range of perhaps 2-10 per day. This would put them in the proper 

 range for a tidal rhythm (~ 4 per day), but to generate a circadian (~ 24 h) 

 rhythm it would be necessary to have a subharmonic resonance of order ^ti- 

 The phenomenon of subharmonic resonance or frequency demultiplication, 

 is a very interesting property of non-linear oscillators which is observed when 

 two or more such oscillators interact by couphng of some kind. This whole 

 question will be treated at some length in the next chapter, but for the present 

 we may note the important fact that very reliable clocks can be obtained by 

 means of frequency demultiplication, clocks whose periodicity is more regular 

 than those of the fundamental oscillations which produce them. Another 

 significant feature of this phenomenon is that the subharmonic oscillation 

 always shows a considerable increase in amplitude over that of the fundamental 

 oscillators, so that a very appreciable amplification can occur. This is an 

 important observation in view of the relatively small oscillatory amplitudes 

 which we have obtained for protein populations. 



There is strong evidence that the circadian clock mechanisms of many 

 different species of organism are " synthesized " by this means from oscillations 

 with a period considerably less than 24 h (see, e.g. Pittendrigh and Bruce, 1 957). 

 It has often been found possible to force circadian systems into light-dark 

 rhythms which are fractions of 24 h, such as 12, 6, and 4 h. This is usually done 

 by subjecting the organism to a light-dark regime based on submultiples of 24, 

 4 h of light being followed by 4 h of darkness, for example. After the artificial 

 regime is stopped and the organism is placed in constant conditions, it will 



