6. THE RELAXATION TIME OF THE EPIGENETIC SYSTEM 87 



continue to show an endogenous rhythm which is the same as that into which 

 it was forced, thus demonstrating the existence of internal oscillations with a 

 period shorter than 24 h. However, it is usually the case that these shorter 

 ''unnatural" rhythms are unstable, and after completing a number of cycles 

 which add up to 24 h, the system reverts to a diurnal rhythm. An interesting 

 exception has been reported in the case of the alga Hydrodictyon, wherein a 

 stable rhythm of growth and photosynthesis can be impressed with a period of 

 17^ h (Pirson, Schon and Doring, 1954). If the period of the fundamental 

 oscillations is 3^ h, then the order of the subharmonic required to generate a 

 17^ h oscillation is \, but there is no definite evidence that this is the case. In 

 Chapter 7 we will investigate in some detail the properties of the coupled non- 

 linear oscillators which arise in this study, and see how the statistical mechanics 

 can be applied to the theoretical analysis of their behaviour. 



On the basis of the assumptions and estimates made above, let us now write 

 out the differential equations of an oscillating feed-back control circuit of the 

 type shown in Fig. 1, giving numerical values to the parameters. This will 

 involve some more very rough estimates, but it suggests that a crude evaluation 

 of some of the microscopic parameters involved in our equations is not out of 

 the question. Consider first the equation 



— - cc^Xi-tii 



A protein population of mean size 24,000 molecules and mean life-time 

 20 h implies that about 1200 molecules are degraded every hour, so 



^. = i|^ = 20 molecules/min 



A protein synthetic time of 5 min gives a,- =^. The steady state value of the 

 mRNA population is then/?,- = 100 molecules. 



The calculations for the equation of mRNA synthesis are not so easy, and 

 we will have to make assumptions about the sizes of metabolite pools and 

 repressor populations to evaluate the parameters. The complete expression 

 for dXiJdt is given by the equation 



dX, _ k\[T;\^UA,] 

 dt l+mA^ + UR^ ' 



as we see from equation (1 1). 



Here /:■ is the rate constant for the synthesis of mRNA, which we have 

 taken to be one molecule per minute, so that k'i = \. The actual units ofk] are 

 1/time. We take {T,]q = 2, assuming only 2 DNA templates of the /th species 

 for mRNA synthesis. [Ai\ is the size of the pool of activated nucleotides, and 

 we will take this to be 100 mRNA "equivalents"; i.e. about 30,000 activated 

 nucleotides (300 nucleotides per messenger of molecular weight about 10^, 

 coding a polypeptide unit of 100 amino acids with a coding ratio of 3 : 1). If 

 there are about 200 active loci with an average rate of synthesis of 1 messenger 



4 



