6. THE RELAXATION TIME OF THE EPIGENETIC SYSTEM 85 



than protein synthesis. The life-time of mRNA in higher organisms is not 

 known with any degree of exactitude for messenger species synthesizing 

 enzymes, as it is for bacteria. In red blood cells synthesizing mainly haemo- 

 globin, however, the messenger molecules must be quite stable with a life-time 

 of several hours, since in the absence of any mRNA synthesis these cells con- 

 tinue to produce proteins for many hours. In Hela cells the half-life of mes- 

 senger molecules is about 3 h (Penman et aL, 1963), so that the evidence 

 certainly indicates a considerably longer life-time for this molecular species in 

 the cells of higher organisms than in bacteria. Let us take an average value of 

 4 h for the life-time of a messenger molecule engaged in the synthesis of an 

 enzyme forming part of a closed control loop. One messenger is then used 

 about 48 times for protein synthesis. This is no more than a reasonable guess, 

 but our present estimates cannot be critical at this stage of our knowledge of 

 molecular processes in cells. A single DNA template could then maintain a 

 maximum of about 240 mRNA molecules if its maximum rate of synthesis is 

 one messenger per minute, as we have assumed above. 



The Estimated Periods of Epigenetic Oscillations 



Suppose, then, that we have a mean population of 100 messenger RNA's 

 of a particular species maintained by two DNA templates which are not func- 

 tioning at full capacity. If the homologous protein species has a turnover rate 

 of 5% per hour, then the mean lifetime of a molecule is about 20 h. The popu- 

 lation of 100 messenger molecules could maintain the protein species at a level 

 of about 24,000 molecules if they are working at the rate of 1 protein molecule 

 synthesized in 5 min (12 molecules/h). Let us now assume that these popula- 

 tions form a closed feed-back control circuit of the type represented by Fig. 1 

 and equations (14), the protein being an enzyme. Assume also that there is an 

 oscillation in the mRNA population which has a mean amplitude of 50 mole- 

 cules, the population varying between 80 and 130 molecules, say, allowing for 

 the asymmetry in the wave form. With 4 h as the mean messenger life-time, we 

 want to know roughly how long such an oscillation might take on the basis of 

 the rates we have assumed, and what size of oscillation it will produce in the 

 protein population. The two DNA templates can produce at most 120 mRNA 

 molecules/h and they must continuously replace degraded messenger. Thus 

 when the mRNA population is at its low value of 80 molecules, it would take 

 the two DNA templates about 1 h, acting near ^-capacity, to increase the 

 mRNA population to 130 molecules, considering continuous messenger 

 degradation and also the fact that the feed-back signal is slowing down mRNA 

 synthesis as the messenger population, and hence the enzyme population, 

 increases. Thus a rough estimate for the time required for the rising phase 

 of the oscillation is about 1 h. The non-linearity of the oscillations is such that 

 the falUng phase is about three times as long as the rising phase in the case of an 

 oscillation of significant amplitude such as we are considering (see Fig. 4), so 

 that the whole oscillation will take about 4 h. With more repression always 

 occurring, the DNA templates may not reach even ^-capacity, and it will take 

 them longer to synthesize the 50 "extra" messengers. Thus the rising phase of 



