Chapter 6 



THE RELAXATION TIME OF THE EPIGENETIC SYSTEM 



The Size of Macromolecular Populations in Cells 



The considerations of this chapter are of crucial importance for the present 

 approach to the dynamics of cellular control mechanisms. The quantitative 

 estimations which we will make will tell us if it is possible to get any kind of 

 regularity in the dynamic behaviour of different species of messenger RNA and 

 protein such as we have assumed so far, or if the sizes of these populations 

 (especially of mRNA) are so small that the assumption of continuity is un- 

 tenable and a stochastic representation is the only reasonable one to con- 

 template. The introduction of stochastic (i.e. random) variables into the 

 present theory would not necessarily alter the fundamental dynamic charac- 

 teristics of the feed-back control devices which we seek to study (their oscil- 

 latory behaviour). In fact Feller (1939) made such a study in connection with 

 Volterra systems and found that the oscillations in prey and predator popu- 

 lations emerged as mean trajectories over the stochastic variations. It would be 

 necessary, however, to consider at length what noise level the biochemical 

 control systems could tolerate and still show some degree of periodic or 

 rhythmic behaviour. That is to say, a fundamental consideration would have 

 to be : How strong must the oscillatory signal be in order to be detected as a 

 periodic variable in the presence of a given noise level in the biosynthetic 

 processes themselves (not in the biochemical environment in which these 

 processes take place) ? This is certainly an important question to answer, but it 

 requires an examination of many aspects of filtering, error correction, and 

 reliability of template synthetic and control processes which are beyond the 

 scope of this study, and for which very few "hard" facts are available. Our 

 procedure has been to assume that regular oscillations occur in the system 

 and to introduce noise as a feature of the biochemical "bath" in which the 

 components are immersed. This attitude is exactly suited to a statistical 

 mechanics, which can be used to determine the sizes of the irregularities or 

 fluctuations occurring in system variables as a result of the noisy bath. Such 

 a procedure is clearly an approximation which can be defended only if the 

 variables are in fact nearly continuous; i.e. if the macromolecular population 

 sizes are fairly large. We must now investigate this question. 



The only good estimates presently available about the sizes of macro- 

 molecular populations in cells are values which have been determined for 

 bacteria, especially for the molecular biologist's friend, Escherichia coli. 

 However, on the basis of these it is possible to make some reasonable guesses 

 about population sizes in cells of protozoa and higher organisms. A recent 

 study by Byrne (1963) shows that there are some 1-6 x 10"* ribosomes in a 



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