6. THE RELAXATION TIME OF THE EPIGENETIC SYSTEM 89 



However it is necessary to obtain an estimate for A:, also. This can only be done 

 by assigning numerical values to the many parameters which enter into the 

 relations between the population of protein molecules, the metabolic feed-back 

 repression signal, and the repressor. One very important parameter involved 

 is the storage capacity of the metabolic pool for the metabolite A/,, denoted by 

 Si. When this is large then most of the time there is no feed-back repression 

 and the /th DNA locus produces mRNA at maximum rate. In this case A:,- 

 tends to be small. However, when S, is small, then for a given mean level of 

 enzyme, 7,, the metabolite tends to spill out of the pool and repress the 

 synthesis of mRNA, in which case k/ is large. Without going into some rather 

 unfruitful speculation about the sizes of the various microscopic parameters 

 involved in the detailed calculation of/:,, an intermediate degree of repression 

 is obtained if we take this quantity to be about 24. 



The final parameter which will be required later is c,-, defined in equations 

 (17) as 



r =^£=1.^ = 10-2 

 Q, 5 480 



These numerical values are extremely crude estimates which cannot be 

 regarded as other than illustrative of the significance of the various parameters 

 in the differential equations. There is no information available about the 

 number of aporepressors which there might be for a particular genetic locus; 

 and all that can be said is that there are enough so that the feedback repression 

 mechanism operated by a metabolite. Mi, can produce continuous control over 

 a considerable range of concentration as shown, for example, by the behaviour 

 of the ornithine transcarbamylase system in response to arginine (Gorini and 

 Maas, 1958). This imphes that the effective repressor concentration can vary 

 continuously over this range so that the population of aporepressors is large 

 enough not to be saturated by feed-back molecules until the metabolite reaches 

 very elevated levels in the cell. However, the figure of 100 suggested is very 

 arbitrary. 



Let us now make use of our estimates about the possible dynamic behaviour 

 of the biochemical control circuits under study, to approach the question of the 

 relaxation time of the epigenetic system. This we must do from a consideration 

 of the rate of change of the distribution function, p, after a small disturbance 

 to the system. Such a disturbance might be, for example, a small change in 

 the supply of amino acids to the cell or cell culture followed by a return to the 

 original conditions. We suppose that immediately after the disturbance is 

 withdrawn the distribution function differs from the original equilibrium 

 function po, by a small quantity, Ap: 



p = po + ^P 



The quantity we are interested in is - dApjdt, the rate at which the effect of the 

 disturbance is annulled. Now for a small enough stimulus the rate of return 



