4. THE DYNAMICS OF THE EPIGENETIC SYSTEM 27 



large or small, depending upon the affinity between T, and /?, but according to 

 the above formulation there will always be some free template present in the 

 system. The probability that the template will be free long enough to synthe- 

 size a complete messenger may be very small, so that the locus may be effec- 

 tively or statistically "shut off"; but even then it will be "leaky". Thus it is a 

 bit misleading to speak of induction and repression as on-off events at the 

 molecular level as is usually done in discussing control of gene activity. It 

 is possible to get switch-like behaviour in biochemical control systems as a 

 result of weak interaction between large numbers of control units; or as a 

 result of specific strong reactions between two or more units, as will be shown 

 later. However, it is difficult to devise an elementary biochemical reaction 

 which could reasonably be assumed to control gene activities in an on-off 

 manner — i.e. to be reversible, but to have only two discrete states, either " on" 

 or "off". One of the results of our analysis will be to show how such discon- 

 tinuous behaviour can arise as a result of interactions between elementary 

 biochemical processes which are themselves continuous, so that biochemical 

 switches need not be assumed to be part of the elementary microscopic 

 machinery of the cell. 



The reaction between templates T,- and the precursors for mRNA synthesis, 

 denoted by the quantity Ai, can be written in the same manner as that between 

 the templates and the repressors : 



Tj+Ai ^ TjAj 



i-i 



and again at equihbrium we have 



[TiA,] lL = Li (5) 



[n{A;\ /_,. 



Assuming first that the operator site on the template is also a coding site for 

 nucleotides so that there is competition between /?,- and some Ai, we can write 

 as the total amount of template present 



m]o = [r,]+[r,.4,]+[r,/?,] (6) 



We can now use equations (4) and (5) to substitute for [r,] and [r,/?,-] in terms 

 of [TiAi\ in equation (6). We get the result 



(7) 



