26 temporal organization in cells 



Control Equations for mRNA Synthesis 



The equations which we consider for messenger RNA synthesis will be of 

 the general form 



^ = <^,(^/, y-. Md -rPiXi, y,, M,) (3) 



Here ^,(A',-, 1^/,M,) is a function describing mRNA synthesis, and i/',(A'/, F;,A/,) 

 represents the rate of its degradation. It is assumed that the kinetics of re- 

 pression of mRNA synthesis by the metabolite M,- are essentially the same as 

 those of enzyme inhibition. This means that we are dealing with a surface- 

 binding phenomenon wherein the repressing molecule or complex combines 

 reversibly with the template and so interferes with its synthetic activity. The 

 template also combines reversibly with the precursors for RNA synthesis, and 

 for the purposes of the present discussion we can assume that the concentration 

 of these precursors is given by a weighted mean value over the different activated 

 nucleotides. Since we are regarding these as parameters in the system, the 

 details of their functional representation is not important. Call the weighted 

 mean Ai. 



Only templates which are free of repressor molecules can function in 

 mRNA synthesis, according to our assumptions. It should be emphasized 

 that surface reactions of macromolecules wherein they form non-covalent 

 complexes with other molecules are always reversible, although if the popu- 

 lations of molecules are very small then it may be unrealistic to assume that 

 well-defined equilibrium constants exist for these reactions. The reaction 

 between a template, 7,-, and a repressor, i?,-, would normally be written in the 

 form 



At equilibrium, classical kinetic theory allows us to write 



mm ^'_,• '''■ ^^^ 



where Ki is the equilibrium constant and square brackets represent concen- 

 trations. We will adopt this procedure in the present analysis, but these equa- 

 tions may not represent very accurately the events which actually take place 

 during the repression of a genetic locus. It is nevertheless important to remem- 

 ber that nearly all of the feed-back repression systems which have been studied 

 so far show continuous control of enzyme level over a range of repressor 

 (metabolite) concentrations. Thus the detailed studies of Gorini and Mass 

 (1958) on the repression of ornithine transcarbamylase by arginine show a con- 

 tinuous response of the enzyme-synthesizing system to the intracellular level of 

 the feed-back signal, suggesting that an equilibrium of some kind is being set up 

 such as that of equation (4), [7,], the effective concentration of template 

 varying inversely with repressor concentration, [/?,]. The value of A',- may be 



