30 TEMPORAL ORGANIZATION IN CELLS 



quantity of activated precursor-template complex which is given by equation 

 (7). Therefore in equation (3) we can write 



cf>^{Xi, M,) = 



l+L,[Ai] + UR,] 



where k] is either a constant or a function of Xj. For the present we will take 

 this to be a constant, which implies that mRNA has no effect upon the rate at 

 which it is synthesized. In this case k] is the rate constant for mRNA synthesis. 

 In Chapter 8 we will consider a modified set of equations in which k] is a func- 

 tion of Xi. We will assume further for the moment that ilji{Xi,M) is also a 

 constant which we call Z?,-. This means that the degradation of mRNA proceeds 

 at a constant rate. Equation (3) can now be written in the form 



dXj _ a-, , (W) 



Here a'i = A:;L,[i,-][r,]o and 5,- = 1 +L,[i,]. If we take [/?,] in this expression 

 to be given by (9), then the equation becomes 



dXj ^ Oi ^ 



dt B'l+KiGiiMi-Si] ' 



However if (10) is taken to give i?,-, then (1 1) becomes 



dXi ^ a](l+K[c7AMi-S,]) ^ 



dt Bi{l+Krai[Mi-Si]) + KiK[ai[Mi-Si][Hi]Q ' 



Now we have the identity 



a](l+K[a^[Mi-Si]) 

 Bi + (5, + UH^^) (7, Kl{M, - 5,] 



a'i ( Ki[Hi\Q , jl 



5, + /^,[//,]o \B;+{Bi+nH],) a^KlWi-S^ 



The differential equation can therefore be reduced to the same functional 

 form with respect to the variable [A/,- 5",] as that obtained when we take 

 [i?,] = CT,[M,— 5,], although the constants will be different in the two cases. 

 Let us therefore write the equation as 



dX j a^ , xj2\ 



dt ~ Bi+niiWi-S] ' 



where the constants a,, Bi, w,-, and bi may be quite complicated functions of 

 more elementary constants if an aporepressor is involved in the repressive 

 activity of the feed-back metabolite M,-. 



