32 TEMPORAL ORGANIZATION IN CELLS 



Solving for M,- at the steady state, we find 



^,M,(C, + /;,M,)-/-,F, = 

 which is a quadratic in Mj. Thus 



Only one root is positive: 



Isihi 



^-^t'-y(' 



4/7,- r, 7/ 



1 



Thus it is clear that quite complicated expressions could be obtained for the 

 functional relation between 7, and Mi if we were to consider in detail the 

 possible effects of metabolites on the activities of enzymes in the system. 

 However, once again we will use the simplest possible analysis in this work and 

 take expression (13) to define the control relations between 7,- and M,. 



Returning to equation (12) and making the substitution of r,y,/5/ for A/,-, 

 we have 



dXi a-f 



dt 



Ai+kiY; 



■bi 



where 



ki — — ^' and Aj = Bj—fjijSj 



Si 



The Properties of the Control Equations 



Taken together with equation (2) for the synthesis of the /th species of 

 protein, the two equations define the motion of a closed control loop such as is 

 represented diagrammatically in Fig. 1. It will be shown that dynamically 

 these equations define an undamped non-linear oscillator. All the more- 

 complicated components to be constructed later will be built up from this one, 

 and they all have the same basic dynamic behaviour, even though essentially 

 new features will emerge as the system gets richer in the pattern of the inter- 

 actions between components. We will now investigate briefly the dynamic 

 behaviour the pair of coupled variables (A',-, 7,), defined by the equations 



dt ~ Ai+kiY- 

 ^^'- y R 



^• 



(14) 



