42 TEMPORAL ORGANIZATION IN CELLS 



in this expression. Equations of this kind continue up to the last two steps, 

 where we have 



dM^,-i ^ A'„,- 1 y,„- 1 Af „,_2 k,„ Y,„ M,„_ i 

 dt K„,- 1 + M^-2 K,„ + M„,_ 1 



dM,„ _ k,„Y,„M,„_i 

 dt K,„ + M„-i 



where c,„ M^ is the rate of withdrawal ofM„, from the pool. Using the assump- 

 tion of steady state kinetics in this metabolic sequence relative to time periods 

 required for changes in the concentrations Yi, Y2,..., F,,,, we have the equations 



K2 + M1 



k2Y2M, k,Y,M2^ 

 K2 + M1 Ki + M2 



km-l Ym-l^m-l k„, Y„,M„,-i 









-C,nM,„ = 



Adding all these equations together, we get the one equation 



CiY, 



CiYi-c,„M,„ = or M„,= 



c. 



This result simply shows us that if we assume that Y^ controls the flow through 

 the metabolic sequence, then M„, is determined linearly by Yy. If we add to 

 our equations terms which represent a small escape of intermediates between 

 steps in the enzyme sequence, thus giving a bit more plausibility to the system, 

 then the result would be 



c,Y,-d 



M.„ = 



c 



m 



where d = ^di 



dj representing the loss in the /th step. Clearly we must assume that d< Ci Yi, 

 for otherwise M,„ = 0; the escape of intermediates is greater than the inflow 

 controlled by Yi in the first step, so no end-product results. 



The pair of equations describing the dynamics of the pair (A'l, Yi) is, 

 according to Fig. 6 and our earlier assumptions 



dXi a, 



dt Bi + mi(M^-SJ 

 dY, 



h 



dt 



= aiA'i-^i 



