46 



TEMPORAL ORGANIZATION IN CELLS 



which led to equation (12), we obtain for the rates of mRNA synthesis the 

 expressions 



dX], ai 



dt 



dj2 



dt 



Bi + mn[Mi- Si]+mi2[M2- S2] 



b. 



B2 + m2x[M i- Sx]+m22{M2- S:;\ 



where the m,y's are parameters analogous to w, in (12). 



Now once again we use the principle of steady state kinetics for the meta- 

 bohc system relative to epigenetic processes so that Mi and M2 can be re- 

 placed by expressions in Yi and Y2. We assume that the relations between 

 these variables are linear, as in (13), so that the differential equations which we 

 get finally for the mutually coupled system depicted in Fig. 2 are : 



(22) 



The equations describing the synthesis of protein are the same as those obtained 

 for the uncoupled system, since no strong interactions have been assumed at 

 the level of protein synthesis. 



The four simultaneous equations in (22) define a single coupled dynamic 

 system for which we will now derive an integral. Let the steady state values be, 

 as before, (pi,qi) and (p2, q2), so that 



Qi 



^1+^11^1+^12^2 

 Qi 



-bx = 



b. = 



Write also 

 and 



^2 + ^21^1 + ^22^/2 



aiPi-^i = 



Qi = Ai+knqi+lcn^2 

 Q2 = A2 + k2\q\ + k22q2 



