4. THE DYNAMICS OF THE EPIGENETIC SYSTEM 



51 



handled by a well-defined dynamics and statistical mechanics, or by an exten- 

 sion of this method by means of functional analysis and the use of other in- 

 variants besides integrals. This procedure allows one to steer a central course 

 between the opposed positions of complete determinism of microscopic or 

 molecular detail in cellular structure and function on the one hand, and com- 

 plete absence of detailed microscopic description on the other. It also, and 

 perhaps more importantly, allows one to make some definite predictions for 

 experimental investigation, and so lays the theory before the sole arbiter of 

 scientific analysis: experimentation. 



Let us now derive an integral for the system represented by Fig. 7. Using 

 the same arguments as those which led to equations (22), we get for the 

 three-component coupled system the set of equations (26). Writing in 

 the usual manner d = ^2 + ^11^1 + ^12^2, ^2 = ^2 + ^21^1 + ^22^2 + ^23^3, 

 Qi = ^3 + A-32^2 + ^33^3, whcre the ^,'s are the steady state values of the 7,'s 

 and so also thep/s of the A','s, we introduce the transformation 



tri 



Vi+yi = ^(A2 + k2l Yi+k22 Y2 + k23 Yi) 



y3+yi = ^(A,+k,2Y2+k,,Y,) 



Xl = Xi-pi, X2 = X2-P2, A-3 = ^^3-/73 



Here the y/s are again auxiliary parameters which will be defined in a moment. 

 The equations now take the form 



dt '\yi+yi I 

 dt \yi-^yi J 



= b ( ^3 I 



\y^+y^ y 



dxi 

 ~dt 



dx2 

 't 



dx^ 

 ~dt 



dy\ vi 



-^ = ^(^'liai'Yl+^12a2^2) 



dyi yi 



■^ = ^(^2iai^l+^22a2-V2 + ^'23a3'Y3) 



dy-x y-x 



-^ = -p-(ki20i2>^'2 + kn<x3^'i) 



(27) 



