52 TEMPORAL ORGANIZATION IN CELLS 



In order that this system be integrable, we require that the cross-coupling 

 terms be equal; i.e. 



7r«i2a2 = 7^-^21 «! 

 7r'^23a3 = 7r«32a2 



Therefore we take 



Vl = 2l^21^32ai, 72 = 22^32^12^2, 73 = 03 ^23 ^12 "3 



as a solution. The equations (27) can then be integrated, with the result 



G(Xi,X2,Xy,yi,y2,y3) = a?/:n/c21^32y + «! a2^12^21^32^1^2 + 



+ al A'22 ^32^12'^ + ^-2 ^3 ^12 ^23 ^32 ^'2 ->^3 + ^3 ^33 ^23 ^12 "y + 



+ ^i[ji-yilog(l+;^i/yi)] + Z>2[j'2-72log(H-j;2/y2)] + 

 + ^3[;^3-y3log(l+;'3/y3)] = constant 



Observe that we return to the differential equations by the familar relations 



dG . _dG 



Xi=-^-, yi-^-, i = 1,2,3 



dyi dXi 



If we were to try to integrate the three-component coupled system in which 

 components 1 and 3 interact by reciprocal repression as well as the other pairs, 

 thus introducing coupling coefficients ki^ and k^i into the equations, then it is 

 readily shown that integrability requires that the coefficients be constrained 

 by the relation 



^11^23^31 = ^21^32^13 



Assuming such a constraint, an integral essentially similar to the above could 

 be obtained but with an added term aia3/:i3A:3iA:2i.XiX3 corresponding to the 

 added interaction. Neither of these procedures is entirely satisfactory how- 

 ever, and it is clear that we are dealing with an inherent hmitation in the classical 

 approach. In going to systems of arbitrary size but with the structure of inter- 

 action represented in Fig. 8, the integral which is obtained has coupling terms, 

 A-1.Y2, A-2.V3, ..., x„-ix„ with Coefficients which get increasingly complicated as 

 n increases, each coefficient containing /; of the coupling parameters kjj. And 

 if we were to allow terms in XiX^, jcia:4, X2A'4, etc., to occur also, then a com- 

 plicated constraint would be placed upon the parameters kjj. 



In the present work these limitations do not present a serious difficulty, for 

 we will not analyse in detail systems more complex than those involving pairs 

 of strongly interacting components. This will allow us to investigate the most 



