48 TEMPORAL ORGANIZATION IN CELLS 



The integral for this system is now obtained in the form 



G(XhX2,yi,y2) = fcnk2i'^V-^ + ki2k2i(xioc2XiX2 + k22kn'4i^+ 



+^lbl-yllog(l+Jl/yl)]+^2[>'2-y2log(l+J2/y2)] (24) 



It is readily verified that the equations (23) are given by the partial differential 

 expressions 



dG . dG 



. _ dG • _dG 



(25) 



Thus the mathematical effect of introducing strong interaction of reciprocal 

 repressive type into the model is the introduction of a coupling term, X1X2, 

 into the integral. This has very important consequences for the behaviour of 

 the system, as we will see in Chapter 7. However, it is the weakness of a theory 

 developed in terms of integrable systems that strong interactions of the re- 

 pressive type considered above must be symmetrical ; i.e. Mi must affect L2 and 

 M2 must affect Lj. The case of asymmetrical action, which seems a very likely 

 possibility in the actual repressive networks of cells, can only be studied in the 

 present theory by approximations to asymmetry, by taking one of the coupling 

 parameters ki2 or k2\ to be very small. 



It is clear now how we can build up more complex networks of compo- 

 nents which interact strongly by reciprocal repression. The next degree of 

 complexity above that represented by Fig. 2 is shown in Fig. 7. However, 

 observe in this figure a further constraint imposed by the condition of inte- 

 grability. Whereas L2 can interact with both L^ and L3 by repression, L^ and 

 L3 cannot interact with each other but only with Z-2. The reason for this con- 

 straint is that the "coupling" constants for repression, which are the para- 

 meters kij, are not symmetrical; i.e. /:,y # kji, i ^j. There is no reason to believe 

 that such symmetry exists, for the affinities of different repressors for different 

 loci are very likely to be quite different. In order to get an integrable system, 

 we must make the cross-coupling terms equal by multiplying by appropriate 

 coefficients similar to yi and 72 of the previous case, and this can only be done 

 if the interaction scheme of Fig. 7 is assumed. This will soon be demonstrated. 



If we go to systems of arbitrary size which have strong repressive coupling 

 between components, then we have a scheme of the type shown in Fig. 8. The 

 structure of the interactions is again restricted by the integrability condition 

 so that only neighbour interactions can occur. This is clearly a condition im- 

 posed by the mathematics and represents a weakness in the present treatment 

 of this problem. If we were to allow an arbitrary pattern of interactions to 

 occur between components with no constraints on the parameters, then it 

 would be necessary to use another analytical procedure altogether, one 



