8. APPLICATIONS AND PREDICTIONS 151 



in terms of the microscopic control mechanisms which form the substructure of 

 the system. Higher-order phenomena of this kind are very likely to constitute 

 an important aspect of cell behaviour, and it will not always be possible to 

 deduce the existence of a microscopic, deterministic mechanism to account for 

 certain cellular phenomena. This observation emphasizes the importance of a 

 statistical approach to the analysis of cellular activities, and the exercise of 

 economy in building a microstructure for macromolecular control mechanisms. 



There is no doubt, however, that the microscopic foundation of the present 

 theory errs on the side of the economy, and that in reality there is a much 

 greater richness of microscopic interaction than we have assumed to exist in 

 constructing our model. To mention one very obvious shortcoming, we have 

 no representation of induction in our system. The closest we have come to a 

 switching circuit of the type which is required to explain induction in particular, 

 and differentiation in general, is the topological discontinuity which arises in 

 relation to the sign of the parametric function A:nA:22-'^ 12^21- We have seen 

 that when this is positive both components 1 and 2 are stable; but when the 

 expression is negative, one or other of the components can under proper 

 conditions be eliminated, leaving a single uncoupled oscillator. Which one of 

 the two disappears from the system is determined by the initial conditions and 

 the parameter values. The coupled system can therefore exist in one of three 

 possible states, only one of which involves the simultaneous presence of both 

 sets of variables. Induction could therefore be regarded as a change of sign 

 of the parametric function (A:n^'22-^l2^2l) from negative to positive, so that 

 instead of having only one component, say number 1, with protein Yi, the 

 system begins to produce both components so that protein Yi and Y2 are 

 simultaneously present. A further change of the function from positive to 

 negative could then cause component 1 to vanish from the system while com- 

 ponent 2 remains, providing that an intermediate inductive state occurs such 

 as to provide the proper initial and parametric conditions for selecting this 

 component. 



However, this is a purely qualitative argument which contributes very little 

 to a real understanding of the inductive process. It is easy enough to produce 

 a model which shows parametric or topological discontinuities, as we have 

 called them. Then by the proper manipulation of parameter values the system 

 will switch from one state to another. The difficulty is to produce a model which 

 switches under an environmental stimulus (a temporary parametric alteration) 

 and then stabilizes itself in the new state by other changes of internal activities 

 so that even when the stimulus is removed the altered state persists. This is not a 

 property of our model, for a change in the sign of the expression kxxkn-kxikii 

 from negative to positive, will always cause the system to return to a state 

 where both components are present. With respect to qualitative or topological 

 features, our dynamic model is a reversible one. The irreversible properties 

 of the system relate solely to the distribution of G throughout the system, the 

 condition of maximum talandic entropy being the equilibrium state where G 

 is equally distributed over all components. In order to have an adequate 

 dynamic representation of induction, it is necessary to have a model which is 



6 



