4. THE DYNAMICS OF THE EPIGENETIC SYSTEM 53 



important properties introduced by strong repressive coupling, and thus to 

 get an idea of the complex behaviour possible in control systems with more 

 complicated patterns of interactions. 



Limitations of the Theory 



It is necessary now to return to a more critical assessment of the biochemical 

 control model which has been constructed. Observe first that we have not yet 

 made use of the full generality of the class of differential equations which was 

 introduced to describe mRNA and protein synthesis, equations (1) and (3). 

 Nothing has been said about systems with self-replicating mRNA species, 

 the possible influence of metabohtes on protein synthesis at the ribosomal 

 level, or the effect of substrates in stabilizing enzymes by reducing their rate of 

 degradation. Such effects can be represented in the equations and, depending 

 upon the assumptions made regarding the kinetics of these processes, we get 

 systems showing a variety of dynamic behaviour. The dynamic and statis- 

 tical consequences of a self-replicating RNA species can be studied by the 

 present approach, and will be investigated in Chapter 8 ; but except for rather 

 special cases, it is not possible to find integrals for systems which have feed- 

 back inhibition of protein synthesis, i.e. control of protein synthesis by feed- 

 back of metabolites to the ribosomes. Such effects tend generally to produce 

 damping in the system. That is to say, the oscillations die out and so there is no 

 integral, which in this study depends upon the occurrence of continuing 

 oscillatory motion. 



This observation brings us face to face with the central weakness of the 

 present "classical" analysis. The oscillations which have been demonstrated 

 to occur in the dynamic system defined by equations (14) persist only because 

 of the absence of damping terms. This is characteristic of the behaviour of a 

 conservative (integrable) system, and it is associated with what has been called 

 weak stabihty. This means that if the system is disturbed slightly it moves to 

 a new trajectory and stays there; it does not return to its original trajectory. 

 A system showing strong stability, however, behaves like a limit cycle, returning 

 to a fixed trajectory after a small disturbance; and if such a system is started 

 at its stationary state, a small disturbance will cause it to oscillate in increasing 

 spirals until it reaches its limiting trajectory or hmit cycle. Such behaviour is 

 non-conservative. However, it is just this kind of behaviour which would be 

 expected in systems where oscillations occur because of time lags in dynamic 

 effects, such as we have discussed in relation to cellular control systems in 

 Chapter 3. The proper representation of such processes is by means of 

 functional equations. Such a representation takes cognizance of the spatial 

 separation of biochemical events in cells, and hence of the dynamic con- 

 sequences of structural heterogeneity. What our present treatment does, in 

 effect, is to substitute for real structure a model whose dynamic behaviour 

 approximates to that which we suspect occurs in the cell. The approximation 

 will be best in the neighbourhood of trajectories which are close to the limit 

 cycles which we expect to occur in cell variables. This approximation allows 



