68 TEMPORAL ORGANIZATION IN CELLS 



hypothesis is the above result which implies that after a suflficient period of 

 time these interactions lead to a condition of equilibrium defined by a uniform 

 distribution of talandic energy throughout the system. This equilibrium is not, 

 of course, thermodynamic equilibrium, since there is a constant flow of matter 

 and physical energy through the system as molecular species are synthesized 

 and degraded. Even at ^ = the system is far from thermodynamic equi- 

 librium, being then in a steady state. In the next chapter we will consider in 

 some detail the relation between these two concepts of equilibrium, and also 

 attempt some estimates about the time required for the epigenetic system to 

 reach equilibrium after a disturbance; i.e. we will attempt to estimate its 

 relaxation time on the basis of its dynamics. These are important considera- 

 tions which bear strongly upon the range of applicability of the present theory, 

 and especially its possible utility in analysing embryological phenomena. 



We may note at this point a very close similarity between the general features 

 of the oscillating biochemical control systems which we are studying and those 

 of generalized Volterra systems, even though the microscopic features of these 

 systems are quite different and depend upon totally different mechanisms. In 

 Volterra's analysis of oscillations in prey-predator systems, stability is depen- 

 dent upon the limited destruction of self-propagating prey species by dependent 

 predatory species, and no negative feed-back devices are introduced. Further- 

 more, there is no complementarity between variables in the system such as we 

 have between protein and mRNA in the cellular control systems, so that all 

 variables behave in roughly the same manner. The behaviour of these " demo- 

 graphic" oscillators has been shown in some detail by Kerner (1957, 1959) 

 in his very fine studies of Volterra systems using the analytical apparatus of 

 statistical mechanics. The present work owes much to the procedures employed 

 by Kerner in his analysis. He showed that the system parameter 6 is, in Volterra 

 systems, a measure of the mean square deviations of the population numbers 

 from steady state values for prey and predator species alike. The condition 

 ^ = again corresponds to the completely quiet state of the system when all 

 variables are at their steady states. However, the actual shape of the oscillations 

 in this case is different from the shape of those arising in the biochemical 

 control systems which we are investigating. Kerner showed that population 

 numbers oscillate in long troughs below their steady state values, with relatively 

 sharp peaks emerging above these values. This is just the opposite of the 

 behaviour of our F,- variables. The difference in the characteristics of these 

 oscillators is important in understanding how the two systems may be expected 

 to differ in response to certain types of environmental stimulus, as is discussed 

 in Chapter 8. Nevertheless, many of the general macroscopic features of the 

 two systems, demographic and epigenetic, show a close similarity. This is 

 hardly surprising insofar as the major dynamic characteristic of both systems 

 is the occurrence of continuing oscillations in system variables. We have here 

 the suggestion that a set of macroscopic parameters may emerge which will be 

 found useful in studying the general "thermodynamic" behaviour of complex 

 oscillatory systems, parameters which describe properties that are to a con- 

 siderable extent independent of the specific microstructure of the system. 



