10 TEMPORAL ORGANIZATION IN CELLS 



systems will refer to the dynamic organization of single cells. This implies a 

 rather restrictive use of the term "epigenetic", but we have in Nanney's (1958) 

 discussion of epigenetic control systems in single cells a very respectable 

 precedent for the use of the term in this context. The applicability of hier- 

 archial analysis to the temporal organization of cells is also brought out 

 clearly by Kacser's (1957) notion of a "hierarchy of the catalysts", again a 

 three-layered structure which recognizes the ordering of intracellular events 

 in time and emphasizes particularly the fact that as one ascends the time scale, 

 from metabolic to genetic, one encounters increasing stability of the catalysts 

 which act as the controlling elements in the systems. However, these concep- 

 tual distinctions are no sooner made than one realizes that all the logically 

 separated systems in cells interact to produce one single integrated structure. 

 The question thus arises how one is to distinguish interactions within a system 

 from those which occur between systems, and we must now try to formalize 

 our notions by introducing a concept which will be of use in a dynamic analysis 

 of intracellular activities. 



The relaxation time of a system is, roughly speaking, the time required for 

 the variables to reach a steady state after a "small" disturbance. Without a 

 fully mathematical description of the system being studied the relaxation time 

 cannot be rigorously defined, for the size of a small disturbance is determined 

 by the mathematical requirement that the perturbation be consistent with a 

 linearization of the system equations in the neighbourhood of a steady state. 

 Students of chemical kinetics are becoming familiar with this concept through 

 the use of perturbation methods for determining the rate constants in steady 

 state reaction systems, and these methods are rapidly being adapted to the 

 field of enzyme kinetics. Applied to other fields this notion allows one to apply 

 a sort of "spectral" analysis to the almost continuous range of time processes 

 which are studied in biology, and thus attempt a more formal distinction 

 between such constructs as epigenetic and genetic systems in single cells than 

 has been done so far. 



The significance of this concept in the present study is the fact that if two 

 systems have very different relaxation times (say one is 100 times larger than 

 the other), then relative to the time required for significant changes to occur 

 in the "slower" system (larger relaxation time), the variables of the "faster" 

 one (shorter relaxation time) can be regarded as being always in a steady state. 

 Therefore only these steady state quantities will enter into the dynamic 

 equations describing the slower system, and a very considerable economy of 

 motional equations can be achieved. On the other hand, the variables of the 

 "slow" system will enter into the equations of motion of the "fast" one as 

 parameters, not as variables. These parameters have a slow rate of change, 

 and the faster system will gradually move in time in response to these slow 

 changes; but for the purpose of studying the short-term dynamics of the fast 

 system, the slowly changing quantities which define the motion of the slow 

 system can be regarded as environmental parameters. A distinction between 

 the two systems in terms of a temporal criterion such as we are attempting, 

 is therefore valid only if the relaxation times of the two systems are sufficiently 



