66 TEMPORAL ORGANIZATION IN CELLS 



system above its ground state (the stationary state) where it is completely quiet. 

 For the variable Xi, ^ is a measure of the sum of the mean square deviation of 

 the variables about their steady states ((A',-;?,)^) and the mean deviation 

 {{Xi-p-^). For the variables 7, the quantity which 6 measures is not familar 

 although one of the terms is again the mean deviation of the variable from its 

 steady state value. These 7, variables show rather unusual behaviour, as will 

 become increasingly apparent as we proceed with our study, and they reflect 

 the major non-linearities in the system. We will find that is a very important 

 parameter in connection with interactions and the stability of temporal organi- 

 zation in the epigenetic system, so that it rightly occupies a central position as 

 the major system parameter, like temperature in physics. 



We can write equations (31) and (32) in slightly different form by observing 

 that 



{Xi-\-p)-- = Ci{Xi+Pi)Xi 



dXi 



and (7, + T,) ^^ = biyi+ r,) 1 1 - j— ^ 



= f,Xi(;r,-/;,) 

 dG 



^ bi{yi+r-;)yi 



Therefore c^X,{X,-p,) = 6 = ^^'^^~^j~ (33) 



which is true for all /. 



What this shows us is that when 6 is not zero (in which case it is positive) 

 the quantities (Xi—pi) and (F,— ^,) are, on the average, more positive than 

 negative, so that Xj and 7, tend to have greater excursions above their steady 

 states than below them. As 6 increases, this behaviour becomes more exag- 

 gerated so that the oscillations become increasingly asymmetrical about the 

 steady states, positive excursions predominating. Furthermore, this asym- 

 metry is greater for the F,'s than for the X^s. This is evident from (33) where 

 we see that (F,-^,) must be proportionately larger than (A',-/?,) in order to 

 balance the effect of the term Aj+kj Y/m the denominator, which is large when 

 y, is large. This observation again shows that the y,'s behave more irregularly 

 than the Xfs, although both reflect the inherent non-linearities of the system. 

 The asymmetry of the oscillations relative to the steady state is shown in Fig. 4, 

 where 7,(0 is seen to dip below the line 7, = ^, in small, sharp troughs, execut- 

 ing a much larger oscillation above the line. The interval when F, < ^, is the 

 period when Xj is rising, which it does rapidly, descending again more slowly 

 during the period when Yi>qi. The converse is true for y,-: when Xi<pi, 



