8. APPLICATIONS AND PREDICTIONS 149 



Since ^^ = XrlPr, this result shows us that if ^ < j3^ then the most probable 

 value of Xr isp;.[l -(d/^r)] which is always less than/?;., the steady state value; 

 but if ^ ^ /3^, then the most probable value of X^ is zero. We have here what was 

 referred to in Chapter 7 as a statistical discontinuity, since it is determined by 

 the size of 6, a macroscopic parameter, relative to ^^. There is no evidence of 

 discontinuous behaviour in the oscillator described by equations (93), so the 

 discontinuity must result from the statistical consequences of weak interaction. 

 In an attempt to explain this result, we may say that when the level of oscillatory 

 excitation as measured by 6 is less than ^„ hence less than oc^Pr, then the steady 

 state value of X^ is large enough so that the self-replication mechanism can 

 compete successfully for precursors for mRNA synthesis and X^ oscillates 

 continuously. However, when the excitation level is greater than oc,p, then the 



Time 

 Figure 10. 



self-replicating process for A'^-synthesis fails to compete successfully for 

 precursors and the rth locus therefore appears to spend most of its time effec- 

 tively shut off. 



The statistical behaviour of an oscillator of this kind was studied in detail 

 by Kerner (1959), and in fact the equations (93) are very similar to those of a 

 Volterra prey-predator pair except for the negative feed-back term. However, 

 the behaviour of the variable X^ is identical with that of a Volterra oscillator; 

 and Kerner has shown that when d^ ^^ the variable X^ appears in sudden, 

 sharp bursts with a somewhat irregular frequency, the variable rising in a steep 

 waveform to a peak and then dropping off again equally rapidly, somewhat as 

 shown in Fig. 10. 



However, when d < jS^, the oscillations have quite a different shape, being 

 more like that shown in Fig. 1 1 . These are rather different from the oscillations 

 which we have described for mRNA population governed by the original 

 equation (14). The most noticeable difference is that the asymmetry with 

 respect to the steady state axis is reversed, the population now spending more 

 time below the steady state than above it. If many mRNA species in cells have 

 the oscillatory characteristics of Fig. 11, then the actinomycin pulsing experi- 

 ment should have the opposite effect of that predicted: the clock should slow 



