108 TEMPORAL ORGANIZATION IN CELLS 



Substituting for the asymptotic values of Z^,. and Z,., this is 



a> 



(yi-^) 



£ = ^'("I'l p-pbilv-\og(l+v)] 





- = ^i^\ 



1 2\27t) 



At V = we have for large 6 the relation 



Therefore the mean frequency of zeros of jv, about its steady state axis varies 

 inversely as the square of the talandic temperature when 6 is large. When 6 is 

 very small, on the other hand, equation (56) shows that the mean frequency 

 of zeros is independent of the talandic temperature. For intermediate values 

 of ^ it is difficult to bring out explicity the functional dependence of aj(>',) on 6, 

 but it can be shown that 9a;/9/S > 0, whence dco/dd < 0. This result is generally 

 true for non-linear oscillations: as the amplitude increases the frequency 

 decreases providing all "microscopic" parameters remain unchanged, whereas 

 for linear oscillators the period is independent of the amplitude. We will have 

 occasion to make use of this result in the next chapter. 



Let us check the units in equation (57). 



bi = _, Ci = - ^, while 6 = — 



-^(y^^ = fJ{Yc^) = f 



which is correct. It is also of some interest to calculate the actual frequencies 

 given by equations (56) and (57), substituting the numerical values estimated 

 for the parameters in the last chapter. Here we had bj =5/12 and r,- = 10^^^ 

 In the limit of very small d, these values give us 



The units are 1 /minutes, so the period of the oscillation is [207t\/( 1 2/5)] minutes, 

 which is about If h. This would therefore represent the lower limit for the 

 period of an oscillator defined by equation (18) and with the parameter values 

 above. With 6 very small the oscillation will not be well defined because of 

 the noise level in the system, so that one would expect to find considerable 

 irregularity in the trajectories. 



In the other limit we use equation (57) to estimate co(y,) when 6 is large, 

 taking d = 100, say. Then we have 



= I ,^ ,^ I oscillations per hour 



