106 TEMPORAL ORGANIZATION IN CELLS 



Taking i^ = 0, we get the result that for very large /S the mean frequency of zeros 

 of J, about its steady state is 



coW - ^ (56) 



Let us check this for proper units, which should be l/T. Since c, = a,A:,/|2/» its 

 units are l/T. \/C, while b, has units CjT. Hence 6,C/ has units IjT^, so that 

 the units of a>0^/) are correct. 



This result can actually be obtained in another way. The limit of large ^ 

 corresponds to very small 6, and this we know to imply that the amplitudes of 

 the oscillation are very small. Therefore in this limit we can linearize the 

 differential equations 



LI +yi 



h = CiXi 



since the variables a-,- and yi will be very small quantities. The resulting 

 equations are 



Xi = -biyi 



h = CiXi 



These are sinusoidal oscillators of the form 



X +biCiXi = 



y +biCiyi = 



The period of this system is iTrj^/ibic), and its frequency is \/(^;C/)/27r, as we 

 have obtained in equation (56). 



Equation (55) gives us some information about how rapidly the mean 

 frequency of zeros drops off about axes displaced from the steady state axis, 

 v- = for very small 6. If we take jS = njbi, where « is a large integer, then the 

 expression becomes 



co{y,-v) ^ VX^)e-"[-iog(i+.)i 

 Vib^cd, 



In 



'(l + v)"e' 



Suppose now n = 100, so that 6 = bJlOO = 1/240 for bj =5/12, referring to the 

 numerical example of the last chapter. Then we may ask how far we must 

 displace the axis from v = in order to decrease the mean frequency of zeros 

 about this axis by \/e. That is to say, we want to find that value of v which 

 satisfies the equation 



i = (1+^)100^-1001' 



e 



or, taking logarithms, 



-1 = -100v+1001og(l+v) 



whence lOOi'-l = 1001og(l+v). 



