D. E. Lea and C. A. Coulson 275 



which relations may be used as a first test of whether the spread of an experimental 

 distribution is comparable with the theoretical spread. 



All the relations in this section are approximations, to be used only when dealing with 

 experiments which lie outside the scope of Table 2. The approximation should not be used 

 for the extreme ends of the distribution, e.g. for values of P r exceeding 0-95 or less than 0-05. 



The estimation of mutation rate from experimental observations 

 m from the mean number of mutants per culture 

 As shown by Luria and Delbruck, the mean and variance of the distribution can be simply 

 calculated, without knowing the distribution p r , as follows: 



While the culture grows from n x to n 1 + dn 1 , the mean number of mutations will be 

 (m/n) dn x (cp. equation (4)), the actual number being distributed in a Poisson distribution 

 about this mean with variance also (m/n) dn x (since the variance of a Poisson distribution 

 is equal to the mean). The contribution to the final number of mutants (when the culture 

 size is n) will be n/n x mutants for each mutation. Thus the contribution to the final number 



of mutants will be distributed about a mean dn, with a variance I — ) — dn,. Thus 



n x n L \n x J n 



the mean of the required distribution is 



f=\ dn,=m log n, (29) 



J l n x n 



and the variance of an individual determination of r will be 



oZ^ri-Y-d^mn* (30) 



J i W n 



We can confirm that our distribution p r yields the same mean and variance. The mean is 



r = Xrp r . 



Since j9 r is the coefficient of x r in the expansion of e _w exp m I r— + .y^ + ••• I (equation (15)), 

 e 

 S ?r af = exp[,„( 1 ^ + 2 4 + ...)]. (31) 



Differentiating 



^f-^m g+f+f +~) exp [m (o+0 + -)]- W 



Inserting x=l, and putting - + - + ... +- = log n and 7-^+^+ ...4=1, we have 



r = T,rp r = e~ m Y*rq r = m log n. 

 Again, multiplying (32) by x 2 and differentiating, 



Sr(r + l)z' ?r = m (x + ^ + £3+...)+m^ 



Inserting x = 1 , Sr(r + l)j r = me m (n + m log 2 n) 



T,r(r + l)p r = mn + m 2 log 2 n. 



* But see Appendix: the correct value is a 2 = 2mn. 



and since q r = z e m p r , we have 



35 



