276 Distribution of numbers of mutants in bacterial 'populations 

 Now the variance cr 2 = 2(r — f) 2 p r =Hr 2 p r — 2r"Zrp r +r 2 Hp r 



= Er (r + 1) p r — r — r 2 . 



Thus o 2 = mn — m\ogn, i.e. o 2 = mn since npl. 



Since f = mlogw, a possible method of determining m (and hence the mutation rate) 

 experimentally would be to divide by log n the mean number of mutants per culture 

 experimentally determined in a batch of N parallel cultures. However, on examination 

 it appears that the precision of the estimate of m given by this method does not increase 

 with increase of N. For it is evident that the total numbers of mutants in batches of 

 N parallel cultures each of size n will be distributed (from batch to batch) in much the 

 same way as the numbers of mutants in parallel cultures of size nN. The mean number of 

 mutations in a culture of size nN will be mN (since the mean number of mutations is 

 proportional to the size of the culture, cp. equation (3)), and hence by application of (30) 

 the variance of the number of mutations in cultures of size nN is mN .nN = mnN 2 . Thus 

 the total number of mutants in a batch of N cultures of size n is distributed from batch to 

 batch with a variance mnN 2 . A fraction 1/N of this total number (i.e. the mean number 

 per culture derived from a count of N cultures) is therefore distributed with variance mn. 



Thus we see that the variance of the mean number r of mutants in N cultures is no smaller 

 than the variance of the number of mutants in an individual culture, which shows that 

 however many cultures are averaged, no improvement in precision is obtained over the 

 use of a single culture selected at random. Consequently, the mean number of mutants per 

 culture is an extremely inefficient statistic from which to calculate the mutation rate. If, 

 nevertheless, this method of estimating m is employed, the variance ( a 2 m ) of the estimate 

 of m will (from (29) and (30)) be 



mn m) 



(ioi^ 2 ' (33) 



independent of the number N of cultures averaged. 



m from proportion of cultures without mutants 

 In view of the unsuitability of r as a means of estimating m from numerical data, Luria 

 and Delbruck proposed its estimation by equating e~ m to the proportion of cultures 

 experimentally determined to be without mutants. In a batch of N parallel cultures, in 

 which the mean number of mutations is m per culture, the expected number of cultures 

 without mutation is Ne~ m , the actual number being distributed about this mean in a 

 binomial distribution having a variance Ne~ m (1 — e~ m ). Thus the variance of the estimate 



e~ m is e~ m (1 —e~ m )/N. Since , = —e m , the corresponding estimate of m has a variance 



(ct^) which is e 2m times as great, i.e. 



•t-V- (34) 



Thus the standard error (a m ) in the estimate of m is given by 



o m jm thus varies with m. It has a minimal value when m = 1-594, when a fraction 0-2032 

 of cultures have no mutants. At m = 1-594, (a m /m) 2 takes the value 1-544/2V. At small or 



36 



