280 Distribution of numbers of mutants in bacterial populations 



Thus S m — ^Jr — - = — S or approximately 8 m E L— = — 8, where we have replaced the 



=—\ of ^— . 



Now from (39) we have 



^=-{x 2 d + x(2cd+l) + c 2 d + c\, 

 dm m 



and x being normally distributed with unit variance about mean zero, 



E[x 2 ] = l and E [x] = 0. 



Thus E 



?\r 



-{d(l+c 2 ) + c}, 

 m 



and so S, 



m d(l+c 2 ) + c' 



The variance of S from batch to batch being l/N, we obtain for the variance( a m 2 ) of m the 

 relation 



fasV-i 1- (43) 



UJ iV{d(l+c 2 ) + c} 2 ' <**' 



with a = 11-6, 6 = 4-5, c = 2-02, d = (l -6 + log m)/a. 



A plot of — ^iV against m as computed by this formula is given in Fig. 3 C. Having 



derived m by the method described in this section, the standard deviation to be ascribed 

 to it is read from Fig. 3 C. 



Maximal likelihood method: large counts 



None of the methods we have so far described is fully efficient statistically. At the 

 expense of somewhat more laborious computation a fully efficient estimate of the mutation 

 rate may be made by employing the method of maximal likelihood. We give two solutions: 

 one for experiments which fall within the range of Tables 1 and 2, i.e. in which most of the 

 cultures have fewer than 64 mutants, which is set out in the next section, and one for 

 experiments falling outside the range of Tables 1 and 2, and for which the approximation 

 that £ is a normal deviate is employed, which is set out in the present section. 



The probability that the number of mutants shall lie between r and r+dr is given 

 approximately (for r not too small) as 



i^*-^)^?*-/* (Say) ' (44) 



(45) 

 (46) 



d 1 df _ dx d 2 x Idx 



Thus a^ l0g ^Ja^ = ~ X d^i + drJmlTr' 



Now x = (-, r^ 7-c) with a = 11-6, 6 = 4-5, c = 2-02, 



\rjm - log m + b J 



and by differentiating we find 



dx . , 9 d x + c d 2 x Idx d 1 



dm m m omor\ or mm 



40 



