MUTATIONS OF BACTERIA 499 



Since we have seen that the mutation hypothesis, in contrast to the hypothe- 

 sis of acquired immunity, predicts a distribution of the number of resistant 

 bacteria with a long tail of high numbers of resistant bacteria, the determina- 

 tion of the variance of the distribution should be helpful in differentiating be- 

 tween the two hypotheses. We may here again determine first the true vari- 

 ance — that is, the variance of the complete distribution — and second the likely 

 variance in a limited number of cultures, by omitting those cases which are 

 not likely to occur in a limited number of cultures. 



The variance may be calculated in a simple manner by considering sep- 

 arately the variances of the partial distributions of resistant bacteria, each 

 partial distribution comprising the resistant bacteria belonging to clones of 

 one age group. The distribution of the total number of resistant bacteria is 

 the resultant of the superposition of these independent partial distributions. 



Each partial distribution is due to the mutations which occurred during a 

 certain time interval dr, extending from (t — r) to (t — r+dr). The average 

 number of mutations which occurred during this interval is, according to 

 equation (3), 



(9) dm = aN T dr = aN t e _T dT. 



These mutations will be distributed according to Poisson's law, so that the 

 variance of each of these distributions is equal to the mean of the distribution. 

 We are however not interested in the distribution of the number of mutations 

 but in the distribution of the number of resistant bacteria which stem from 

 these mutations at the time of observation — that is, after the time interval t. 

 Each original mutant has then grown into a clone of size e T . The distribution 

 of the resistant bacteria stemming from mutations occurred in the time inter- 

 val dr has therefore an average value which is e T times greater than the average 

 number of mutations, and a variance which is e 2r times greater than the vari- 

 ance of the number of mutations. Thus we find for the average number of 

 resistant bacteria: 



dp = aN t dr, 



and for the variance of this number 



var dp = aNtCdr. 



From this variance of the partial distribution, the variance of the distribution 

 of all resistant bacteria may be found simply by integrating over the appro- 

 priate time interval — that is, either from time t to time o (r from o to t), if 

 the true variance is wanted, or from time t to time t (r from o to t — 1 ), if the 

 likely variance in a limited number of cultures is wanted. In the first case we 

 obtain: 



(10) var p = aNtCe* — 1). 

 In the second case we obtain: 



(10a) var r = aN t [e (t-to) — 1 J. 



