MUTATIONS OF BACTERIA 497 



average number of bacteria belonging to clones of one age group is considered. 

 In other words, at the time of observation we shall have, on the average, as 

 many resistant bacteria stemming from mutations which occurred during the 

 first generation after the culture was started as stemming from mutations 

 which occurred during the last generation before observation, or during any 

 other single generation. 



On the other hand, for small mutation rates it is very improbable that any 

 mutation will occur during the early generations of a single or of a limited num- 

 ber of experimental cultures. It follows that the average number of resistant 

 bacteria derived from a limited number of experimental cultures will, probably, 

 be considerably smaller than the theoretical value given by equation (6), and, 

 improbably, the experimental value will be much larger than the theoretical 

 value. The situation is similar to the operation of a (fair) slot machine, where 

 the average return from a limited number of plays is probably considerably 

 less than the input, and improbably, when the jackpot is hit, the return is 

 much bigger than the input. 



This result characterizes the distribution of the number of resistant bacteria 

 as a distribution with a long and significant tail of rare cases of high numbers of 

 resistant bacteria, and therefore as a distribution with an abnormally high vari- 

 ance. This variance will be calculated below. 



For such distributions the averages derived from limited numbers of samples 

 yield very poor estimates of the true averages. Somewhat better estimates of 

 the averages may in such cases be obtained by omitting, in the calculation of 

 the theoretical averages, the contribution to these averages of those events 

 which probably will not occur in any of our limited number of samples. We 

 may do this, in the integration leading to equation (6), by putting the lower 

 limit of integration not at time zero, when the cultures were started, but at a 

 certain time t , prior to which mutations were not likely to occur in any of our 

 experimental cultures. We then obtain as a likely average r of the number of 

 resistant bacteria in a limited number of samples, instead of equation (6), 



(6a) r = (t - t )aN t . 



It now remains to choose an appropriate value for the time interval t — to. 



For this purpose we return to equation (4), in which it was stated that the 

 average number of mutations which occur in a culture is equal to the mutation 

 rate multiplied by the increase of the number of bacteria. Let us then choose 

 to such that up to that time just one mutation occurred, on the average, in a 

 group of C similar cultures, or 



1 = aC(N t0 - No). 



In this equation we may neglect No, the number of bacteria in each inoculum, 

 in comparison with Nt , the number of bacteria in each culture at the critical 

 time t . We may also express N to in terms of N t , the number of bacteria at the 

 time of observation, applying equation (1): 



N t e- 



(t-to) 





