470 HOWARD B. NEWCOMBE 



survival of one of the bacteria resulting from a division. In a population of N 

 viable bacteria, one division will raise the number from N to 2NS viable in- 

 dividuals, and, where A is the true rate of mutation, will result in the produc- 

 tion of ANS/ln 2 viable mutant individuals. If the increase in population is 

 determined from assays of viable bacteria, the apparent mutation rate, a, 

 during this division will be (In 2) (ANS)/(ln 2) (2NS-N), which reduces to 

 AS/(2S— 1). The true mutation rate may thus be obtained from the apparent 

 rate by the formula: 



A = a(2S - 1)/S. (6) 



The relationship between the true mutation rate and an apparent rate, ai, 

 obtained by utilizing the increase in total population as distinct from viable 

 population, may be derived in a similar manner and is expressed by the for- 

 mula: 



A = ai /S. (7) 



Where estimates of mutation rate have been made in both ways, that is, 

 using both viable population increase and total population increase, the value 

 of S may be determined from the apparent mutation rates a and ai. The for- 

 mula for this as obtained from formulas (6) and (7) by simultaneous solution 

 and is: 



S = (a + ai)/2a. (8) 



The true mutation rate also may be obtained from the values a and ai, 

 using equation (8) and substituting for S. This yields the formula: 



A = 2a-ai(a + ai). (9) 



These formulae may be applied to the data from figure 1 in order to deter- 

 mine whether the high apparent mutation rate during the first few divisions 

 could be due to a low survival. It will be noted that points on figure 1 obtained 

 by estimating growth from viable numbers of bacteria do not differ greatly 

 from those obtained by estimating growth from total bacteria. This suggests 

 that mortality due to division is not excessive, and we may calculate the chance 

 of survival of the individual products of one division, by using formula (8). 

 Thus, over the period from a twofold to a tenfold increase in population, the 

 rates estimated from the two curves would be approximately 2.6 X10 -8 and 

 1.4X10~ 8 , respectively. From these two values the calculated survival is 0.75. 

 Therefore the high rate of appearance of resistant clones during the early 

 divisions on a solid medium discussed earlier cannot be due to an excessive di- 

 vision mortality. 



It should also be noted that the estimate of rate which would have been 

 obtained over this period of growth by either method, had survival been unity, 

 is given by formula (9) and is 1.9X10 -8 . This latter figure, however, has little 

 bearing on the present discussion, as it is in any case intermediate between the 

 estimates of rate obtained from numbers of resistant clones and from numbers 

 of resistant individuals. 



72 



