270 Distribution of numbers of mutants in bacterial populations 



the mutation occurs at such a time that it gives rise finally to 2 mutants, in one-eighth at 

 such a time that it gives rise to 4 mutants, and so on. 



According to this argument, a mutation is necessarily represented, by the time the 

 culture has reached size n, by a clone of 1 or 2 or 4 or 8, etc., organisms, there being no 

 intermediate numbers. This would be so if divisions in a clone were synchronous. It is 

 probably true that clones of 3, 5 or 7 cells will be less common than clones of 2, 4 or 8 cells 

 (cp. Adolph & Bayne-Jones, 1932), but rather than make the extreme assumption that 

 only integral powers of 2 are to be considered it is probably preferable to neglect this fact 

 and to assume that the frequency of clones of different sizes is a smooth function of 

 clone size. 



A clone which, by time t, contains v mutants will have originated when the number of 

 bacteria in the culture was about njv. We thus replace the subdivision of the growth of the 

 culture into generations by subdivision into intervals in which the population increased 

 from \n to n, from \n to \n, from \n to \n, and so on, and suppose that a mutation which 

 occurred while the population increased from n/(v + 1) to njv is, by the time the population 

 has grown to n, represented by a clone of v mutants. Now, of those cultures in which 

 exactly one mutation has occurred, the proportion in which the mutation occurred while 



the culture grew from n/(v + 1) to njv is I I =— — (cp. equation (4)). This may 



be represented as the coefficient of x v in the generating function 



1.2 2.3 * v(v + l) 



Considering now all those cultures in which exactly j independent mutations occurred, 

 the fraction of cultures in which the final number of mutants is v is, from (23), evidently 

 the coefficient of x v in the expansion of 



t_x_ tf_ a?_ y 

 \l.2 + 2.3 + 3.4 + "'7 ' 



Now if m is the mean number of mutations per culture, the proportion of cultures in 

 which exactly j mutations occurs is e~ m — . Thus the proportion of all cultures in which the 

 final number of mutants is r is the coefficient of x r in the expansion of 

 f° / x x 2 a; 3 V m mi 



Thus p r , the probability of a culture having r mutants, m being the mean number of 

 mutations per culture, is the coefficient of x r in the expansion of 



e-exp[m( I ^ + ^+...)] 



in agreement with equation (15). 



Arithmetical procedure 

 By means of the recurrence relation (9) and the boundary values (10) a table of values 

 of Cj r has been computed for all (integral) values of j from 1 to 36 and of r from 1 to 64, 

 subject to r^j. Equation (18) was employed as a check on the arithmetic at ^=63. Cj r 



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