496 S. E. LURIA AND M. DELBRUCK 



may be used to determine the mutation rate. It is only necessary to determine 

 the fraction of cultures showing no mutation in a large series of similar cultures. 

 This fraction p , according to theory, should be: 



(5) Po = e~ m . 



From this equation the average number m of mutations may be calculated, 

 and hence the mutation rate a from equation (4). 



Let us now turn to the discussion of the distribution of the number of 

 resistant bacteria. 



The average number of resistant bacteria is easily obtained by noting that 

 this number increases on two accounts — namely, first on account of new muta- 

 tions, second on account of the growth of resistant bacteria from previous 

 mutations. During a time element dt the increase on the first account will be, 

 by equation (3) : adtN t . N t , the number of bacteria present at time t, is 

 given by equation (1). The increase on the second account will depend on the 

 growth rate of the resistant bacteria. In the simple case, which we shall 

 treat here, this growth rate is the same as that of the sensitive bacteria, and 

 the increment on this account is p dt, where p is the average number of re- 

 sistant bacteria present at time t. We have then as the total rate of increase 

 of the average number of resistant bacteria dp/dt = aN t +p and upon integra- 

 tion 



(6) p = taNt 



if we assume that at time zero the culture contained no resistant bacteria. 



It will be seen that the average number of resistant bacteria increases more 

 rapidly than the total number of bacteria. Indeed the fraction of resistant 

 bacteria in the culture increases proportionally to time. This, as pointed out 

 in the introduction, is a distinguishing feature of the mutation hypothesis 

 but unfortunately, as will be seen in the sequel, is not susceptible to experi- 

 mental verification due to statistical fluctuations. 



The resistant bacteria in any culture may be grouped, for the purpose of 

 this analysis, into clones, taking together all those which derive from the 

 same mutation. We may say that the culture contains clones of various age 

 and size, calling "age" of a clone the time since its parent mutation occurred 

 and "size" of a clone the number of bacteria in a clone at the time of observa- 

 tion. It is clear that size and age of a clone determine each other. If, in par- 

 ticular, we make the simplifying hypothesis that the resistant bacteria grow 

 as fast as the normal sensitive strain, the relation between size and age will be 

 expressed by equation (1), with appropriate meaning given to the symbols. 



The relation implies that the size of a clone increases exponentially with its 

 age. On the other hand, the frequency with which clones of different ages may 

 be encountered in any culture must decrease exponentially with age, according 

 to equations (3) and (1). 



Combining these two results — namely, that clone size increases exponen- 

 tially with clone age and that frequency of clones of different age decreases 

 exponentially with clone age — we see that the two factors cancel when the 



