ACTIVE FROM INACTIVATED BACTERIOPHAGE 101 



In a mixture of phage with bacteria, however, there is a distribution of the 

 number of phage particles infecting individual bacteria. With relatively good 

 approximation — see Appendix — this distribution can be considered as a 

 Poisson distribution. If x is the average number of particles adsorbed per 

 bacterium, the fraction of bacteria with k particles is: x k e~ x /kl 



The fraction of bacteria receiving k particles which carry a full complement 

 of non lethal units is then: 



X [1 " (1 " er*/»)*]» (2) 



kl 



Finally, the fraction of the total bacterial population which receives all units 

 in a non-lethal form is the sum of the expression (2) for all possible values of k: 



Z = it -^7" t 1 - (1 - e-*/»)*]». (3) 



fc=0 kl 



This expression embodies the following consequences of our hypothesis: 



(a) No full complement of active units can be present in uninfected bacteria 

 (*=0). 



(b) Of the bacteria with one phage particle (k = 1), only those with an active 

 particle fulfill the requirement for active phage production (Z = xe~ x e~ r ) . 



(c) Any bacterium that receives at least one active particle fulfills the 

 requirement for active phage production, whether it also receives inactive 

 particles or not. For each unit of that particle, 1 — e -r/n = 0; hence, 

 (l- e -'-/»)*=0, and [l-(l-e-' /n )*]" = 1. 



(d) For any given value of r>0, Z increases with increasing x, that is, the 

 probability of having a full complement of active units increases as the number 

 of particles adsorbed per bacterium increases. 



(e) For any given value of x, Z diminishes with increasing r, that is, the 

 probability of having all active units diminishes as the dose of radiation 

 increases. 



For the purpose of comparison with data from different experiments, it is 

 more convenient to eliminate from the computation those bacteria that receive 

 either zero or one phage particle, since they are not expected to contribute 

 to reactivation. This is done by using instead of Z the expression 



z = Z -^ [1 - (1 - ef/")*]«. (4) 



fc=2 kl 



2 represents the fraction of bacteria in the total population that have two or 

 more phage particles, which together contain a full complement of active units. 

 Since the fraction m of bacteria with two or more phage particles ("multiple- 

 infected bacteria") is 



m = 1 - (.v + l)e~* (5) 



the multiple-infected bacteria receiving a full complement of active units 

 represent a fraction 



274 



