100 S. E. LURIA AND R. DULBECCO 



quantities by making a number of simple assumptions. We shall first develop 

 this simple theory and then describe the experiments by means of which it was 

 tested. 



Theory 



We shall assume that in each particle of a given phage there exist n "units" 

 (or "loci") each capable of undergoing a lethal mutation when exposed to 

 ultraviolet light. Since one effective hit is sufficient to inactivate a phage par- 

 ticle, a lethal mutation can be defined as an effective hit, that is, as an altera- 

 tion of one unit which makes the phage particle unable to initiate by itself the 

 production of active phage in a bacterium. A particle may undergo more than 

 one lethal mutation, and mutations will be distributed at random and independ- 

 ently among the different units, the distribution depending only on the 

 sensitivity of each unit. 



We shall now make the assumption that the sensitivity of all units is the 

 same, and show later that this assumption, if incorrect, only requires a nu- 

 merical correction which does not invalidate the applicability of the theory. 



Our next assumption is that active phage cannot be produced in a bacterium 

 unless the infecting particle or particles, taken as a group, contain at least one 

 copy of each unit in non-lethal form. This assumption is an essential feature of 

 the theory, and corresponds to treating each unit as a discrete, material, in- 

 dependent hereditary unit endowed with genetic continuity and individuality. 

 An inactive unit cannot be replaced by copies of different units. This assump- 

 tion implies that production of active units cannot result from the cooperation 

 of two or more lethal units, but only from actual reproduction of active units. 



When a population consisting of M phage particles is irradiated with a given 

 dose, there will be produced in each particle, on the average, r lethal mutations, 

 or a total of MX^ mutations in the whole population. Since M particles contain 

 MXn units, each unit will receive on the average Mr/Mn = r/n lethal muta- 

 tions. 



A given unit, taken at random, will have a probability e"'''" of not having a 

 lethal mutation, and a probability (1 — e"'"''") of having at least one. 



We ask next: what is the probability that each of the n units is present in at 

 least one non-lethal copy in a group of k particles that enter a bacterium? 

 According to our assumptions, this probability should represent an upper limit 

 for the probability that a bacterium produces active phage. 



If a bacterium is infected by k particles, the probability that a given unit 

 is lethal in all of them is: (1 — e~'"'")'', and the probability that it is non-lethal 

 in at least one of them is: 1 — (1 — e"""'")^. 



The probability that at least one non-lethal copy of each of the n units is 

 present in the k particles is the product of the probabilities referred to the 

 individual units. Since we have assumed equal sensitivity for all units — that 

 is, r/n constant for all units for each value of r — the product will be 



[1 - (1 - e-"-/")^]". (1) 



The expression (1) represents the probability that a bacterium infected by 

 k particles receives at least one full non-lethal complement of the n units. 



254 



