BACTERIPHAGE GENETICS 291 



Geiringer, in a series of papers (1944, 1945, 1948, 1949), developed the 

 mathematical theory of populations in which such repeated matings occur. 

 She derived the general formula for the situation in which any number of 

 genetically different types interact pairwise for an arbitrary number of 

 generations as well as the more limited special cases which are of 

 particular mterest to the biologist. One of the most important aspects of her 

 treatment is that it is completely independent of the nature of the elementary 

 act by which recombinants are formed. It is not necessary to specify how 

 many particles are produced in the mating act or whether the parental types 

 are conserved or whether reciprocal recombinants are formed in the same 

 event. 



It is only necessary to determine the over-all probability that a particular 

 genetic marker is separated from its neighbors in one generation. Thus, in a 

 mating event between a particle of the type abc and one of the type a^b'^c'^, 

 the theory makes use of the probability of forming the recombinants a+6c, 

 ab^c, and abc^; it assumes that these probabilities are equal to those for 

 forming ab'^c^, a^bc^, and a+6+c. Geiringer shows how these probabilities are 

 related to the more usual genetic parameters giving the probabihty of recom- 

 bination between each pair of markers under various assumptions as to the 

 interference between adjacent recombinational events. 



Visconti and Delbriick (1953) applied the methods of Geiringer to an 

 analysis of phage genetics, and they independently derived the formula for 

 a cross involving three genetic markers. This formula [Equation 5 (Geiringer, 

 1945) and Equation 8 (Visconti and Delbriick, 1953)] gives the expected 

 frequencies of each of the possible types in the progeny as a function of the 

 inputs of each of the parental types and four additional parameters, namely, 

 the probabilities that each of the markers becomes separated from its neigh- 

 bors in one mating and the number of rounds of mating. The three prob- 

 abilities involved here can then be related to recombination probabilities 

 between pairs of markers if an additional assumption is made as to the inter- 

 ference between adjacent recombinational events. 



Symonds (1953) expanded the analysis to show that if the number of 

 matings in which each particle or its ancestors takes part is distributed at 

 random, then for two- or three-factor crosses a simple modification of the 

 Geiringer formulas should represent the results. 



Visconti and Delbriick showed that a good fit could be obtained with the 

 experimental data for the phages T2 and T4 by making the following assump- 

 tions: (a) there is complete mixing of the genetic structures (which they 

 called vegetative phage) in a mating pool within the infected bacterium; (b) 

 there is repeated pairwise mating of these vegetative phage which find their 

 partners in a random process; and (c) the genetic material of the phage does 

 not separate into individual hnkage groups which assort independently of 



