The expression of the asynaptic gene is highly variable, bivalents per cell ranging between 0 and 10. Swaminathan and Murty (1959) made the interesting observation that although variation in bivalent frequency follows binomial or Poisson distribution when the mean value per cell is high, marked deviation from binomial distribution can be noted when this value is low and approaches half of the potential number of bivalents. This was explained on the assumption that certain pairs of homologous chromosomes entered into bivalent association more frequently than others. These workers based their conclusion on analysis of Beadle's (1933) data on asynaptic maize as well as data on asynapsis in other organisms. The present study was undertaken to examine the situation more critically and determine as far as possible the cause of the deviation. The recent data of Miller (1963) were analyzed for the purpose.
Method of analysisIn this species with 2n=20, the expectations for the frequency of varying number of bivalents can be obtained from the expansion of the binomial (p+q)10, where p=the coefficient of synapsis or the probability that a pair of homologues would enter into synapsis (=one-tenth of the mean number of bivalents per cell), and q=the probability that a given pair would show asynapsis and equals (1-p). In case all homologous pairs within a meiocyte and all meiocytes behave alike (or if 'p' varies but slightly), observed frequencies should not differ significantly from these expected values. Deviation from binomial distribu tion may result under two different situations and correspondingly two models can be set up as follows depending on: (1) differential behaviour of homologous pairs within a meiocyte or 2) differences between cells within a population.Model 1. Assuming the first situation, suppose there are two groups within each meiocyte with n1 and n2 chromosomes (so that nl+n2=10) with two different values of 'p' (and correspondingly with two different values of 'q'). Let these values be p1, p2 and q1, q2. It can be proved by assigning different numerical values to n1, n2; p1, p2; and q1, q2 that 1) the deviation would follow unimodal distribution, 2) the frequencies at the extremes would be less than those expected from binomial distribution, and 3) the frequencies in the middle would be higher than those expected from binomial distribution.Model 2. According to this model, the population of meiocytes (=N) may comprise groups (say N1+N2=N) such that each of N, cells has p, and q1 as coefficients of synapsis and asynapsis respectively and each of N2 cells has p2 and q2 as the same coefficients. It can be proved again by assigning different values that the deviation according to this model would be characterized by the following: 1) Frequencies at the ends would be more than those expected from binomial distribution. 2) Frequencies in the middle would be corre