The mathematics of mosaic analysis. I: The relationship between sturts and distance.

In mosaic fate mapping the fraction of mosaics in which two structures are of different genotype is calculated. This frequency of separation has been called a "distance" and the units of this distance are called "sturts". The fundamental assumption of fate mapping is that the frequency of separation increases continuously with the actual distance between the anlage for these structures on the blastoderm. This paper shows that the frequency of separation does not increase beyond a certain value. --For the current theory to work as proposed, each mosaic animal must be half mutant and half normal. This is rarely the case in collections of mosaics. It has been thought that if some flies are less than half mutant and others more than half, these two types would introduce compensating errors in mapping distance. We show that this is not true and describe the nature of the errors introduced. It is probable that these errors are the main reason that mapping distances reported from different sets of mosaics have not been reproducible. This paper presents methods for the proper handling of data from mosaics with different amounts of mutant tissue.--We prove here that for mosaics with an arbitrary fraction of mutant tissue (m), the largest frequency of separation that can occur is 2m. We prove that sturts underestimate actual distance on the blastoderm by a factor of r/m, where r is the radius of the mutant patch, and that sturts give no information on distances greater than 2r. This, and not double crossing over, is the reason for the nonadditivity of sturts and the shrinking of large distances in sturt measures. Sturtoids overestimate distances by a factor of 1/(2r) and also give no information on distances over 2r. This paper gives formulae for correctly estimating distance when using a collection of mosaics with varying amounts of mutant tissue. We also describe the nature of the errors introduced by convoluted or elongate mosaic boundaries and by multiple mosaic patches.