The Mathematics of Mosaic Analysis

Mosaic fate mapping requires first a measurement of the frequency of separation (by genotype) of two structures and then a conversion of this frequency of separation to distance (WYMAN and THOMAS 1982). If the genotype of two structures is visible, the frequency of separation (sturt distance) may be directly obtained. If the genotype is not visible (e.g. , for behavioral foci) then the frequency of separation (sturt distance) itself must be calculated. The formulae introduced by HOTTA and BENZER (1972) for calculating frequency of separation are appropriate only for a set of mosaics in which each fly has half normal and half mutant tissue. Using these formulae for a set of mosaics with a different fraction of mutant tissue can give enormously incorrect results.--In this paper we use intuitive lines of reasoning to obtain simple formulae for frequencies of separation that are algebraically equal to the more elaborate HOTTA and BENZER (1972) formulae.-We show that when calculating sturt distances, data from a collection of mosaics with a range of malenesses, even if the average maleness is s, cannot be lumped together. We prove that applying any formula appropriate for m = ‘/z to a set of mosaics all of maleness m, and thsn to a set of maleness I-m, and then averaging the two results, does give the correct value for sturt distances. In this way all the mapping distances may be obtained.-Another method for locating foci is called “contour mapping”. We show that the currently available contour formulae are inaccurate. We suggest that contour maps be drawn using the accurate sturt distances. T H E previous paper (WYMAN and THOMAS 1982, hereafter called paper I) developed formulae to determine the distance between two structures when their frequency of separation (by genotype) in a series of mosaics is known. This paper develops formulae to determine the frequency of separation fo r structures (foci) whose genotype is not visible. If the genotypes of two structures are directly visible (e.g. , cuticular structures marked with forked or yellow) determining the frequency of separation is simply a matter of counting (GARCIABELLIDO and MERRIAM 1969). Structures whose genotype is directly observable are called landmarks ( HOTTA and BENZER 1972). Often, the genotypes of structures of interest are not directly observable. This is the case with internal structures which may be the cause of, for example, a behavioral defect. The ‘‘focus” for a behavioral mutation is defined as the tissue which must be mutant for the fly to express mutant behavior. If there is a single Genetics 100: 677-696 April, 1982 6 78 R. J. WYMAN AND L. SALKOFF structure which is the focus, the phenotype signals the genotype of the structure and its location can be mapped in the same way as any anatomical landmark. However, it may happen that a behavior is governed by two bilateral sites, and that only one (domineering case) or both (submissive case) foci must be mutant for the behavior to be mutant. These are called interacting foci. The genotypes of the foci cannot be scored directly; only the phenotype of the whole animal can be scored. Since the genotype of each focus cannot be uniquely determined, the investigator does not immediately know how often a given focus and a given landmark are of different genotypes. In general, this frequency of separation (or sturt distance) cannot be determined. However, if certain symmetries are present, a solution can be found. HOTTA and BENZER (1972) considered the case where the mosaic borderline cuts each blastoderm in half, so that half of each embryo is normal and half is mutant. Given this symmetry, and the right-left symmetry of the embryo, and assuming that the fate map far the mutant is the same as for the normal, they derived formulae for estimating the sturt distance between an anatomical landmark and the behavioral focus (AT) and between the right and left behavioral foci @). These formulae have been extremely useful in neurogenetics, but they are somewhat complex and do not lend themselves to an intuitive understanding. In this paper we use very simple intuitive lines of reasoning to suggest simple formulae and then show that these formulae are mathematically equal to the HOTTA and BENZER (1972) formulae. The simplified means of analysis developed here allows us to derive accurate methods for dealing with the more difficult problem of fate mapping mosaics which have unequal amounts of mutant and normal tissue. FORMULAE FOR 50/50 MOSAICS Focus-to-focus distance-p: If a behavioral focus is domineering, the only way for a mosaic to have normal behavior is for both foci to be of normal genotype. The two foci are always of the same genotype if #' is zero; this happens when the two foci coincide at the midline. The genotype of this midline point would be normal in half the mosaics and mutant in half; thus, the behavior would be normal in half the mosaics and mutant in half. Using a to denote the fraction of flies with normal behavior, if = 0, then a = x. If the focus were not on the midline, the two foci wouId be less likely to both be of the same (normal) genotype and so fewer than half the flies would have normal behavior. The farther the foci are from the midline, the greater the probability that the mosaic dividing line falls between the two and so fewer flies will have normal behavior. This decrease below i /z in the fraction of flies with normal behavior should be a measure of how far the focus has moved from the midline. MATHEMATICS O F MOSAIC ANALYSIS 6 79 If ( a ) denotes the fraction of mosaics with normal behavior then (1/2 a ) is the decrease. In fact, it will be shown below that (1/2 a ) is precisely the sturt distance of the focus from the midline. Twice this number ( 1 2a) is the distance between the two foci. It will be shown in APPENDIX 1 that this new formula : fJ'=1-2a Domineering ( 1 ) is algebraically equivalent to the original HOTTA and BENZER (1972) formula: boo -~ ] +AA'[ b10--1o ] (2) fj '=(l-AA')[ aoo + boo ail + 611 alo + blo By the same type of argument as above, may be determined for submissive foci. In this case, the fly exhibits mutant behavior only i f both foci are of the same genotype. Thus, the fraction of flies with mutant behavior (b) decreases as the foci are farther apart. The same logic as above results in: ff'z1-2b Submissive (3) This formula is equivalent (APPENDIX 1) to the HOTTA and BENZER (1972) formula: (4) boo ] + A2 [ aio bio ] -~ a11 fJ'= ( 1 A T ) [ all + bll aoo iboo alo f blo The decision as to whether a domineering or submissive model fits the data can be taken simply by noting whether the majority of mosaics have normal or mutant behavior. If more than half the mosaics have normal behavior only a submissive model can fit; if more than half have mutant behavior only a domineering model will work. Note that for a given mosaic collection a + b = 1, i.e., every fly must be either normal or mutant. Because of this, formulae 1 and 3 can be written in several useful and symmetric ways. Landmark-to-focus distance-*: For calculating the sturt distance from an anatomical landmark to a behavioral focus, mosaic flies are scored for their behavior and for the genotype of the right and left landmarks. The fraction of mosaics with normal (a) or mutant (b) behavior and with landmarks that are both normal (subscript 11) both mutant (subscript 00), or one normal and one mutant (subscript IO) are counted. 680 R. J. W Y M A N A N D L. SALKOFF If the landmarks exactly coincide with the foci of the behavioral mutation (Af = 0) , the genotype of the behavioral foci will be the same as the genotype of the landmarks. In the submissive case, the behavior would be mutant if, and only if. both landmarks were mutant. In other words, no mosaics could fall into classes a,,, b,, and b,,. As the behavioral foci move away from the landmarks, the probability that the genotype of the foci differs from that of the landmarks increases. This allows some mosaics to fall into classes aoo, b,, and b,,,. These are the mosaics in which both landmarks are mutant, yet at least one focus is normal (a,,), or where both foci are mutant yet one (bl,) or both (bIl) landmarks are normal. The fraction of mosaics falling into these classes should be a measure of the distance from the landmarks to the behavioral focus. It will be shown below that, in fact, this fraction (aoo + b,, + b,") is exactly equal to the sturt distance from the landmark to the behavioral focus. It will be shown in APPENDIX 1 that Af = aoo + bii f bm Submissive ( 5 ) is algebraically equivalent to the original HOTTA and BENZER (1972) formula: The same logic may be applied to the domineering case. In this case, if the landmarks coincided with the behavioral foci (Af = 0) , the behavior would be normal if and only if both landmarks were normal. Boxes aoo, b,, and alo would be zero. Af = acre + b,, + alcl Domineering (7) This formula is equivalent to the original HOTTA and BENZER (1972) formula: Note that the formulae for Af for the domineering and submissive cases differ by only one term. The submissive formula includes b,, while the domineering formula includes a,,. ANALYSIS BY CONFIGURATTON Insight into the meaning of these formulae as well as proof of their validity can be obtained by examining diagrams of mosaic blastoderms. We use diagrams like those of HOTTA and BENZER (1972) in which the right and left landmark, and the right and left behavioral focus are denoted by circles; filled circles denote normal genotype, open circles denote mutant genotype (Figures 1, 2, 3). There are 16 different configurations that the four points may take when each one may MATHEMATICS O F MOSAIC ANALYSIS 681 rJ-J