Genetics and Molecular Biology · Unit 2: Mapping and gene function

Three-Point Crosses: Gene Order and Double Crossovers

A test cross that follows three linked genes at once gives eight classes of offspring, and those eight counts tell you the order of the genes, the distance across each interval, and whether one crossover makes another nearby less likely. This lesson works one such cross from raw counts to a map, then asks why a centimorgan is not a fixed length of DNA. Afterwards you can order three genes, calculate interval distances, coincidence and interference, and say where crossovers in humans and mice tend to fall.

Watch first A Middle Gene Reveals Double Crossovers · 2:36

Two crossovers between the same pair of genes cancel out. If you score only those two genes, an offspring with two crossovers looks exactly like one with none.

In linkage and mapping you measured the distance between two genes by counting recombinant offspring. That count has a blind spot. A crossover is an exchange of segments between two non-sister chromatids in meiosis (see chromosomes and meiosis). One crossover between two genes recombines them. A second crossover between the same two genes puts them back together. The gamete has been rearranged twice and looks parental. This lesson shows how a third gene, placed between the other two, makes those hidden events visible, and what the counts then tell you about order, distance and the spacing of crossovers.

Why add a third gene: the middle marker catches the doubles

A locus is the place a gene occupies on a chromosome, and the versions of a gene are its alleles. A test cross mates an individual of unknown genotype with one homozygous recessive at the loci being followed. The tester’s gametes carry only recessive alleles, so every offspring’s phenotype shows directly which alleles came in the gamete from the other parent. A test cross is a way of reading gametes.

Follow three linked loci in one test cross and each gamete can carry one of eight allele combinations. Those combinations fall into four pairs:

  • Parentals: no crossover between any of the three loci. Two classes, the largest.
  • Single crossovers in the first interval: two classes.
  • Single crossovers in the second interval: two classes.
  • Double crossovers: one crossover in each interval. Two classes, usually the smallest.

The analogy is a sentence copied from two drafts, switching drafts at random. Check only the first and last words and you see a switch only when there was an odd number of them. Check a word in the middle as well and you catch the pair. The middle marker does that for chromosomes.

Why are doubles the rarest? If crossovers in the two intervals happened independently, the chance of both would be the product of the two chances, by the product rule from probability and crosses. The product of two small fractions is a smaller fraction.

Reading the eight classes: find the middle gene first

Worked problem. A female heterozygous for three recessive mutations, p, q and r, carries p q r on one chromosome and + + + on the other (+ is the normal allele). She is test-crossed to a p q r / p q r male. Assume that every class survives equally well, that phenotypes are scored without error, and that her phase (which alleles sit together on each chromosome) is known. Here are the 1,000 progeny, written with the genes in alphabetical order, not map order:

  • p q r: 350
      • +: 342
    • q +: 53
  • p + r: 55
  • p q +: 96
      • r: 92
    • q r: 7
  • p + +: 5

Check the total first: 350 + 342 + 53 + 55 + 96 + 92 + 7 + 5 = 1,000.

Step 1: the parentals. The two largest classes are p q r and + + +. They match the mother’s two chromosomes, as they should.

Step 2: the doubles and the middle gene. The two smallest classes are + q r and p + +. Match each with the parental it most resembles. + q r against p q r: only p differs. p + + against + + +: only p differs. A double crossover swaps only the gene that lies between the two crossovers, so p is the middle gene. The order is q, p, r.

Notice what did not decide it. The middle of the written name is q, and q is an outer gene. The order on the page is a listing convention; the doubles tell you the order on the chromosome.

Three-point order finder: which gene is in the middle?⚙ original · interactive
Candidate gene order, left to right

    A double crossover swaps only the gene between the two crossovers, and doubles are usually the rarest classes; check them against the parentals, and against viability, before trusting rarity alone. The counts are the lesson’s 1,000 progeny, assumed equally viable and scored without error.

    Step 3: rewrite in map order. Put every class in the order q p r:

    • Parentals: q p r (350) and + + + (342)
    • Singles between q and p: q + + (53) and + p r (55)
    • Singles between p and r: q p + (96) and + + r (92)
    • Doubles: q + r (7) and + p + (5)

    In q + +, the chromosome starts as the q p r parental and switches to the + + + one between q and p. In q p +, the switch comes between p and r. Every class now has one clear reading.

    A caution. Rarest usually means double, but a class can also be small because those offspring survive poorly. Sturtevant reported reduced viability for one of his wing mutations in 1913. Check the doubles against the parentals, as in Step 2, before you trust rarity alone.

    Distances, coincidence and interference

    Step 4: distances. A double crossover is a crossover in each interval, so it counts in both:

    distance (cM) = 100 x (single crossovers in the interval + double crossovers) / total

    • q to p: (53 + 55 + 7 + 5) / 1,000 = 120 / 1,000 = 12.0 cM
    • p to r: (96 + 92 + 7 + 5) / 1,000 = 200 / 1,000 = 20.0 cM
    • q to r: 12.0 + 20.0 = 32.0 cM

    Now score only q and r, as a two-point cross would. The doubles q + r and + p + carry the parental q-r combinations, so they drop out: (53 + 55 + 96 + 92) / 1,000 = 296 / 1,000 = 29.6%. The two-point estimate is short by 2.4 points, which is the 12 doubles counted twice (24 / 1,000). Leaving the doubles out of the interval counts is the commonest error in these problems.

    Even 32.0 cM is a lower bound. Two crossovers inside the q-p interval, with no marker between them, still escape detection. That is why Sturtevant preferred to build maps from short adjacent intervals.

    Step 5: are doubles as common as chance predicts? If the intervals behaved independently:

    expected doubles = (interval 1 fraction) x (interval 2 fraction) x total = 0.12 x 0.20 x 1,000 = 24

    Twelve were observed. The coefficient of coincidence is observed doubles over expected doubles, and interference is one minus that:

    coincidence = 12 / 24 = 0.50 interference = 1 - 0.50 = 0.50

    An interference of 0.5 means half the expected doubles are missing. This is positive interference: a crossover in one interval makes a crossover in the neighbouring interval less likely. Sturtevant showed in 1913 that doubles were scarcer than chance predicts, and in 1915 he named the effect interference, crediting Hermann Muller with suggesting the name. Interference tends to be strongest for short neighbouring intervals; for intervals far apart, coincidence approaches 1. Interference of 1 means no doubles at all. Interference can also be negative, with more doubles than expected, and that has been reported in some fungal and phage crosses, so do not call a coincidence above 1 impossible.

    Interference is a property of a pair of intervals in a given organism, not a constant. In the plant Arabidopsis, one 2018 study measured an interference of 0.64 for one pair of intervals, which fell to 0.34 when the dosage of a meiotic protein called HEI10 was raised. And in a 2016 sequencing study of Drosophila female meioses, noncrossover gene conversions, which exchange a short patch without exchanging the flanking DNA, did not show interference at all.

    Design problem

    In another three-point test cross of 500 progeny, the two intervals are 6 cM and 10 cM, and no double-crossover progeny appear. A classmate concludes that interference here is complete. Is that justified? What would you do to decide?

    One way to do it

    Start with the expectation if the intervals were independent: 0.06 x 0.10 x 500 = 3 doubles.

    Three is a small number, and small counts scatter. Treat the number of doubles as a Poisson count with mean 3, a good approximation for a rare event in many trials. The chance of seeing none is

    P(0) = e^-3 = about 0.05

    So even with no interference at all, about one cross in twenty this size would show zero doubles. (The exact binomial, (1 - 0.006)^500, gives 0.049.) Zero is suggestive but is not strong evidence of complete interference, and it cannot distinguish complete interference from, say, an interference of 0.5, which predicts 1.5 doubles.

    To decide, score more progeny. With 5,000 progeny you expect 30 doubles under independence, and the chance of zero becomes e^-30, about 10^-13. If you still see none, interference is close to complete over these intervals. If you see about 15, interference is about 0.5.

    Before blaming interference, check that the double-crossover classes survive and can be scored, since a missing class can be a viability problem. And remember why interference is expected here: these intervals are short and adjacent, where one crossover most strongly reduces the chance of another. Over widely separated intervals the coincidence would be near 1.

    Maps versus sequence: where crossovers really fall

    A centimorgan is a frequency: 1 cM means that, for a short interval, about 1 gamete in 100 is recombinant. It is not a length. Sturtevant said in 1913 that map spacing need not reflect physical spacing, and molecular maps have shown how far apart the two can be.

    In humans the genome-wide average is close to 1 cM per million base pairs. That is a rough average. In mice and humans most crossovers fall in hotspots, short stretches of DNA where recombination is far more frequent than in the DNA around them. In 2010 three groups linked hotspot position to the gene PRDM9. Its protein binds particular DNA sequences through a row of zinc fingers and places a methyl mark on histone H3 nearby; breaks made by the enzyme Spo11 near such sites start recombination (see DNA repair and recombination). Change the zinc fingers and the hotspots move. This holds for humans and mice. It does not hold for all animals: dogs lack a working PRDM9, and birds appear to lack the gene, and their hotspots tend to lie near promoter-like features such as transcription start sites and CpG islands, as they do in budding yeast, which has far more recombination per kilobase than humans.

    The map also depends on who makes the gametes. The human female map is longer than the male map, by about 1.6 to 1.7 times over the genome as a whole. Drosophila males go further: they have no meiotic crossing over. That is why Drosophila mapping crosses put the triple heterozygote in the female, as in the fly question at the end of this lesson. Near centromeres, crossovers are suppressed in many organisms, including Drosophila and Arabidopsis. And in Drosophila females, crossover rates measured in 100-kilobase windows varied strongly along the chromosomes and between strains.

    So a genetic map and a DNA sequence agree on order and disagree on spacing. Use the map to predict how often genes travel together. Use the sequence to say how much DNA lies between them.

    How we measure it

    Three-point test cross

    An individual heterozygous at three linked loci is crossed to a tester homozygous recessive at all three. The tester adds only recessive alleles, so each offspring's phenotype reads out one gamete of the heterozygote, and the eight classes can be counted directly.

    Cytological markers on chromosomes

    A chromosome is made visibly different from its partner, for example by a knob at one end or a piece exchanged with another chromosome. Scoring both the visible chromosome and linked genes in the progeny tests whether genetic crossing over goes with a physical exchange of chromosome parts.

    Dense SNP genotyping of progeny

    Single-nucleotide differences between the two parental chromosomes are read in many offspring, or in the genomes of single offspring. Each switch from one parent's sequence to the other's marks a crossover to within the spacing of the markers, which places crossovers on the DNA sequence itself.

    Finish line

    Test yourself

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    Practice and MCAT-style, with explained answers.

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    Complementation and Gene Function: From Beadle and Tatum to Benzer

    Two pure-breeding white-eyed flies can have red-eyed offspring. What does that tell you about their genes?

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    PRDM9 zinc fingers on hotspot DNA — the protein that chooses where crossovers start
    PRDM9 zinc fingers on hotspot DNA — the protein that chooses where crossovers start ● LIVE · RCSB PDB 5EGB
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    Loading live structure 5EGB from RCSB PDB…

    A row of zinc fingers wraps along the major groove, each finger reading a few bases. A change in the fingers changes the sequence bound, which is how PRDM9 variants move hotspots.

    Drag to rotate · scroll or pinch to zoom · right-drag to pan
    PRDM9 methyltransferase domain — the mark that flags a hotspot
    PRDM9 methyltransferase domain — the mark that flags a hotspot ● LIVE · RCSB PDB 4IJD
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    Loading live structure 4IJD from RCSB PDB…

    This domain trimethylates lysine 4 of histone H3 on nucleosomes near the bound site. Look for the pocket where the lysine side chain enters.

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    Topoisomerase VI A subunit — the archaeal relative of Spo11, which breaks DNA to start recombination
    Topoisomerase VI A subunit — the archaeal relative of Spo11, which breaks DNA to start recombination ● LIVE · RCSB PDB 1D3Y
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    Loading live structure 1D3Y from RCSB PDB…

    Spo11, the enzyme that makes meiotic double-strand breaks, resembles this subunit. Two copies form a dimer with a groove wide enough for duplex DNA.

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    A four-way DNA junction — the Holliday junction at the heart of a crossover
    A four-way DNA junction — the Holliday junction at the heart of a crossover ● LIVE · RCSB PDB 1DCW
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    Loading live structure 1DCW from RCSB PDB…

    Four double helices meet at one point, stacked in two pairs. How such a junction is cut decides whether the flanking DNA is exchanged.

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    Alfred H. Sturtevant

    Alfred H. Sturtevant

    In 1913 ordered six X-linked factors of Drosophila in a line from their crossover frequencies, built the map from short adjacent intervals, and used crosses following three or more factors to detect double crossovers, which came out rarer than chance predicted.

    Sturtevant worked in Thomas Hunt Morgan's laboratory at Columbia University, and Morgan supplied the flies for his mapping paper. The idea under test was that the frequency of crossing over between two linked factors reflects how far apart they lie. If that is right, frequencies should add: if A to B is 1 unit and B to C is 30, then A to C should be near 31 or 29, depending on the order. Sturtevant took six factors on the X chromosome of Drosophila and found that, in the main, they fitted a single line. Two of them never separated in his data, so he placed them at one point. He built the map from short adjacent intervals, which he judged more reliable than long ones, and he examined crosses following three or more factors, most of them from Morgan's laboratory, which showed that double crossing over occurs. He cautioned that the spacing on his map need not match spacing on the chromosome. The paper appeared in 1913; he completed his doctorate with Morgan in 1914. Its arithmetic is the arithmetic of this lesson.

    ✓ verbatim Sturtevant, Journal of Experimental Zoology 14:43–59 (1913); E. B. Lewis, Alfred Henry Sturtevant, National Academy of Sciences Biographical Memoirs (1998).
    Hermann J. Muller

    Hermann J. Muller

    In a four-part paper of 1916 on the mechanism of crossing-over, analysed how a crossover in one region of a Drosophila chromosome affects crossing over in the neighbouring region, the effect called interference.

    Muller was another member of Morgan's group at Columbia, and with Morgan, Sturtevant and Calvin Bridges he wrote The Mechanism of Mendelian Heredity, published in 1915. The next year The American Naturalist carried his paper The mechanism of crossing-over, in four parts. Sturtevant had seen that double crossovers were scarce, and in 1915 he named the effect interference, crediting Muller with suggesting the name and with helping to find the phenomenon. Muller's 1916 paper took up the mechanism: he proposed that the stiffness of a chromosome makes it hard to bend back on itself to form a second crossover close to the first. The ratio of observed to expected double crossovers is now called the coefficient of coincidence. Muller later turned to mutation, and in 1946 received the Nobel Prize in Physiology or Medicine for showing that X-rays cause mutations. Three-point data thus became a measure of how crossovers are spaced along a chromosome, as well as of order and distance.

    ✓ verbatim Muller, The mechanism of crossing-over, American Naturalist 50 (1916), parts I to IV; Morgan, Sturtevant, Muller & Bridges, The Mechanism of Mendelian Heredity (1915).
    Bernard de Massy

    Bernard de Massy

    With Frédéric Baudat and colleagues, reported in 2010 that the zinc-finger protein PRDM9 is a major determinant of where meiotic recombination hotspots lie in humans and mice.

    Crossovers in humans and mice cluster in short stretches of DNA called hotspots, and a 13-base sequence motif had already been found enriched at many human hotspots. What read the motif was unknown. In 2010 Frédéric Baudat, Bernard de Massy and colleagues, including Molly Przeworski and Graham Coop, reported three lines of evidence from the Institut de Génétique Humaine in Montpellier and collaborating groups. Two mouse strains that use different hotspots carry different versions of the zinc-finger array of PRDM9, the part of the protein that binds DNA. The common human version of PRDM9 was predicted to recognise the 13-base motif, and binding was confirmed in vitro. And people carrying different zinc-finger variants differed in which hotspots they used across the genome. In the same issue of Science, Simon Myers, Peter Donnelly and colleagues, and Emil Parvanov, Petko Petkov and Kenneth Paigen, reached PRDM9 by other routes. Together the papers gave a molecular reason why, in these mammals, a centimorgan is not a fixed length of DNA.

    ✓ verbatim Baudat et al., Science 327:836–840 (2010); Myers et al., Science 327:876–879 (2010); Parvanov, Petkov & Paigen, Science 327:835 (2010).
    Test yourself
    Question 1 of 8 Score 0 / 0

    In a three-point test cross of linked genes, which offspring classes are usually the double crossovers?
    Sources & Integrity

    Every structure, slide, simulation, and clip on this page is listed below with the source it came from. None of the 3D structures is drawn by hand or by AI: each one loads by accession ID from RCSB PDB or PubChem. This panel is generated from the lesson's own data file, so what it lists is what the page loads.

    Molecular structures live data

    • PRDM9 zinc fingers on hotspot DNA — the protein that chooses where crossovers start — RCSB PDB 5EGB · coordinates CC0 · Human PRDM9 allele A zinc-finger domain bound to a recombination hotspot DNA sequence, X-ray crystal structure (Genes Dev., 2016)
    • PRDM9 methyltransferase domain — the mark that flags a hotspot — RCSB PDB 4IJD · coordinates CC0 · Methyltransferase (PR/SET) domain of human PRDM9, X-ray crystal structure
    • Topoisomerase VI A subunit — the archaeal relative of Spo11, which breaks DNA to start recombination — RCSB PDB 1D3Y · coordinates CC0 · DNA topoisomerase VI A subunit, X-ray crystal structure (EMBO J., 1999)
    • A four-way DNA junction — the Holliday junction at the heart of a crossover — RCSB PDB 1DCW · coordinates CC0 · Four-way junction formed by an inverted-repeat DNA sequence, X-ray crystal structure (Proc. Natl. Acad. Sci. USA, 2000)

    Simulations

    • Three-point order finder: which gene is in the middle? · Original interactive — Biochemistrypedia (Deppmann Lab).

    Scientist profiles & quotes

    • Alfred H. Sturtevant — Sturtevant, Journal of Experimental Zoology 14:43–59 (1913); E. B. Lewis, Alfred Henry Sturtevant, National Academy of Sciences Biographical Memoirs (1998).
    • Hermann J. Muller — Muller, The mechanism of crossing-over, American Naturalist 50 (1916), parts I to IV; Morgan, Sturtevant, Muller & Bridges, The Mechanism of Mendelian Heredity (1915).
    • Bernard de Massy — Baudat et al., Science 327:836–840 (2010); Myers et al., Science 327:876–879 (2010); Parvanov, Petkov & Paigen, Science 327:835 (2010).

    Quotes are verbatim from the cited source. Profile text is original to this site and was checked against the sources cited.

    Animated explainer

    A Middle Gene Reveals Double Crossovers · narration: an AI narrator voice (Gemini, Charon) reading narration written from this lesson's text · animation: AI-generated in code (Manim) by Claude, a schematic with no drawn molecular structures, plots computed from their equations · captions from the narration script.

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    integrity Machines check the mechanics. The build confirms that every local file exists, that every question's answer key points at a real choice, and that any AI-generated scientific figure names the person who checked it; a weekly job re-resolves every accession ID against its database. Questions are original; MCAT-style items are written for this site and are not real AAMC questions. Banned sources (the Tymoczko textbook, publisher decks, copyrighted figures) are not used.

    Like I'm five

    Imagine two strings of beads, one red and one blue, that swap pieces. If they swap twice, the two ends look the same as before, and you would never know a swap happened unless you look at a bead in the middle. Scientists put a marker in the middle so they can see the double swaps. They also found that one swap makes another swap nearby less likely.
    High school

    Genes on the same chromosome stay together unless a crossover swaps pieces between the two copies. To map three genes, scientists cross a parent carrying different versions of all three with a parent carrying only recessive versions, and count the eight kinds of offspring. The commonest kinds have no swap. The rarest have two swaps, one on each side of the middle gene, and the middle gene is the only one that looks swapped. Counting how often each gap is crossed gives distances in centimorgans. These distances measure how often crossovers happen, not how much DNA lies between the genes.
    College

    In a three-point test cross, a triple heterozygote of known phase is crossed to a triple recessive tester, so each progeny class is one gamete type. The two largest classes are parental and the two smallest are usually double crossovers; the locus that differs between a double and its matching parental is the middle locus. Each interval's distance is (single crossovers in that interval + double crossovers)/total x 100, so the doubles are counted in both intervals. A two-point count across the outer loci misses the doubles and underestimates the distance. Expected doubles under independence are the product of the two interval frequencies times the total; the coefficient of coincidence is observed over expected, and interference is 1 minus coincidence. Genetic distance is a crossover frequency, and in humans and mice crossovers concentrate in hotspots placed largely by PRDM9.
    Graduate

    Map distance in centimorgans is 100 times the expected number of crossovers per gamete in an interval, while recombination frequency reports only an odd number of exchanges and saturates at one half. Three-point mapping recovers doubles that span the middle marker but still misses even numbers of exchanges inside a single interval, so long intervals are underestimated. Positive interference is usually described for crossovers from the interfering pathway; noncrossover gene conversions in a 2016 sequencing study of Drosophila female meioses did not show it, and in Arabidopsis raising the dosage of HEI10 lowered interference in one pair of intervals from 0.64 to 0.34, which fits a proposed coarsening model of crossover spacing.

    Where crossovers fall is a separate question from how they are spaced. In humans and mice PRDM9 binds a sequence motif through its zinc fingers, trimethylates histone H3 on lysine 4 nearby, and so marks sites where Spo11 makes breaks. Dogs lack a functional PRDM9 and birds appear to lack the gene; in those animals, as in budding yeast, hotspots tend to sit near promoter-like features. An open question is why PRDM9 binding sites, and so hotspots, turn over so fast in evolution, given that the motif a hotspot needs is erased by the very recombination it starts.
    Two factors at one point

    Sturtevant's 1913 map had six factors, but only five places on it. Two of the factors had never been separated by crossing over in his crosses, so he drew them at a single point. He did not claim they were one gene; he reported that his data could not tell them apart. The same paper warns that the spacing of factors on the map need not show their spacing on the chromosome. Both cautions still apply. A map is only as fine as the number of recombinants scored, and its units are crossover frequencies, not lengths.

    — Sturtevant, Journal of Experimental Zoology 14:43–59 (1913).
    Seeing the exchange in maize

    Maps assumed that crossing over is a physical exchange between chromosomes. Harriet Creighton and Barbara McClintock at Cornell tested it in maize, using a chromosome 9 that carried a visible knob at the end of its short arm and, on the long arm, an exchanged piece of chromosome 8. They followed linked genes on the same chromosome. When the genes recombined, the visible chromosome parts were exchanged too. Critics later objected that the data were few, and in 1935 the authors published a larger confirmation. The link between a genetic crossover and a physical one held.

    — Creighton & McClintock, Proc. Natl. Acad. Sci. USA 17:492–497 (1931).
    A motif that erases itself

    A 13-base motif had been associated with the activity of about 40% of human hotspots. Myers and colleagues reported that the motif does not work in chimpanzees, and that it is being removed from the human genome by what they called self-destructive drive. When a break starts at a hotspot, repair copies the partner chromosome, so an allele that attracts breaks tends to be replaced in the gamete. Hotspots therefore wear themselves out. The authors pointed to PRDM9, whose zinc fingers change fast, as a likely source of new ones.

    — Myers et al., Science 327:876–879 (2010).

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