Gene maps, trihybrid crosses and pedigree puzzles

Three genes at once

With two genes, a test cross gives four classes. With three linked genes, it gives eight classes. They tell you three things at the same time: the order of the genes, the distances, and the double crossovers.

A three-point test cross

A plant is heterozygous for three linked genes and is crossed with a plant that is homozygous recessive for all three, abc over abc. We write the heterozygote as ABC over abc. The 1000 offspring come in eight classes.

• The two commonest classes, ABC and abc, are the parental classes. No crossover is needed to make them.
• The two rarest classes, AbC and aBc, are the double crossovers. A double crossover is rare because two crossovers have to happen together.
• The other four classes are single crossovers.

Which gene is in the middle?

Compare a double crossover with a parental class. AbC has the same A and C as ABC, but a different B. The gene that has switched is the one in the middle. So the order is A, B, C.

The distances

A recombination frequency counts every offspring that has a recombinant arrangement for a pair of genes.

• A to B: the single crossovers between A and B (45 + 45) and the double crossovers (5 + 5) are 100 of 1000, which is 10 map units.
• B to C: (95 + 95) + (5 + 5) = 200 of 1000, which is 20 map units.
• A to C by direct count: 90 + 190 = 280 of 1000, which is 28 map units. The double crossovers do not count here, because two crossovers cancel. The map distance of A to C is 10 + 20 = 30 map units.

Coincidence

If the two crossovers were independent, the double crossovers would be $0.10 \times 0.20 = 0.02$, which is 2 per cent. The real number is 1 per cent. The ratio of the real to the expected number is the coefficient of coincidence, $\tfrac{1}{2} = 0.5$. Interference is $1 - 0.5 = 0.5$. The crossover in one interval makes a crossover in the next interval less likely.

You should see that a longer interval means more single crossovers, and that a low coefficient of coincidence leaves few double crossovers.

Three genes on different chromosomes

For AaBbCc × AaBbCc with independent genes, use the product rule.

• 8 kinds of gamete, 64 boxes, 27 genotypes and 8 phenotypes.
• The phenotypes come in the ratio 27 : 9 : 9 : 9 : 3 : 3 : 3 : 1.
• The chance of exactly two of the three traits being dominant is $3 \times \left(\tfrac{3}{4}\right)^2 \times \tfrac{1}{4} = \tfrac{27}{64}$. The 3 counts the choices of which trait is recessive.

A pedigree puzzle

A woman's brother has haemophilia. Their mother must be a carrier, so the woman is a carrier with a chance of $\tfrac{1}{2}$. She has two healthy sons. Does this change the chance that she is a carrier? Yes. A carrier has a healthy son with a chance of $\tfrac{1}{2}$ each time, but a woman who is not a carrier has healthy sons for certain.

• Carrier and two healthy sons: $\tfrac{1}{2} \times \left(\tfrac{1}{2}\right)^2 = \tfrac{1}{8}$.
• Not a carrier and two healthy sons: $\tfrac{1}{2} \times 1 = \tfrac{1}{2}$.

The chance that she is a carrier, now that we know about the sons, is $\dfrac{1/8}{1/8 + 1/2} = \dfrac{1}{5}$, or 20 per cent. This kind of updating is the conditional probability that genetic counsellors use.

Try it in Python

The first program works out a three-point cross. The second works out the pedigree puzzle with exact fractions.

Practise with new numbers

Common mistakes

• Counting the double crossovers once. They must be counted in both intervals.
• Choosing the parental classes by the letters. Parental classes are the commonest two, and not always ABC and abc.
• Taking the middle gene from the commonest class. The middle gene comes from comparing the rarest class with the parental ones.
• Forgetting the information in a pedigree. A healthy son tells you something about the mother.

Remember these

• A three-point cross has 8 classes: 2 parental (commonest), 4 single crossovers and 2 double crossovers (rarest).
• The gene that switches in a double crossover is in the middle.
• Coefficient of coincidence is the observed double crossovers divided by the expected ones. Interference is 1 minus that.
• Healthy children can lower the chance that a mother is a carrier.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. The distance from A to B is 12 and from B to C is 8. The coefficient of coincidence is 1. What share of offspring are double crossovers?

2. Which classes are the double crossovers in a three-point cross, and what do they show?

3. AaBbCc is crossed with AaBbCc. What is the chance of an offspring that is aabbcc?

4. How many kinds of gamete does an ABC over abc plant make if there is no crossing over at all?

5. Why does a healthy son lower the chance that his mother is a carrier?

References

• Clark M. A., Douglas M. and Choi J. (2018). Biology 2e, OpenStax, chapter 13.1.
• NCERT (2023). Biology, Textbook for Class XII, chapter 4 (linkage, recombination and sex-linked inheritance).