Cousins, a Carrier Test and a Residual Risk
Salma and Omar are first cousins. They are both unaffected but have a son with a rare autosomal recessive condition, confirmed by enzyme assay. No one else in either family is affected.
Salma's brother Ahmed and Omar's sister Fatima are both unaffected and plan to marry. Both take a DNA test that detects the two most common disease-causing variants in this population; the test is known to identify about 90% of carriers. Both receive a negative result.
In the general population the carrier frequency for this condition is about 1 in 25, so a child born to two unrelated people has a risk of roughly 1 in 2500.
The couple ask the counsellor, "So our children are safe now?"
1. Explain why Salma and Omar must both be carriers, and why Ahmed's chance of being a carrier is taken as 1/2 rather than 2/3.
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Obligate carriers. The condition is autosomal recessive, so an affected child must have two copies of the mutant allele, one inherited from each parent. The son is affected, therefore both Salma and Omar, who are unaffected, must each carry one mutant allele: they are obligate heterozygotes.
Ahmed's risk. Salma's mutant allele must have come from one of her two parents; that parent is a carrier and the other parent is, with overwhelming probability, homozygous normal, since the allele is rare. Each child of such a mating has a 1/2 chance of inheriting the allele, so Ahmed's probability of being a carrier is 1/2.
The tempting figure of 2/3 applies only when both parents are known carriers and the child is known to be unaffected - then the possible genotypes of an unaffected child are AA, Aa, aA, giving 2/3. That is not the situation here: nothing has established that both of Ahmed's parents are carriers, so 2/3 is an overestimate. The same reasoning gives Fatima a 1/2 chance of being a carrier.
2. Calculate the risk to a child of Ahmed and Fatima before and after the negative DNA tests. Show each step.
Read the worked answer
Before testing. For an affected child, both parents must be carriers and both must pass on the mutant allele:
- P(Ahmed is a carrier) = 1/2
- P(Fatima is a carrier) = 1/2
- P(both transmit the allele | both are carriers) = 1/4
Risk = 1/2 × 1/2 × 1/4 = 1/16, about 6.3% - roughly a 1 in 16 chance for each pregnancy, which is far higher than the 1 in 2500 for an unrelated couple, and reflects the shared grandparents.
After testing. The test misses about 10% of carrier variants, so a negative result does not mean "not a carrier"; it reduces the probability:
- P(Ahmed is a carrier | negative) = 1/2 × 0.10 = 1/20
- P(Fatima is a carrier | negative) = 1/2 × 0.10 = 1/20
- Risk = 1/20 × 1/20 × 1/4 = 1/1600 ≈ 0.063%
Interpretation. The risk has fallen about 100-fold, from 1/16 to 1/1600, but it is still somewhat above the 1/2500 background figure, because the family history remains relevant.
3. The counsellor says the couple's children are not "safe", only at lower risk. Justify this and explain two reasons why the test cannot guarantee that neither parent is a carrier.
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Why "lower risk" is the accurate phrase. Risk is a probability per pregnancy, not a property that can be switched off. The calculation gives 1/1600 for each child, not zero, and each pregnancy is an independent event: two unaffected children do not reduce the risk for a third.
Reason 1 - incomplete variant coverage. The test detects only the two commonest variants and is validated to find about 90% of carriers. Carriers with a different variant - a rare missense change, a small deletion, a splice-site change deep in an intron, or a large rearrangement - will test negative while still carrying the allele. A negative result therefore shifts the probability from 1/2 to about 1/20; it does not eliminate it.
Reason 2 - the family's variant may not be one of those tested. The test's 90% figure is a population average. If the affected son's variants had been characterised first, the couple could be tested specifically for the family's own variants, which would make a negative result far more informative. Without that, a negative result may simply mean the family carries an untested variant.
Wider counselling points. Consanguinity raises the risk of other recessive conditions not covered by this test, and the couple should also be told what the 1/1600 figure means in practice, including the option of testing the affected child to define the family variants.