Sex and Linked Traits

Genetics and Heredity Fundamentals

Course 1 · Chapter 7 · Sex and Linked Traits

Until now every gene sat on a chromosome that comes in two matching copies. One pair of chromosomes is different. In humans it decides whether a person is female or male, and any gene on it follows its own rules. In this chapter we count the children of a cross with the same enumeration tools as before, then turn the question around: given a family tree, what can we say about the gene? All the programs here are Python 3.

What was run, and what was not

Every program in this chapter was run and the output shown is what it printed. The biology is written from general knowledge; items marked (check) are the ones to verify first. Where a program uses random numbers it fixes the seed, so the same output appears each time. The chapter describes the usual human pattern; other species do it differently (check).

The Sex Chromosomes

Humans have 46 chromosomes in 23 pairs (check). Twenty-two pairs are the same in everyone and are called autosomes. The last pair is the sex chromosomes, named X and Y. A typical female has two X chromosomes (XX) and a typical male has one X and one Y (XY) (check). The Y is much smaller than the X and carries far fewer genes (check). A gene on the Y called SRY has a large part in starting male development (check).

Meiosis (Genetics and Heredity Fundamentals 4) makes the next step easy. A mother is XX, so every egg carries an X. A father is XY, so half of his sperm carry an X and half a Y. Whether a child is a girl or a boy depends on which sperm wins. We can enumerate it exactly as we did for a Punnett square in Genetics and Heredity Fundamentals 1.

from itertools import product
from collections import Counter

def sex(pair):
    return "female" if pair == "XX" else "male"

kids = Counter(sex(a + b) for a, b in product("XX", "XY"))
print(kids)
    

Counter({'female': 2, 'male': 2})

Two of the four cells are female and two are male, a 1:1 ratio. The mother's two X chromosomes are identical in this simple count, so each of the four cells is a pairing of one egg type with one sperm type. The father alone decides the child's sex, because the mother can only give an X. In real populations the ratio at birth is close to, but not exactly, equal (check).

XX and XY are the usual pattern, not the only one

Some people have a different number of sex chromosomes, and sex in the wider sense (body, hormones, identity) is not fully captured by chromosomes alone (check). This chapter only follows the chromosomes, because that is what decides how a gene on them is passed on.

Why Sons and Daughters Differ

The X chromosome carries many genes that have nothing to do with sex, such as genes needed for seeing certain colours or for making blood clot (check). A daughter has two copies of each of these genes, one on each X, so everything from Genetics and Heredity Fundamentals 1 applies to her: a recessive allele is hidden if the other copy is the dominant one. A son has only one X, which he always gets from his mother. His Y does not carry a matching copy of most X genes (check). So whatever allele sits on his one X shows, whether it is dominant or recessive. There is nothing to hide it.

The result is called X-linked recessive inheritance (check). A recessive allele on the X is expressed:

in a son, if his one X carries it;
in a daughter, only if both her X chromosomes carry it.

A daughter with one copy of the allele and one of the normal allele is a carrier: she does not show the trait herself but can pass the allele on (check). Writing the alleles as H (normal) and h (the recessive one), the possible people are:

Person	X chromosomes	Look
daughter	`H H`	unaffected
daughter	`H h`	carrier, unaffected
daughter	`h h`	affected
son	`H` (and a Y)	unaffected
son	`h` (and a Y)	affected

A classic textbook example is red-green colour vision deficiency, where the genes involved lie on the X (check). It is much more common in men than in women, which is exactly what the table predicts: a man needs one copy of the allele, a woman needs two (check).

Counting Children

The program below crosses a mother with the pair (m1, m2) on her two X chromosomes and a father with one X allele. Each child is one egg (one of the mother's two X chromosomes) combined with one sperm: an X from the father (a daughter) or his Y (a son). Notice that a son takes only the mother's allele; the father's X never reaches a son.

from itertools import product
from collections import Counter

def child_look(sex, x_alleles):
    # x_alleles: the X chromosomes the child has, "H" = normal, "h" = recessive
    if sex == "male":
        return "affected" if x_alleles == ("h",) else "unaffected"
    n = x_alleles.count("h")
    return {0: "unaffected", 1: "carrier", 2: "affected"}[n]

def cross(mother, father_x):
    # mother: two X alleles, e.g. ("H", "h"); father_x: his one X allele
    out = Counter()
    for m, f in product(mother, ("X", "Y")):
        if f == "X":
            out["daughter " + child_look("female", tuple(sorted((m, father_x))))] += 1
        else:
            out["son " + child_look("male", (m,))] += 1
    return out

for name, mom, dad in [("carrier mother x unaffected father", ("H","h"), "H"),
                       ("unaffected mother x affected father", ("H","H"), "h"),
                       ("carrier mother x affected father", ("H","h"), "h")]:
    print(name)
    for k, v in sorted(cross(mom, dad).items()):
        print("   ", k, v)
    

carrier mother x unaffected father
    daughter carrier 1
    daughter unaffected 1
    son affected 1
    son unaffected 1
unaffected mother x affected father
    daughter carrier 2
    son unaffected 2
carrier mother x affected father
    daughter affected 1
    daughter carrier 1
    son affected 1
    son unaffected 1
    

Read the three crosses in turn. Each has four equally likely cells.

Carrier mother, unaffected father: half the sons are affected (they got her h), and no daughter is affected, because each daughter also gets the father's normal X. Half of the daughters are carriers.
Unaffected mother, affected father: no child is affected, but every daughter is a carrier, because each gets her father's one X, which is h. Every son is unaffected, because he gets no X from his father.
Carrier mother, affected father: now a daughter can be affected, since she may get h from both parents.

These crosses make a pattern worth remembering: an affected father cannot pass the trait directly to a son, because he gives a son his Y, not his X. The trait can skip a generation through a carrier daughter. This is sometimes described as passing from grandfather through a daughter to a grandson (check).

Simulating Families

Enumeration gives exact fractions. A simulation shows how those fractions look in real families. The program below makes 40,000 children of a carrier mother and an unaffected father, with a fixed seed, and reports the share of sons and of daughters who are affected.

import random
from collections import Counter

def child(rng, mother, father_x):
    sex = rng.choice("XY")
    m = rng.choice(mother)
    if sex == "Y":
        return "son", m == "h"
    return "daughter", (m, father_x).count("h") == 2

rng = random.Random(11)
c = Counter()
for _ in range(40000):
    s, aff = child(rng, ("H", "h"), "H")
    c[(s, aff)] += 1
for s in ("son", "daughter"):
    tot = c[(s, True)] + c[(s, False)]
    print(s, tot, c[(s, True)], round(c[(s, True)] / tot, 3))
    

son 19935 9998 0.502
daughter 20065 0 0.0
    

The columns are the group, the number of children in it, how many were affected, and the fraction. About half the sons are affected (0.502, close to 1/2) and none of the daughters, matching the first cross above. The sex split is also near even: 19,935 sons and 20,065 daughters. Because the sex of each child is a separate chance, a single small family can look very different from these averages, as we saw with sample size in Genetics and Heredity Fundamentals 1.

Reading a Pedigree

Real families are not crossed on purpose. What we have is a pedigree: a family tree that records who is affected (check). Geneticists read pedigrees backwards, asking which patterns of inheritance could produce the one they see. The two clues we have found give a way to do this for an X-linked recessive trait:

Flagging: an affected father with an affected son is a warning sign, since the son's X came from his mother and not from the father. It can still happen if the mother is a carrier or affected, but that is a less likely explanation, so a careful analysis looks at the mother too (check).
Ruling in: a son who is affected must have got the allele from his mother, so she is a carrier or affected herself.

A pedigree is a good fit for a dictionary: each person maps to their father, mother and sex. The program below applies the two clues in their simplest form. If a father and son are both affected, it flags the pair as a pattern that does not fit the simple model.

# pedigree: name -> (father, mother, sex)
pedigree = {
    "Ann":   (None, None, "F"),
    "Bob":   (None, None, "M"),
    "Cara":  ("Bob", "Ann", "F"),
    "Dan":   ("Bob", "Ann", "M"),
    "Eli":   (None, None, "M"),
    "Fay":   ("Eli", "Cara", "F"),
    "Gus":   ("Eli", "Cara", "M"),
}
affected = {"Dan", "Gus"}

def father_to_son(pedigree, affected):
    # an affected father with an affected son: a warning sign for a trait on X
    hits = []
    for name, (dad, mom, sx) in pedigree.items():
        if sx == "M" and name in affected and dad in affected:
            hits.append((dad, name))
    return hits

def must_be_carrier(pedigree, affected):
    # mothers of affected sons carry the allele (X-linked recessive, check)
    carriers = set()
    for name, (dad, mom, sx) in pedigree.items():
        if sx == "M" and name in affected and mom is not None:
            carriers.add(mom)
    return sorted(carriers)

print(father_to_son(pedigree, affected))
print(must_be_carrier(pedigree, affected))
pedigree["Hal"] = ("Dan", "Fay", "M")
affected.add("Hal"); affected.add("Dan")
print(father_to_son(pedigree, affected))
    

[]
['Ann', 'Cara']
[('Dan', 'Hal')]
    

The first line is an empty list: in this family no affected father has an affected son, so the pedigree does not contradict an X-linked recessive trait. The second line lists the mothers who must be carriers: Ann (mother of the affected Dan) and Cara (mother of the affected Gus). Cara herself shows no trait here, which is just what a carrier looks like. Then we add Hal, son of Dan, and mark him affected. The last line reports the pair (Dan, Hal), a father and son who are both affected, which this simple model flags (a carrier mother could explain it, and so could other patterns, such as a gene on an autosome, so the mother needs checking and other patterns need considering) (check).

A pedigree rules patterns out more than it proves them

A check like this can reject a pattern, but passing it does not prove the trait is X-linked. Several patterns can produce the same small family, and real pedigrees also include new mutations, incomplete penetrance (a person with the genotype who does not show the trait), and mistaken records (check). Genetic counsellors use pedigrees as a first step, not as a final answer (check).

Challenges

Exercise 1

For each mother genotype HH, Hh, hh and each father (H or h), enumerate the children with itertools.product. Print the fraction of sons that are affected and the fraction of daughters that are affected, as fractions.Fraction values. Which pairings can have an affected daughter?

📄 View solution

Exercise 2

A carrier mother and an unaffected father have three children. Simulate 20,000 such families with random.Random(4), count the families with at least one affected son, and compare the fraction with the exact value 1 - (3/4)³ as a Fraction.

📄 View solution

Exercise 3

Write check(pedigree, affected) that returns a message: either that the pedigree does not fit the simple X-linked recessive model, naming the affected father and son, or that it is consistent, listing the mothers of affected sons. Test it on a pedigree of eight people with three different sets of affected people.

📄 View solution

Quick Reference

Sex chromosomes: typical female XX, typical male XY; the father decides the child's sex, 1 : 1
X-linked recessive: a son needs one copy of the allele, a daughter needs two
Carrier: a daughter with one copy; unaffected, but can pass it on
Fathers and sons: a son gets his father's Y, never his X, so no direct father-to-son passing
Pedigree: a family tree used to rule patterns in or out; it rarely proves one
Check: the chromosome facts, the colour vision example, and the pedigree caveats

Coming next

Genetics and Heredity Fundamentals 8 moves from families to whole populations, and asks how allele frequencies stay the same or change.