Math of
Genetics
Mary Simpson
MATH 150
Objectives
 Understanding how to find the probability of genetic
outcomes for situations involving:
 Multiple Traits
 Linkage
 Incomplete Dominance
 Codominance
 Multiple Allelism
 Understanding Hardy Weinberg Equations in relation
to population genetics
Flashback to High School
Biology!
 Genetics: the study of the inheritance of traits
 Gene: a section of DNA that influences the heredity
of a trait
Flashback to High School
Biology!
 Genetics: the study of the inheritance of traits
 Gene: a section of DNA that influences the heredity
of a trait
 Chromosome: dense coils of DNA that contain
multiple genes
 Allele: denotes different versions of the same gene
Flashback to High School
Biology!
 Genetics: the study of the inheritance of traits
 Gene: a section of DNA that influences the heredity
of a trait
 Chromosome: dense coils of DNA that contain
multiple genes
 Allele: denotes different versions of the same gene
 Gregor Mendel was a pioneer in genetics
Mendelian Genetics
 Gregor Mendel (1822-
1884)
 Studied the inheritance
of traits in pea plants
Mendelian Genetics
 Gregor Mendel (1822-
1884)
 Studied the inheritance
of traits in pea plants
 Mendel looked for
patterns in the
inheritance traits from
parents with specified
traits
How Genes Are Inherited
 The average human had 46 chromosomes (2 sets of
23)
How Genes Are Inherited
 The average human had 46 chromosomes (2 sets of
23)
 Half of these chromosomes come from the mother
and half from the father (1 set from each parent)
How Genes Are Inherited
 The average human had 46 chromosomes (2 sets of
23)
 Half of these chromosomes come from the mother
and half from the father (1 set from each parent)
 Because there are two sets of chromosomes, a person
inherits two copies of each gene
How Genes Are Inherited
 The average human had 46 chromosomes (2 sets of
23)
 Half of these chromosomes come from the mother
and half from the father (1 set from each parent)
 Because there are two sets of chromosomes, a person
inherits two copies of each gene
 A person has two alleles for each trait that interact,
resulting in the expressed trait
Inheritance of Single Traits
 Dominant Trait: if a gene for the dominant trait
(called a dominant allele) is present, it will be
expressed
 Usually expressed with an uppercase letter (ex. A)
 Recessive Trait: this trait will only be expressed in the
absence of a dominant allele
 Usually expressed with a lowercase letter (ex. a)
Inheritance of Single Traits
 Dominant Trait: if a gene for the dominant trait
(called a dominant allele) is present, it will be
expressed
 Usually expressed with an uppercase letter (ex. A)
 Recessive Trait: this trait will only be expressed in the
absence of a dominant allele
 Usually expressed with a lowercase letter (ex. a)
 Genotype: the combination of two alleles (ex. Aa)
 Phenotype: the trait expression that results from a
genotype
Inheritance of Single Traits

Dominant Trait: if a gene for the dominant trait (called a dominant allele) is
present, it will be expressed


Usually expressed with an uppercase letter (ex. A)
Recessive Trait: this trait will only be expressed in the absence of a dominant
allele

Usually expressed with a lowercase letter (ex. a)

Genotype: the combination of two alleles (ex. Aa)

Phenotype: the trait expression that results from a genotype

Homozygous: genotype with two copies of the same allele (ex. AA, aa)

Heterozygous: genotype with one dominant allele and one recessive allele
(ex. Aa)
Punnett Squares
 To form a punnett square, form a grid with the
paternal genotype on the top and the maternal
genotype down the left side
Punnett Squares
 To form a punnett square, form a grid with the
paternal genotype on the top and the maternal
genotype down the left side
 In the center sections of the table, combine the
paternal and maternal alleles to create all possible
genotypes for the offspring
Punnett Square Example
 If we have a mother with genotype aa and a father
with genotype Aa
 The punnett square would look as follows:
a
A
a
a
Punnett Square Example
 If we have a mother with genotype aa and a father
with genotype Aa
 The punnett square would look as follows:
a
a
A
A
A
a
a
a
Punnett Square Example
 If we have a mother with genotype aa and a father
with genotype Aa
 The punnett square would look as follows:
a
a
A
Aa
Aa
a
aa
aa
Punnett Square Example
 If we have a mother with genotype aa and a father
with genotype Aa
 The punnett square would look as follows:
a
a
A
Aa
Aa
a
aa
aa
Genotypic Ratio: a ratio
of the number of
possible outcomes of
each genotype (in this
example 1:1)
Phenotypic Ratio: ratio
of the number of
outcomes that will result
in different phenotypes
(in this example 1:1)
Practice Problem
 The allele for dark hair (B) is dominant and the allele
for light hair (b) is recessive
 If a female with genotype Bb and a male with
genotype Bb mate, what are the chances that they will
have a light haired offspring?
Practice Problem
 The allele for dark hair (B) is dominant and the allele
for light hair (b) is recessive
 If a female with genotype Bb and a male with
genotype Bb mate, what are the chances that they will
have a light haired offspring?
B
B
b
b
Practice Problem
 The allele for dark hair (B) is dominant and the allele
for light hair (b) is recessive
 If a female with genotype Bb and a male with
genotype Bb mate, what are the chances that they will
have a light haired offspring?
B
b
B
B
B
b
b
b
Practice Problem
 The allele for dark hair (B) is dominant and the allele
for light hair (b) is recessive
 If a female with genotype Bb and a male with
genotype Bb mate, what are the chances that they will
have a light haired offspring?
B
b
B
BB
Bb
b
Bb
bb
Practice Problem
 The allele for dark hair (B) is dominant and the allele
for light hair (b) is recessive
 If a female with genotype Bb and a male with
genotype Bb mate, what are the chances that they will
have a light haired offspring?
B
b
B
BB
Bb
b
Bb
bb
To have light hair the genotype
must be bb
There is only a 1/4 chance of
that, therefore the chance is
25%
Inheritance of Two Traits
 Looking at the inheritance of two traits is called a
dihybrid cross
Inheritance of Two Traits
 Looking at the inheritance of two traits is called a
dihybrid cross
 To set up the punnett square you have to look at all
possible combinations of maternal and paternal DNA
Inheritance of Two Traits
 Looking at the inheritance of two traits is called a
dihybrid cross
 To set up the punnett square you have to look at all
possible combinations of maternal and paternal DNA
 You use those 4 combinations from each parent to set
up the punnett square
Practice Problem
 We will look at the inheritance of brown and black fur
and coarse and soft fur in hamsters
 Brown fur (B) and soft fur (S) are dominant
Practice Problem
 We will look at the inheritance of brown and black fur
and coarse and soft fur in hamsters
 Brown fur (B) and soft fur (S) are dominant
Practice Problem
 We will look at the inheritance of brown and black fur
and coarse and soft fur in hamsters
 Brown fur (B) and soft fur (S) are dominant
 If the mother has genotype BBss and the father has
genotype BbSs, what is the chance that an offspring
will have brown coarse fur?
Practice Problem Cont.
 If the mother has genotype Bbss and the father has
genotype BbSs, what is the chance that an offspring
will have brown coarse fur?
BS
Bs
Bs
bs
bs
Bs
bS
bs
Practice Problem Cont.
 If the mother has genotype Bbss and the father has
genotype BbSs, what is the chance that an offspring
will have brown coarse fur?
BS
Bs
bS
bs
Bs
Bs
Bs
Bs
Bs
Bs
Bs
Bs
Bs
Bs
bs
bs
bs
bs
bs
bs
bs
bs
bs
bs
Practice Problem Cont.
 If the mother has genotype Bbss and the father has
genotype BbSs, what is the chance that an offspring
will have brown coarse fur?
BS
Bs
bS
bs
Bs
BBSs
BBss
BbSs
Bbss
Bs
BBSs
BBss
BbSs
Bbss
bs
bBSs
bBss
bbSs
bbss
bs
bBSs
bBss
bbSs
bbss
Practice Problem Cont.
 If the mother has genotype Bbss and the father has
genotype BbSs, what is the chance that an offspring
will have brown coarse fur?
BS
Bs
bS
bs
Bs
BBSs
BBss
BbSs
Bbss
Bs
BBSs
BBss
BbSs
Bbss
bs
bBSs
bBss
bbSs
bbss
bs
bBSs
bBss
bbSs
bbss
Phenotypic Ratio 6:6:2:2
Practice Problem Cont.
 If the mother has genotype Bbss and the father has
genotype BbSs, what is the chance that an offspring
will have brown coarse fur?
BS
Bs
bS
bs
Bs
BBSs
BBss
BbSs
Bbss
Bs
BBSs
BBss
BbSs
Bbss
bs
bBSs
bBss
bbSs
bbss
bs
bBSs
bBss
bbSs
bbss
Phenotypic Ratio 6:6:2:2
Practice Problem Cont.
 If the mother has genotype Bbss and the father has
genotype BbSs, what is the chance that an offspring
will have brown coarse fur?
BS
Bs
bS
bs
Bs
BBSs
BBss
BbSs
Bbss
Bs
BBSs
BBss
BbSs
Bbss
bs
bBSs
bBss
bbSs
bbss
bs
bBSs
bBss
bbSs
bbss
Phenotypic Ratio 6:6:2:2
Out of the sixteen possible genetic combinations, 6 result
in brown, coarse fur
6/16= .375 = 37.5%
Linkage
 Linked genes are those found on the same
chromosome
Linkage
 Linked genes are those found on the same
chromosome
 This means that these traits should not follow the
same pattern of inheritance because the traits cannot
be independently assorted into gametes
Linkage
 Linked genes are those found on the same
chromosome
 This means that these traits should not follow the
same pattern of inheritance because the traits cannot
be independently assorted into gametes
 In terms of a punnett square, having two linked traits
would be treated like having a single trait
Linkage
 Linked genes are those found on the same
chromosome
 This means that these traits should not follow the
same pattern of inheritance because the traits cannot
be independently assorted into gametes
 In terms of a punnett square, having two linked traits
would be treated like having a single trait
 Mendel was lucky that each of the traits he studied
had genes that were not linked
Incomplete Dominance
 Incomplete dominance means that the dominant allele
will not completely dominant the recessive allele
Incomplete Dominance
 Incomplete dominance means that the dominant allele
will not completely dominant the recessive allele
 In many cases this means that heterozygous
individuals will have intermediate phenotypes
Incomplete Dominance
 Incomplete dominance means that the dominant allele
will not completely dominant the recessive allele
 In many cases this means that heterozygous
individuals will have intermediate phenotypes
 This will not alter genotypic ratios, but it will alter
phenotypic ratios
Practice Problem
 The allele for white flowers (R) is dominant, but it’s
dominance incomplete
 The allele for red flowers (r) is recessive
Practice Problem
 The allele for white flowers (R) is dominant, but it’s
dominance incomplete
 The allele for red flowers (r) is recessive
 What are the possible phenotypes of the offspring of
two plants with genotypes Rr and Rr?
Practice Problem
 The allele for white flowers (R) is dominant, but it’s
dominance incomplete
 The allele for red flowers (r) is recessive
 What are the possible phenotypes of the offspring of
two plants with genotypes Rr and Rr?
R
R
r
r
Practice Problem
 The allele for white flowers (R) is dominant, but it’s
dominance incomplete
 The allele for red flowers (r) is recessive
 What are the possible phenotypes of the offspring of
two plants with genotypes Rr and Rr?
R
r
R
RR
Rr
r
Rr
rr
Practice Problem
 The allele for white flowers (R) is dominant, but it’s
dominance incomplete
 The allele for red flowers (r) is recessive
 What are the possible phenotypes of the offspring of
two plants with genotypes Rr and Rr?
R
r
R
RR
Rr
r
Rr
rr
RR will have white
flowers
rr will have red flowers
Rr will have pink
flowers (intermediate
between white and red)
Practice Problem
 If we mated two of that same type of flowers with the
genotypes, RR and Rr, what is the probability that the
offspring will have pink flowers?
Practice Problem
 If we mated two of that same type of flowers with the
genotypes, RR and Rr, what is the probability that the
offspring will have pink flowers?
R
R
r
R
Practice Problem
 If we mated two of that same type of flowers with the
genotypes, RR and Rr, what is the probability that the
offspring will have pink flowers?
R
R
R
RR
RR
r
Rr
Rr
Practice Problem
 If we mated two of that same type of flowers with the
genotypes, RR and Rr, what is the probability that the
offspring will have pink flowers?
R
R
R
RR
RR
r
Rr
Rr
2/4 or 50%
chance
Codominance
 Codominance: when heterozygotes have the
phenotypes associated with each allele (because both
alleles are dominant)
Codominance
 Codominance: when heterozygotes have the
phenotypes associated with each allele (because both
alleles are dominant)
 The best example is blood type
 There are three alleles for blood type (IA, IB, i)
Codominance
 Codominance: when heterozygotes have the
phenotypes associated with each allele (because both
alleles are dominant)
 The best example is blood type
 There are three alleles for blood type (IA, IB, i)
 IA and IB are codominant, so if a person has genotype
IAIB, they will have type AB blood
 IAi, results in type A, IBi in type B and ii in type O
Practice Problem
 What are the possible blood types of offspring of
parents with genotypes IAi and IBIB
Practice Problem
 What are the possible blood types of offspring of
parents with genotypes IAi and IBIB
IB
IA
i
IB
Practice Problem
 What are the possible blood types of offspring of
parents with genotypes IAi and IBIB
IB
IB
IA
IAIB
IAIB
i
IBi
IBi
Practice Problem
 What are the possible blood types of offspring of
parents with genotypes IAi and IBIB
IA
i
IB
IB
IAIB
IAIB
IBi
IBi
IAIB will result in type AB
IBi will result in type B
Practice Problem
 What is the chance that a mother with genotype IBi
and a father with genotype IAi will have a child with
type O blood?
Practice Problem
 What is the chance that a mother with genotype IBi
and a father with genotype IAi will have a child with
type O blood?
IB
IA
i
i
Practice Problem
 What is the chance that a mother with genotype IBi
and a father with genotype IAi will have a child with
type O blood?
IB
i
IA
IAIB
IAi
i
IBi
ii
Practice Problem
 What is the chance that a mother with genotype IBi
and a father with genotype IAi will have a child with
type O blood?
IB
i
IA
IAIB
IAi
i
IBi
ii
1/4 or 25%
Multiple Gene Inheritance
 Multiple Gene Inheritance: there is more than one
gene that controls the expression of a trait
Multiple Gene Inheritance
 Multiple Gene Inheritance: there is more than one
gene that controls the expression of a trait
 Example: Pepper Color
 Pepper color is controlled by two different genes
 The first gene controls the expression of red pigment
 The dominant allele (R) indicates the presence of red
pigment
 The recessive allele (r) indicates the absence of red
pigment
Multiple Gene Inheritance
 Multiple Gene Inheritance: there is more than one
gene that controls the expression of a trait
 Example: Pepper Color
 Pepper color is controlled by two different genes
 The first gene controls the expression of red pigment
 The dominant allele (R) indicates the presence of red
pigment
 The recessive allele (r) indicates the absence of red
pigment
 The second gene controls the expression of either green
(G) or yellow (g) pigment
Multiple Gene Inheritance
 If red pigment is expressed, the pepper will be red,
regardless of the second gene.
Multiple Gene Inheritance
 If red pigment is expressed, the pepper will be read,
regardless of the second gene.
 If the red pigment is absent, you must look to the
second gene to determine color
Multiple Gene Inheritance
 If red pigment is expressed, the pepper will be red,
regardless of the second gene.
 If the red pigment is absent, you must look to the
second gene to determine color
 What would the color of a pepper with the genotype
Rrgg be?
Multiple Gene Inheritance
 If red pigment is expressed, the pepper will be read,
regardless of the second gene.
 If the red pigment is absent, you must look to the
second gene to determine color
 What would the color of a pepper with the genotype
Rrgg be?
 Red
Multiple Gene Inheritance
 If red pigment is expressed, the pepper will be red,
regardless of the second gene.
 If the red pigment is absent, you must look to the
second gene to determine color
 What would the color of a pepper with the genotype
Rrgg be?
 Red
 What about rrGg
Multiple Gene Inheritance
 If red pigment is expressed, the pepper will be read,
regardless of the second gene.
 If the red pigment is absent, you must look to the
second gene to determine color
 What would the color of a pepper with the genotype
Rrgg be?
 Red
 What about rrGg
 Green
Hardy Weinberg Principle
 Looks at the frequency of alleles in a population
 The Principle makes several important assumptions:
 There is not natural selection regarding the gene in
question
 There is no genetic drift
 There is no gene flow
 There is no mutation
 Random mating with respect to the gene in question is
occurring
Hardy Weinberg Principle
 Hardy Weinberg Equation:
 p2 + 2pq + q2 = 1
 p+q=1
Hardy Weinberg Principle
 Hardy Weinberg Equation:
 p2 + 2pq + q2 = 1
 p+q=1
 p=allele frequency of the dominant allele
 q=allele frequency of the recessive allele
Hardy Weinberg Principle
 Hardy Weinberg Equation:
 p2 + 2pq + q2 = 1
 p+q=1
 p=allele frequency of the dominant allele
 q=decimal version of the recessive allele
 p2 is the frequency of the homozygous dominant
genotype
 q2 is the frequency of the homozygous recessive
genotype
 2pq is the frequency of the heterozygous genotype
Genes that the Hardy Weinberg
Equilibrium Applies To
 Tongue Rolling (dominant)
Genes that the Hardy Weinberg
Equilibrium Applies To
 Tongue Rolling (dominant)
 Free (dominant) v. Attached (recessive) Earlobes
Genes that the Hardy Weinberg
Equilibrium Applies To
 Tongue Rolling (dominant)
 Free (dominant) v. Attached (recessive) Earlobes
 Hand Clasping
 Left thumb over right (dominant)
 Right thumb over left (recessive)
Genes that the Hardy Weinberg
Equilibrium Applies To
 Tongue Rolling (dominant)
 Free (dominant) v. Attached (recessive) Earlobes
 Hand Clasping
 Left thumb over right (dominant)
 Right thumb over left (recessive)
 Widow’s Peak (dominant)
Genes that the Hardy Weinberg
Equilibrium Applies To
 Tongue Rolling (dominant)
 Free (dominant) v. Attached (recessive) Earlobes
 Hand Clasping
 Left thumb over right (dominant)
 Right thumb over left (recessive)
 Widow’s Peak (dominant)
 Mid-Digital Hair (dominant)
Using the Hardy Weinberg
Equations
 If the frequency of the recessive allele for sickle cell
anemia is .4 in a population of 100,000
 The dominant allele has a frequency of .6
 Individuals that are heterozygous for this allele have a
higher resistance to malaria
 How many members of the population would have the
increased resistance to malaria?
Using the Hardy Weinberg
Equations
 If the frequency of the recessive allele for sickle cell
anemia is .4 in a population of 100,000 people
 The dominant allele has a frequency of .6
 How many members of the population would have the
increased resistance to malaria?
 Heterozygous Frequency = 2pq
Using the Hardy Weinberg
Equations
 If the frequency of the recessive allele for sickle cell
anemia is .4 in a population of 100,000 people
 The dominant allele has a frequency of .6
 How many members of the population would have the
increased resistance to malaria?
 Heterozygous Frequency = 2pq
 2pq = 2 * 0.4 * 0.6 = .48
Using the Hardy Weinberg
Equations
 If the frequency of the recessive allele for sickle cell
anemia is .4 in a population of 100,000 people
 The dominant allele has a frequency of .6
 How many members of the population would have the
increased resistance to malaria?
 Heterozygous Frequency = 2pq
 2pq = 2 * 0.4 * 0.6 = .48
 48,000 people would have increased malaria resistance
Homework
1. What is the probability that a father with genotype
Hhpp and a mother with genotype HHPp will have
offspring that have the dominant phenotype for both
traits?
2. If the allele frequency for blue eyes in a population is
0.35 and that allele is recessive, what is the frequency
of heterozygous individuals in the population?