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Math of Genetics
Mary SimpsonMATH 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?
Phenotypic Ratio 6:6:2:2
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?
Phenotypic Ratio 6:6:2:2
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?
Phenotypic Ratio 6:6:2:2
Out of the sixteen possible genetic combinations, 6 result in brown, coarse fur
6/16= .375 = 37.5%
BS Bs bS bs
Bs BBSs BBss BbSs Bbss
Bs BBSs BBss BbSs Bbss
bs bBSs bBss bbSs bbss
bs bBSs bBss bbSs bbss
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 flowersrr will have red flowersRr 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 IB
IA
i
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
IB IB
IA IAIB IAIB
i 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 i
IA
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
Homework1. 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?
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