CHAPTER 3: Mendel's Methods and Statistics

Mendel researched pea genetics in the nineteenth century.
His work was published in 1866.
- In 1857, Mendel lived in an Augustinian monastery.
- Peas were available in many easily distinguishable varieties.
  - Character is a detectable inheritable feature of an organism.
  - Trait is a variant of an inheritable character.
  - Character (Trait)
  - For peas:
  - Flower color (purple or white)
  - Flower position (axial or terminal)
  - Seed color (yellow or green)
  - Seed shape (round or wrinkled)
  - Pod shape (inflated or constricted)
  - Pod color (green or yellow)
  - Stem length (tall or dwarf)
- Mendel started his experiments with true-breeding plant varieties, which he hybridize (cross-pollinated) in experimental crosses.
- The true-breeding parental plants of such a cross are called the P generation (parental).
His peas had many different characters available to study, each with distinct observable alternative traits.
Because peas can self-fertilize, it was possible to find and use pure breeding varieties.
Peas could also be forced to self-fertilize or to cross-breed.
- True-breeding is when plants always yield progeny with the same appearances for a certain trait. These are called the P generation.
- A plant with round seeds that always yields round seeded progeny when crossed to others within its group or with itself is true-breeding.
Hybrids form when true-breeding plants with different traits for the same character cross.
- If a cross is designed to observe the behavior of one character, it is called a monohybrid cross.
- If a cross is designed to observe the behavior of two characters, it is called a dihybrid cross.
- A dihybrid is a plant that had parents which differed in two characters: height and color for example.
- P generation organisms with different traits are crossed to generate the F₁.
- The F₁ hybrids are crossed to generate the F₂ generation; F₂ are crossed to generate F₃.
Dominant and recessive traits: one dominant allele present is enough to get the corresponding phenotype; recessive appearance requires the presence of 2 copies of the recessive allele.
When looking at the separation of two alleles, one of Mendel’s laws, Random Segregation of alleles, can be observed by following the movement of alleles into gametes. Another law, Independent Assortment, can be observed in dihybrid or greater crosses.
- Alleles for different characters segregate independent of each other. This is independent assortment.
- This is the law that has useful exceptions as we will see in the mapping section in another chapter.
Probability and statistics are essential to understanding genetics.
- Random events are independent of one another.
- The outcome of a random event is unaffected by the outcome of previous such events.
- For example, it is possible that five successive tosses of a normal coin will produce five heads; however, the probability of heads on the sixth toss is still 1/2.
- The rules of probability can be used to solve complex genetics problems. For example, Mendel crossed pea varieties that differed in three characters (trihybrid crosses).
  - Question: What is the probability that a trihybrid cross between two organisms with the genotypes AaBbCc and AaBbCc will produce an offspring with the genotype aabbcc?
  - Answer: Because segregation of each allele pair is an independent event, we can treat this as three separate monohybrid crosses:
    - Aa X Aa: probability for aa offspring equals 1/4.
    - Bb X Bb: probability for bb offspring equals 1/4.
    - Cc X Cc: Probability for cc offspring equals 1/4.
  - The probability that these independent events will occur simultaneously is the product of their independent probabilities (rule of multiplication). So the probability that the offspring will be aabbcc is: 1/4 aa X 1/4 bb X 1/4 cc = 1/64
Mendelian Inheritance in Humans
- Mendelian inheritance in humans is difficult to study.
- Human generation time is about 20 years.
- Humans produce relatively few offspring compared to almost all other species.
- Well-planned breeding experiments are impossible.
Human Pedigrees
- Current understanding of Mendelian inheritance in humans is gained by analysis of family pedigrees or the results of matings that have already occurred.
- A Pedigree is a family tree that diagrams the relationships among parents and children across several generations and that shows the inheritance pattern of a particular phenotypic character.
- For example, wooly hair results from a dominant allele W, thus homozygous dominant (WW) and heterozygous (Ww) individuals have this trait.
  - Homozygous recessive (ww) individuals have normal (wild type) hair.
  - If the man at the top of the pedigree has normal hair, his genotype must be ww.
  - If his wife had wooly hair and three of their six children had normal hair, she must have been a heterozygote for wooly hair. ( If she had been homozygous, all the children would have had wooly hair).
  - If a grandson with wooly hair marries a woman with non-wooly hair and they plan on having three children, what would be the probability that all three children will have wooly hair?
  - Since the man is heterozygous (Ww) and the wife is homozygous (ww), each child has a 1/2 probability of inheriting the wooly allele.
  - Using the rule of multiplication, the probability of these independent events occurring together is:
  - 1/2 X 1/2 X 1/2 = 1/8.
- This type of analysis is an important tool for geneticists, especially if the allele being analyzed causes a disabling or lethal disorder.

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MENDELIAN GENETICS

In 1865, Gregor Mendel became the father of classical genetics.

His research was ignored, forgotten and then rediscovered at the turn of the twentieth century.

He suggested two laws: Random Segregation and Independent Assortment.
He suggested that units of inheritance exist in pairs, segregate independently during gamete formation, and one from each parent form new pairs in the offspring.
He crossed purebred peas, which differed in clearly observable traits.
Purebred original stocks are called the P₁ generation.
First generation hybrid crosses are called F₁ for filial generation 1.
- Second generation, F₁ crossed with F₁ (in peas, self-fertilized) are called F₂ for filial generation 2. (See Figure 3.1)
- F₁ crosses are also called hybrid crosses.
- When just one character and two traits are under consideration, it is called a monohybrid cross.
- When two characters are being considered, it is called a dihybrid cross.
Dominant traits are those exclusive traits that appear in the F₁ generation.
Recessive traits are the traits that are hidden or masked in the F₁ generation.
Alleles are different forms of a gene, each diploid individual has two, each gamete has one, the union of gametes re-establishes a pair.
The genotype is the actual genes in the individual or the alleles the individual has in its genome.
The phenotype is the physical appearance or biochemical activity caused by the genotype.
- The genotype could be Aa; the phenotype would be A, or what A makes the organism look or function like.
- A genotype of aa would have the phenotype a or what a makes the organism look like.

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Crossing

The genotype AA produces gametes with A or A.
The genotype Aa produces gametes with either A or a.
The genotype aa produces gametes with a or a.
Homozygous is when the diploid individual has a pair of identical alleles (AA or aa) for the character.
Heterozygous is having a pair of different alleles (Aa).

Proportions of progeny (offspring from the cross) will be close to those expected, but not necessarily exactly the expected.
- It is like tossing a coin. If you toss a coin ten times, the expected value of 5 heads and 5 tails will only be observed sometimes.

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The Punnett Square

The Punnett square is a useful device to recombine alleles of gametes and generate all possible outcomes to determine there relative proportions.
For any cross, Punnett square(s) can be used, but their best contribution can be realized if used on one character at a time.
In the below square, a simple monohybrid cross is diagrammed. The males generate two kinds of gametes, those with an A allele and those with an a. The same is true for the females.

Female -> Male	A	a
A	AA	Aa
a	Aa	aa

The outcome is a genotypic ratio of 1AA : 2Aa : 1aa.
- To express as proportions, simply add the values of the ratios and use the total as the denominator.
- In the above example: 1 + 2 + 1 = 4. The demoninator would be 4.
- The proportions would be 1/4 AA, 2/4 (or 1/2) Aa and 1/4 aa.
- The percentages would be the proportions times 100. For example, 1/4 X 100 = 25%.
Genetics includes an element of detective style investigation. As the course progresses, we will learn how to determine the probability of genetic diseases, DNA fingerprinting methodologies and their use in crime scene investigation and more. The first simple detective case presented in a typical genetics course is called the test cross.
If a plant or animal has the dominant phenotype but the genotype is unknown, how can we determine its genotype?
To accomplish this, use the test cross. The unknown with a dominant phenotype is crossed or mated to homozygous recessive individuals.

Female -> Male	a	a
A	_ _	_ _
?	_ _	_ _

The Punnett square above diagrams the problem. The genotype of the homozygous recessive is know and is aa. One of the alleles of the unknown must be A, because it has the dominant phenotype. This is what is known even before knowing the phenotypes of the progeny:

Female -> Male	a	a
A	Aa	Aa
?	_ _	_ _

The progeny will either consist of one or two different phenotypes depending on whether the unknown is homozygous for A or is heterozygous. Two different phenotypes would be possible only if the ? was an a. If homozygous, all progeny would be the same dominant phenotype. Use the Punnett square above to verify this statement. (Also see Figure 3.2)

A back cross is when progeney are crossed with one of the parental types.

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Genotypes to Phenotypes
A 1AA : 2Aa : 1aa genotypic ratio is a 3 : 1 phenotypic ratio, if the gene's alleles follow the dominant/recessive mode of inheritance.
Another very useful way to think about the phenotypic ratio is that it is a 3A_ : 1aa ratio. The blank is either allele.
When observing the progeny of an F₁ hybrid cross, phenotypes will be all that can be observed. Any progeny with the dominant trait has the genotype of A_; any with the recessive trait will be aa.

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Dominant/Recessive is not always the mode of inheritance

An example is partial dominance (also called incomplete dominance).

Characteristic of partial dominance is an intermediate expression: Red (AA) X White (aa) yields Pink (Aa).
In this case, the genotypes can be determined by observing the phenotypes.
The genotype AA would be red and the phenotype of Aa would be pink.
Another important indicator of whether a gene exhibits incomplete dominance is the ratio of progeny found in the F₂ generation, which is 1:2:1 if incomlete dominance is the mode of inheritance.
In the example above the ratio would be 1 red : 2 pink : 1 white. (1AA : 2Aa : 1aa)

Female -> Male	A	a
A	AA Red	Aa Pink
a	Aa Pink	aa White

See Figure 3.10 for a more visual appealing example of incomplete dominance.

Another exception is codominance where both alleles are expressed.

M / F	A	a
A	AA Black	Aa Black & White
a	Aa Black & White	aa White

Why are there dominant and recessive alleles? What is the molecular basis of dominance?
- Usually, dominant alleles are functional forms of the gene, whereas recessive alleles are non-functional forms of the gene. Often genes code for enzymes. If the gene is functional, the enzyme it codes for works. If non-functional, then the enzyme either fails to work or it is not synthesized.
- Therefore, recessive traits tend to be the lack of something. For example, the lack of pigment might mean the color white is observed instead of the dominant brown color.
How can partial dominance be explained? What is the molecular explanation for partial dominance?
- Snapdragons are the commonly used textbook example of partial dominance. Red, pink and white snapdragon flowers are commonly seen.
- The red flowered plants are homozygous for the red allele. White flowered plants are homozygous for the white allele. Pink flowered plants are heterozygous.
- Red flowered plants produce twice the amount of pigment than pink. White flowered plants produce no red pigment.
- In the typical dominant /recessive mode of inheritance, one functional copy of the gene, such as that found in a heterozygote, is sufficient to generate a phenotype that is indistinguishable from the homozygous dominant individuals.
- In snapdragons and other cases of partial dominance, a dose difference is observable.
What molecular explanation describes codominance?
- The ABo blood types of humans are the commonly used textbook example of codominance.
- However, only the A and B blood types demonstrate the codominant mode of inheritance.
- Ignoring the o, an individual could be either A, AB or B blood type. The A blood type in this hypothetical would be AA. The B would be BB and the heterozygous individuals would be AB.
- Both the A and B alleles are functional. However, they code for different forms of a cell surface antigen, which is distinguishable by testing with antibodies.
- Humans also have an MN blood antigen system, which is codominant.
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Answers to a few frequently asked questions
Q: What is meant by wild type allele? A: The most commonly found phenotype and the allele that causes it.
Q: What is a mutant type allele? A: Deviants from the wild type phenotype, even if they are observed in the wild, and the allele that causes it.
Q: Why are there two determinants, gene copies or alleles per individual? A: Because in humans, other animals and in the plants first studied, the diploid form of the organism was studied, and the diploid has two copies of each type of chromosome. The determinants are at specific locations on a certain chromosome. We humans get one of each chromosome from our mother and one from our father (ignoring for the moment, the sex chromosomes).

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Multiple Alleles

Why should there be two versions of each gene? It would be expected that in some species with few individuals, only one version of a given gene might exist.
There might also be many forms of a gene.
In humans, it is the fact that there are many forms of certain cell surface antigens that make organ transplantation such a problem.
Blood type in humans is an example of codominance, dominance and multiple alleles. If we represent the ABo blood type system using corresponding symbols for the alleles, such as A, B, and o; there are three different alleles for the same gene.
- Having more than two different alleles is having multiple alleles for the gene. Individuals will have just two alleles: AA, AB, BB, Ao, Bo, or oo.
- Each individual will still only have just two copies of the gene, one from mother and one from father.

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Pedigree Analysis

You might have seen a pedigree, if you are an animal breeder. They are useful for following what is called the bloodline.
Pedigrees are also useful for following genetic diseases within families.
- The most common use is for diseases associated with the X chromosome.
- This is because males have just one X and are affected by any defect found on the X they get.
Pedigrees are used by geneticists to trace the appearance of genes through several generations.
- Symbols are used for male/female, affected/unaffected, and normal/abnormal. (see Figure 3.13)
- Consanguineous matings are those between relatives. This causes a marked increase in homozygosity. (see Figures 3.15 and 3.16)
- Learning to understand pedigree analysis is best done by studying many different charts, and by solving associated problems.

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Independent Assortment: Dihybrid and Multi-hybrid Crosses

Independent assortment can be observed when studying more than one character at a time.
Dihybrid cross is a mating between parents that are heterozygous for two characters (dihybrids).
- For example, Mendel began his experiments by crossing true-breeding parent plants that differed in two characters such as seed color (yellow or green) and seed shape (round or wrinkled).
- The allele for yellow seeds (Y) is dominant to the allele for green (y), and round (R) is dominant to wrinkled (r).
- If plants homozygous for round yellow seeds (RRYY) are crossed with plants homozygous for wrinkled green seeds (rryy), the resulting F₁ dihybrid progeny would be heterozygous for both traits (RrYy) and have round yellow seeds, the dominant phenotypes.
- The F₁ progeny self-pollinate. Two alternate hypotheses predict different outcomes:
  - Hypothesis 1: If the two characters segregate together, the F₁ hybrids can only produce two classes of gametes (RY and ry) that they received from the parents, and the F₂ progeny will show a 3:1 phenotypic ratio.
  - Hypothesis 2: If the two characters segregate independently, the F₁ hybrids will produce four classes of gametes (RY, Ry, rY, ry), and the F₂ progeny will show a 9:3:3:1 ratio.
  - Experiment: Mendel performed a dihybrid cross by allowing self-pollination of the F₁ plants (RrYy X RrYy).
  - Results: Mendel categorized the F₂ progeny and determined a ratio of 315:108:101:32, which approximates 9:3:3:1. These results were repeatable. Mendel performed similar dihybrid crosses with all seven characters in various combinations and found the same 9:3:3:1 ratio in each case.
  - Conclusion: The experimental results supported the hypothesis that each allele pair segregates independently during gamete formation
Mendel's law of independent assortment states that each allele pair segregates independently of other gene pairs during gamete formation.
Mendel observed that different genes assort independently of each other. It was as if each gene had its own chromosome. (Chromosomes had not yet been discovered when Mendel was alive. The link between chromosomes and genes had not yet been made.)
- Mendel's law of independent assortment is not always correct.
- Studying deviations from this law is how chromosome mapping is done, and will be covered in a future chapter.
- In this chapter, further discussion and problems solving algorithms are presented as if Mendel's Law of Independent Assortment is always correct.

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The Dihybrid Cross

If a plant with the alleles Aa and Bb, typically written as AaBb for genetics problem solving, was selfed (made to self-polinate) the gametes produced would be:

Male Part of the Plant	B	b
A	AB	Ab
a	aB	ab

This Punnett square is for the male part of the plant.

Female Part of the Plant	B	b
A	AB	Ab
a	aB	ab

This is the Punnett square for the female part of the plant.

These are the allelic combinations that will be found in the gametes, their proportions and what will merge to form the progeny. Below is the Punnett square for the merging of male and female gametes.

Female -> Male	AB	Ab	aB	ab
AB	AABB	*AABb*	*AaBB*	AaBb
Ab	AABb	*AAbb*	*AaBb*	Aabb
aB	*AaBB*	*AaBb*	*aaBB*	aaBb
ab	AaBb	*Aabb*	*aaBb*	aabb

These would be the genotypes of the progeny. Some cells have the same genotypes, and must be grouped and added to determine the ratio.

See Figures 3.3 and 3.4
See Figure 3.6 for predicting the out come for a two gene backcross.
See Figures 3.8 and 3.9 for the Punnett square probability method.
Using Punnett squares for more than one gene at a time can get confusing, quickly. Crosses that are more complex than dihybrid or even dihybrid crosses can be more easily solved by using the Punnett square for each gene, separately. The results can then be combined using a matrix approach.
This approach is simple, far faster, easier to learn and can be used for any problem involving more than one gene.
- An everyday example would be Dr. Herr's wardrobe for work. The money I have saved by bringing my lunch to work each day, delivering phone books and finding super sales on shirts and pants, I now have an extensive wardrobe, which includes seven shirts and seven pairs of pants. I now have forty-nine different outfits!
- If I just have one red, two white and one purple shirt; and, green pants and red pants, how many distinctly different outfits do I have, and what is the probability of me wearing each outfit by random chance?

Shirts	.	1 red	2 white	1 purple	.	.
Pants	.	1 green	1 red	.	.	.
Outfits	1 red shirt, green pants	2 white shirt, green pants	1 purple shirt, green pants	1 red shirt, red pants	2 white shirt, red pants	1 purple shirt, red pants

Dr. Herr's atypical but styling wardrobe generates forty-nine different outfits for work. This matrix demonstrates how. Just expand the rows to include seven shirts, and seven pants.

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Use of the Matrix System

Many students are already experience with the use of a matrix for problem solving.

Our numerical system can be visualized using a matrix, as can the genetic code and the binary system used in computers.

				The Decimal System							Number of symbols per row
1^st Symbol	0	1	2	3	4	5	6	7	8	9	10
2^nd Symbol	0	1	2	3	4	5	6	7	8	9	10
						00-99	Maximum 2-digit numbers				100

The ten symbols of the decimal system generate 100 possible two digit numbers (00-99); three digits generates 1000 possible numbers.

Binary System			Number of Symbols
1st Symbol	0	1	2
2nd Symbol	0	1	2
Maxium 2-digit numbers (00-11)			4

With the 2 characters of the binary system: 0 and 1, which can also be conceptualized as off and on, four characters can be generated.

0		1
0		1
00	01	10	11

If you study how the bottom values were generated from the top two, you will understand how to use the matrix system.

0				1
0				1
00		01		10		11
0				1
000	001	010	011	100	101	110	111

Each additional character multiplies the possible combinations. There are two, one character; four, two character; and, eight, three character combinations.

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Solving Genetics Problems with the Matrix System

If a plant with the alleles Aa and Bb, typically written as AaBb for genetics problem solving, was selfed (made to self-polinate) what would be the expected genotypic ratio of the progeny?
- This time, instead of using the Punnett square method for the dihybrid, we will use the matrix system.
- The genes are treated separately using a Punnett square for each, and then recombined using the matrix.

Female> Male	A	a
A	AA	Aa
a	Aa	ab

This Punnett square is for the A and a alleles.

Female> Male	B	b
B	BB	Bb
b	Bb	bb

This Punnett square is for the B and b alleles.

The figure above displays the genotypic ratio outcome of this F₁ dihybrid cross. Click the image above or this link to see the animation that generated this image.

No need to find similar cells and add them, the matrix system calculates the ratios; each entry is unique.
- A major advantage of using this approach is that it initiates a methodology based on simplification.
- I call it "Refuse to be confused".
If the above problem was a trihybrid F₁ cross the results of another monohybrid cross would be combined with the results of the dihybrid cross above. Imagine that the third gene and pair of alleles were C and c. The solution to the problem of the genotypes expected in the F₂ generation would continue with the work on the monohybrid cross of Cc with Cc. Then, these results need to be placed into the matrix.

Female> Male	C	c
C	CC	Cc
c	Cc	cc

This is the Punnett square for the C and c alleles.

The figure above displays the genotypic ratio outcome of this F₁ trihybrid cross. Click the image above or this link to see the animation that generated this image.

Each gene is treated as a separate cross. This can be done even when the problem is presented with their genetic combinations.
An example problem: Progeny of repeated crosses between two mice yielded 6 with long tails and white coat color, 6 with long tails and brown coat color, 2 with short tails and white coat color, and 2 with short tails and brown coat color. What were the genotypes of the parents, assuming a dominant/recessive mode of inheritance?
To solve this problem, study each gene separately. One gene controls tail length. Among the progeny 12 had long tails and 4 had short tails. Arbitrary symbols can be applied to represent the alleles. The symbol S can be used for long tails and s for short.

Step 1.

Parent 1> Parent 2	?	?
?	_ _	_ _
?	_ _	ss

What is known is the genotype of the short tailed mice. It is ss.

Step 2.

Parent 1> Parent 2	?	s
?	_ _	_ _
s	_ _	ss

Each parent must have at least one s allele. From this knowledge, the Punnett square can be partially filled in.

Step 3.

Parent 1> Parent 2	S	s
?	_ _	_ _
s	Ss	ss

At least one of the parents must have had the dominant allele for long tail length because progeny were observed with this trait. With this knowledge, two more cells can be filled.

Step 4.

Parent 1> Parent 2	S	s
S	SS	Ss
s	Ss	ss

With one cell for the parents' genotype left to fill, the most reasonable answer should be chosen. The ratio for tail length is about 3:1 so S is the most reasonable answer.

The same procedure is used for the coat color gene. If w is the allele for white and W for brown, one parent was Ww, the other was ww.
Putting this information back together, one parent was SsWw and the other was Ssww.

Using the Matrix for Phenotypic Ratios

The figure above displays the phenotypic ratio outcome of an F₁ dihybrid cross. Click the image above or this link to see the animation that generated the image.

A matrix can be used to quickly calculate phenotypic ratios.

Care and modifications must be observed when using the matrix on phenotypes when epistasis, described below, is involved.

A combination of characters with incomplete and complete modes of inheritance can be easily resolved as the animation below demonstrates.

The figure above displays the phenotypic ratio outcome of an F₁ dihybrid cross when one character demonstrates incomplete dominance. Click the image above or this link to see the animation that generated this image.

Another example, a trihybrid F₁ cross:

9Red Tall : 3White Tall : 3Red Short : 1White Short

3Round : 1Wrinkled

27Red Tall Round : 9White Tall Round : 9Red Short Round : 3White Short Round :

9Red Tall Wrinkled : 3White Tall Wrinkled : 3Red Short Wrinkled : 1White Short Wrinkled

Step 1: Determine the gametes for the cross, treating each gene independently.

Step 2: Cross each gene, singularly.

Step 3: Add one result at a time.

Also see Figure 3.5

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Epistasis: When Genes Affect Each Other

(This material will be covered again in Chapter 4)

Webster's New World Dictionary definition for epistasis is: "the suppression of gene expression by one or more other genes".
- This definition provides a sense of what epistasis is; however, the gene expression might not be and often is not suppressed, but the effect of the gene on the phenotype of the organism is altered or interfered with by another gene.
- Another textbook definition is: "an interaction between genes, in which the presence of a particular allele of one gene determines whether another gene will be expressed".
- This definition is closer to the actual events that usually occur in epistasis. However, there seems to be two different meanings for the term expression in genetics. According to one definition, a gene is expressed when it is used to generate an RNA molecule and then a protein. The word usage in these definitions is different. It implies that expression is when a gene reveals itself in an observable phenotypic trait.
- Often, the gene that is masked still codes for an enzyme. However, the enzyme lacks its substrate because another gene that would have produced it is defective.
- By definition, the epistatic gene masks the expression of the hypostatic gene.
In the preceding sections, genes affected just one character. An allele was associated with a trait.
- In fact there are cases where one gene can affect many traits. This is called pleotrophy.
- In the discussion of epistasis in a general genetics course including this one, the influence of one gene on the phenotypic effects of another is studied. All follow the dominant/recessive mode of inheritance. You can imagine that in nature much more complex examples of epistasis exist.
Obviously, there is a biochemical basis for epistasis. It is enjoyable to hypothesize about the nature of the biochemical basis of a case involving epistasis. This is possible once the ratio has been determined.
- The most basic genetic crosses used to study epistasis are F₁ dihybrid. The F₂ generation's phenotypic ratios deviate from the 9:3:3:1 expected from genes following the dominant recessive mode of inheritance.
- A F₂ phenotypic ratio of 12:3:1 is an example of one involving epistasis. A grouping of one or more of the ratios occurs. In this case it is the grouping of 9 and 3.
Examples of deviations from the normal phenotypic ratio indicative of epistasis:
- 9 : 7 is a variation of the normal ratio. (9 : 3 : 3 : 1)
- 9 : 3 : 4 is a variation of the normal ratio. (9 : 3 : 3 : 1)
- 15 : 1 is a variation of the normal ratio. (9 : 3 : 3 : 1)
- 9 : 6 : 1 is a variation of the normal ratio. (9 : 3 : 3 : 1)
Here is an example of a problem that generates a 9:7 phenotypic ratio in the F₂ progeny, and a biochemical hypothesis to explain how. Assume we have two different strains of mice, which have alleles for black and white coat color, but for different genes. The F₁ dihybrid population is all black. If individuals homozygous for either recessive allele are white then the outcome of a dihybrid cross would generate a ratio indicative of epistasis.
- (Alone) Gene 1: 3 black : 1 white would be expected from a monohybrid F₁ cross.
- (Alone) Gene 2: 3 black : 1 white would be expected from a monohybrid F₁ cross.
- But, an F₁ cross for the dihybrids, i.e., CcPp X CcPp would yield:

3 Black : 1 White (cc)

3 Black : 1 White (pp)

9 Black : 3 White (cc) : 3 White (pp): 1 White (ccpp)

or 9 Black :7 White

An example of how a biochemical pathway could yield these results is diagrammed below.

	cc		pp
	No or faulty enzyme	No Precursor 2 made	No or faulty enzyme	No black pigment made
Precursor 1	=>	Precursor 2	=>	Black Pigment
	^ Functional Enzyme	Precursor 2 made	^ Functional Enzyme	Black pigment made
	^ CC or Cc		^ PP or Pp

I have created a series of animations involving the painting of a car to represent a biochemical pathway for an epistasis that yields a ratio of 9:3:4. Click the image below to enter the page with this series. Use the scroll to see many of the possible genotypes and their effects.

How about a 15 : 1? Propose a molecular mechanism?

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Epistasis Problem Solving

As stated before, F₂ ratios indicative of epistasis in a genetics course are deviations from 9:3:3:1.
A 9:3:3:1 ratio can be represented by alleles. It is two 3 to 1 ratios.

Female>
Male

A a

A A_ A_

a A_ ab
A 3:1 is also a 3A_ to 1aa ratio.

A 9:3:3:1 phenotypic ratio is also a 9A_B_ to 3A_bb to 3aaB_ to 1aabb ratio.

	3A_	1aa
	3B_	1bb
9A_B_	3aaB_	3A_bb	1aabb

Many textbook genetics problems on the topic of epistasis involve dihybrid crosses to homozygotes.
The problem first gives the observed F₂ phenotypic ratio, which is a derivative of the 9:3:3:1 ratio.

The cross then continues that F₁ are crossed to homozygotes. The individual Punnett squares would look like this:

Female> Male	a	a
A	Aa	Aa
a	aa	aa

Female> Male	b	b
B	Bb	Bb
b	bb	bb

Combined:

	1Aa	1aa
	1Bb	1bb
1AaBb	1aaBb	1Aabb	1aabb

Depending on what F₂ phenotypic ratio was provided by the question, the phenotypic ratio of the test cross-like cross will deviate from that not involving epistasis.

9	A_B_
3	A_bb
3	aaB_
1	aabb

If 15:1, then 3:1 would be the correct answer to what the expected ratio would be for the above example.

9	A_B_	Green
3	A_bb	Green
3	aaB_	Green
1	aabb	Yellow

If 9:6:1, then 1:2:1 would be the correct answer.

9	A_B_	Black
3	A_bb	Grey
3	aaB_	Grey
1	aabb	White

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Recessively Inherited Human Disorders

A recessive allele that causes a disorder is usually a defective version of the normal allele.
Defective alleles that cause disorders code for either a malfunctional protein or none at all.
Heterozygotes can be phenotypically normal, if one copy of the normal allele is sufficient to code for the needed quantities of the protein.
Recessively inherited disorders range in severity from nonlethal traits to lethal diseases.
Since these disorders are caused by recessive alleles:
- Heterozygotes (Aa) can be phenotypically normal and act as carriers, making transmittion of the recessive allele to their offspring possible.
- The vast majority of people afflicted with recessive disorders are born to normal parents, both of whom are carriers.
- The probability is 1/4 that a mating of two carriers (Aa X Aa) will produce a homozygous recessive zygote.
- The probability is 2/3 that a normal child from such a mating will be a heterozygote, or a carrier.
Genetic disorders are not usually distributed evenly among all racial and cultural groups due to the different genetic histories of the world's people. Three examples of such recessively inherited disorders are cystic fibrosis, Tay-Sachs disease and sickle-cell anemia.
- Cystic fibrosis is the most common lethal genetic disease in the United States, striking 1 in every 2,500 Caucasian. It is much rarer in other races.
  - The frequency of this allele of the gene in the United States population is 2%.
  - The probability is 1/4 that a mating of two carriers (Aa X Aa) will produce a homozygous recessive zygote.
  - The probability is 2/3 that a normal child from such a mating will be a heterozygote, or a carrier.
  - The dominant allele codes for a membrane protein that pumps chloride ions out of cells. These pumps are lacking or are defective in recessive homozygotes, so chloride accumulates abnormally in the cells causing the osmotic uptake of water from the surrounding mucus. Disease symptoms result because the thickened mucus builds up in the pancreas, lungs, digestive tract, and other organs.