<-chapter_02|Chapter 02^table_of_contents|Table of Contents^chapter_04|Chapter 04->
Chapter 03: Defining %%genes%% by segregation patterns
===== Introduction =====
The classical definition of a gene is based on Mendel's Laws of Inheritance. Most genetics textbooks teach Mendelian genetics as a matter of course ("because it's how we've ALWAYS done things, duh!"), but there is an inherent danger in doing so. Mendel's view of genetics is very limiting because of what was known about genetics and biology in his day, and it is not uncommon for students that spend a lot of effort learning Mendel to become trapped into the same limiting world view.
We discuss basic Mendelian genetics here in this chapter for several reasons. First, there is some value in understanding Mendel's analysis because it helps in further understanding meiosis ([[chapter_01|Chapter 01]]), an understanding of which is essential for learning all genetics. Second, we use classical Mendelian genetics in a more experimentally practical way as a vehicle to try to wean students off bad habits in learning genetics, including Mendelian notation and Punnett squares. We will deliberately choose Drosophila for our examples instead of pea plants (as is traditional when learning Mendelian genetics) to learn and use Drosophila genetic notation (what we refer to as fractional notation in this book), which is generally preferable when discussing more advanced diploid genetics. Third, it is instructive to learn about Mendel's definition of a gene to see how ideas about genes have changed as scientists have learned more about genes.
Classical Mendelian genetics can be studied using yeast, but there are more powerful tools for analysis of yeast genetics (tetrad analysis) that will be discussed in [[chapter_13|Chap. 13]] and [[appendix_a|Appendix A]]. Mendelian genetics is more commonly used for analyzing obligate diploid organisms.
===== Obligate diploids =====
Some organisms, such as baker's yeast //Saccharomyces cerevisiae// introduced in [[chapter_02|Chap. 02]], can exist as either haploid or diploid cells. But some organisms are obligate diploids, meaning that their cells exist only as diploids (except for their gametes and other rare exceptions). Humans and other mammals, for instance, are obligate diploids. For this chapter, we will focus on the fruit fly //Drosophila melanogaster//, because it is a well characterized model genetic organism (Fig. {{ref>Fig1}}). We can do genetic experiments with Drosophila, something that is not easily or ethically done with humans. Let's consider the genetics of diploid organisms:
{{:drosophila_life_cycle.jpg?400|}}
The Drosophila life cycle. The organism exists as an obligate diploid; only the gametes (the eggs and sperm) are haploid. When a zygote forms through fertilization, the diploid state is reconstituted. Source: [[https://schoolbag.info/biology/concepts/56.html|schoolbag.info]]. Licensing: [[https://creativecommons.org/licenses/by-sa/3.0/|CC BY-SA 3.0]].
A zygote is a cell that is formed through the fertilization of gametes (sperm and egg cells). The genotype of the zygote will depend on which alleles are carried by the gametes that form it. Let's consider a gene "$A$" for which two alleles exist, $A$ and $a$. Two parents, both of which are $\frac{A}{a}$ heterozygotes, mate to form offspring, or progeny:
Possible combinations of allele combinations in a cross between two heterozygous parents. This method of analyzing genotypes from a genetic cross is called a Punnett square. Although it seems like a simple method to use, it becomes unwieldy when more than one gene is involved. Punnett squares are strongly discouraged in this course (see [[chapter_03#Fractional_notation_in_genetic_crosses|below]]), and we will avoid it as much as possible in future examples.
When heterozygotes mate, their offspring may have different phenotypes: If $A$ is dominant to $a$, the two possible phenotypes will be the phenotype of $\frac{a}{a}$ (the recessive phenotype) or the phenotype of $\frac{A}{A}$ and $\frac{A}{a}$ (the dominant phenotype). [[chapter_03#Fractional_notation_in_genetic_crosses|See below]] for notes on how we write genotypes similar to mathematical fractions.
When we do genetic crosses in breeding experiments, it is important to know the genotypes of the parents. But as you can see from the example above, progeny from the cross with the dominant trait could have either $\frac{A}{A}$ or $\frac{A}{a}$ as their genotypes. Without knowledge of what gene $A$ is, we can only make inferences as to an organism's genotype by looking at its phenotype. One way to be more certain about genotypes is to start with populations that we know to be homozygous for a particular gene. One way to do this is to keep inbreeding individuals (that is, breeding siblings or close relatives together for multiple generations) until all crosses among related individuals always produce identical or nearly identical offspring. This generates a true-breeding strain. We can assume that individuals from a true-breeding population are homozygous for most genes (or homozygous mutant if they are bred for that particular mutant phenotype). Laboratory wildtype strains are assumed to be homozygous wildtype for nearly all genes.
===== An example =====
Say we have a true breeding line of mutant Drosophila fruit flies. These flies are paralyzed compared to wildtype flies that have normal mobility. We name this mutant $shibire$ (hiragana: しびれ), which means "numb" in Japanese. We assume there is a gene (the $shibire$ gene, abbreviated $shi$) that is mutated in this strain, and we assume it has the genotype $\frac{shi^-}{shi^-}$, with the "-" superscript indicating that this is a mutant allele. The "+" sign is usually used to indicate a wildtype allele. Drosophila mutants and genes are traditionally named after mutant phenotypes, and Drosophila scientists can be very creative in naming their mutants((Some examples of creative Drosophila mutants include: $ether\text-a\text-go\text-go$ (a mutant than twitches its legs when exposed to ether), $kenny$ (a mutant that dies easily), $tinman$ (mutant that does not have a heart), $cheap \ \ date$ (a mutant that is sensitive to ethanol), $swiss \ \ cheese$ (brain is full of holes), etc. )). For now, we will use these +/- symbols as superscript. Later, we will use a more streamlined set of symbols.
We can use some of the same ideas presented in [[chapter_02|Chapter 02]] to analyze $shibire$. We can first test to see whether the mutant $shi^-$ allele is dominant or recessive by crossing true-breeding $shi^-$ flies to true-breeding wildtype ($shi^+$) flies. For simplicity, we don't consider the sex of the mating flies (for now):
$$\begin{aligned} P: \frac{shi^-}{shi^-} &\times \frac{shi^+}{shi^+}\\&\downarrow \\F1: &\frac{shi^-}{shi^+} \end{aligned}$$
Genetic cross between $shibire$ and wild type flies.
The offspring from a cross are known as the F1 (this stands for first filial generation). Geneticists often joke that their children are their F1s. In Fig. {{ref>Fig2}} , the F1 flies are heterozygous and appear like wildtype. Therefore, the $shi^–$ allele is recessive (relative to the wildtype).
===== Fractional notation in genetic crosses =====
Table {{ref>Tab1}} above shows a genetic cross and analysis of its outcomes using a Punnett square. Many beginning students like using the Punnett square because it is intuitive, and it is also usually how they learned it in high school or earlier. As you start to learn more advanced genetic concepts, it is important to learn how to write genetic crosses for diploid organisms and their outcomes using "fractional notation". An example is given in Fig. {{ref>Fig2}} above. Genotypes are written similar to mathematical fractions, with the genetic contributions of each parent written in the "numerator" and "denominator". The "$\times$" (pronounced "cross") symbol is used to indicate a cross, or a mating event. Parents are indicated using the "P" symbol (sometimes P0 is used), and subsequent generations of offspring use the symbols F1, F2, etc. At this point it may not be clear why "fractional notation" is better than using a Punnett square. You will soon see that Punnett squares work very poorly when crosses get more complex, and they also work poorly for thinking about crossing over, which we will see in [[chapter_05|Chapter 05]]. You can write genotypes in fractional notation either vertically as shown in Fig. {{ref>Fig2}}, or you can also write then horizontally (e.g., $shi^-$/$shi^-$) when typing, for instance (although you can also type vertical fractions now with most modern word processing/typesetting software such as Microsoft Word or [[https://www.latex-project.org/|$\LaTeX$]]). When writing horizontally, you can include parentheses or brackets to help reduce ambiguity: for example, ($shi^-$)/($shi^-$) or ($shi^-$/$shi^-$). In general, the vertical method is better and usually preferred.
===== Complementation testing in obligate diploids =====
We can also use complementation testing ([[chapter_02|Chapter 02]]) to analyze other Drosophila mutants that have similar phenotypes to ask if those mutants are allelic with other mutants with similar phenotypes. Say we have isolated a different paralyzed Drosophila mutant that we temporarily call $par$. We start with a true breeding $par^-$ strain (i.e., we can assume that its genotype is $\frac{par^-}{par^-}$) that we mate to wildtype. We find that the $par$ mutation is not expressed in the F1 heterozygotes (i.e., the F1 heterozygotes are not paralyzed) and therefore is recessive.
Since both the $par^-$ and $shi^-$ mutations are recessive, we can do a complementation test (Figure {{ref>Fig3}}). For this test, we cross a true breeding $par^–$ strain to a true breeding $shi^–$ strain. Just like we saw in [[chapter_02|Chapter 02]], the phenotype of the F1 progeny tells us if $shi$ and $par$ are allelic to each other. The possible outcomes and interpretations are shown in Table {{ref>Tab2}}.
$$\begin{aligned} \frac{par^-}{par^-} &\times \frac{shi^-}{shi^-}\\&\downarrow \end{aligned}$$
progeny must inherit both $shi^-$ and $par^-$ alleles
A complementation test for //Drosophila// $par$ and $shi$ mutants.
^ possible F1 phenotype ^ complementation? ^ explanation ^ inferred genotype ^
| not paralyzed | $shi^–$ and $par^–$ complement | $par^–$ genotype can supply function missing in $shi^–$ and vice versa | $\frac{par^-}{par^+} \cdot \frac{shi^+}{shi^-}$ |
| paralyzed | $shi^–$ and $par^–$ do not complement | $par^–$ has lost function needed to restore $shi^–$; the $par^-$ mutant contains a mutant allele of $shi$ | $\frac{shi^-}{shi^-}$ |
Possible outcomes and inferred genotypes from Fig. {{ref>Fig3}}. The "dot" separating the two “fractions” is an informal symbol to separate two different gene symbols where their linkage is unknown. For now, it's just a temporary symbol used to separate two different gene symbols. We will learn about linkage in [[chapter_05|Chapter 5]].
If $par^-$ and $shi^-$ complement, this means we can think of the parents in Figure {{ref>Fig3}} as having the genotypes $\frac{par^-}{par^-} \cdot \frac{shi^+}{shi^+}$ and $\frac{par^+}{par^+} \cdot \frac{shi^-}{shi^-}$. On the other hand, if $par^-$ and $shi^-$ do not complement, this implies that $par^-$ and $shi^-$ are mutant in the same gene (i.e., $par = shi$). Since $shibire$ has already been named and $par$ is just a temporary name, we preferably write the outcome of the cross as $\frac{shi^-}{shi^-}$ instead of $\frac{par^-}{shi^-}$.
===== Mendel's First Law of Segregation (monohybrid cross) =====
In the above discussion, we are assuming that the true-breeding $shibire$ line contains a mutation in a single gene (the $shibire$ gene). However, it is entirely possible that there are mutations in multiple genes, all of which (either singly or in combination) cause the paralyzed phenotype we observe. For instance, it is possible that two genes, $shi1$ and $shi2$, are mutant in our true-breeding $shibire$ line and it is the combinatorial effect of two mutations that is causing paralysis. In this case we can re-write the cross in Fig. {{ref>Fig2}} as:
$$ \begin{aligned} P: \frac{shi1^-}{shi1^-} \cdot \frac{shi2^-}{shi2^-} &\times \frac{shi1^+}{shi1^+} \cdot \frac{shi2^+}{shi2^+} \\ &\downarrow \\ F1: \frac{shi1^-}{shi1^+} &\cdot \frac{shi2^-}{shi2^+} \end{aligned} $$
A cross between a true-breeding paralyzed fly and wild type, except that there are two paralysis-causing recessive mutations in the mutant instead of one (compare to Figure {{ref>Fig2}}).
Provided that the $shi1$ and $shi2$ mutations are recessive, the F1 offspring in Figure {{ref>Fig4}} will have the same non-paralyzed phenotype as the F1 offspring in Figure {{ref>Fig2}}. In other words, crossing a paralyzed mutant to wildtype (the experiments being done in Figures {{ref>Fig2}} and {{ref>Fig4}}) tells us that the mutation(s) is recessive, but does not (and cannot) tell us how many different mutations are contributing to the phenotype. Similarly, the complementation test (Figure {{ref>Fig3}}) tells us that "$par$" and $shi$ are different, but it does not (and cannot) tell us whether the paralysis in a $shibire$ mutant is the result of one mutation or more.
To answer the question of whether the paralysis phenotype of a $shibire$ mutant is caused by a single mutation or not, we need to do a different experiment involving the F2 generation. From Figure {{ref>Fig2}} above, we can take the heterozygote F1s and perform a sibling cross (or sib cross). Look more carefully at how alleles will segregate in a cross between heterozygotes:
$$ F1: \frac{shi^+}{shi^-} \times \frac{shi^+}{shi^-} $$
A sibling cross where male and female F1 flies from the Figure {{ref>Fig2}} are crossed to each other.
What is the probability of a paralyzed fly in the next (F2) generation? The probability is calculated as:
$$ p(a)=\frac{n_a}N $$
where $n_a$ = number of outcomes that satisfy condition $a$, and $N$ = total number of possible outcomes (of equal probability). Probability problems can be solved by accounting for every outcome, but usually it is easier to combine probabilities.
$p$(paralyzed F2 fly) = $p$(inherit $shi^-$ from mother and inherit $shi^-$ from father)
We have not yet discussed linkage (Chapters [[chapter_04|4]] and [[chapter_05|5]]), but Mendel's original findings showed that each parent contributes one of two alleles from each gene to the zygote. For our purposes here we can use the product rule, which lets us calculate the probability $p$ of events $a$ and $b$ both happening:
$p$($a$ and $b$) = $p(a) \times p(b)$
Note the product rule only applies if events $a$ and $b$ are independent of each other, which is the case here since the allele from mother does not affect the allele from the father.
$p$(inherit $shi^–$ from mother) = $\frac{1}{2}$
$p$(inherit $shi^–$ from father) = $\frac{1}{2}$
$p$(paralyzed) = $p$(inherit from both mother and father) = $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$
There are only two possible outcomes from our cross; either the progeny are paralyzed, or they are not paralyzed (wildtype). Since all possible probabilities must add up to 1, we can easily calculate the probability of obtaining progeny that are not paralyzed as follows:
Each individual Drosophila mating event can produce several hundred offspring. With large numbers of progeny, the ratio of phenotypes will mirror the probabilities of the outcomes. Thus, in the F2 generation the phenotypic ratio for not paralyzed: paralyzed = $\frac{3}{4}$: $\frac{1}{4}$ = 3:1. The genotypes are:
A 3:1 phenotypic ratio among the F2 progeny in a breeding experiment shows that alleles of a single gene are segregating. Any result other than 3:1 tells us that something other than a single gene is segregating.
This actually constitutes our second definition of a gene: genes are units of inheritance that follow Mendel's Laws. A phenotype is determined by a single gene if it displays a 3:1 dominant to recessive ratio in a monohybrid cross. Historically, this was the first and original definition of a gene developed by Gregor Mendel in the 1860s. Mendel was able to detect gene segregation of single genes in pea plants because he looked at simple traits and started with true breeding strains. That genes with two alleles segregate in this way is often described as Mendel's First Law.
===== Mendel's Second Law of Independent Assortment (dihybrid cross) =====
Now let’s look at the properties of gene segregation when two different genes are involved. Here we introduce a new Drosophila mutant called $vestigial$ ($vg$ for short). $vg$ mutant flies have misshapen wings compared to wildtype (Figure {{ref>Fig6}}); the vg mutant allele we are talking about is also recessive.
{{ :vestigial_dorosphila.png?400 |}}
Here, we introduce a new way to write genotypes. Instead of writing $shi^+$ or $vg^+$ for wildtype alleles of $shibire$ or $vestigial$, we simply use the symbol "+" for all wildtype alleles. In fact, this symbol is so useful that sometimes we just call wildtype flies "+" (when we say it out loud, we just say "plus"). When necessary, we can always use the superscript + to emphasize wildtype, but geneticists are a lazy bunch, and we hate writing unnecessary symbols. Another new guideline is we use the gene symbol to indicate a mutant allele. So instead of writing $shi^-$ and $vg^-,$ we can simply write $shi$ and $vg$ without the "-" superscript. As with "+", we can always include "-" when we need to emphasize mutant alleles.
We cross true breeding $shi$ and $vg$ mutants together to see what happens:
$$ P: \frac{shi}{shi} \cdot \frac{+}{+} \times \frac{+}{+} \cdot \frac{vg}{vg}$$
Crossing two mutants with different phenotypes: $shi$ and $vg$. The left parent is paralyzed with normal wings; the right parent is not paralyzed but has shriveled wings.
As before, we ignore the sex of the parents for now. A few notes about how we wrote the cross above:
* The $\frac{+}{+}$ in the genotype of the first parent indicates that it is homozygous for $vg^+$.
* The $\frac{+}{+}$ in the genotype of the second parent indicates that it is homozygous for $shi^+$.
* Since Drosophila are obligate diploids, $shi$ implies $\frac{shi}{shi}$. A really lazy geneticist will just write: $shi \times vg$. This means exactly the same thing as what is written in Figure {{ref>Fig7}}, but we will try to write as clearly as we can in this book without being too lazy.
Since both the parents are true breeding, the gametes from the first parent will all be ($shi$ $\cdot$ +) and from the second parent (+ $\cdot$ $vg$). These gametes will then give an F1 generation whose genotype are all $\frac{shi}{+} \cdot \frac{+}{vg}$. It's also OK to write this as $\frac{shi}{+} \cdot \frac{vg}{+}$, although it's customary to write the gamete contributions from each parent on the "numerator" and "denominator" in a consistent way.
Mendel discovered that alleles of different genes segregate independently of each other (he did not know about the exception, which is when genes are linked; see [[chapter_05|Chapter 05]]). If $shi$ and $vg$ segregate independently of each other the same way as Mendel observed, we can calculate the probabilities of all possible F2 phenotypes resulting between a sib cross between F1 individuals. Mendel called this kind of cross a dihybrid cross.
$$F1: \frac{shi}{+} \cdot \frac{+}{vg} \times \frac{shi}{+} \cdot \frac{+}{vg}$$
A sib cross from Figure {{ref>Fig7}}. The entire set of crosses from Figures {{ref>Fig7}}-{{ref>Fig8}} and Table {{ref>Tab3}} from P to F2 is called a dihybrid cross.
The possible F2 outcomes are given in Table 3.3 and can be calculated also using the product rule, since our assumption is that $shi$ and $vg$ segregate independently of each other:
A Mendelian dihybrid cross, Drosophila style. The probability of $shi$ and $vg$ phenotypes on their own are based on Mendel's First Law [[chapter_03#Mendel's_First_Law_Of_Segregation|discussed above]].
The ratio of the four different possible F2 phenotypes shown in Table {{ref>Tab3}} is $\frac{9}{16}:\frac{3}{16}:\frac{3}{16}:\frac{1}{16}$, or 9:3:3:1. Beginning geneticists will often solve the problem above by figuring out all the genotypes first using a 4x4 Punnett square (which also assumes independent assortment). You are strongly discouraged from continuing to use Punnett squares (although since there are 16 different possible genotypes, a 4x4 Punnett square can be helpful in figuring those out).
===== The test cross =====
Calculating the probabilities (and determining all 16 possible genotypes) for the outcomes of a dihybrid cross is not difficult, but it is complex. A more convenient way to look at segregation of two genes is by a test cross of the F1 heterozygote to a homozygous recessive tester individual (assuming such a thing exists in your lab):
A test cross. Note that "test cross" is a special term; you should not use the phrase "test cross" to describe any generic cross you might be conducting to test a hypothesis.
The four possible gamete genotypes from the F1 progeny will be ($shi \cdot +$), ($+ \cdot vg$), ($shi \cdot vg$), and
($+ \cdot +$), and the probability of obtaining any one of those is $\frac{1}{4}$. However, the homozygous recessive tester can only produce ($shi \cdot vg$) gametes. In other words, the probability $p$ of the tester producing a ($shi \cdot vg$) gamete is $p=1$. Thus, the possible F2 outcomes for this cross are:
A test cross, Drosophila style. The term parental means that the F2 phenotypes resemble those of the parents in Cross 3.4, whereas recombinant means that it is different than those parents. Other synonyms for recombinant include non-parental and crossover class (see [[chapter_05|Chapter 05]]).
Note that the frequency of the different F2 phenotypes from the test cross is exactly the same as the frequency of the different gametes that can form from the F1 heterozygote. This makes interpreting test crosses easier than a dihybrid cross, especially when doing a real-life experiment. In this case we can easily observe that each F2 progeny receives either the $shi$ or + allele and receives either the $vg$ or + allele. The test cross clearly shows gene segregation for each gene. Also note that in a test cross, you will get two kinds of outcomes: parental types, which resemble the original parents; and recombinant types, which do not resemble the parents (Figure {{ref>Fig7}}). This becomes very convenient when we look at mapping in [[chapter_05|Chapter 05]].
No matter if we do a dihybrid cross between F1 heterozygotes (and get a 9:3:3:1 ratio) or if we do a test cross with F1 heterozygotes (and get a 1:1:1:1 ratio), either outcome shows that the two genes $shi$ and $vg$ segregate independently of each other. That is, the segregation of alleles for one gene does not affect that of another gene. Mendel called this independent assortment, and this phenomenon is often referred to as Mendel's Second Law.
===== Perspectives on Mendel's Laws =====
For now, it seems like regardless of whether you look at it from a 9:3:3:1 "dihybrid cross" perspective or a 1:1:1:1 "test cross" perspective, Mendel's Second Law is not as useful as Mendel's First Law. The First Law gives you information about whether a mutant phenotype is caused by mutation in a single gene – this seems to be useful. The Second Law appears to describe a phenomenon (alleles for different genes segregate independently of each other) but doesn't seem to give useful information about the genes themselves. In Chapters [[chapter_04|04]] and [[chapter_05|05]], you will see that the ideas underlying the Second Law set the stage for understanding linkage, which provide a new definition of genes based on position.
Historically, Mendel's First and Second Laws also set the stage for chromosome theory, which we discuss in [[chapter_04|Chapter 04]]. Chromosomes behave in meiosis the same way that Mendel showed genes to behave. This is now a good time to review [[chapter_01|Chapter 01]]. Each gamete receives only one of the two homologous chromosomes from its mother cell, a behavior that is analogous to segregation of alleles of a single gene (Mendel's First Law). Furthermore, the relative orientation of different homologous chromosome pairs (tetrads) at the first meiotic cell division is random, which is analogous to independent assortment of two different genes (Mendel's Second Law). To scientists in the early 20th century when chromosomes were just recently discovered and Mendel was just being re-discovered, this correlation strongly suggested that genes are physically located on chromosomes. In [[chapter_04|Chapter 04]], we will see the experimental evidence that supported this idea.
You might also be wondering at this point: what if two genes happen to be on the same chromosome? We address this later in Chapters [[chapter_04|04]] and [[chapter_05|05]]. Mendel got lucky - the genes he chose to study were all unlinked to each other. If he had chosen genes that were linked to each other (closely positioned on the same chromosome) he may not have been able to draw the same conclusions that he did regarding his Second Law.
===== Application of Mendel's Laws to a modern problem =====
Let's step outside the box a little and see how Mendel's Laws can be applied to a very interesting problem in the evolution of domestic corn (maize). Domestic corn is derived from its wild progenitor, a plant called teosinte (Figure {{ref>Fig10}}). Native Americans in what is now Mexico likely started to domesticate teosinte around 6000 years ago.
{{ :teosinte_and_modern_corn_comparison_3745571067_.jpg?400 |}}
Figure 3.3. Teosinte (left) vs. maize, or modern corn (right). Source: [[https://www.nsf.gov|National Science Foundation]]. Credit: Nicole Rager Fuller. Licensing: Public domain.
There is no historical record of how the breeding was done to produce maize, but there is a genetic record of the differences between teosinte and maize recorded in the genomic differences between these two species. Maize and teosinte can be crossed to give viable progeny, which can then be self-crossed. Because plants are hermaphroditic, a self-cross is basically the same thing as a sib cross:
P: teosinte $\times$ maize
$\downarrow$
F1: all the same and unlike either parent
$\downarrow$
F2: 50,000 progeny
Teosinte and maize breeding experiment.
Of the 50,000 F2 plants that are produced, around 100 (or 1 in 500) look like teosinte and around 100 (also 1 in 500) look like maize. The remaining F2 plants look like neither maize nor teosinte. How many genes contribute to the differences between the two kinds of plants? Let’s designate the genes that differ as $A, B, C, D$ ... etc.
For each gene let's imagine there are two alleles: the allele present in teosinte and the allele present in maize. For the $A$ gene we will designate these alleles $A_T$ and $A_M$ respectively. For the $B$ gene there will be alleles $B_T$ and $B_M$, and so on, for all the genes that are different.
Let’s follow the $A$ gene through the cross between maize and teosinte:
$$\begin{aligned}P: \frac{A_T}{A_T} &\times \frac{A_M}{A_M} \\ &\downarrow \\ F1: &\frac{A_T}{A_M} \end{aligned}$$
Codominant F1, which resembles neither parent.
Because the F1 don't look like either parent, let's assume that the alleles are codominant. That is to say, the phenotype of heterozygotes is different than either homozygote. This is not the only possible explanation, but for now let's just keep it simple.
With regard to the phenotype conferred by the $A$ gene, $\frac{1}{4}$ of the F2 progeny will look like maize, $\frac{1}{4}$ will look like teosinte, and $\frac{2}{4}$ will look like neither (see Exercise 2 at the end of the chapter). By a similar logic, with regard to the phenotype conferred by the $B$ gene, $\frac{1}{4}$ of the F2 progeny will look like maize, $\frac{1}{4}$ will look like teosinte, and $\frac{2}{4}$ will look like neither. Using the product rule, we can calculate what proportion of F2 progeny will phenotypically be maize-like with regards to two genes, i.e., the A and B genes:
$p(\frac{A_M}{A_M}$ and $\frac{B_M}{B_M}) = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$
Similarly, for three genes the probability will be $\frac{1}{64}$ ($\frac{1}{4} \times \frac{1}{4} \times \frac{1}{4}$ or $4^{-3}$). For four genes it will be $\frac{1}{256}$ (or $4^{-4}$), and for five genes it will be $\frac{1}{1024}$ (or $4^{-5}$). Since around $\frac{1}{500}$ of the F2 progeny look like maize, the conclusion is that approximately 4 to 5 genes differ between wild corn (teosinte) and domestic corn (maize). Using modern molecular genetics, it has been confirmed that there are about five genes with significantly different alleles between maize and teosinte. Several of these genes have been located using mapping methods.
===== Closing thoughts =====
If you revisit [[chapter_01|Chapter 01]] at this time, you will see that both Mendel's First and Second Laws relate directly to meiosis. The patterns of allele segregation as described by Mendel match nearly perfectly with the patterns of chromosome segregation in meiosis. It is very important for students of genetics to know that Mendel's view of genetics helps us understand the relationship between segregation patterns of genes and meiosis. However, it is equally important for students to realize that Mendel made these discoveries in the 1860s, and in many ways Mendel's Laws are very outdated both in terms of their conclusions and how they are described. In fact, continuing to call his discoveries "Laws" is a misnomer reflecting a great deal of European cultural bias; his view of heredity is much too simplistic, and there are many exceptions to Mendel's Laws as he described them, such as linkage, codominance, epistasis, epigenetic inheritance, allelic series, etc., most of which we don't cover in detail in this book (many of these concepts make more sense once you stop thinking like Mendel and start thinking about genes from a molecular perspective anyway). Once you reach [[chapter_06|Chapter 06]], it's important to relate everything you learn about genetics to the physical definition of a gene and chromosomes, rather than view Mendel as dogma. View Mendel as a beginner’s learning tool instead.
===== Questions and exercises =====
Exercise 1: See Figure {{ref>Fig3}} above. What would the F2 outcomes be from an F1 dihybrid cross if $shi$ and $par$ were allelic? What if they were not allelic? Remember that the $shi$ and $par$ mutants have identical phenotypes!
Exercise 2: This question relates to the [[chapter_03#application_of_mendel's_laws_to_a_modern_problem|teosinte experiment]]. In the F2 generation arising from self-crossing the F1, the ratio for the $A_M$ and $A_T$ alleles segregating is $\frac{A_M}{A_M}∶\frac{A_M}{A_T}∶\frac{A_T}{A_T} = 1:2:1$. Using what you have learned about how a single gene should segregate in a dihybrid cross, write out the cross in fractional notation and show how this ratio is derived using probability calculations.