10.1 Mendel's Laws and Probability

Mendel's laws form the foundation of genetics. By studying pea plants, Gregor Mendel discovered that traits are passed down through discrete units we now call genes. His principles of segregation and independent assortment explain why offspring look the way they do, and they give us tools to predict the probability of specific traits appearing in the next generation.

Mendel's Laws

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Gregor Mendel's Experiments

Gregor Mendel was an Austrian monk who conducted breeding experiments on pea plants in the 1850s and 1860s. He chose pea plants because they reproduce quickly, can self-fertilize or be cross-pollinated by hand, and have clearly distinct trait variations with no blending.

Mendel tracked seven traits across multiple generations, including:

• Flower color (purple vs. white)
• Seed shape (round vs. wrinkled)
• Seed color (yellow vs. green)
• Plant height (tall vs. short)

Each trait had exactly two contrasting forms, which made patterns easier to spot. His careful record-keeping and large sample sizes (he grew roughly 29,000 pea plants) allowed him to identify mathematical ratios in inheritance, something no one had done before.

Law of Segregation

Every organism carries two alleles for each gene, one inherited from each parent. Mendel's Law of Segregation says that during gamete formation (meiosis), those two alleles separate so that each gamete carries only one allele for each trait.

Here's how it works:

1. A parent cell has two alleles for a trait (e.g., $Pp$ for flower color).
2. During meiosis, the chromosome pairs separate.
3. Each resulting gamete gets just one allele (either $P$ or $p$, not both).
4. At fertilization, two gametes combine, restoring the pair (e.g., the offspring might be $Pp$, $PP$, or $pp$).

This is why offspring aren't just clones of one parent. They receive a random allele from each parent, creating new allele combinations every generation.

Law of Independent Assortment

Mendel's Law of Independent Assortment states that alleles for different traits are sorted into gametes independently of one another. In other words, inheriting one allele for seed shape has no effect on which allele you inherit for flower color.

This law holds true when genes are located on different chromosomes (or far apart on the same chromosome). Genes that are close together on the same chromosome can be inherited together, a concept called linkage, but that's beyond Mendel's original model.

Two types of crosses demonstrate these principles:

• Monohybrid cross: Tracks one trait. For example, crossing a purple-flowered plant ($Pp$) with a white-flowered plant ($pp$). This tests the Law of Segregation.
• Dihybrid cross: Tracks two traits simultaneously. For example, crossing a round, yellow pea plant ($RrYy$) with a wrinkled, green pea plant ($rryy$). This tests independent assortment because you can see whether the two traits sort together or separately.

Mendel's dihybrid crosses produced a characteristic 9:3:3:1 phenotypic ratio in the $F_2$ generation, confirming that the traits assorted independently.

Alleles and Genotypes

Dominant and Recessive Alleles

Alleles are different versions of the same gene. For any given trait, you carry two alleles, and those alleles can interact in a dominant/recessive relationship.

• A dominant allele is expressed in the phenotype whenever it's present. You only need one copy. It's represented by a capital letter (e.g., $P$).
• A recessive allele is expressed only when two copies are present (homozygous recessive). It's represented by a lowercase letter (e.g., $p$).

In Mendel's pea plants, purple flower color ($P$) is dominant over white ($p$), and round seed shape ($R$) is dominant over wrinkled ($r$). A plant with genotype $Pp$ looks purple because the dominant allele masks the recessive one.

Genotypes and Zygosity

Your genotype is the specific pair of alleles you carry for a trait. There are three possible configurations:

• Homozygous dominant ($PP$): Two dominant alleles. Phenotype = dominant trait.
• Heterozygous ($Pp$): One dominant, one recessive. Phenotype = dominant trait (the dominant allele masks the recessive).
• Homozygous recessive ($pp$): Two recessive alleles. Phenotype = recessive trait.

Notice that $PP$ and $Pp$ produce the same phenotype. This is why you can't always determine genotype just by looking at an organism. A purple flower could be $PP$ or $Pp$. To figure out which, you'd need a test cross (crossing the organism with a homozygous recessive individual and observing the offspring ratios).

Phenotypes and Punnett Squares

Phenotypes and Genetic Expression

A phenotype is the observable characteristic of an organism, such as purple vs. white flowers or round vs. wrinkled seeds. Your phenotype results from your genotype, though environmental factors can also play a role.

The key relationship: genotype determines phenotype, but the same phenotype can come from different genotypes. Two purple-flowered plants might have different genotypes ($PP$ vs. $Pp$), yet they look identical.

Punnett Squares and Probability

A Punnett square is a grid that helps you predict the probability of offspring genotypes and phenotypes. Here's how to set one up:

1. Determine each parent's genotype (e.g., both are $Pp$).
2. List one parent's possible gametes across the top of the grid.
3. List the other parent's possible gametes down the left side.
4. Fill in each cell by combining the allele from the top with the allele from the side.
5. Count the resulting genotypes and convert to ratios or percentages.

For a cross between two heterozygous parents ($Pp \times Pp$):

This gives a genotypic ratio of 1 $PP$ : 2 $Pp$ : 1 $pp$, and a phenotypic ratio of 3:1 (three dominant phenotype to one recessive phenotype).

Each cell in the Punnett square represents an equally likely outcome, so the probability of any single genotype is the number of cells with that genotype divided by the total number of cells. In this example, the probability of a homozygous recessive offspring ($pp$) is $\frac{1}{4}$, or 25%.

For dihybrid crosses, the same logic applies but with a 4×4 grid (16 cells), since each parent produces four types of gametes. The expected phenotypic ratio from a $RrYy \times RrYy$ cross is 9:3:3:1.