# 2.4 A Dihybrid Cross Showing Mendel’s Second Law (Independent Assortment)

Mendel found that each locus had two alleles that segregated themselves during the creation of gametes. He wondered whether dealing with multiple traits at a time would affect this segregation, so he created a **dihybrid cross**. The distribution of offspring from his experiments led him to formulate **Mendel’s Second Law**, the **Law of Independent Assortment**, which states that the segregation of alleles at one locus will not influence the segregation of alleles at another locus during gamete formation — the alleles segregate independently. Next, we will discuss how he came to this understanding, given that independent assortment occurs.

# Mendel’s Second Law

To analyze the simultaneous segregation of two traits at the same time in the same individual, he crossed a pure-breeding line of green, wrinkled peas with a pure-breeding line of yellow, round peas. This produced F_{1} progeny that had all yellow and round peas. They were called **dihybrids** because they carried two alleles at each of the two loci (**Figure 2.4.1**).

From **Figure 2.4.1,** we know that yellow and round are dominant, and green and wrinkled are recessive. If the inheritance of seed colour was truly independent of seed shape, then when the F_{1} dihybrids were crossed to each other, a 3:1 ratio of one trait should be observed within each phenotypic class of the other trait (**Figure 2.4.1**). Using the product law, we would therefore predict that if ¾ of the progeny were yellow, and ¾ of the progeny were round, then [latex]\frac{3}{4}\times\frac{3}{4}=\frac{9}{16}[/latex] of the progeny would be both round and yellow (**Table 2.4.1**).

Likewise, [latex]\frac{3}{4}\times\frac{1}{4}=\frac{3}{16}[/latex] of the progeny would be both round and green. And [latex]\frac{3}{4}\times\frac{1}{4}=\frac{3}{16}[/latex] of the progeny would be both wrinkled and yellow. And [latex]\frac{1}{4}\times\frac{1}{4}=\frac{1}{16}[/latex] of the progeny would be both wrinkled and green. So by applying the product rule to all of these combinations of phenotypes, we can predict that if the two loci assort independently in a **9:3:3:1** phenotypic ratio among the progeny of this dihybrid cross, if certain conditions are met (see section below). Indeed, 9:3:3:1 is very close to the ratio Mendel observed in his studies of dihybrid crosses, leading him to formulate his Second Law, the **Law of Independent Assortment**.

**Table 2.4.1 ***Phenotypic Classes Expected in Monohybrid and Dihybrid Crosses for Two Seed Traits in Peas*

**Frequency of Phenotypic Crosses Within Separate Monohybrid Crosses**

- Seed shape: [latex]\frac{3}{4}\text{ round}[/latex], [latex]\frac{1}{4}\text{ wrinkled}[/latex]
- Seed colour: [latex]\frac{3}{4}\text{ yellow}[/latex], [latex]\frac{1}{4}\text{ green}[/latex]

**Frequency of Phenotypic Crosses Within a Dihybrid Cross**

[latex]\begin{array}{lllll}\frac{3}{4}\text{ round}&\times&\frac{3}{4}\text{ yellow}&=&\frac{9}{16}\text{ round & yellow}\\\frac{3}{4}\text{ round}&\times&\frac{1}{4}\text{ green}&=&\frac{3}{16}\text{ round & green}\\\frac{1}{4}\text{ wrinkled}&\times&\frac{3}{4}\text{ yellow}&=&\frac{3}{16}\text{ wrinkled & yellow}\\\frac{1}{4}\text{ wrinkled}&\times&\frac{1}{4}\text{ green}&=&\frac{1}{16}\text{ wrinkled & green}\end{array}[/latex]

The 9:3:3:1 phenotypic ratio that we calculated using the product rule could also be obtained using Punnett Square (**Figure 2.4.2**). First, we list the genotypes of the possible gametes along each axis of the Punnett Square. In a diploid with two heterozygous genes of interest, there are up to four combinations of alleles in the gametes of each parent. The gametes from the respective rows and column are then combined in the each cell of the array. When working with two loci, genotypes are written with the symbols for both alleles of one locus, followed by both alleles of the next locus (e.g., *AaBb*, not *ABab*). Note that the order in which the loci are written does not imply anything about the actual position of the loci on the chromosomes.

To calculate the expected phenotypic ratios, we assign a phenotype to each of the 16 genotypes in the Punnett Square, based on our knowledge of the alleles and their dominance relationships.

In the case of Mendel’s seeds, any genotype with at least one *R* allele and one *Y* allele will be round and yellow. We can represent all of four of the different genotypes shown in these cells with the notation (*R_Y*_), where the blank line (__), means “any allele”. The three genotypic classes that have at least one R allele and are homozygous recessive for *y* (i.e., *R_yy*) will have a round, green phenotype. Conversely, the three classes that are homozygous recessive *r*, but have at least one *Y* allele (*rrY_*) will have wrinkled, yellow seeds. Finally, the rarest phenotypic class of wrinkled, green seeds is produced by the doubly homozygous recessive genotype, *rryy*, which is expected to occur in only one of the sixteen possible offspring represented in the square.

Take a look at the following video, *Dihybrid Cross Explained*, by Nicole Lantz (2020) on YouTube, on some worked examples of Dihybrid crosses.

# Assumptions of the 9:3:3:1 Ratio

Both the product rule and the Punnett Square approaches showed that a 9:3:3:1 phenotypic ratio is expected among the progeny of a dihybrid cross such as Mendel’s *RrYy* × *RrYy*. In making these calculations, we assumed that:

- Alleles at each locus segregate independently of the alleles at the other;
- One allele at each locus is completely dominant (the other recessive); and
- Each of four possible phenotypes can be distinguished unambiguously, with no interactions between the two genes that would interfere with determining the genotype correctly.

For simplicity, most student examples involve easily scored phenotypes, such as pigmentation or other changes in visible structures. However, keep in mind that the analysis of segregation ratios of any two marker loci can provide insight into their relative positions on chromosomes.

# Deviations from the 9:3:3:1 Phenotypic Ratio

There can be deviations from the 9:3:3:1 phenotypic ratio. These situations may indicate that one or more of the above conditions has not been met. Modified ratios in the progeny of a dihybrid cross can, therefore, reveal useful information about the genes involved. One such example is linkage.

**Linkage** is one of the most important reasons for distortion of the ratios expected from independent assortment. Two loci show linkage if they are located close together on the same chromosome. This close proximity alters the frequency of allele combinations in the gametes. We will return to the concept of linkage later on. Deviations from 9:3:3:1 ratios can also be due to interactions between genes, such as epistasis, duplicate gene action and complementary gene action.

**Media Attributions**

Figure 12 03 02 by Rye et al. (2016), CNX OpenStax, CC BY 4.0, via Wikimedia Commons**Figure 2.4.1****Figure 2.4.2**Independent Assortment & Segregation by Giac83 (derivative) from original work by Mariana Ruiz Villarreal (LadyofHats), public domain, via Wikimedia Commons

**References**

Rye, C., Wise, R., Jurukovski, V.,DeSaix, J., Choi, J., & Avissar, Y. (2016, October 21). Figure 12.5 A test cross can be performed to… [digital image]. CNX *OpenStax Biology *(Chapter 12). https://openstax.org/books/biology/pages/12-2-characteristics-and-traits

Giac83. (2009, February 14). Independent assortment & segregation [digital image]. Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Independent_assortment_&_segregation-it.svg (original by Ladyofhats)

Nicole Lantz. (2020, April 17). Dihybrid cross explained [Video file]. YouTube. https://www.youtube.com/watch?v=fe5kSSs83qc

**Long Description**

**Figure 2.4.1**A Punnett square represents a dihybrid cross, using green and yellow seeds which are either round or wrinkled to demonstrate. The parental genotypes are pure bred lines, YYRR and yyrr, which are yellow round seeds and green wrinkled seeds respectively. The F1 generation is all heterozygous for both traits. The F2 generation is shown, with the production of sixteen offspring, being produced in the typical dihybrid ratio of 9:3:3:1. [Back to Figure 2.4.1]**Figure 2.4.2**Details of segregation and independent assortment are illustrated by green and yellow pods, which are either smooth or constricted to represent the variety of phenotypes appearing in the F2 generation of a dihybrid cross. These occur in the typical 9:3:3:1 phenotypic ratio. [Back to Figure 2.4.2]