Polygenic Inheritance: How Our Traits Can Vary Across a Continuum

Idea creds @leucine thanks for asking me to explain something genetics and mathy it reminded me of this lol.

Polygenic inheritance, also known as quantitative inheritance, is the genetic phenomenon that creates a continuum of phenotypes. In other words, polygenic inheritance is the reason there is a gradient of skin colors or heights or hair colors, instead of a set number of possible phenotypes.

Polygenic inheritance is unique in that it involves multiple genes (poly=multiple, gen=genes, ic=involved). Normally, we deal with monogenic inheritance, which just involves the alleles of a single gene. It gives you the typical monohybrid cross:

You might now be thinking of a dihybrid cross as a possible example of polygenic inheritance:

BUT NO! Notice in the dihybrid cross above that each gene controls for a different trait. In polygenic inheritance, multiple genes control expression of the same trait.

You might now be thinking of multiple alleles, which is the thing that occurs with blood types where there are more than two options (more than A and a, for example) to fill in the two allele positions for the genotype (like IA, IB, or i, which are three options instead of two).

BUT NO again! In polygenic inheritance, there are multiple genes at play, not multiple alleles. Each gene still only has two options to fill it (A or a, for example), but there are multiple genes controlling the same trait.

Look at this picture of polygenic inheritance, where the A, B, and C genes are controlling the color trait. Notice that dominant alleles confer darker color, while recessive ones don't add anything. Each gene has the same degree of influence on the final phenotype.

TIME FOR MATH! Let's arrange the color result ratios in a bar graph:

What's the significance of the ratio, 1:6:15:21:15:6:1? It's a row on Pascal's triangle!

The color example involved three genes, and a three-gene system corresponds to this ratio. But using some math, we can find the ratios corresponding to any number of genes involved in a system.

The 2n+1 rule relates the number of genes in a polygenic system to the number of possible phenotypes: it states that, assuming "n" is the number of genes involved, "2n+1" describes how many phenotypes can result from this polygenic system. For example, in our three-gene example above (n=3), there were seven possible colors (2x3+1=7). Seven is also the number of terms in our Pascal ratio (and this is obvious, if you think about what those numbers are describing).

To solidify these concepts, let's try a problem together:

Here’s a visual representation of the problem:

We need to:

• match the scenario given to a row on Pascal's triangle,

• try to determine how many cm each dominant allele confers,

• convert this into a linear model,

• and use this model to determine what fraction of F2 is 28cm.

Step 1 - match the scenario given to a row on Pascal's triangle.

We're given ratios of the extreme phenotypes (4/1000). In a Pascal ratio, these extremes would correspond to the "1"s on the first and last terms. Simplifying the fraction and approximating yields about 1/256 as the extremes' fraction of the total.

Now we know the total is 256. The row on Pascal's triangle whose terms add up to 256 has this ratio: 1:8:28:56:70:56:28:8:1.

Step 2 - try to determine how many cm each dominant allele confers.

To find how many cm each dominant allele confers, we need to examine the row of Pascal we identified. It has nine terms, which means there are nine phenotypes total, including 12 and 40. Additionally, it was given that 24 is the average. (The math doesn't work perfectly, but approximations are ok for data-based problems.)

Step 3 - convert this into a linear model.

Place the data into a table of values, as shown:

Approximate the slopes (again, can be very rough due to experimental errors affecting the data):

Step 4 - use this model to determine what fraction of F2 is 28cm.

Based on the table of values, it becomes clear that five dominant alleles are needed to create a height of 28cm. The corresponding Pascal term, describing the proportion of offspring that are 28cm, is 56.

THAT'S THE ANSWER Y'ALL!

Note 1: I explained this concept from the USABO problem-solving pov. However, it's important to appreciate exceptions like the fact that multiple alleles and polygenic inheritance can coexist, or that each gene involved may confer a different amount of influence on the final phenotype.

Note 2: There are shortcuts to the Pascal method: using nCr, you can easily get specific terms from the triangle. Here's a resource to learn more about that: https://ptri1.tripod.com/. I challenge y’all to derive shortcut formulas for the whole problem-solving process outlined in this article using the info contained in the linked article!

To be honest I have no resources for y'all on this one other than the math link above - I kinda just figured everything out from a million places lol. Method works tho trust :D And lmk if u have any questions abt it, I’d be happy to explain it.

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