How to calculate dog breed percentage

James Round for Quanta Magazine

Introduction

Our June Insights puzzle featured Dax, a designer puppy who combines the handsome wolflike face of a Siberian husky with the size and fluffiness of a Pomeranian. When humans breed such combinations of two different purebred dog lines, they hope to produce offspring with an admixture of the visual form, temperament, and other characteristics — the phenotype — of both breeds.Is it possible to quantify this proportion? You could assign a weight to every possible phenotypic characteristic and add them up, but this would be far too subjective. Instead, breeders focus on the genotype and perform a simple calculation to determine the proportion of genetic material contributed by each ancestral line.This mathematical exercise yields an objective and seemingly precise number for each breed's genetic contribution; however, as we will see, when we examine how genetic material actually flows down through generations, it becomes clear that even this process is far too complicated to be captured precisely in an individual case.

Puzzle 1

Given that a cross between two purebreds is nominally considered to have an equal genetic mixture of both breeds, how is Dax's unusual genetic makeup produced? What is the smallest number of generations required to produce his genetic makeup to the nearest percentage point? (You must start with purebreds and cross their offspring only with each other or with purebred Poms or huskies.)

Here is a possible family tree for the 56-44 mixture.

Samuel Velasco/Quanta Magazine

Introduction

There are 16 dogs in the founding generation (generation 0), eight males and eight females, and no matings occur in any of the generations between individuals with a common ancestor.nbsp; As a result, each founding individual contributes exactly the same amount of genetic material to all descendants belonging to the same generation as one another.The fraction for the fourth generation is 1/16, and since there are nine huskies and seven Poms to begin with, our target individual, Dax, must be 9/16 (56.25%) husky and 7/16 (43.75%) Pom.

This is a simple mathematical exercise involving the powers of two. If you only want to breed for one generation, you only need two purebred individuals, each of whom contributes one-half (50%) of their genetic material to the offspring.You can produce individuals that are 0%, 50%, or 100% husky by selecting 0, 1, or 2 huskies in generation 0.

If you can breed for two generations without interbreeding, you can produce 0%, 25%, 50%, 75%, or 100% huskies by selecting the appropriate number of huskies and Poms in generation 0.

In the case of three generations, the "least count of huskyness" (or Pomness) available to you is 1/8 (12.5%), which means that you can produce individuals with 0%, 12.5%, 25%, 37.5%, 50%, 62.5%, 75%, 87.5%, or 100% of either breed.

The lowest count with four generations is 1/16 (6.25%), and we can achieve the required 56-44 proportion by doing a 9-7 split in generation 0.

Of course, since Dax is more than 50% husky, there must be more purebred huskies than Pomeranians in the entire family tree, and there are 16 huskies but only 11 Pomeranians. However, if you look only at how the huskyness proportion of 56% is achieved (eliminating extraneous purebred-purebred matings), it requires only two purebred huskies (one parent, contributing 50% huskyLook at the family tree but only look at the mixed, non-purebred individuals to see this. Because each parent has a 50% genetic contribution, Dax's immediate purebred parent (in generation 3) cannot be a Pom.It must be a husky who mates with an individual who is 87.5% Pom; to achieve this high proportion of Pomness from a 50-50 cross breed in generation 1, we need two successive matings with purebred Poms in generations 1 and 2.

This and the following puzzles were expertly solved by and.

Puzzle 2

If Dax had a cousin, Max, who was 60% husky and 40% Pomeranian, how many generations would it take to produce a dog like Max using the same rules as in Puzzle 1? In general, how many generations are required to produce a Pomsky with any given percentage of "huskyness" to the nearest integer?

Let's start with the second question and use it to answer the first: To produce any given percentage of huskyness to the nearest integer, you need a contribution "least count" of less than 1% so that no integers are skipped.This means that more than 100 purebred individuals are required in the founding generation; the smallest power of two greater than 100 is 27, which equals 128.To get 60% huskiness, find 60% of 128, which is 76.8, and choose the nearest integer to it, which is 77. With 77 huskies and 51 Poms in generation 0, we can produce a person who is 77/128 = 60.16% husky.Because 77 is an odd number, we can't reduce it to a lower power of 2 in the denominator, so 5 or 6 generations won't work.

To calculate the number of generations required to breed an individual with any percentage p of huskyness, create a fraction with the integer closest to 128 p/100 as the numerator and 128 as the denominator, and simplify the fraction. The denominator will be of the form 2n where n is the number of generations.This method was similar to the one described by.

However, because 128 is greater than 100, some integer percentages can be obtained in more than one way from 128 purebreds; for example, to obtain a husky percentage of 20, you can use either 26 (20.3%) or 25 (19.5%) huskies in generation 0.The former fraction can be reduced to 13/64 and thus achieved in six generations, whereas the latter cannot, so we need to add one more step to your algorithm to ensure you have the fewest number of generations.If 128 p/100 rounds to an odd integer, see if the even integer obtained by rounding in the opposite direction also yields the desired percentage; if so, choose the even number and reduce it to obtain the fewest number of generations.

Puzzle 3

The total amount of genetic material in a genome can be calculated by counting the number of DNA base pairs in it. All female dog cells, with the exception of egg cells, have two copies of each chromosome, totaling approximately 5 billion DNA base pairs.Female offspring receive an equal genetic contribution from their father's sperm, including all non-sex chromosomes and another X chromosome, with 50% passing through the egg. However, for male offspring, sperm carries the remaining chromosomes as well as the Y chromosome, which has approximately 100 million fewer base pairs than the X chromosome. Based on this information, what proportion of their genome do male dogs actually inherit from their mother? How does this affect the answer to Puzzle 1?

As explained, male dogs have 4.9 billion DNA base pairs, 2.4 billion of which are inherited from their father.

As a result, male dogs inherit 2.5 billion base pairs (51%) from their mother and 49% from their father, implying that a male dog like Dax would have about 1% more huskyness than we previously calculated if his mother was a husky and 1% less huskyness if his father was a husky.

Because Dax is a male, the previously described family tree would result in 57% huskyness if his mother is a husky or 55% huskyness if his father is a husky; to achieve 56% huskyness, we will have to target 55% or 57% using our previous technique.

The former is reachable in the sixth generation by calculating the following: 55% of 128 rounds down to 70, giving the fraction 70/128, which reduces to 35/64. 64 is 26.If Dax's mother is a husky, he can reach 56% huskyness in the sixth generation.

The fraction for 57% huskyness is 73/128, which cannot be reduced and will take seven generations to eliminate.

Within a single generation, this 1% deviation from 50% in the amount of genetic material inherited from a male's mother and father applies, but there is another phenomenon — "meiotic recombination" — that causes a far greater disparity in the amount of genetic material that two siblings can inherit from their grandfather or grandmother.The effect of this, as explained by Ty Rex, is that "two siblings, i.e., with the same parents, can share widely differing amounts of DNA: from as little as 37% to as much as 62%." The same phenomenon exists in all mammals, though the spread may be slightly smaller in dogs because they have many more pairs of chromosomes (39) than humans (23), which may dampen the effects of recombination somewhat.

What this means for Puzzle 1 is that the family tree we calculated (with the maleness correction we made in Puzzle 3) does not guarantee that Dax will have the advertised husky-Pomeranian percentage split in reality; all we can say is that the average genetic endowment of a group of dogs bred in exactly the same way as Dax will be 56% husky and 44% Pomeranian.

Yes, heredity is far more subtle than the simple Mendelian mathematical rules we learned in school, and Mendel's genius lay in deducing the simple foundational rules that allow us to begin thinking intelligently about heredity.Even if we consider the genotype alone, there is plenty of complexity to be added, and we haven't even touched on how the genotype creates the phenotype — how a single gene can orchestrate the development of a complex organ like the eye, while polygenic traits like height can be affected by hundreds of genes.As a result, even though the father's contribution to the male genotype is less than the mother's, a small number of genes on the Y chromosome can produce maleness, which is a significant phenotypic change.

Ty Rex also recommended Carl Zimmer's recent book She Has Her Mother's Laugh, which is about "the history and remarkable facts of heritability, including'mosaics,' and the mother for whom multiple DNA tests insisted she was not related to her own children!"

Question 1

COVID-19 is known to be especially lethal to the elderly, and the CDC data below provides a breakdown of approximately 70,000 COVID deaths in the United States.It demonstrates that people aged 85 and up are the most vulnerable.There were more males than females in this data, though the male-to-female death ratio was around 55:45, which is slightly lower than the rest of the world.

However, as people get older, the number of years they can expect to live decreases. Can you guess which age group in the above table has lost the most years of life due to COVID?To calculate this accurately, you must use actuarial tables such as this one, which shows that a 62-year-old man has an average life expectancy of another 20 years, whereas an 87-year-old can only expect to live another five years. Does the final answer surprise you?

and ran this calculation, the number of years lost in each age range is shown below.

It turns out that 65- to 74-year-olds have lost the most life years to COVID-19, but 55- to 64-year-olds and 75- to 84-year-olds are close behind, and 45- to 54-year-olds have lost nearly as many life years as the over-85 group.So the societal loss of person-years caused by COVID-19 is just as high in middle-aged people as it is in the very elderly! It goes without saying: Stay safe, everyone!

For Question 2 of this column, I presented a speculative hypothesis that attempted to explain why mammals rely entirely on sexual reproduction, despite the fact that some reptiles can reproduce asexually. Based on this idea, I asked readers to discuss: Could pandemics be the reason that mammals reproduce sexually rather than asexually?

I didn't get much of a response, so this hypothesis may need to be refined and revisited another time.

I'd like to congratulate Ty Rex and Jonathan Vercruysse on winning this month's prizes, and thank you all for your contributions to this discussion.

Next month, expect new Insights.

By Pradeep MutalikPradeep Mutalik

Puzzle Columnist

July 31, 2020

biologyCOVID-19geneticsinfectious diseaseInsights puzzlepuzzlessexAll topicsCOVID-19geneticsinfectious diseaseInsights puzzlepuzzlessexAll topics

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