main | Arslan Zaidi

Why we expect genetic differences between populations to be zero

Sun, 14 Mar 2021 00:00:00 +0000

We expect the genetic difference between two (or an arbitrary number of) groups that diverged from some common ancestral population to be zero under neutrality. This is a well-known result that I feel like isn’t discussed enough, especially in discussions around mean differences in traits such as IQ and educational attainment. I can’t comment on the validity of these phenotypes and I don’t (and will not) wade into loaded conversations about how much of the variance underlying these traits is heritable and in which direction. However, I sense that there are misconceptions among students about the expectation of the mean genetic difference underlying a neutrally evolving phenotype, and I’m casually here to set the record straight.

So what do we mean when we say that the expectation of the mean difference between populations is exactly zero? Does that mean that we don’t expect to see any differences between populations? As we will see, these two questions are asking different things.

The expected genetic value in a population

To answer these questions, we must first talk about genetic value and figure out what it’s expectation is in a single (randomly mating) population. Let’s begin (as we do) with a generative model for the phenotype:

\[y = g + e\]

where $g = \sum_{i = 1}^M \beta_i x_i$ is the genetic value (summed across $M$ loci) and $e$ is random (environmental) error. I like to think of the genetic value as the expected phenotype of an individual given their genotype ($x$): $\mathbb{E}(y|x)$. You can also (informally) think of it as the phenotype of an individual that can be predicted from their genotype.

Assuming only additive effects, we can write the genetic value of each genotype at the $i^{th}$ (biallelic) locus ($x_i \in \{0,1,2\}$) as $\mathbb{E}(y|x_i = 0) = 0$, $\mathbb{E}(y|x_i = 1) = \beta_i$, and $\mathbb{E}(y|x_i = 2) = 2\beta_i$ ¹. From this, we can get the mean genetic value in the population by marginalization. Let’s denote the frequency of the $i^{th}$ variant in the population as $f_i$. Then,

\[\begin{aligned} \mathbb{E}(y) = & \mathbb{E}\{\mathbb{E}(y|x_i)\} \\ = & \mathbb{E}(y|x_i = 0)\mathbb{P}(x_i = 0) + \mathbb{E}(y|x_i = 1)\mathbb{P}(x_i = 1) + \\ & \mathbb{E}(y|x_i = 2)\mathbb{P}(x_i = 2) \\ = & 0(1 - f_i)^2 + \beta_i\{2 f_i (1 - f_i)\} + 2\beta_i(f_{i}^2) \\ = & 2\beta_i f_i - 2\beta_i f_i^2 + 2\beta_i f_i ^2 \\ = & 2\beta_i f_i \end{aligned}\]

Where $\mathbb{P}(x_i = 0)$ is the frequency of individuals carrying 0 copies of the (effect) allele in a randomly mating population, $\mathbb{P}(x_i = 1)$ is the frequency of individuals carrying 1 copy and so on. Aggregating this across multiple independent (unlinked) causal loci gives us the mean genetic value in a single population: $\mathbb{E}(y) = 2\sum_{i=1}^M\beta_i f_i$. To demonstrate this, note that $\mathbb{E}(y|x_i) = \beta_ix_i$ and $\mathbb{E}(y|x) = \sum_i^M \beta_ix_i$. Therefore,

\[\begin{aligned} \mathbb{E}(y) = & \mathbb{E}\{\mathbb{E}(y|x)\} \\ = & \mathbb{E}\left( \sum_{i= 1}^M \beta_i x_i \right) = \sum_{i = 1}^M \beta_i \mathbb{E}\left( x_i \right) \\ = & \sum_{i = 1}^M \beta_i 2 f_i = 2 \sum_{i = 1}^M \beta_i f_i \end{aligned}\]

From this it follows that the mean genetic value in populations 1 and 2 are $\mathbb{E}(g_1) = 2\sum_{i = 1}^M\beta_i f_{i}^{(1)}$ and $\mathbb{E}(g_2) = 2\sum_{i = 1}^M\beta_i f_{i}^{(2)}$, respectively, where $f_{i}^{(1)}$ and $f_{i}^{(2)}$ are the allele frequencies in populations 1 and 2, respectively. Note, that I’m assuming that the causal effect size is the same across populations. The expected difference in genetic values between the two populations is then $2\sum_{i=1}^M\beta_if_{i}^{(1)} - 2\sum_{i = 1}^M\beta_if_{i}^{(2)} = 2\sum_{i = 1}^M\beta_i(f_{i}^{(1)} - f_{i}^{(2)})$. In this expression, $\beta_{i}$s are fixed and $f_{i}^{(1)} - f_{i}^{(2)}$ are random (over the evolutionary process), which means that to understand the expected mean difference in genetic value between populations, we need to work out the expected difference in allele frequency.

Expected difference in allele frequency between populations

For two (randomly mating) populations that have diverged from a common ancestral population ², the frequency of a neutral allele in the $j^{th}$ population at the $i^{th}$ locus approximately follows a normal distribution:

\[f_{i}^{(j)} \sim \mathcal{N}(f_i ~, ~ f_i(1-f_i)F)\] Where $f_i$ is the frequency in the ancestral population and $F$ is a measure of drift since the populations split. $F \approx \frac{t}{2N_e}$ where $t$ is the number of generations since the split and $N_e$ is the (diploid) effective population size. You can also think of $F$ as the $F_{st}$ between the two populations (more on this some other time). The derivation for this is complicated and not super important for this post but I’ll demonstrate that it holds with simulations.

library(dplyr)
library(ggplot2)
library(data.table)

Write a function to simulate drift in a population given some starting frequency, the effective population size, and the number of generations for which we want the simulation to run.

#write a binomial sampling function to simulate drift in a population
let.it.drift <-function(f0, g, Ne){
  #f0 is  starting frequency in the ancestral population
  #make sure g and Ne are integers
  for(i in 1:g){
    nalleles = rbinom(Ne, 2, f0) #sample Ne diploid individuals
    f0 = mean(nalleles)/2 #calculate frequency in the population
  }
  return(f0) # return final frequency
}

Now simulate the drift in the allele frequency at 1,000 independent loci, each starting with an ancestral frequency of 0.5. We’ll simulate a population of size 1,000 individuals and see how the frequency changes as we increase the divergence time between the two populations.

#simulate drift in 1000 loci for increasing number of generations of drift (5-50)
p.exp = sapply(seq(5,50,10),
               function(x){
                 replicate(1000, let.it.drift( f0 = 0.5, g = x, Ne = 1000))})

p.exp = data.table(p.exp)
colnames(p.exp) = as.character(seq(5,50,10))
p.exp$rep = c(1:1000)

# melt to long format for plotting in panels
mp.exp = reshape2::melt(p.exp, id.vars="rep", 
              variable.name = "generations",
              value.name = "frequency")

#calculate the expected and observed mean and variance in allele frequency for each value of the number of generations
mp.exp.summary = mp.exp %>%
  group_by(generations)%>%
  summarize(mean.f = mean(frequency),
            var.f = var(frequency)) %>%
  mutate(g = as.numeric(as.character(generations)),
         exp.var = 0.5*(1 - 0.5)*(g/2000))

## `summarise()` ungrouping output (override with `.groups` argument)

#plot!
ggplot(mp.exp, aes(frequency))+
  geom_histogram()+
  facet_grid(generations~.)+
  theme_classic()+
  geom_vline(xintercept = 0.5, 
             color = "Red", 
             linetype= "dashed")+
  geom_vline(data = mp.exp.summary, 
             aes(xintercept = mean.f), 
             color = "blue",
             linetype= "dashed")+
  geom_text(data = mp.exp.summary, 
            aes(x = 0.65, y = 100, 
                label = paste("obs. var", 
                              round(var.f,4))),
            color = "blue")+
  geom_text(data = mp.exp.summary, 
            aes(x = 0.65, y = 200, 
                label = paste("exp. var",
                              round(exp.var,4))),
            color = "red")+
  labs(x = "Frequency",
       y = "Count")

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

Notice a few things:

The observed mean (red line) and variance (red text) in allele frequency closely match those predicted from the model above, even for populations (of 1,000 individuals) that diverged 45 generations ago.
Even though the allele frequency at any single locus can vary substantially from the ancestral frequency, the mean of the distribution (representing the expected allele frequency) does not change regardless of how many generations of drift we simulate.
However, what does increase as a function of the number of generations is the variance (or spread) of the allele frequency distribution. In other words, even though we expect the allele frequency to remain the same (regardless of the number of generations), the possible values that the frequency can reach (in either direction) gets larger and larger with every generation.

Because the expected frequency in both populations is the same, i.e., $\mathbb{E}(f_i^{(1)}) = \mathbb{E}(f_i^{(2)}) = f_i$, the expected difference in allele frequency between the two populations $\mathbb{E}(f_i^{(1)} - f_i^{(2)}) = \mathbb{E}(f_i^{(1)}) - \mathbb{E}(f_i^{(2)}) = 0$. And the expected mean difference in genetic value is $\mathbb{E}(g_1 - g_2) = 2\sum_{i = 1}^M\beta_i\mathbb{E}(f_{i}^{(1)} - f_{i}^{(2)}) = 0$.

In other words, we expect the genetic difference underlying a phenotype between populations to be zero. Does that mean that we don’t expect to see any differences between populations? No, because even though the expected difference in frequency is zero, the expected absolute difference is not zero. Let me illustrate this by simulating drift in two populations that have evolved independently since their divergence from a common ancestral population with allele frequency of 0.5.

#simulate drift in 1000 loci for increasing number of generations of drift (5-50)
#pop 2 (we will use p.exp as pop1 frequencies)
p.exp2 = sapply(seq(5,50,10),
               function(x){
                 replicate(1000, let.it.drift( f0 = 0.5, g = x, Ne = 1000))})

p.exp2 = data.table(p.exp2)
colnames(p.exp2) = as.character(seq(5,50,10))
p.exp2$rep = c(1:1000)

# melt to long format for plotting in panels
mp.exp2 = reshape2::melt(p.exp2, id.vars = "rep", 
              variable.name = "generations",
              value.name = "frequency")

colnames(mp.exp)[3] = "freq1"
colnames(mp.exp2)[3] = "freq2"

mp.exp = merge(mp.exp, mp.exp2, by = c("rep","generations"))

#calculate the expected and observed mean and variance in allele frequency for each value of the number of generations
mp.exp.summary = mp.exp %>%
  group_by(generations)%>%
  summarize(mean.abs.diff = mean(abs(freq1 - freq2)))

## `summarise()` ungrouping output (override with `.groups` argument)

#plot!
ggplot(mp.exp, aes(abs(freq1 - freq2)))+
  geom_histogram()+
  facet_grid(generations~.)+
  theme_classic()+
  geom_vline(data = mp.exp.summary, 
             aes(xintercept = mean.abs.diff), 
             color = "blue",
             linetype= "dashed")+
  labs(x = bquote("Absolute difference in allele frequency : abs("~f^(1)~"-"~f^(2)~")"),
       y = "Count")

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

The figure above shows that the expected absolute difference in allele frequency between the two populations is not zero. In fact, it increases as a function of divergence time (blue line in the plot above). In other words, the more diverged the populations, the more we can expect there to be allele frequency differences between them.

It follows then that we also expect the absolute difference in genetic values between populations to also increase with divergence time.

A simpler way to say this is that for two populations that diverged from some common ancestor, we do expect there to be genetic differences underlying a neutrally evolving trait. BUT, we don’t know which direction this difference will take. Putting it differently, we don’t know whether population 1 or 2 should have a higher mean genetic value for the trait. This is one thing that needs to be kept in mind when discussing phenotypic differences between populations.

Can we not infer this from the direction of the phenotypic difference? Say if population 1 has a higher phenotypic value than populaion 2, can we say anything about the direction of underlying genetic difference? No. so far I’ve only discussed the variation in the genetic component ($g$ in our phenotype model). The direction of phenotypic difference is also driven by the environment, which could be different between the two populations. More on this some other time but in conclusion:

The expectation of the genetic difference underlying a trait between populations is zero.
But this does not mean that there won’t be any genetic difference between populations. In fact, we expect this difference to increase with divergence time.
However, we cannot predict (for a neutral trait) which direction the difference is going to take, i.e, we cannot say if population 1 or 2 will have a higher genetic value.

I should point that these conclusions are only applicable to neutral traits. Under directional selection, we do expect systematic differences in mean genotypic value between populations. However, as mentioned earlier, this cannot be inferred directly from differences in phenotypic means between populations. And even differences in genetic means between populations may not be due to selection. More on this at some other undecided time in the future…

Acknowledgements

Thanks to Trang Le for giving feedback on an earlier draft of this post.

You can take a look at my post on narrow-sense heritability for intuition.↩︎
This model works well for recently diverged populations where time is measured on the scale of the effective population size. Thus, the model can accommodate long divergence times (in actual number of generations) for large populations.↩︎

Heritability (3): Non-additive effects dominance

Wed, 24 Oct 2018 16:44:00 -0400

Dominance is a just a fancy-schmancy way of saying that the phenotypic effects of alleles at a single locus are not additive 💁‍♂️. Ok, but what does that mean?

To illustrate, let’s go back to the example of differences in milk-yield among a bunch of cows 🐮🐮(Fig. 1). As I mentioned in a previous post, some of these differences are due to genetic differences among the cows and some are due to environmental differences. For example, there could be (average) differences between the milk yield of cows with AA genotype (30 liters) vs cows with Aa genotype (20 liters) vs cows with aa genotype (10 liters). In this example, each additional A allele results in an (average) increase of 10 liters of milk (Fig. 1A). This may not always be the case if, for example, cows with the Aa genotype and AA genotype have the same average milk yield: 30 liters (Fig 1B). If this is the case, we say that the effect of alleles at this locus are non-additive or that there is dominance at this locus.

Fig. 1: A) Additivity is when each additional allele adds a fixed value to the phenotype. B) Things are not that symmetric when there’s dominance. In this example, the average phenotype is the same for the AA and Aa genotypes.

When you were learning genetics in school 🏫, you probably came across terms like “dominant” and “recessive”, which are often used to refer to Mendelian traits such as the presence/absence of cleft chin 🍑 in humans. These terms are ok for simple Mendelian traits but quite limiting for quantitative traits . Also limiting are terms like complete dominance, a special case of dominance in which the phenotypic distributions of AA and Aa genotypes are indistinguishable (Fig. 1B). There is a more flexible way to describe dominance: d.

The dominance deviation, d, describes the degree (and direction) in which genotypic values deviate from additivity.

🛑✋ Sidebar: It’s useful to define genotypic value here as it is an important concept in quantitative genetics. The genotypic value is the average phenotype of a specific genotype. For example, the genotypic value of the Aa genotype in Fig. 1A is 20 liters. So even though some individuals carrying the Aa genotype have higher or lower milk yields because of environmental differences among them, they all yield, on average, 20 liters of milk. Thus, the genotypic value is supposed to represent the ‘true’ phenotypic effect of the genotype. It is often convenient to represent genotypic values as deviations from the mean of the population. For example, the genotypic value of the aa genotype in Fig. 1A can be written as 10 - 20 = -10 (instead of 10) if we assume the average milk yield in the population is 20 liters. This is mostly for convenience sake in calculations so don’t worry too much about this. Sidebar over.👍

Now take a look at Fig. 2 and give it some time.🤔

Fig. 2: Different values of d describe different cases of dominance. The scale represents the genotypic values, where +1 and -1 refer to the genotypic values of AA and aa, respectively, expressed as scaled deviations from the mean. The red line, which represents the scaled dominance deviation, shows the extent to which the genotypic value of the heterozygote Aa deviates from what we expect had the allelic effects been completely additive.

Can you see how the dominance deviation flexibly and quantitatively describes the different cases of dominance (including no dominance!)? The meaning of the different values of d are further explained below:

d = 0: no dominance. The allelic effects are completely additive and the average phenotype of Aa is smack dab in the middle between AA and aa.

d = 1: complete dominance. AA has the same genotypic value as Aa. Note that this is the same as the example of milk yield in Fig. 1B.

0 > d > 1: partial dominance. The genotypic value of Aa lies somewhere between that of AA and aa. As d gets closer to 1, the genotypic value of Aa gets closer to that of AA.

d > 1: overdominance. This is when the genotypic value of Aa is greater than that of either homozygote. So, for example, the average milk yield of cows carrying the Aa genotype might greater than the average milk yield of cows carrying the AA or aa genotype. This is sometimes referred to as hybrid vigor.

d < -1: underdominance. The genotypic value of Aa is less than that of either homozygote.

🚨There is one common misconception regarding dominance that I want to clear up: dominance does not tell us anything about how frequent the allele is in the population. For example, the A allele in Fig. 1B, even though it is ‘dominant’, might only be present at 10% frequency in the population.

Dominance is only one example of non-additive effects of alleles on the phenotype. So far I have only talked about the phenotypic effects of alleles at a single locus. What happens if we consider alleles at two or more loci? We’re faced with non-additive effects of alleles across loci. Non-additive effects of alleles at two or more loci are called epistasis and I plan to talk about this next time. See you in a couple of months…📆

What your genetic ancestry test can and cannot tell you about your genealogical ancestors

Wed, 17 Oct 2018 00:00:00 +0000

The topic of this post was not on my radar 📡 but I don’t live under a rock and I’ve been following the recent developments with Elizabeth Warren and her much publicized ancestry results. Political drama aside, there are concerns over what such results mean (and don’t mean) for one’s identity and genealogical history. While many geneticists and anthropologists have weighed in on the matter, their discussions are not very accessible for people who are not statistical geneticists. So I figured I would take a crack at explaining it in simpler terms.

Quick run down: Ms. Warren is somewhat Native American. The analysis was conducted by a well-known geneticist who knows his sh*t. While I cannot find the % of Native American ancestry in his report, one can wager a very rough guess of 0.1% - 1.6% based on the simple fact that the number of ancestors doubles in every previous generation: you have two parents, four grandparents, eight great grandparents, 16 great great grandparents and so on. Based on a more sophisticated algorithm, it is estimated that Elizabeth Warren had an ancestor who was 100% Native American 6-10 generations ago. Ten generations ago, you, she, and I had 2¹⁰ (1,024) ancestors! Some of these passed on their genetic material to you and some did not (in fact many did not).

Some questions surrounding Warren’s ancestry results are: Does her ancestry make her Native American? What if her ancestry results had shown that she didn’t have any Native American ancestry?

I want to try to answer these questions by illustrating how chromosomal segments are inherited over a small number of generations forward in time. Take a look at Fig. 1 which shows a pedigree on the left (the genealogical history of a family). On the right I’m showing a single possibility of how chromosomal segments in this family might be inherited.

Fig. 1: Difference between genealogical ancestry and genetic ancestry. A family tree (the genealogical ancestry) is shown on the left where the squares represent males and circles represent females. To the right is one possible way in which chromosomes can be inherited by family members. The numbers with % signs refer to the % of blue ancestry in each child.

In the first generation, two people of different ancestries (blue and yellow) get together and have two kids, Child 1 and Child 2. Each parent gets to pass only 50% of their genetic material (only counting autosomes for simplicity) to their offspring. At every independent location on a chromosome (shown as segments), only one of two alleles can be passed on by each parent (two because chromosomes come in pairs). Which allele gets passed on is determined by a random process (independent assortment). In Fig. 1, the blue mother can only pass blue alleles to her kids and the yellow father can only pass yellow alleles to his kids. As a result, both children in the 2nd generation will carry exactly 50% of the “blue” ancestry. So far so good?

Now Child 1 goes off and has two kids (Child 3 and Child 4) with another person of yellow ancestry. Child 1 can pass either a blue allele or a yellow allele (picked at random) at each segment to her children because she has both blue and yellow chromosomes. Because the “choice” of alleles in each segment is random, Child 3 inherits her mother’s yellow allele in the 1st segment, blue allele in the 2nd segment, blue again in the 3rd segment, yellow in the 4th, and blue in the 5th. Child 3 cannot inherit blue alleles from her dad, who is 100% yellow, giving her a total of 30% blue ancestry. Meanwhile, Child 4 happens to receive a slightly different configuration from her mother (blue, yellow, blue, yellow, yellow), adding up to a total of 20% blue ancestry. Even though we expect Child 3 and Child 4 to inherit 25% blue ancestry (it gets diluted by 1/2 in every generation - see Fig. 2), it doesn’t mean that’s going to happen. Because which allele the mother picks to give to her children is random, Child 3 and Child 4 can carry different % of blue ancestry. This difference between the expectation and what we actually observe can increase with every subsequent generation. For example, Child 5 and Child 6 have 20% and 0% blue ancestry in their genome, respectively, even though we expect both of them to have 12.5% blue ancestry.

Fig. 2: The % of blue ancestry is expected to decrease every generation by 1/2 (red line and numbers on top of the red line). However, this is only the expectation. In other words, if you had many many families at every generation from the same pedigree, you would see that the blue ancestry would decrease, on average, by 1/2 in every generation. However, this % can be very different between two siblings as shown in Fig. 1. The dotted lines show the range of % blue ancestry that different siblings in each generation can have. It’s also clear that the blue ancestry can be lost in some siblings (lower dotted line touches the 0% mark). The code for simulation and plot is available here.

What is even more interesting is that eventually you could have someone without any blue ancestry at all (Child 6). This raises an important question: Does Child 6 have a blue ancestor? The answer is: yes!

So, if someone has Native American ancestry, they necessarily would have had to have an ancestor who, at some point in the past, was 100% Native American. However, just because someone does not have any Native American ancestry does not mean that they did not have a Native American ancestor (triple negative was unavoidable 🤕).

What does this mean for Elizabeth Warren? Did she have a Native American ancestor in the past? Yes, but that’s not saying a lot. There are many people in the world walking around without any Native American ancestry even though they had Native American ancestors at some point in the past, as the example in Fig. 1 illustrates. Does that make them Native American? Is she Native American? This is not a question genetics can answer as a person’s identity depends on their culture, geography, customs, and family history. Forcing genetics to answer such questions presents more questions than answers: How much Native American ancestry do you have to have to be called Native American? How far back would you have had to have a Native American ancestor? Answers to these questions will be driven by inherent biases that are influenced by our political and social inclinations. But they are certainly not scientific answers.

Further reading: If you want a more detailed understanding of the difference between genetic and genealogical ancestry, be sure to check out Graham Coop’s blog post

—— UPDATE 10/18/2018: ——-

Technical clarification on Fig. 2: The red line (mean) and dashed lines (range = max - min) are based on a simulation of 1,000 independent loci and 1,000 predigrees. The simulations are meant to illustrate the concept that the observed % ancestry can deviate greatly from expectation. In reality, there are many more loci in the human genome, and the patterns of inheritance are much more complex because of linkage and recombination. Because of these reasons, it is unlikely that the ancestry is going to be lost in two generations as Fig. 2 currently shows ¹.

This clarification was prompted by someone in the comments section on the previous website.↩︎

Heritability (2): narrow and broad sense heritability

Fri, 04 May 2018 16:24:08 -0400

Quick recap: I mentioned last time that heritability = Vg / Vp, where Vg is the genetic variance among individuals and Vp is the total phenotypic difference among individuals. Furthermore, Vp = Vg + Ve, where Ve is the environmental variance contributing to phenotypic differences.

There are two different types of heritability: broad-sense heritability and narrow-sense heritability. What we’ve been discussing so far is broad-sense heritability: the proportion of phenotypic variance that can be explained by ALL genetic differences among individuals. Geneticists often like to split Vg, the genetic variance, into two: additive genetic variance and non-additive genetic variance. There is a reason for this split (besides a masochistic need to make life harder for everyone 🤦🏽‍♂️), which I will talk about later but first let’s talk about what these things are.

Additive genetic variance first since that’s easier. Suppose there is a gene called A (very creative, I know 😛) involved in milk production in cows 🐮. Further suppose that a cow can have the genotype AA, Aa, or aa at gene A because cows are diploid. Cows with genotype aa produce, on average, 10 liters of milk everyday. I don’t know if that’s a normal amount or not so don’t judge me if you’re a farmer 👨🏻‍🌾 and you’ve chanced upon this blog. Cows with a single A allele in their genotype (i.e. genotype Aa or aA) produce, on average, 20 liters of milk and cows with genotype AA produce 30 (Fig. 1).

Figure 1: Cows with different genotypes can have different means for milk yield. This variation is due to a combination of genetic and environmental differences. However, among cows with the same genotype, differences in milk yield are purely due to environmental differences. There are a couple of interesting things happening here (if we play fast and loose with the word ‘interesting'😅)

There are differences in milk yield among cows of the same genotype even though there are no genetic differences among them (Vg = 0) (Fig. 1). Remember from the last post that these differences arise solely because of environmental differences. This is why I keep saying things like “cows with a single A allele produce, on average, 20 liters of milk” as there are some that might produce 22 and some that might produce 18, because of small environmental differences. Note also that this means that heritability among cows of the same genotype is 0. Ok, I think I’ve hammered 🔨 this point hard enough. The differences among groups of cows with different genotypes (AA vs Aa vs aa) arise not only due to environmental differences, but also due to genetic differences (i.e. Vg > 0). Because Vg > 0, Heritability > 0 and we can say that milk yield is heritable in this specific herd of cows (Fig. 1).

It seems that each additional A allele that a cow carries, on average, increases the milk production by 10 liters (aa = 10, Aa = 20, AA = 30, Fig. 1). There is no dominant or recessive allele, despite my poor use of the letters. We say that the effect of A (and a if we look at it as decreasing milk yield) is additive. If one of the alleles were dominant/recessive, we would say the allelic effects are non-additive because we couldn’t just figure out the average milk yield by adding up the number of A alleles. (Side exercise: if A is the dominant allele and a is recessive, what would the average milk yields be for the three genotypes?).

The last point can be extended from one gene/ locus to two (or more) genes/loci. Different alleles at the two loci might contribute additively to milk yield (Fig. 2). This additivity has everything to do with the difference between narrow-sense and broad-sense heritability. If broad-sense heritability is the proportion of phenotypic variance that is due to ALL genetic effects (additive + non-additive), narrow-sense heritability is the proportion of phenotypic variation only due to additive genetic effects. What are non-additive effects? Why are they important? Why does this distinction matter? What is the meaning of life 🌄🤔? There’s a whole post about this (ok, not the last question), so stay tuned 😎🍿.

Figure 2: The effect of alleles on milk yield (numbers) in this example is additive. Doesn’t matter if we look at each gene separately, or combine effects across them. Primary source: Lynch M, Walsh B. 1998. Genetics and analysis of quantitative traits. Sunderland, MA: Sinauer Associates, Inc.

Heritability (1): How 'genetic' is a trait?

Wed, 25 Apr 2018 13:19:16 -0400

One of the reasons I’ve come to write about this is because recently I’ve been asked a lot about “how genetic is this trait or that trait?”. And I find myself responding with another question (yeah I’m that guy , get over it): “What do you mean by genetic?”. I ask this because I think there is often a confusion between the term ‘genetic’ and ‘heritable’, and that distinction is extremely important to understand.

Maybe you’ve heard this before: Every trait is genetic. There are genes involved in the development of every trait. Nothing pops out of the environment (unless it’s a spray tan or something?🤔). Is height genetic? Yes. Is hair color genetic? Yes. If you go work out a lot🏋🏽‍♂️, are your bulging biceps genetic? Yes, because there are genes that respond to the body’s need by triggering a cascade of biochemical processes that result in muscle production. The question, “How genetic?”, is not a legit question. What one really means to ask is “How heritable?”.

Cutting to the chase and then I’ll explain: Heritability is the proportion of variation in a trait (Vp) explained by genetic differences among individuals (Vg).

$$Heritability = \frac{V_g}{V_p}$$

$V_p = V_g + V_e$. This is the classic equation which, in english, reads: Variation in the phenotype/trait can be explained by the sum of the variation in the genotype ($V_g$) and variation in the environment ($V_e$), where environment is everything that is not ‘genetic’. It becomes clear from the definition of heritability that it cannot be defined for a trait in a single individual. That’s not a thing. It can only be defined for differences between two or more individuals.

Let’s expand on this. Imagine there are 10 cows 🐮☘️ grazing on a pasture. They’re all clones of each other, i.e., they are all exactly the same genetically. If you milk all of them, the milk yield 🥛 will vary among them (Figure 1). How much it varies (the variance) is Vp. Is the milk yield of a cow genetic? There are genes involved in milk production, so yeah. Is the milk yield among the cows heritable? Nope. Think about it. Since the cows are clones of one another, there are no genetic differences among them (i.e. $V_g = 0$ 👌🏽) . So any differences in milk yield have to be because of the environment ($V_p = V_e$). Maybe they were grazing on different regions of the pasture or maybe some of them were being tipped more than others 🙌🐄 (it’s a real thing). The heritability of milk yield among the cows is 0.

Figure 1: Variation in milk yield in a hypothetical herd of cloned cows. Since there is no genetic variation among cows, heritability of milk production in THIS set of cows is zero.

There are some other things about the concept of heritability that make it tricky. One thing that I’ll mention now and leave the rest for some other time is that heritability is not a fundamental property of the trait itself. It is context specific and is a property of the sample you are looking at. If I tried to measure the heritability of milk yield in another set of 10 cows grazing in the same pasture, who are genetically different from one another, the heritability of milk yield will likely not be zero since $V_g > 0$.

So when most people ask “how genetic is skin pigmentation” or “how genetic is intelligence” (that’s a whole can of worms that I hope to address at some point), they’re really asking “how heritable are they?”.