Welcome back. This week again we're going to be talking about heritability. And in the first module I use Mendelian inheritance really as a vehicle to, to begin to introduce some concepts and terminology that I'll be using throughout the course. In the second module for this week, I'm going to begin talking about what is sometimes called Galtonian inheritance or more typically, Quantitative Genetics. Now Quantitative Genetics itself is a very complex subfield within Genetics. And what I'm hoping to do over the next couple of modules is to give us enough of an introduction to the field of Quantitative Genetics that we'll be able to interpret the Behavioral Genetic literature that we're going to look at a little bit later in the course. When Mendel's laws were rediscovered in 1905 at the beginning of the 20th century there actually ensued a major debate about the applicability of those laws. And the debate really centered around this issue, what Mendel studied was the inheritance of particulate characteristics, a yellow versus a green pea or a round versus a wrinkled pea. And what other people had been studying when they'd been studying the inheritance of phenotypes or characteristics, from one generation to the next. When Galton and his students were studying the inheritance of characteristics the characteristics they were studying were not particulate. They were like the, like things like height or weight, that were more continuously distributed. And Galton and his disciples thought that Mendelian inheritance couldn't possibly apply to characteristics like this, because what Mendel was talking about were these particulate characteristics. And so therein ensued a rather major debate. And one of the conclusions that, that many early geneticists arrived at was that were actually may be two forms of inheritance. On the one hand, Mendelian inheritance, that governed the transmission of particulate characteristics, and on the other hand, Galtonian Inheritance, that which governed the inheritance of quantitative characteristics like weight or height. In 1918, the famous statistician R.A. Fisher actually resolved this debate. And he resolved it by showing that really Mendelian inheritance could account for the inheritance of quantitative characteristics. And he did this in a very famous paper that he published. But the gist of the paper was, assuming that genes, and I, I understand that, I haven't precisely defined gene yet. That will be next week. But what, what Fisher postulated is that genes could have a quantitative effect on a phenotype. So let's go through what Fisher developed mathematically. Suppose we had a locus, and recall a locus where a gene is located in, in the genome, suppose we have a locus, and we'll call it the A locus. And there are two alleles for this locus, two alternative forms that the gene could take at this particular location. We'll call them the A1 and A2 locus. And suppose that the effect of inheriting the A1 allele increased your height by 1/2 of a centimeter. Whereas if you inherit the A2 allele, it decrements your height by 1/2 of a centimeter. Here height is a quantitative phenotype, which quantitative geneticists usually denote by the letter P. And in this case with this very simple, two allele, one locus system, we could say that your height is a function of the overall average height in the population plus the effect of this A locus. And what would that look like? Well, we'd have three possible genotypes with two alleles. And, if you are the heterozygote, that is you inherit and A1 and an A2 allele, then those two effects will cancel out and your height would be the average. But two A1 alleles, would give you a half a centimeter timed two, or one centimeter increase over the average, whereas two A2 alleles, one centimeter less than the average. So, with one locus, two alleles, we get three, actually three quantitative values. Now suppose, as Fisher did, that there was a second loci, a B locus. And the B locus was like the A locus. So we can say there's a B1 allele and a B2 allele. And if you inherit a B1 allele you get 1/2 centimeter more in height. Now with two loci, we're going to get a distribution of five phenotypic values. Suppose we add in a third locus, a C locus that looks like the, that has the same quantitative effect. It's a different location in the genome, but it has the same quantitative effect as the A and B locus. And those effects are adding up over loci. Then now with three loci, we can generate 7 phenotypic values. And you get this symmetric distribution. And if you think this through, with four loci, if there is a D locus with four loci, we would get nine values. And five loci we would get 11 values. And if we carried this on ad infinitum, what we would end up with is a continuous distribution. When we add up loci, each of which has a quantitative effect, and there are many, many loci, we get what Fisher called the Polygenic Model. Polygenic being Greek: poly for many, genic for genes. The key characteristics of the polygenic model are these. First, that there are a large number of contributing loci or genes. Loci and genes are being used synonymously here. So there's a large number of contributing loci. That's what Fisher postulated in his mathematical development of this model. What exactly is large? That's always hard to say in these mathematical developments. But later in this course when we begin to talk about schizophrenia and intelligence, we'll get a better sense for what large actually is. It's probably on the order of hundreds or maybe even thousands of contributing loci. But in general, we think of the polygenic model not of two or three or four or five contributing loci, but a large number, hundreds or thousands. The second key assumption that Fisher made in developing this model is that each one of these loci, loci had a small and equal effect on the phenotype. And we can see why you would need that. Suppose there was a locus out there, let's say the X locus, such that when you had inherited the X1 allele at the X locus, all of a sudden your height is boosted by ten centimeters, not a half centimeter. Then the distribution of height isn't going to look nice and continuous like this. You're going to have a whole bunch of people out here and then you're going to have some people up here that get that X1 locus X1 allele and have that ten centimeters increment in height. So to get this continuous distribution we need to assume, or Fisher assumed, that each locus contribution was small and roughly equal. And then finally that the locus effects add up over loci and that gives you a continuous distribution. Although, Fisher might not have been, in, in the mathematical development might not have been explicit about this, one thing that we presume is that as compared to classical Mendelian inheritance, Polygenic inheritance, we assume, that the environmental effects are much more fundamental. Likely to be much more important. And the reason to assume that is that the primary purpose of genes isn't to give you a half a centimeter, centimeter of height or to decrement your height by a half a centimeter. The primary function of genes are far removed from how tall you are or how extraverted you are or whether or not you're schizophrenics. And that being far removed from the phenotypes we're going to be dealing with, we presume that because of the remoteness of the primary gene function there are many more opportunities for environmental influences. So, in general, when we talk about polygenic traits, we tend to emphasize, or we tend to want to make sure that we think about how the environment might be affecting those traits. And you'll see how the biometricians do that in a, in a little bit here. So, what Fisher did back in 1918 is he assumed that we, if we had multiple loci, we could actually show that what Galton was studying really could be accounted for by basic Mendelian laws of inheritance, that we didn't need to have two different types of inheritance. One of the things to further develop this particular Polygenic Model that we would like is an index of the extent of which people differ on a quantitative trait in a population. And if you recall from your statistics, we actually already have such an index. And the index is called the variance of a distribution. The variance is a measure of the degree to which scores in a distribution actually differ from one another. It's a summary measure. We're not going to actually compute variances in this course, but we're going to use the concept of variance as a summary index of individual differences in a distribution to really develop the, the notion of heritability. So I just went through three distributions here when I was developing Fisher's Polygenic Model. One locus, we got three values. Two loci, we got five values. Three loci, we got seven phenotypic values. And clearly there's more individual differences here than there. And actually if we, if we computed the variance that I did that for these three distributions, we see that, that the variance goes up as the distribution of individual differences increases. So there's more variance here, and this happens to be what the variance is, than there is here. Variance is a very convenient measure of individual differences in a population. Biometrics is a field that really was founded by Galton to try to understand the origin of individual differences in quantitative traits. Fundamental to Biometrics is to try to identify which genetic and environmental factors contribute to the phenotypic variance in a trait. And typically, at least at the initial stages, what biometricians do is they take a phenotypic variance, and again, the phenotypic variance is an index of the extent to which we differ from one another on a particular phenotype, and we, and they decompose it into a portion associated with genetic factors, the genetic variance, and a portion associated with environmental factors, environmental variance. We need to go through some mathematics here to really understand what biometricians were trying to do and ultimately to understand really what heritability is. So little notation here, the phenotypic variance is often denoted as V for variance sub P for phenotype. And it's decomposed into a genetic part plus an environmental part. At this very early stage, we can actually define the heritability coefficient, the heritability, and I'm going to come back to this repeatedly over the next couple of modules, but the heritability is the proportion of phenotypic variance that can be attributed to genetic factors. So as a proportion it's going to vary from zero, genetic factors account for none of the phenotypic variance, to 100% genetic factors that count for all of the phenotypic variance or individual differences in the trait. Complimentary to that is what we might call the environmentality, which is 1 minus the heritability. Now it turns out that not only do the biometricians decompose the phenotypic variance, but they can actually decompose the genetic variance and the environmental variance, as well. And again, to understand heritability we need to understand what they're doing when they're decomposing these various variant components. The genetic variance in the biometrician's formulation can be decomposed in two parts: an additive genetic part and a non-additive genetic part. These are the symbols for those two components. What is additive genetic variance and non-additive genetic variance? Well, on the next slide I try to explain this to you. Additive genetic variance, denoted by A, which would actually be pretty fundamental to what we'll be talking about in, in, in other phenotypes in this course, additive genetic variance corresponds to the effects of individual alleles that are added up across loci. And fundamental to the notion of what is additive is the notion is as we add these alleles up or their effects up over loci, the effect of the second allele doesn't depend upon what you had at the first allele. So the B1 allele's effect is giving you 1/2 cm increase in height regardless of what your A1 allele is, or whether or not you have an A1 allele. That's the fundamental notion of additivity. We can add up the B1 effect without worrying about what you had at the A locus or the C locus or the D locus. In contrast with the non-additive genetic effect is it corresponds to the effects of the alleles that depend upon the allele's other low side. So, for example, here, if the B1 allele increased your height by 1 cm, if it was paired with the A1 allele, but it didn't increase your height at all if paired with the A2 allele. We might imagine that if you had a lot of alleles that contributed to being taller, that there would be some synergy among those alleles so the effects might be greater, that there might be some interaction. What non-additive genetic effects are, is beginning to get at those genetic interactions, that the allele effects might depend on one another. Similarly, although I think probably more easily understood, the environmental effect, or variance component, can be decomposed in the basic biometrical formulation. The total environmental effect is decomposed into a portion that's, that's called the shared environmental effect, which is denoted by C, and the non-shared environmental effect, which is typically denoted by E. What are these two components? The shared environmental component is the environmental, it corresponds to environmental effects that individuals growing up in the same home share. And because they share those effects, they can potentially lead to their phenotypic similarity. They would correspond to things like the income level of the home, how parents approach child rearing, the neighborhood you live in, the schools you go to, if you, if siblings go to the same school and they usually do, and the consequences, the environment consequences of, if you have parent with psychopathology. Those would correspond to what biometricians would call shared environmental effects. They're denoted by C, which seems a little bit odd, but they're, rather than S, but the, it's an, it's an old symbol. And it actually refers to cultural effects. That's why it's, the letter C was used. In contrast, nonshared environmental effects, denoted by E, correspond to environmental effects that individuals growing up in the same home do not share. And because they don't share those environmental effects, those are the potential source for their phenotypic dissimilarity or differences. So to think about what nonshared environmental factors might be, think about the environmental factors if you have siblings or maybe some of you have twins. Think about the environmental factors you and your twin or your sibling didn't share when you grew up together. They might be your peer group. They might be they might be accidents that happened to you. Maybe one of you was in a, a serious motor vehicle accident, the other wasn't. Or they might be, actually parents have general practices that they apply in the home, but they also can differentially treat individuals in the home. That differential parental treatment would be considered a non-shared environmental effect. So this next table is a bit complicated, but what I'm trying to do here is just summarize what the biometricians have done in their decomposition of the variance, of the phenotypic variance. And recall that phenotypic variance is a measure of individual differences. One of the things we're trying to understand in this course is why we differ from one another. So we're trying to understand the sources of phenotypic variance. There are genetic factors, additive and non-additive, environmental contributions, shared and non-shared. This is a brief description of those. These are the symbols I used in the formula. Sometimes it's convenient to use the raw symbols. But sometimes it's more convenient to think of these as a proportion of variance. And again, a proportion is something that goes anywhere from 0 to 100%. I've highlighted three of these in red, because what you'll see is what we're going to converge on is that, at least at the initial stages of analyzing phenotypes, behavioral geneticists focus on these three sources of variants. Additive genetic factors or A, shared environmental factors or C, non-shared environmental factors or E, sometimes called ACE, A, C, and E. Next time actually we'll develop or talk about how biometricians try to go about estimating the heritability coefficient.
