# 3.3: Simple Quantitative Genetics Models for Brownian Motion

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## Brownian motion under Genetic Drift

The simplest way to obtain Brownian evolution of characters is when evolutionary change is neutral, with traits changing only due to genetic drift (e.g. Lande 1976). To show this, we will create a simple model. We will assume that a character is influenced by many genes, each of small effect, and that the value of the character does not affect fitness. Finally, we assume that mutations are random and have small effects on the character, as specified below. These assumptions probably seem unrealistic, especially if you are thinking of a trait like the body size of a lizard! But we will see later that we can also derive Brownian motion under other models, some of which involve selection.

Consider the mean value of this trait, $\bar{z}$, in a population with an effective population size of *N*_{e} (this is technically the variance effective population )^{2}. Since there is no selection, the phenotypic character will change due only to mutations and genetic drift. We can model this process in a number of ways, but the simplest uses an "infinite alleles" model. Under this model, mutations occur randomly and have random phenotypic effects. We assume that mutations are drawn at random from a distribution with mean 0 and mutational variance *σ*_{m}^{2}. This model assumes that the number of alleles is so large that there is effectively no chance of mutations happening to the same allele more than once - hence, "infinite alleles." The alleles in the population then change in frequency through time due to genetic drift. Drift and mutation together, then, determine the dynamics of the mean trait through time.

If we were to simulate this infinite alleles model many times, we would have a set of evolved populations. These populations would, on average, have the same mean trait value, but would differ from each other. Let’s try to derive how, exactly, these populations^{3} evolve.

If we consider a population evolving under this model, it is not difficult to show that the expected population phenotype after any amount of time is equal to the starting phenotype. This is because the phenotypes don’t matter for survival or reproduction, and mutations are assumed to be random and symmetrical. Thus,

\[ E[\bar{z}(t)] = \bar{z}(0) \label{3.1}\]

Note that this equation already matches the first property of Brownian motion.

Next, we need to also consider the variance of these mean phenotypes, which we will call the between-population phenotypic variance (*σ*_{B}^{2}). Importantly, *σ*_{B}^{2} is the same quantity we earlier described as the “variance” of traits over time – that is, the variance of mean trait values across many independent “runs” of evolutionary change over a certain time period.

To calculate *σ*_{B}^{2}, we need to consider variation within our model populations. Because of our simplifying assumptions, we can focus solely on additive genetic variance within each population at some time *t*, which we can denote as *σ*_{a}^{2}. Additive genetic variance measures the total amount of genetic variation that acts additively (i.e. the contributions of each allele add together to predict the final phenotype). This excludes genetic variation involving interacions between alleles, such as dominance and epistasis (see Lynch and Walsh 1998 for a more detailed discussion). Additive genetic variance in a population will change over time due to genetic drift (which tends to decrease *σ*_{a}^{2}) and mutational input (which tends to increase *σ*_{a}^{2}). We can model the expected value of *σ*_{a}^{2} from one generation to the next as (Clayton and Robertson 1955; Lande 1979, 1980):

\[ E[\sigma_a^2 (t+1)]=(1-\frac{1}{2 N_e})E[\sigma_a^2 (t)]+\sigma_m^2 \label{3.2}\]

where t is the elapsed time in generations, *N*_{e} is the effective population size, and *σ*_{m}^{2} is the mutational variance. There are two parts to this equation. The first, \((1-\frac{1}{2 N_e})E[\sigma_a^2 (t)]\), shows the decrease in additive genetic variance each generation due to genetic drift. The rate of decrease depends on effective population size, *N*_{e}, and the current level of additive variation. The second part of the equation describes how additive genetic variance increases due to new mutations (*σ*_{m}^{2}) each generation.

If we assume that we know the starting value at time 0, *σ*_{aStart}^{2}, we can calculate the expected additive genetic variance at any time *t* as:

\[ E[\sigma_a^2 (t)]={(1-\frac{1}{2 N_e})}^t [\sigma_{aStart}^2 - 2 N_e \sigma_m^2 ]+ 2 N_e \sigma_m^2 \label{3.3}\]

Note that the first term in the above equation, \({(1-\frac{1}{2 N_e})}^t\), goes to zero as *t* becomes large. This means that additive genetic variation in the evolving populations will eventually reach an equilibrium between genetic drift and new mutations, so that additive genetic variation stops changing from one generation to the next. We can find this equilibrium by taking the limit of Equation \ref{3.3} as *t* becomes large.

\[\lim_{t → ∞}E[σ_a^2(t)] = 2N_eσ_m^2 \label{3.4}\]

Thus the equilibrium genetic variance depends on both population size and mutational input.

We can now derive the between-population phenotypic variance at time *t*, *σ*_{B}^{2}(*t*). We will assume that *σ*_{a}^{2} is at equilibrium and thus constant (equation 3.4). Mean trait values in independently evolving populations will diverge from one another. Skipping some calculus, after some time period *t* has elapsed, the expected among-population variance will be (from Lande 1976):

\[ \sigma_B^2 (t)=\frac{t \sigma_a^2}{N_e} \label{3.5} \]

Substituting the equilibrium value of *σ*_{a}^{2} from equation 3.4 into equation 3.5 gives (Lande 1979, 1980):

\[ \sigma_B^2 (t)=\frac{t \sigma_a^2}{N_e} = \frac{t \cdot 2 N_e \sigma_m^2}{N_e} = 2 t \sigma_m^2 \label{3.6}\]

Thie equation states that the variation among two diverging populations depends on twice the time since they have diverged and the rate of mutational input. Notice that for this model, the amount of variation among populations is independent of both the starting state of the populations and their effective population size. This model predicts, then, that long-term rates of evolution are dominated by the supply of new mutations to a population.

Even though we had to make particular specific assumptions for that derivation, Lynch and Hill (1986) show that Equation \ref{3.6} is a general result that holds under a range of models, even those that include dominance, linkage, nonrandom mating, and other processes. Equation \ref{3.6} is somewhat useful, but we cannot often measure the mutational variance *σ*_{m}^{2} for any natural populations (but see Turelli 1984). By contrast, we sometimes do know the heritability of a particular trait. Heritability describes the proportion of total phenotypic variation within a population (*σ*_{w}^{2}) that is due to additive genetic effects (*σ*_{a}^{2}):

\[h^2=\frac{\sigma_a^2}{\sigma_w^2}.\]

We can calculate the expected trait heritability for the infinite alleles model at mutational equilibrium. Substituting Equation \ref{3.4}, we find that:

\[ h^2 = \frac{2 N_e \sigma_m^2}{\sigma_w^2} \label{3.7}\]

So that:

\[ \sigma_m^2 = \frac{h^2 \sigma_w^2}{2 N_e} \label{3.8} \]

Here, *h*^{2} is heritability, *N*_{e} the effective population size, and *σ*_{w}^{2} the within-population phenotypic variance, which differs from *σ*_{a}^{2} because it includes all sources of variation within populations, including both non-additive genetic effects and environmental effects. Substituting this expression for *σ*_{w}^{2} into Equation \ref{3.6}, we have:

\[ \sigma_B^2 (t) = 2 \sigma_m^2 t = \frac{h^2 \sigma_w^2 t}{N_e} \label{3.9}\]

So, after some time interval *t*, the mean phenotype of a population has an expected value equal to the starting value, and a variance that depends positively on time, heritability, and trait variance, and negatively on effective population size.

To derive this result, we had to make particular assumptions about normality of new mutations that might seem quite unrealistic. It is worth noting that if phenotypes are affected by enough mutations, the central limit theorem guarantees that the distribution of phenotypes within populations will be normal – no matter what the underlying distribution of those mutations might be. We also had to assume that traits are neutral, a more dubious assumption that we relax below - where we will also show that there are other ways to get Brownian motion evolution than just genetic drift!

Note, finally, that this quantitative genetics model predicts that traits will evolve under a Brownian motion model. Thus, our quantitative genetics model has the same statistical properties of Brownian motion. We only need to translate one parameter: *σ*^{2} = *h*^{2}*σ*_{w}^{2}/*N*_{e}^{4}.

## Brownian Motion under Selection

We have shown that it is possible to relate a Brownian motion model directly to a quantitative genetics model of drift. In fact, there is some temptation to equate the two, and conclude that traits that evolve like Brownian motion are not under selection. However, this is incorrect. More specifically, an observation that a trait is evolving as expected under Brownian motion is not equivalent to saying that that trait is not under selection. This is because characters can also evolve as a Brownian walk even if there is strong selection – as long as selection acts in particular ways that maintain the properties of the Brownian motion model.

In general, the path followed by population mean trait values under mutation, selection, and drift depend on the particular way in which these processes occur. A variety of such models are considered by Hansen and Martins (1996). They identify three very different models that include selection where mean traits still evolve under an approximately Brownian model. Here I present univariate versions of the Hansen-Martins models, for simplicity; consult the original paper for multivariate versions. Note that all of these models require that the strength of selection is relatively weak, or else genetic variation of the character will be depleted by selection over time and the dynamics of trait evolution will change.

One model assumes that populations evolve due to directional selection, but the strength and direction of selection varies randomly from one generation to the next. We model selection each generation as being drawn from a normal distribution with mean 0 and variance *σ*_{s}^{2}. Similar to our drift model, populations will again evolve under Brownian motion. However, in this case the Brownian motion parameters have a different interpretation:

\[ \sigma_B^2=(\frac{h^2 \sigma_W^2}{N_e} +\sigma_s^2)t \label{3.10}\]

In the particular case where variation in selection is much greater than variation due to drift, then:

\[σ_B^2 ≈ σ_s^2 \label{3.11}\]

That is, when selection is (on average) much stronger than drift, the rate of evolution is completely dominated by the selection term. This is not that far fetched, as many studies have shown selection in the wild that is both stronger than drift and commonly changing in both direction and magnitude from one generation to the next.

In a second model, Hansen and Martins (1996) consider a population subject to strong stabilizing selection for a particular optimal value, but where the position of the optimum itself changes randomly according to a Brownian motion process. In this case, population means can again be described by Brownian motion, but now the rate parameter reflects movement of the optimum rather than the action of mutation and drift. Specifically, if we describe movement of the optimum by a Brownian rate parameter *σ*_{E}^{2}, then:

\[σ_B^2 ≈ σ_E^2 \label{3.12}\]

To obtain this result we must assume that there is at least a little bit of stabilizing selection (at least on the order of 1/*t*_{ij} where *t*_{ij} is the number of generations separating pairs of populations; Hansen and Martins 1996).

Again in this case, the population is under strong selection in any one generation, but long-term patterns of trait change can be described by Brownian motion. The rate of the random walk is totally determined by the action of selection rather than drift.

The important take-home point from both of these models is that the pattern of trait evolution through time under this model still follows a Brownian motion model, even though changes are dominated by selection and not drift. In other words, Brownian motion evolution does not imply that characters are not under selection!

Finally, Hansen and Martins (1996) consider the situation where populations evolve following a trend. In this case, we get evolution that is different from Brownian motion, but shares some key attributes. Consider a population under constant directional selection, *s*, so that:

\[ E[\bar{z}(t+1)]=\bar{z}(t) + h^2 s \label{3.13}\]

The variance among populations due to genetic drift after a single generation is then:

\[ \sigma_B^2 = \frac{h^2 \sigma_w^2}{N_e} \label{3.14}\]

Over some longer period of time, traits will evolve so that they have expected mean trait value that is normal with mean:

\[ E[\bar{z}(t)]=t \cdot (h^2 s) \label{3.15}\]

We can also calculate variance among species as:

\[ \sigma_B^2(t) = \frac{h^2 \sigma_w^2 t}{N_e} \label{3.16}\]

Note that the variance of this process is exactly identical to the variance among populations in a pure drift model (equation 3.9). Selection only changes the expectation for the species mean (of course, we assume that variation within populations and heritability are constant, which will only be true if selection is quite weak). Furthermore, with comparative methods, we are often considering a set of species and their traits in the present day, in which case they will all have experienced the same amount of evolutionary time (*t*) and have the same expected trait value. In fact, equations \ref{3.14} and \ref{3.16} are exactly the same as what we would expect under a pure-drift model in the same population, but starting with a trait value equal to \(\bar{z}(0) = t \cdot (h^2 s)\). That is, from the perspective of data only on living species, these two pure drift and linear selection models are statistically indistinguishable. The implications of this are striking: we can never find evidence for trends in evolution studying only living species (Slater et al. 2012).

In summary, we can describe three very different ways that traits might evolve under Brownian motion – pure drift, randomly varying selection, and varying stabilizing selection – and one model, constant directional selection, which creates patterns among extant species that are indistinguishable from Brownian motion. And there are more possible models out there that predict the same patterns. One can never tell these models apart by evaluating the qualitative pattern of evolution across species - they all predict the same pattern of Brownian motion evolution. The details differ, in that the models have Brownian motion rate parameters that differ from one another and relate to measurable quantities like population size and the strength of selection. Only by knowing something about these parameters can we distinguish among these possible scenarios.

You might notice that none of these “Brownian” models are particularly detailed, especially for modeling evolution over long time scales. You might even complain that these models are unrealistic. It is hard to imagine a case where a trait might be influenced only by random mutations of small effect over many alleles, or where selection would act in a truly random way from one generation to the next for millions of years. And you would be right! However, there are tremendous statistical benefits to using Brownian models for comparative analyses. Many of the results derived in this book, for example, are simple under Brownian motion but much more complex and different under other models. And it is also the case that some (but not all) methods are robust to modest violations of Brownian motion, in the same way that many standard statistical analyses are robust to minor variations of the assumptions of normality. In any case, we will proceed with models based on Brownian motion, keeping in mind these important caveats.