# Basic Modelling in Quantitative Genetics

I am pretty bad in thinking quantitative genetics models. I am trying to get some basic understanding of modelling the evolution of a quantitative trait. I am therefore asking for help to analyze a very simple model. I welcome any explanation of some other classic model of the evolution of quantitative traits.

Scenario

Consider a haploid population of constant size $N$. The fitness of the individuals is determined exclusively by a single quantitative trait $z$. $l$ loci codes for this trait. The genetic value at each locus adds up to give the quantitative trait $z$. The mutation rate at each locus is $\mu$. A mutation changes the genetic value at any given locus by $+ \Delta z$ with probability $\frac{1}{2}$ and by $- \Delta z$ with probability $\frac{1}{2}$. If you prefer to consider the effects of a mutation to be drawn from a normal distribution with mean $=0$ and variance $= \sigma^2$, please feel free to do so. The fitness $w(z)$ of an individual is given as a gaussian function of its quantitative trait $z$

$$w(z)=\exp\bigl(-Vz^2\bigr),$$

where $V$ is the strength of selection. The optimal phenotype is $0$, that is $w(0)=1$. The larger is $V$, the more severe it is to be at a given distance to the optimal phenotype. If you want to assume that the fitness $w(z)$ is some other function of the trait $z$, such as the even more simple $w(z)=z$, please feel free to do so. For simplicity, we assume no environmental variance on the quantitative trait.

Questions

What is the equilibrium mean fitness $\bar w$ of the population?

What is the equilibrium genetic variance (=phenotypic variance) for fitness $w$ and for phenotype $z$?

• What have you gotten, so far? This reads like a homework question. – blep May 6 '15 at 0:51
• I would have love receiving this kind of homework. It is not a homework question. Because I was trained in Europe, the last homework I had to do was probably during the first year of my bachelor degree or even in High school. For the moment, I pretty much developed the basis of a model (presented in the post) to think the evolution of quantitative trait but this is it, I fail to bring it any further to answer my question(s). – Remi.b May 6 '15 at 2:06
• @dd3 This is a little complicated than a usual homework. – WYSIWYG May 30 '15 at 14:22

Trait z is represented by k genes: z1....zk (I am using k instead of l because the former is visually differentiable from 1 ). For simplicity let's assume that there is only one mutable site in a gene. So a mutation can impart a change of $\pm \Delta z$. Starting from initial state of z at zero, the system will proceed to equilibrium where the rate of forward mutation would be same as that of backward mutation; since the rates are same the deterministic steady state of z should be zero.
 Mean fitness = mean forward mutation — mean backward mutations = 0
You can also model the mutation events as a simple random walk. The mean of that is 0 and the variance is n. But what about large n; I am not sure. You should read about random walks or ask in CrossValidated. With $t\to \infty$, you would get a lot of variants but variants would be filtered out by selection (as the optimum is at 0). I think selection can be incorporated in the random walk with death rate being proportional to the distance from 0. But again I do not have much experience with such hybrid models.