Difference between revisions of "Drummond:PopGen"

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(Evolution is linear on a log-odds scale)
(Diffusion approximation: Added statistical analysis section.)
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==Diffusion approximation==
 
==Diffusion approximation==
 
Insert math here.
 
Insert math here.
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==Statistical analysis of relative growth rates==
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We have three strains, <math>i</math>, <math>j</math> and <math>r</math>, where <math>r</math> is a reference strain.
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Strains <math>i</math> and <math>j</math> have fitness <math>w_i = e^{r_i}</math> and <math>w_j=e^{r_j}</math>.  Define the selection coefficient <math>s_{ij} = \ln \frac{w_i}{w_j} = r_i - r_j</math> as usual.
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We have data consisting of triples (number of generations, number of cells of type <math>i</math>, number of cells of type <math>r</math>).
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What is the best estimate, and error, on <math>s_{ij}</math>?
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===Model===
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Given
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Let <math>\Pr(s_{ij}=t) = \mathcal{N}(t;\mu_{ij}, \sigma^2_{ij})</math>.
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===Maximum-likelihood approach===
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===Bayesian approach===

Revision as of 07:43, 22 April 2010

Introduction

Here I will treat some basic questions in population genetics. For personal reasons, I tend to include all the algebra.

Per-generation and instantaneous growth rates

What is the relationship between per-generation growth rates and the Malthusian parameter, the instantaneous rate of growth?

Let be the number of organisms of type at time , and let be the per-capita reproductive rate per generation. If counts generations, then

and

Now we wish to move to the case where is continuous and real-valued. As before,

but now

where the last simplification follows from L'Hôpital's rule. Explicitly, let . Then

The solution to the equation

is
Note that the continuous case and the original discrete-generation case agree for all integer values of . We can define the instantaneous growth rate for convenience.

Continuous rate of change

If two organisms grow at different rates, how do their proportions in the population change over time?

Let and be the instantaneous rates of increase of type 1 and type 2, respectively. Then

With the total population size
we have the proportion of type 1
Define the fitness advantage
Given our interest in understanding the change in gene frequencies, our goal is to compute the rate of change of .

This result says that the proportion of type 1 changes most rapidly when and most slowly when is very close to 0 or 1.

Evolution is linear on a log-odds scale

The logit function , which takes , induces a more natural space for considering changes in frequencies. Rather than tracking the proportion of type 1 or 2, we instead track their log odds. In logit terms, with ,

This differential equation has the solution

showing that the log-odds of finding type 1 changes linearly in time, increasing if and decreasing if .

Diffusion approximation

Insert math here.

Statistical analysis of relative growth rates

We have three strains, , and , where is a reference strain. Strains and have fitness and . Define the selection coefficient as usual. We have data consisting of triples (number of generations, number of cells of type , number of cells of type ).

What is the best estimate, and error, on ?

Model

Given Let .

Maximum-likelihood approach

===Bayesian approach===