Drummond:PopGen

(Difference between revisions)
 Revision as of 09:36, 15 July 2008 (view source) (→Continuous rate of change)← Previous diff Revision as of 15:37, 15 July 2008 (view source)Next diff → Line 1: Line 1: {{Drummond_Top}} {{Drummond_Top}}
+ ==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== ==Per-generation and instantaneous growth rates==

- Let $n_i(t)$ be the number of organisms of type $i$ at time $t$, and let $R$ be the ''per-capita reproductive rate'' per generation.  If $t$ counts generations, then + What is the relationship between per-generation growth rates and the Malthusian parameter, the instantaneous rate of growth? +

+

+ Let $n_i(t)$ be the number of organisms of type $i$ at time $t$, and let $R$ be the ''per-capita reproductive rate per generation''.  If $t$ counts generations, then :$n_i(t+1) = n_i(t)R\!$ :$n_i(t+1) = n_i(t)R\!$ and and Line 60: Line 66: is is :$n_i(t) = n_i(0) e^{t\ln R} = n_i(0) R^{t}.\!$ :$n_i(t) = n_i(0) e^{t\ln R} = n_i(0) R^{t}.\!$ - Note that the continuous case and the original discrete-generation case agree for all values of $t$.  We can define the ''instantaneous rate of increase'' $r = \ln R$ for convenience. + Note that the continuous case and the original discrete-generation case agree for all values of $t$.  We can define the ''instantaneous growth rate'' $r = \ln R$ for convenience.

==Continuous rate of change== ==Continuous rate of change== - +

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

+

Let $r_1$ and $r_2$ be the instantaneous rates of increase of type 1 and type 2, respectively.  Then Let $r_1$ and $r_2$ be the instantaneous rates of increase of type 1 and type 2, respectively.  Then :${dn_i(t) \over dt} = r_i n_i(t).$ :${dn_i(t) \over dt} = r_i n_i(t).$

the drummond lab

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 ni(t) be the number of organisms of type i at time t, and let R be the per-capita reproductive rate per generation. If t counts generations, then

$n_i(t+1) = n_i(t)R\!$
and
$n_i(t) = n_i(0)R^t.\!$

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

$n_i(t+1) = n_i(t)R\!$
but now
 $n_i(t+\Delta t)\!$ $=n_i(t)R^{\Delta t}\!$ $n_i(t+\Delta t) - n_i(t)\!$ $= n_i(t)R^{\Delta t} - n_i(t)\!$ $\frac{n_i(t+\Delta t) - n_i(t)}{\Delta t}$ $=\frac{n_i(t)R^{\Delta t} - n_i(t)}{\Delta t}$ $\frac{n_i(t+\Delta t) - n_i(t)}{\Delta t}$ $=n_i(t) \frac{R^{\Delta t} - 1}{\Delta t}$ $\lim_{\Delta t \to 0} \left[{n_i(t+\Delta t) - n_i(t) \over \Delta t}\right]$ $=\lim_{\Delta t \to 0} \left[ n_i(t) \frac{R^{\Delta t} - 1}{\Delta t}\right]$ $\frac{d n_i(t)}{dt}$ $=n_i(t) \lim_{\Delta t \to 0} \left[\frac{R^{\Delta t} - 1}{\Delta t}\right]$ $\frac{d n_i(t)}{dt}$ $=n_i(t) \ln R\!$

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

 $\lim_{\Delta t \to 0} \left[{R^{\Delta t} - 1 \over \Delta t}\right]$ $= \lim_{\epsilon \to 0} \left[\frac{R^{\epsilon} - 1}{\epsilon}\right]$ $=\lim_{\epsilon \to 0} \left[\frac{\frac{d}{d\epsilon}\left(R^{\epsilon} - 1\right)}{\frac{d}{d\epsilon}\epsilon}\right]$ $=\lim_{\epsilon \to 0} \left[\frac{R^{\epsilon}\ln R}{1}\right]$ $=\ln R \lim_{\epsilon \to 0} \left[R^{\epsilon}\right]$ $=\ln R\!$

The solution to the equation

$\frac{d n_i(t)}{dt} = n_i(t) \ln R$
is
$n_i(t) = n_i(0) e^{t\ln R} = n_i(0) R^{t}.\!$
Note that the continuous case and the original discrete-generation case agree for all values of t. We can define the instantaneous growth rate r = lnR for convenience.

Continuous rate of change

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

Let r1 and r2 be the instantaneous rates of increase of type 1 and type 2, respectively. Then

${dn_i(t) \over dt} = r_i n_i(t).$
With the total population size
$n(t) = n_1(t) + n_2(t)\!$
we have the proportion of type 1
$p(t) = {n_1(t) \over n(t)}$
$s \equiv s_{12} = r_1 - r_2\!$
 ${\partial p(t) \over \partial t}$ $= {\partial \over \partial t}\left({n_1(t) \over n(t)}\right)$ $= {\partial n_1(t) \over \partial t}\left({1 \over n(t)}\right) + n_1(t){-1 \over n(t)^2}{\partial n(t) \over \partial t}$ $= {\partial n_1(t) \over \partial t}\left({1 \over n(t)}\right) + n_1(t){-1 \over n(t)^2}\left({\partial n_1(t) \over \partial t} + {\partial n_2(t) \over \partial t}\right)$ $= {r_1 n_1(t) \over n(t)} - {n_1(t) \over n(t)^2}\left(r_1 n_1(t) + r_2 n_2(t)\right)$ $= {r_1 n_1(t) \over n(t)} - {n_1(t) \over n(t)^2}\left(r_1 n_1(t) + (r_1-s)(n(t)-n_1(t))\right)$ $= {r_1 n_1(t) \over n(t)} - {n_1(t) \over n(t)^2}\left(r_1 n(t) -s n(t) + s n_1(t))\right)$ $= {n_1(t) \over n(t)^2}\left(s n(t) - s n_1(t))\right)$ $= s{n_1(t) \over n(t)}\left(1 - {n_1(t) \over n(t)}\right)$ $= s p(t)(1-p(t))\!$
The logit function $\mathrm{logit} (p) = \ln {p \over 1-p}$, which takes $p \in [0,1] \to \mathbb{R}$, induces a more natural space for considering changes in frequencies. In logit terms,