# Difference between revisions of "Drummond:PopGen"

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{{Drummond_Top}} | {{Drummond_Top}} | ||

<div style="width: 750px"> | <div style="width: 750px"> | ||

+ | ==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== | ||

<p> | <p> | ||

− | Let <math>n_i(t)</math> be the number of organisms of type <math>i</math> at time <math>t</math>, and let <math>R</math> be the ''per-capita reproductive rate'' | + | What is the relationship between per-generation growth rates and the Malthusian parameter, the instantaneous rate of growth? |

+ | </p> | ||

+ | <p> | ||

+ | Let <math>n_i(t)</math> be the number of organisms of type <math>i</math> at time <math>t</math>, and let <math>R</math> be the ''per-capita reproductive rate per generation''. If <math>t</math> counts generations, then | ||

:<math>n_i(t+1) = n_i(t)R\!</math> | :<math>n_i(t+1) = n_i(t)R\!</math> | ||

and | and | ||

Line 60: | Line 66: | ||

is | is | ||

:<math>n_i(t) = n_i(0) e^{t\ln R} = n_i(0) R^{t}.\!</math> | :<math>n_i(t) = n_i(0) e^{t\ln R} = n_i(0) R^{t}.\!</math> | ||

− | Note that the continuous case and the original discrete-generation case agree for all values of <math>t</math>. We can define the ''instantaneous rate | + | Note that the continuous case and the original discrete-generation case agree for all integer values of <math>t</math>. We can define the ''instantaneous growth rate'' <math>r = \ln R</math> for convenience. |

</p> | </p> | ||

+ | ==Continuous rate of change== | ||

+ | <p> | ||

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

+ | </p> | ||

+ | <p> | ||

Let <math>r_1</math> and <math>r_2</math> be the instantaneous rates of increase of type 1 and type 2, respectively. Then | Let <math>r_1</math> and <math>r_2</math> be the instantaneous rates of increase of type 1 and type 2, respectively. Then | ||

:<math>{dn_i(t) \over dt} = r_i n_i(t).</math> | :<math>{dn_i(t) \over dt} = r_i n_i(t).</math> | ||

With the total population size | With the total population size | ||

− | :<math>n(t) = n_1(t) + n_2(t)</math> | + | :<math>n(t) = n_1(t) + n_2(t)\!</math> |

we have the proportion of type 1 | we have the proportion of type 1 | ||

:<math>p(t) = {n_1(t) \over n(t)}</math> | :<math>p(t) = {n_1(t) \over n(t)}</math> | ||

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|- | |- | ||

| | | | ||

− | |<math>= {\ | + | |<math>= {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)</math> |

+ | |- | ||

+ | | | ||

+ | |<math>= {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)</math> | ||

+ | |- | ||

+ | | | ||

+ | |<math>= {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)</math> | ||

+ | |- | ||

+ | | | ||

+ | |<math>= {n_1(t) \over n(t)^2}\left(s n(t) - s n_1(t))\right)</math> | ||

+ | |- | ||

+ | | | ||

+ | |<math>= s{n_1(t) \over n(t)}\left(1 - {n_1(t) \over n(t)}\right)</math> | ||

+ | |- | ||

+ | | | ||

+ | |<math>= s p(t)(1-p(t))\!</math> | ||

|} | |} | ||

+ | This result says that the proportion of type 1, <math>p</math>, changes most rapidly when <math>p=0.5</math> and most slowly when <math>p</math> is very close to 0 or 1. | ||

+ | |||

+ | ==Evolution is linear on a log-odds scale== | ||

+ | The logit function <math>\mathrm{logit} (p) = \ln {p \over 1-p}</math>, which takes <math>p \in [0,1] \to \mathbb{R}</math>, 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 <math>L_p(t) \equiv \mathrm{logit} (p(t))\!</math>, | ||

+ | |||

+ | :{| | ||

+ | |<math>{\partial L_p(t) \over \partial t} </math> | ||

+ | |<math>= {\partial \over \partial t}\left(\ln {p(t) \over 1-p(t)}\right)</math> | ||

+ | |- | ||

+ | | | ||

+ | |<math>= {\partial \over \partial t}\left(\ln {n_1(t) \over n_2(t)}\right)</math> | ||

+ | |- | ||

+ | | | ||

+ | |<math>= {\partial \over \partial t}\left(\ln {n_1(0) \over n_2(0)} e^{st}\right)</math> | ||

+ | |- | ||

+ | | | ||

+ | |<math>= s. \!</math> | ||

+ | |} | ||

+ | |||

+ | This differential equation <math>L_p'(t) = s</math> has the solution | ||

+ | |||

+ | :<math>L_p(t) = L_p(0) + st\!</math> | ||

+ | |||

+ | showing that the log-odds of finding type 1 changes linearly in time, increasing if <math>s>0</math> and decreasing if <math>s<0</math>. | ||

==Diffusion approximation== | ==Diffusion approximation== | ||

+ | Insert math here. | ||

+ | |||

+ | ==Statistical analysis of relative growth rates== | ||

+ | We have three strains, <math>i</math>, <math>j</math> and <math>r</math>, where <math>r</math> is a reference strain. | ||

+ | 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. | ||

+ | We have data consisting of triples (<math>g=</math>number of generations, <math>n_i=</math>number of cells of type <math>i</math>, <math>n_r=</math>number of cells of type <math>r</math>). | ||

+ | We have data consisting of pairs (<math>g=</math>number of generations, <math>p_{ir}= n_i/n_r</math>) where <math>n_i</math>=number of cells of type <math>i</math> and <math>n_r=</math>number of cells of type <math>r</math>. | ||

+ | |||

+ | What is the best estimate, and error, on <math>s_{ij}</math>? | ||

+ | |||

+ | ===Model=== | ||

+ | Assuming exponential growth, <math>\ln p_{ir} = </math> | ||

+ | |||

+ | Let <math>\Pr(s_{ij}=t) = \mathcal{N}(t;\mu_{ij}, \sigma^2_{ij})</math>. | ||

+ | |||

+ | ===Maximum-likelihood approach=== | ||

+ | Add text. | ||

− | == | + | ===Bayesian approach=== |

+ | Add text. |

## Latest revision as of 18:40, 28 March 2011

## 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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n_i(t)}**
be the number of organisms of type **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle i}**
at time **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle t}**
, and let **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle R}**
be the *per-capita reproductive rate per generation*. If **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle t}**
counts generations, then

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n_i(t+1) = n_i(t)R\!}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n_i(t) = n_i(0)R^t.\!}**

Now we wish to move to the case where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle t}**
is continuous and real-valued.
As before,

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n_i(t+1) = n_i(t)R\!}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n_i(t+\Delta t)\!}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle =n_i(t)R^{\Delta t}\!}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n_i(t+\Delta t) - n_i(t)\!}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle = n_i(t)R^{\Delta t} - n_i(t)\!}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{n_i(t+\Delta t) - n_i(t)}{\Delta t}}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle =\frac{n_i(t)R^{\Delta t} - n_i(t)}{\Delta t}}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{n_i(t+\Delta t) - n_i(t)}{\Delta t}}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle =n_i(t) \frac{R^{\Delta t} - 1}{\Delta t}}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \lim_{\Delta t \to 0} \left[{n_i(t+\Delta t) - n_i(t) \over \Delta t}\right]}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle =\lim_{\Delta t \to 0} \left[ n_i(t) \frac{R^{\Delta t} - 1}{\Delta t}\right]}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{d n_i(t)}{dt}}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle =n_i(t) \lim_{\Delta t \to 0} \left[\frac{R^{\Delta t} - 1}{\Delta t}\right]}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{d n_i(t)}{dt}}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle =n_i(t) \ln R\!}**

where the last simplification follows from L'Hôpital's rule. Explicitly, let **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \epsilon=\Delta t}**
. Then

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \lim_{\Delta t \to 0} \left[{R^{\Delta t} - 1 \over \Delta t}\right]}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle = \lim_{\epsilon \to 0} \left[\frac{R^{\epsilon} - 1}{\epsilon}\right]}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle =\lim_{\epsilon \to 0} \left[\frac{\frac{d}{d\epsilon}\left(R^{\epsilon} - 1\right)}{\frac{d}{d\epsilon}\epsilon}\right]}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle =\lim_{\epsilon \to 0} \left[\frac{R^{\epsilon}\ln R}{1}\right]}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle =\ln R \lim_{\epsilon \to 0} \left[R^{\epsilon}\right]}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle =\ln R\!}**

The solution to the equation

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{d n_i(t)}{dt} = n_i(t) \ln R}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n_i(t) = n_i(0) e^{t\ln R} = n_i(0) R^{t}.\!}**

*instantaneous growth rate*

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle r = \ln R}**for convenience.

## Continuous rate of change

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

Let **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle r_1}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle r_2}**
be the instantaneous rates of increase of type 1 and type 2, respectively. Then

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle {dn_i(t) \over dt} = r_i n_i(t).}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n(t) = n_1(t) + n_2(t)\!}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle p(t) = {n_1(t) \over n(t)}}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle s \equiv s_{12} = r_1 - r_2\!}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle p(t)}**.

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle {\partial p(t) \over \partial t}}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle = {\partial \over \partial t}\left({n_1(t) \over n(t)}\right)}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle = {\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}}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle = {\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)}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle = {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)}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle = {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)}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle = {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)}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle = {n_1(t) \over n(t)^2}\left(s n(t) - s n_1(t))\right)}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle = s{n_1(t) \over n(t)}\left(1 - {n_1(t) \over n(t)}\right)}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle = s p(t)(1-p(t))\!}**

This result says that the proportion of type 1, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle p}**
, changes most rapidly when **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle p=0.5}**
and most slowly when **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle p}**
is very close to 0 or 1.

## Evolution is linear on a log-odds scale

The logit function **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathrm{logit} (p) = \ln {p \over 1-p}}**
, which takes **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle p \in [0,1] \to \mathbb{R}}**
, 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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle L_p(t) \equiv \mathrm{logit} (p(t))\!}**
,

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle {\partial L_p(t) \over \partial t} }****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle = {\partial \over \partial t}\left(\ln {p(t) \over 1-p(t)}\right)}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle = {\partial \over \partial t}\left(\ln {n_1(t) \over n_2(t)}\right)}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle = {\partial \over \partial t}\left(\ln {n_1(0) \over n_2(0)} e^{st}\right)}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle = s. \!}**

This differential equation **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle L_p'(t) = s}**
has the solution

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle L_p(t) = L_p(0) + st\!}**

showing that the log-odds of finding type 1 changes linearly in time, increasing if **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle s>0}**
and decreasing if **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle s<0}**
.

## Diffusion approximation

Insert math here.

## Statistical analysis of relative growth rates

We have three strains, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle i}**
, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle j}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle r}**
, where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle r}**
is a reference strain.
Strains **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle i}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle j}**
have fitness **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle w_i = e^{r_i}}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle w_j=e^{r_j}}**
. Define the selection coefficient **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle s_{ij} = \ln \frac{w_i}{w_j} = r_i - r_j}**
as usual.
We have data consisting of triples (**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle g=}**
number of generations, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n_i=}**
number of cells of type **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n_r=}**
number of cells of type **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle r}**
).
We have data consisting of pairs (**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle g=}**
number of generations, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle p_{ir}= n_i/n_r}**
) where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n_i}**
=number of cells of type **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n_r=}**
number of cells of type

What is the best estimate, and error, on **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle s_{ij}}**
?

### Model

Assuming exponential growth, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \ln p_{ir} = }**

Let **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Pr(s_{ij}=t) = \mathcal{N}(t;\mu_{ij}, \sigma^2_{ij})}**
.

### Maximum-likelihood approach

Add text.