Simulating Allele Counts in a population using R

This post is inspired by the Week 7 lectures of the Coursera course "Introduction to Genetics and Evolution" (I highly recommend this course for anyone interested in genetics, BTW.) Professor Noor uses a Univ Washington software called AlleleA1 for trying out scenarios.

We can just as well use R to get an intuitive feel for how Alleles and Genotypes propagate or die out in populations.

Basic Scenario

There are N individuals in an isolated island. Say, we are interested in two specific Alleles (Big "A", or small "a"). This in turn means that they can have 3 types of genotypes: AA, Aa or aa. The individuals mate in pairs, and produce two offspring and die out. (Thus the total population remains the same generation after generation.)
The genotype of the offspring depends on those of the parents. A 'gamete' has only one parental allele, depending on what the parent's genotype was. AA type parent can only product gamete type A, aa parent can only produce gamete type a, but Aa can produce either type of gamete.

A Punnett square of parents gametes to offspring's genotypes. 

  | A  | a

 ---------------- 

A | AA | Aa 

a | Aa | aa 

With these simple rules, we can use R Simulation scripts to observe what happens to the Allele Frequencies over generations. (The goal here is to learn to use R for Monte Carlo simulations.)

Writing the R Script from scratch

 I toyed around with the idea of using character strings for the genotypes and the alleles. But then I realized that are only three types and I could just as easily represent them with the numbers 1, 2, 3 as a simple R vector.


With that done, we can write very simple functions for the procreation process.
With these useful functions, we can take one generation and produce another, 2 offspring for each set of 2 parents.
Putting it all together to generate multiple trials:

We also need to compute the Allele counts for each generation, and for plotting I use ggplot.



Using this simple Monte Carlo "toy" we can develop quite a bit of intuition.



For small starting populations, either the big A or the small a allele takes over the entire population fairly quickly. Given large enough number of generations, invariably one of the alleles gets wiped out.














As one example, we can see that even a small increase in the probability of Allele A to be 0.53 (up from 0.5) makes it take over quite dramatically.

Conversely, setting it to any value under 0.5 means that the Big A allele gets wiped out of the entire population.

The entire R script can be found here. You can download the code and try playing with various starting scenarios, changing the starting population counts, generations and probabilities.


References:

1. Coursera.org (Introduction to Genetics and Evolution by Md. Noor, Week 7 lectures) 
2. AlleleA1 software at Univ Washington

Simulating Population Growth in Cities Using R

R is great for anyone who wants to get started on learning Simulation. (Both Discrete Event or Agent-based, with stochastic elements in the process.)

This post is inspired by Matt Asher's "quick-and-dirty" R simulation work on Population Growth. Matt uses it to create aRt. I felt that his core idea provided a very good framework to introduce various concepts of simulation.

The Basic Idea behind this Simulation

We take an area (it could be a country, state, city) that is made up of units (cells) that can be settled (occupied).
We allow the initial 'pioneers' to settle first. (Seeding the cells)
We then establish some rules for future residents to settle. (Settling Rules)
Plot the area, and collect some stats to study the growth of the population under different starting conditions and different settling rules.

Pseudo-Code for the Simulation

Define parameters: Area Width, Height

Number of Pioneers P and future settlers S

for each pioneer p{ 

  settle p according to seeding rules in a valid cell in the area

}

for each settler S{

  Find an unoccupied Cell.

  Settle S in that cell, if the settling criteria permits it

}

Print Iterations Stats
Print out the Plot of the Area

A Few Changes from Matt's R code

1. I switched from his matrix to a dataframe. I also switched from his image to ggplot.
2. Delineated the initial parameters, seeding and settling clearly so that anyone who downloads the code can experiment
3. Created a few different Seeding Functions for the pioneers (beyond random seeding)
4. Matt allowed settling if the new cell was adjacent to an occupied cell. I played around with a few different population settling functions.  (rectilinear, diagonal etc).
5. Added a function to collect a few statistics regarding population growth
6. Ran the simulation a few 100 times (without plotting) to compare different growth and settling schemes

Color Coding the Population

• Black cells : Unoccupied
• Blue cells: Pioneers
• Orange cells: Settlers who found a cell in 1-2 attempts
• Yellow cells: Settlers who found a cell after more than 2 scouting attempts

The full R code I used for these R can be found here.

Here are some runs with a 30x30 area.

Here's what one run looked like. (Blue cells are the initial pioneers. The other colors represent the number of steps taken for them to find a home cell.)

Trying Out Different Seeding Schemes (for the pioneers)

As opposed to starting out with randomly strewn cells, what if we started out with a city downtown (a densely populated central rectangle) and let the settlers follow them?



I tried a rectangluar ring (annular cells for pioneers) starting point to see how the population grew inside and outside. So I used this seeding function:

One final seeding scheme I tried was to have two thick bands (columns of pioneer cells) and to let the population grow from there. Seeding function:

 

 

 

 

 

 

Different Population Growth Schemes

The default settling rule that Matt Asher uses is a very good one, one that I like. If a given cell is adjacent to any occupied cell (one of its 8 neighbors) then it becomes a valid cell to settle down in.

Just for experimentation, I tried a few other schemes.

1. Rectilinear Growth

Rule: You can only settle down in the 4 cells that are due N, S, East or West of an occupied cell.

2. Diagonal Growth

Rule: A settler can occupy any cell that is diagonally adjacent to another occupied cell.

The code can be found here.

 

 

 

 

 

 

Collecting Stats

Now that we have all the functionality that we wanted, it is time to start collecting some statistics for each run ('replication'). There are many possibilities for what we could collect in a population simulation.
Here are a couple:

• How many settlers found a 'home cell?'
• How many scouting attempts did each settler make before they found a home?

Making Multiple Replications

 

>st
  Iter FoundHome NumSettlers  Percent
1    1       243         350 69.42857
2    2       214         350 61.14286
3    3       213         350 60.85714
4    4       258         350 73.71429
5    5       235         350 67.14286

The full R code I used for these runs can be found here. Feel free to give it a spin and try out your own settling/growth schemes.
Reference: Matt Asher's post on his StatisticsBlog