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Tag Archives: breeding

Bane of the Garden Gnomes

Last week, we discussed the use of the Hardy Weinberg equations to estimate the rate of change in population under conditions of extreme selection, i.e. total elimination of one phenotype. This is essentially the goal of any sort of eugenics program. As an example of a way that this kind of policy could creep into culture, we watched GATTACA. Besides, it’s just a good film.

The purpose of the Hardy Weinberg equations is to model conditions under which allele frequencies can NOT change from one generation to the next. Therefore, it is evident that these are exactly those conditions that are responsible for allele frequency changes.

These conditions are:

  1. No Mutation
  2. No Selection (survival)
  3. No Sexual Selection
  4. No Genetic Drift –due to occasional fluctuations occurring by chance
  5. No Gene Flow – immigration / emigration

In order to prevent the random changes in allele populations stipulated in #4, we also need a sufficiently large population, where sufficient is likely definable by someone with better probability-computing skills than my own. (I feel like going off half-cocked on notions of probability and finite vs infinite time, but I’ll spare you).

Anyway, if we know something about the population, we might be able to work out the allele frequencies and then compute our theoretical proportions for the next generation from the equations…

p+q = 1,

where p and q are the frequencies of the (only) two alleles we are calculating.

and

p2+2pq+q2 = 1

where each unit above represents the proportion of that genotype.

Mathematically, these equations provide insight into how rapidly the rate of an allele in a population could be eliminated if reproduction was prevented in a specific group. (This sounds completely esoteric without using an example, so let’s come up with one…)

A Healthy Gnome Couple

A Healthy Gnome Couple

Imagine a population of fictional creatures – Garden Gnomes.

These gnomes have a recessive allele that makes them susceptible to a fungal disease. We’ll call the two alleles for this trait H – hearty (resistant) and h– weak (susceptible)

There was recently a new law passed amongst the gnomes forbidding susceptible gnomes from breeding (let’s imagine that the H allele is apparent by a normal complexion and the h allele is apparent by a jaundiced complexion. Like susceptibility to disease, jaundice only appears in the homozygous recessive (hh) gnomes.)

Imagine a population starting with equal allele frequencies, p=q=0.5.

p2+2pq+q2 = 1

will give us genotype frequencies of:

25% HH   + 50% Hh + 25%hh = 1

for the present generation.

Now, if we start our draconian, anti-jaundiced gnome policy and prevent breeding of these individuals, then this generation‘s breeding population only consists of the HH and Hh gnomes, where only the heterozygotes will contribute the h allele to the next generation.

If we call the next generation q1, we can estimate the new proportion of the q allele in the population as the frequency of the heterozygote over the total population excluding the hh gnomes:

No wonder they want to get rid of these guys

No wonder they want to get rid of these guys

After one generation, the frequency of the H allele is now 67%.

Since the same process would occur generation after generation (as long as the law was in place – and followed), we can determine the frequency of q at any generation, where n is the generation number.

  1. From this information, try calculating the frequency of both alleles after the policy has been in place for 5 generations.
  2. How long will it take to completely eliminate the h allele?
  3. How would this change if the susceptible (h) allele is dominant?

 
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Posted by on April 28, 2014 in Education, Uncategorized

 

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Eugenics in film: GATTACA

Tomorrow in class, where we have recently been discussing Mendelian Genetics and its twisted perversion,Eugenics; we will be watching the dystopian film, GATTACA. The story is good enough, but what I find compelling is the way that society has become the way it is. The population has been recently ‘improved’ by the production (?) of ‘designer babies‘. The method seems very much like one that I can honestly imagine working its way into present society. These children aren’t fabricated, they’re yours. Only – just the best parts of you.

Society fell for Eugenics once – and not just Hitler. I know that’s where your mind is going. But there were plenty of Eugenics believers here in the USA as well. Just ask this happy family:

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They’re smiling because they’ve just won the ‘medium family size’ medal for fittest family at the 1927 Kansas Free Fair.

It was a time when Mendelian genetics was coming to be understood in principle by a wider audience following the work’s ‘rediscovery’ by Hugo de Vries and Carl Correns in 1900. The main idea behind Eugenics was that better people could be made through selective breeding of only the right kinds of folks. The term Eugenics was coined by Sir Frances Galton, who actually a great thinker contributing several key ideas in the field of statistics and inventing the sciences of meteorology and psychometrics. His books, Hereditary Genius (1869) and Essays on Eugenics (1909) lay the groundwork for thinking about which traits are inherited and which are learned in humans. In exploring the idea of hereditary greatness, he also explores the hereditary of less desirable genes. 

What he concluded was that great, geniuses like himself simply aren’t having enough children while the lowly dregs of humanity were breeding like bunnies. Well, there’s a couple of ways to put an end to that nonsense. 

Here is an excerpt from a Scientific American editorial of the time (1911) lauding Galton’s ideas: 

ADA JUKE is known to anthropologists as the “mother of criminals.” From her there were directly descended one thousand two hundred persons. Of these, one thousand were criminals, paupers, inebriates, insane, or on the streets. That heritage of crime, disease, inefficiency and immorality cost the State of New York about a million and a quarter dollars for maintenance directly. What the indirect loss was in property stolen, in injury to life and limb, no one can estimate.

Suppose that Ada Juke or her immediate children had been prevented from perpetuating the Juke family. Not only would the State have been spared the necessity of supporting one thousand defective persons, morally and physically incapable of performing the functions of citizenship, but American manhood would have been considerably better off, and society would have been free from one taint at least.

The Free Kansas Fair of 1927 had more than just pretty families. It also proposed just how even prettier families could show up in the years to come:

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Why is Blind in quotes? Is that, perhaps, a suggestion? Or is it just poor grammar?

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Do you suppose ‘Pauperism’ is dominant or recessive? Either way, it’s bad. How can they go around having no money like that? Have they no shame?

 
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Posted by on April 23, 2014 in Uncategorized

 

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Drift

ImageIn population genetics there are two equations that allow us to estimate the frequency of alleles within a population and also to estimate the number of homozygotes vs heterozygotes for a recessive trait. These equations are known today as the Hardy-Weinberg equations because they were simultaneous proposed by two independent scientists. Like many equations, they assume a model that is not exactly reflective of the real world, however they do lend us an understanding of the rules of the system.

The two equations are:

q + p = 1

q2 +2pq +q2 = 1

It’s that easy. In each of these equations p stands for the frequency of one allele in a population and q stands for the frequency of the other allele. Assuming there are only two alleles, they must add up to 100%, represented by the decimal number 1 here.

In order to use these equations, certain conditions must be adhered to.

  1. No gene flow (immigration / emigration)
  2. No sexual selection
  3. No survival selection
  4. No mutations
  5. No genetic drift

 The last one is the one that has been interesting me lately.

 What is genetic drift? What it describes are statistical anomalies, like a run of ‘Red’ on the Roulette Wheel or an unexpectedly long string of ‘Heads’ when tossing a coin.

 What happens during genetic drift is that one allele becomes favored just because of such a statistical swing. But unlike roulette or coin tosses, when an allele loses out for a number of generations, it stands a diminishing chance of being seen again. The statistical anomaly becomes ‘hard-coded’ and self-reinforcing, such that eventually alleles disappear.

The key is that small samples allow genetic drift to happen more often, while larger populations tend to not see this occur. Using out coin toss example, if you toss a coin ten times, it is not especially surprising when you get 8 ‘heads’ and 2 ‘tails’. Whereas, in a toss of 1000 coins, getting 800 ‘heads’ is nearly inconceivable.

I encountered this while coding a genetics simulation program (note: my simulation uses a Wright-Fisher model that has distinct, non-overlapping generations).  I wrote the program and started testing it by allowing random breeding to occur over 100 generations or so. I started using only 100 animals in my simulation, but regularly saw one allele outcompete all others, meaning that the population had lost diversity.

Below is an example with 100 organisms with four alleles for the gene breeding randomly for 200 generations.

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I was sure it was a problem with my algorithm. Then I started increasing the number of animals and the ‘problem’ went away.

Here’s a second experiment at the other end of the spectrum using 50,000 animals also with four alleles breeding for 200 generations. I’ve forced Excel to graph this out on the same axis.

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All this, just to demonstrate to myself that the prohibition against genetic drift is actually another way of saying, “This only works with large populations.” 

What interested me is how to know whether your population is large enough to ‘resist’ genetic drift. And, how quickly will genetic drift drive alleles to fixation / loss?

“The expected number of generations for fixation to occur is proportional to the population size, such that fixation is predicted to occur much more rapidly in smaller populations.”

Not surprisingly, there is an equation designed to predict the time (# of generations) before an allele is lost by drift.

The expected time for the neutral allele to be lost through genetic drift can be calculated as

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where T is the number of generations, Ne is the effective population size, and p is the initial frequency for the given allele.

(this section is informed greatly by the work of Otto and Whitlock at the University of Columbia, Vancouver. ) 

Sometimes having a computer simulation comes in handy to help get a better look at how these rules apply given different populations. I’d like to get this simulation built into a simple app for either desktop or mobile device to make public, but I have been having a lot of difficulty making the leap from a program running in the console to something worth sharing.

 
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Posted by on August 8, 2013 in Uncategorized

 

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