# Question for Dennis about population genetics

Sorry @Swamidass, I was at the beach for a few days and only had sporadic internet access.

Can you show the math or link to it?

Going from a continuous equation (like the Schrodinger Equation of the diffusion equation) to a discrete equation is extremely well-studied, because you can’t solve continuous differential equations on a digital computer! So every continuous equation that is ever solved computationally is first transformed into a discrete equation (e.g. by putting it on a lattice). This subject will be treated in virtually any textbook on differential equations.

Going from discrete to continuous is more complicated, but you can still do it. Here’s a derivation for turning a random walk into the diffusion question:
http://nebula.physics.uakron.edu/dept/faculty/jutta/modeling/diff_eqn.pdf
I don’t think it’s always possible, but diffusion seems very likely to yield similar results if it’s cast into a continuous form.
-Neil

Well, yes I know that too. I was asking for a clear example of how this diffusion model could be made to reproduce 1/f exactly. If you do not have it, that is fine. But it looked like you might.

Oh, I have no idea. I just made the analogy to diffusion because it’s what I’m familiar with.

I don’t think it makes much sense to intentionally call these things “non-Darwinian” … as distinct from something that involves natural selection.

Natural selection applies pressure to every trait and behavior that a creature’s biology and neural system allows.

It’s equivalent to saying that when God arranges for a rain storm, via natural laws controlling the water cycle, that the rainstorm is supernatural.

Don’t misconstrue my comments. I personally accept that God does make supernatural or miraculous rainstorms. But just because God intends something doesn’t automatically make it miraculous.

For each little segment you can estimate a time to the most recent common ancestor of that segment, based on the number of differences in it.

Could you elaborate on this? Wouldn’t around 1.2 M mutations date back to around 200,000 years ago regardless of whether there was a single ancestral pair or a large population at that time? Or are you saying that you’d only look at alleles that had achieved a certain frequency in the population?

I am no help with the math, but perhaps what you are looking for could be found in this paper:

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2842629/

Take two chromosomes from an idealized, freely mixing population of size N; ignore recombination. The number of differences between them provides an approximate clock to the time they shared a common ancestor. The probability that they shared a common ancestor (i.e. are siblings) in the previous generation is 1/2N; in the jargon, this is the probability that the lineages coalesce in that generation. Looking back in time, the probability of lineages that have not yet coalesced sharing an ancestor is 1/2N in every generation. If N is 10,000, then there is a small probability of coalescence every generation, and looking across the population as a whole there will be a wide range of coalescence times, and thus degrees of genetic difference.

Now if the population plummeted as some point in the past to a single pair, then the probability of coalescence in that generation rises to 0.25. One quarter of all lineage pairs coalesce at that time, and their mutational clocks will all report the same time.

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Let me rephrase that to see if I understand. Imagine an original human pair living around 200,000 years ago with chromosome 1s given by (A,B) and (C,D). Each chromosome would differ from the others at around 1.8M b.p. If we assume that this pair gives rise to some N=10,000 population, then all the members of that population would have one of only 10 possible genotypes (A,A), (A,B), (A,C), (A,D), (B,B), (B,C), etc…

Now let’s assume that mutations can occur at some constant rate. We’d presumably still be able to classify a mutated chromosome as coming from the original A, B, C, or D chromosome. So, for example, after 1 generation, we’d see a chromosome A’ which would differ from A at only 50 bp, but would differ from B,C, and D by ~1.8M bp. So let’s say we run the tape forward 1,000 generations. We could pick two A-type chromosomes at random, call them A’ and A’’. They should differ at around 100,000 bp, which we could use as a molecular clock to correctly determine that a single-pair had existed 1,000 generations ago. Is that correct? Or course, this method relies crucially on being able to pick out two A-type chromosomes, since comparing an A-type and B-type chromosome would give a much older age.

The first question that comes to mind is how long it takes for the ‘no crossover’ assumption to cause serious problems. If a lot of crossover occurs, then it will no longer be at all possible to look at a particular chromosome E and say whether it belongs to A, B, C, or D. So how often does crossover occur and how easy is it to correct for it? Can we look at really small segments of the chromosome to avoid the problems it causes?

@nashenvi

Wow… this is excruciating.

I think you should start over.

1. @Swamidass has done a very good job of showing that genealogy runs more true than genetics.

2. And @DennisVenema has done a good job showing that a population founded by a 2 individuals 6000 years ago cannot possibly create the diversity that we find in the current human population.

3. And finally, @Swamidass has done an even better job showing that with not much effort, a mating couple from 6000 years ago could easily join with hundreds of other pairs as Universal Common Ancestors (i.e., Adam/Eve, and many other couples can be shown - - with almost virtually complete reliability - - to be the genealogical ancestor of every single human alive today).

So… what would you like to adjust or fiddle with using the three premises that Swami and Dennis have provided for us?

George, do you understand my question? Do you see why linkage disequilibrium is crucial to measuring ancestral population sizes in this way?

If @DennisVenema agrees with you, @nashenvi, then i’ll agree with you.

If @Swamidass agrees with you, I might too.

Agree with me on what?

“Do you see why linkage disequilibrium is crucial to measuring ancestral population sizes in this way?”

Ah, so you’re not sure if linkage disequilibrium is needed to determine time to common ancestor using particular segments of DNA. Well, I’m fairly certain that it is. We’ll see what @glipsnort thinks.

(note: I’m not a biologist- I keep prefacing these posts like this as they are highly technical)

So, earlier in this thread, @Swamidass showed a paper that literally showed there never could have been less than 29 people at any given time. And then the size of the human population needs to be much grater to retain the DRBJ polymorphisms.

Also not sure if you’ve read this thread but you might find it useful:

Right.

Almost. Most of the time they’ll differ by about 100,000 bp, but some will differ by a lot less; those are the ones that have a more recent common ancestor than A. [quote=“nashenvi, post:72, topic:36348”]
Or course, this method relies crucially on being able to pick out two A-type chromosomes, since comparing an A-type and B-type chromosome would give a much older age.
[/quote]
You don’t have to pick out the A’s. Just take all pairs of chromosomes and look at the distribution of differences. You’ll find a wide range, but with a big bump around 800,000 bp. [quote=“nashenvi, post:72, topic:36348”]
The first question that comes to mind is how long it takes for the ‘no crossover’ assumption to cause serious problems. If a lot of crossover occurs, then it will no longer be at all possible to look at a particular chromosome E and say whether it belongs to A, B, C, or D. So how often does crossover occur and how easy is it to correct for it? Can we look at really small segments of the chromosome to avoid the problems it causes?
[/quote]
Depends on the size of the population and how crossing over is distributed. In humans (as in most mammals), recombination is highly concentrated at hot spots. As a result, the typical size of a region with little or no detectable recombination in good-sized sample set of chromosomes is something like 10 or 20 thousand base pairs. For a single pair of chromosomes, the stretches will be longer.

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“As a result, the typical size of a region with little or no detectable recombination in good-sized sample set of chromosomes is something like 10 or 20 thousand base pairs. For a single pair of chromosomes, the stretches will be longer.”

Very helpful. So assuming a region with neutral mutation and no crossover, what are some studies which look at the distribution of bp differences?

See here, here, and here. I don’t keep up with that field much these days, so there may have been developments in the last few years that I’m unaware of.

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Great references… I found this abstract to be quite compelling!

Nat Genet. 2011 Sep 18;43(10):1031-4. doi: 10.1038/ng.937.

Bayesian inference of ancient human demography from individual genome sequences.

Gronau I1, Hubisz MJ, Gulko B, Danko CG, Siepel A.

Author information

Abstract
“Whole-genome sequences provide a rich source of information about human evolution. Here we describe an effort to estimate key evolutionary parameters based on the whole-genome sequences of six individuals from diverse human populations. We used a Bayesian, coalescent-based approach to obtain information about ancestral population sizes, divergence times and migration rates from inferred genealogies at many neutrally evolving loci across the genome.”

“We introduce new methods for accommodating gene flow between populations and integrating over possible phasings of diploid genotypes. We also describe a custom pipeline for genotype inference to mitigate biases from heterogeneous sequencing technologies and coverage levels.”

"Our analysis indicates - -
o that the San population of southern Africa diverged from other human populations approximately 108-157 thousand years ago,

o that Eurasians diverged from an ancestral African population 38-64 thousand years ago,

o and that the effective population size of the ancestors of all modern humans was ∼9,000."

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