I need to create a graphic like this which shows both patrilineal and matrilineal descent and can be used to show that mtDNA Eve and Y-Adam are not necessarily contemporaries.
You mean like this?
Figure 1. Genealogical ancestry is not genetic ancestry. Illustrating the story in the text, we show a cartooned pedigree, a genealogy, from past (top) to present (bottom). Squares and circles denote men and women, respectively, with lines indicating parentage. Red and blue individuals are those in the genetic lineages to a single ancestor, Mito-Eve and Y-Adam, respectively. In contrast, every individual with a black border is a common genealogical ancestor of all those in recorded history (grey box). The Scriptural Adam and Eve (the black box and square) are created from the dust and a rib less than 10,000 years ago, have no parents, are in the Garden of Eden (black box), and are genealogical ancestors of everyone in history. This story is entirely consistent with the genetic data (see http://peacefulscience.org/genealogical-science/ for more information)
Or like this?
Or figure 1 in this paper?
Wow, very nice.
What I was thinking specifically was something in between…like, 16 across and 20 down…in a gif or interactive form that shows both maternal and paternal descent and allows someone to select either way.
If you make a diagram of a coalescent genealogy in a single population it won’t quite look like the ones above. In a coalescent in a random-mating population each gene copy comes from a random one in the previous generation, and those draws are independent (with replacement) from one copy to another in the current generation. The result shows a lot more criss-crossing of lines than in the above diagrams. The current ones implicitly assume that adjacent individuals are more likely to mate. This could be more realistic or less realistic, depending on the situation. Here is a link to the a diagram of coalescence in a random-mating population, built up generation by generation. (See also Chapter 26 of my book on phylogenies):
It is certainly true that this doesn’t depict panmictic mating. Rather the populations mating is geographically restricted on a 1-dimensional map (left to right). This is a valid population genetics model too.
The problem with random mating is that it is impossible to display all relationships cleanly. The 1d geography makes the figure nice and clean. Real populations are on a geography too, so this in some ways is more realistic then a panmictic population.
For the observations we are a drawing, the results are the same either way.
Would you refine any of my qualifications here @Joe_Felsenstein?
I agree with your qualifications. As I said in the comment
As a non-scientist just trying to grasp the gene pool stuff, I have a question…Does the same math/models apply to inbreeding? There are several biblical references to incestuous scenarios, just curious if that has any impact or changes timing significantly. As I look at the models, they all seem to imply a single generation progression, when we know from bible stories that there was also cross generational breeding and same family breeding.
@Mark10.45, these are excellent question. We are fortunate to have a giant of population genetics here to answer them if he chooses, @Joe_Felsenstein. I expect he will correct any mistakes I make in my brief response.
You are entirely correct that this is a simplified model. In science, it is common to work from simple model, and then explore how adding complexity changes the questions we are studying. In this case, we are considering these questions:
Did mt-Eve and y-Adam live at the same time and location as a paired couple?
Does the answer change if, actually, population arises from a single couple in the distant past, without any interbreeding with others?
The answers to these questions are “very unlikely” and “no,” respectively, no matter what population genetics model you use.
@Joe_Felsenstein’s slides are using a Wright-Fisher (WF) model, which has non-overlapping generations and panmictic (anyone can breed with anyone). This model gives these same answers. Notably, incest is possible in this model, and in fact expected.
My figures use a 1D-WF model, which also has non-over-lapping generations, but restricts breeding to a 1D geography. This increases the amount of inbreeding. It produces the same answers as panmictic-WF.
You asked about the effect of inbreeding. As you can see, inbreeding is already taking place in these two models. It does not, however, change the answers to the questions we are considering here. The degree of inbreeding can affect other questions, such as increasing or decreasing the expected time to y-Adam and mt-Eve (and the effective population size), but that is not what we are considering here.
You asked about cross-generation breeding. As I understand it (@glipsnort and @Joe_Felsenstein can correct me), results from a WF are provably close to a model with overlapping generations, at least for questions such as these. At the same time, there are times when such a model is employed. We do not expect it to change these answers in any way.
So yes, these are simplified models, but they are chosen to match reality with respect to the questions we care about. The panmictic-WF model is the simplest mathematically, and the 1D-WF is the simplest visually.
Depends on what you mean by inbreeding. Usually one means mating of individuals who are more related to each other than is average in the population. Say someone marrying their cousin. But in a small population everyone becomes related, distantly. Coalescent trees based on random mating will reflect that amount of inbreeding.
I have three technical questions @Joe_Felsenstein:
What do we call a WF model modified to have overlapping generations?
What proofs of convergence are there between models with and without overlapping generations?
Is there a mathematical estimate for the inbreeding coefficient of a panmictic population of a given size?
1 Cor 5:1 - It is actually reported that there is sexual immorality among you, and such sexual immorality as is not even named among the Gentiles—that a man has his father’s wife!
Genesis 19:34 - It happened on the next day that the firstborn said to the younger, “Indeed I lay with my father last night; let us make him drink wine tonight also, and you go in and lie with him, that we may preserve the lineage of our father.
1 Peter 3:20 - who formerly were disobedient, when once the Divine longsuffering waited in the days of Noah, while the ark was being prepared, in which a few, that is, eight souls were saved through water.
1 Kings 11:3 - And he had seven hundred wives, princesses, and three hundred concubines; and his wives turned away his heart. (Regarding Solomon)
The first was Paul writing to Corinth on the seemingly normal practice of sexual immorality with mom or step-mom (may or may not have led to pregnancy). The Genesis verse is about Lot after the destruction of Sodom and everyone in it, so complete re-population. Noah’s case was a matter of re-population also, though starting with four non-related couples. In Solomon’s case, probably more an issue of by the third generation, everyone is related…so, I meant close relations, and what I was thinking was that the mutation rates would be different with close family ties.
I’ll stick with John 8:32. In population genetics the quantity of more interest is the average coefficient of kinship between pairs of copies of genes in a population. In a population, inbreeding inevitably accumulates because even if everyone is terribly virtuous and obeys all the divine rules, you inevitably end up being distantly related to the whole population, and so there are a lot of twelfth-cousin marriages, etc. The degree of inbreeding rises steadily.
The inbreeding in particular families owing to incestuous matings is different – as soon as the offspring marry somebody from outside the family, the inbreeding level of their offspring then drops back to the average for the whole population. So however much the Bible gets worked up over it, it is not a long-term issue.
The overlapping-generations counterpart of the WF model is called the “Moran model”, after PAP Moran who introduced it. In a simple haploid version. at each step one individual in the population is chosen to reproduce an offspring, and one is chosen to die. If there are N haploid individuals, N of these steps is a generation, so the time scale is in 1/N-ths of a generation. (Note that this has overlapping generations, but does not have age-dependent birth or death). Wikipedia: “Moran process”,
There is Sewall Wright’s concept of “effective population size” or “effective population number”. You look for a formula that will show you the size of an idealized simple model (for example, a single-sex random mating population) that has the same rate of increase of inbreeding or the same rate of increase of variance of gene frequency around its starting frequency. There is a literature on the case of overlapping generations (Nei and Imaizumi, Heredity 1966, me, Genetics 1971, Hill Theor. Pop. Biol. 1974) on this that results in useable formulas. By considering limiting cases as N gets large, one can also ask about the diffusion processes that the two cases (WF and overlapping) approach, and it turns out that the effective population size formulas work – they are the same when matched by effective population size.
Sewall Wright did this in about 1932. Basically one gets a rise in the average inbreeding coefficient of 1/(2N) per generation. Or to be more precise, of 1/(2N) of the distance from the current average F to 1. i.e. F(t) = 1 - (1 - 1/(2N))^t. My online text Theoretical Evolutionary Genetics covers all that in chapter VI. It also shows how to do the calculation when N varies through time (another 1930s result of Sewall Wright).