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Reconstructing Expected Mutations

HAM Surname DNA Project
copyright 2006
by Dave Hamm
Novi, MI


Introduction
Theory
Expected Mutations
Credits
  Introduction:      

I received an email from Laverne Piatt of Ontario, OH regarding how many mutations she should expect to see for an ancestor that was born in the 1670's.  Good question.  Aside from the normal probability curves, can we answer that with something more direct, and easy to calculate?

One question that remains in the back of my  mind is how will I be able to tell if or when one of the HAM groups might match the DNA of one of our famous European cousins? 

I have been working along the lines of DNA reconstruction, just in the event that some day some genetic genius figures out a way to extract enough DNA from the body of Odon's crystallized bones.

Presuming that at least one branch of our HAM lines descend from him, for example, how will we know if any of our HAM groups would descend from him?  

Well, if the data was good enough, then we would be looking for things like haplotype group, and the number of matching markers. But, if the data is not good enough, then my question becomes, how can we reconstruct what his DNA should have looked like?

Along those lines, I took a stab at answering the question of how many mutating markers should we expect among any of his descendants that might be living today?

Given the number of generations ago, and the number of markers tested, how would we calculate the expected number of mutations for Odon, or say perhaps Jerome HAM, given the birth date and the number of markers tested?  O.K., in the event that you are not so much interested in either of them, but just want to figure out how many mutations you might expect to see from your newly discovered-long-lost DNA cousin? 

Do we have a general procedure for this (that might be easy to use)? 

    Theory:

There should be an equation somewhere that does this, but where?

  For the heck of it, I grabbed Ken Nordtvedt's equation, and  pretty much butchered his intent, but it  does give me an easy to calculate "estimate" of mutating markers out of the number of total markers. 

I probably would have flunked Nordtvedt's pop quiz, but this is what I used:

    http://www.northwestanalysis.net/Iweb6.jpg

   which is briefly explained here:

    http://www.northwestanalysis.net/Demograph.htm

 - and tried to reduce it to give me a simple equation to give me an estimation of the expected number of mutating markers.
Nordtvedt has, roughly  the change in Average Squared Difference (dASD) is proportional to roughly 2 times the mutation rate (m)
times the change in generations (dG):
 
            dASD/dG ~= 2m

For the purposes of illustration, Nordtvedt uses dASD to be about 1, and mutation rate at about  1/400.
So, then all  we need to solve for is the number of changing markers, di, or roughly proportional to dASD.

   di  ~=  N*[SQRT(dG*2*m)]     (in very simplified terms)

where
    di = number of changing markers
    N = number of markers tested
    dG = the number of Generations to MRCA, or TMRCA in generations
    m  = mutation rate of 1/400

If we have a TMRCA of 1670, then the number of generations from the 1670 date:   G = (2006 - 1670 )/25 = 13
And say that the number of markers has been given at 67 markers.

That gives:

    67*[SQRT(2*13/400)] ~= 17     mutating markers (out of 67)
    37*[SQRT(2*13/400)] ~=   9     mutating markers
    25*[SQRT(2*13/400)] ~=   6     mutating markers

NOTE:   Aside from me not fully expanding Nordtvedt's equations, any experts that may be reviewing my crude interpretation (of Nordtvedt's equations) can easily see that my interpretation self implodes after some 400 generations, or some 10,000 years in the past (400*25).  Perhaps this may be an indication of an implied limit within the equations pertaining to the haplotype group, and this limit is of course, proportional to known mutation rates.
   Expected Mutations:

Anyway, to get on with it, let's do a few estimated mutations that we might expect.

Presuming that I have the equations roughly correct, Odon would have been born in about say,

1150 (I don't have the exact date handy). That gives the number of generations as:

 G = (2006 - 1150)/25 = 34 generations
 
 From the equations below, descents that have been DNA tested today should show this many mutating markers:

    67*[SQRT(2*34/400)] ~=   28  mutating markers
    37*[SQRT(2*34/400)] ~=   15  mutating markers
    25*[SQRT(2*34/400)] ~=   10  mutating markers

 That is, descendants of Odon IV living today would show 15/25, 22/37, or 39/67 markers that match.

 As another example, estimating that Jerome HAM was born in about 1630, the equation becomes:

 G = (2006 - 1630)/25 = 15 generations

    67*[SQRT(2*15/400)] ~=   18  mutating markers
    37*[SQRT(2*15/400)] ~=   10  mutating markers
    25*[SQRT(2*15/400)] ~=    7  mutating markers

 Or, I would expect descendants of Jerome HAM living today to share 18/25, 27/37, or 49/67 matching markers.

 - Dave
   Credits:

   I would like to thank Laverne Piatt of Ontario, Ohio for asking the question.
   I would like to thank Ken Nordtvedt for posting his theory to the internet, and for not taking me to the woodshed for pretty much butchering his equations.
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