Probability, genetic disorder related

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fluidistic

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Note: not homework; thread moved to biology
1. The problem statement, all variables and given/known data

It is not a textbook problem, but a real life scenario, as such I am not sure the solution exists.
A child is born with a rare genetic condition (1 chance in a million). There is about 1/3 chance it comes from a mutation that occurred only in him and 2/3 chance that it comes from his mother (carrier). I want to calculate the probability that the sister of the mother of this child is a carrier of that genetic disorder. But I have more information that do impact on such a probability: the grand mother of that child has had 3 children. One is a boy (no genetic disorder), 2 are girls. Both of these girls got 1 (son) child, 1 ill, the other safe.
The genetic disorder is recessive and on the X chromosome, hence only boys do display signs of the condition, while women are carriers (theoretically they could be affected but this has never occurred as far as we know) but show no sign.
Also note that a carrier has 25% chances to give birth to an ill son, 25% to a non ill son, 25% to a non carrier daughter and 25% to a carrier daughter.

2. Relevant equations
Bayes theorem?

3. The attempt at a solution
Only thoughts for now. I was wondering whether I could apply Bayes theorem several times in a row to solve the problem, or whether it is more complex than that, or whether we have not enough information to solve it.
If I start with a first estimation, the mother of that child has 66% of being a carrier, so the grand mother has, at first order of estimation, (2/3)^2=4/9=44.4% chances of being a carrier. However the fact that the grand mother got a son without this genetic disorder should lower that probability, and similarly for the fact that the other daughter has had a non ill son.
In the end it looks like I'm seeking to calculate the probability that the mother "A" is a carrier, given that her sister "B" has 2/3 chances to be a carrier.

Lastly, if I ever get that result, I wish to combine it with another totally independent "test" that claims that the mother "A" has 5% chances to be a carrier. How would that number lower given the prior calculations.
 
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epenguin

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1. The problem statement, all variables and given/known data
It is not a textbook problem, but a real life scenario, as such I am not sure the solution exists.
A child is born with a rare genetic condition (1 chance in a million). There is about 1/3 chance it comes from a mutation that occurred only in him and 2/3 chance that it comes from his mother (carrier). I want to calculate the probability that the sister of the mother of this child is a carrier of that genetic disorder. But I have more information that do impact on such a probability: the grand mother of that child has had 3 children. One is a boy (no genetic disorder), 2 are girls. Both of these girls got 1 (son) child, 1 ill, the other safe.
The genetic disorder is recessive and on the X chromosome, hence only boys do display signs of the condition, while women are carriers (theoretically they could be affected but this has never occurred as far as we know) but show no sign.
Also note that a carrier has 25% chances to give birth to an ill son, 25% to a non ill son, 25% to a non carrier daughter and 25% to a carrier daugh'

2. Relevant equations
Bayes theorem?

3. The attempt at a solution
Only thoughts for now. I was wondering whether I could apply Bayes theorem several times in a row to solve the problem, or whether it is more complex than that, or whether we have not enough information to solve it.
If I start with a first estimation, the mother of that child has 66% of being a carrier, so the grand mother has, at first order of estimation, (2/3)^2=4/9=44.4% chances of being a carrier. However the fact that the grand mother got a son without this genetic disorder should lower that probability, and similarly for the fact that the other daughter has had a non ill son.
In the end it looks like I'm seeking to calculate the probability that the mother "A" is a carrier, given that her sister "B" has 2/3 chances to be a carrier.

Lastly, if I ever get that result, I wish to combine it with another totally independent "test" that claims that the mother "A" has 5% chances to be a carrier. How would that number lower given the prior calculations.
The mother of the affected son is almost 100% sure to be a carrier.
Her sister is 50% likely to be a carrier.
So any son of hers is 25% likely to be affected.
If intending to have children she should certainly seek genetic counselling beforehand.
Likely this is a known mutation responsible, and if it is that, it will probably be testable whether she is a carrier or not.
If she is found to not be a carrier she can have children without any more worries than the other 999,999 out of 1,000,000; if she is a carrier she can be advised on how she can be helped and followed to avoid having an affected son.

I think there is at least one mistake in your reasoning above – there is notThe one in three chance that this son's mutation is new, there is only a one in 1 million chance. You are probably confusing with a third of all cases being a new mutation, something quite different. Anyway the important thing is the sister needs to take the actions I mentioned before ever conceiving.
 

fluidistic

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The mother of the affected son is almost 100% sure to be a carrier.
Her sister is 50% likely to be a carrier.
So any son of hers is 25% likely to be affected.
I am not sure where you've drawn these conclusions. From the information I gave, the probability of a de novo mutation is about one third. Thus, the sister has "only" 2/3 probability to be a carrier, not 100%.
Source: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5118406/.
The Source said:
Based on the large study by Hichri et al. [18] about two-thirds of Lowe cases are transmitted by maternal carriers.
,
The Source said:
Lowe syndrome/Dent-2 disease can be attributed to a de novo variant in around one-third of all cases.
in agreement with what I wrote:
fluid the stick said:
There is about 1/3 chance it comes from a mutation that occurred only in him and 2/3 chance that it comes from his mother (carrier).
 

epenguin

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OK I have thought more about it now and I do not stand by most of what I said. This question needs to go into the biology section, geneticists are used to this kind of calculation and the problems are biological, and what assumptions can be made.

You say this is a real life problem. The doctors or counsellors will no doubt want to investigate for information more complete than what we have, they can identify the molecular DNA change that has caused the affliction in the son, then they can establish whether the mother's sister has that gene damage or not. If you are saying this is Lowe's syndrome, I see there is a specific known gene involved, all those putatively bearing the mutant gene, the grandmother if alive and other female relatives, can be checked I guess in any reasonably modern health system. If she has it then there is a very great risk for having children. If not I guess there is no more risk than anyone else, however we cannot give medical advice here and this should not be taken as such.

As the scientific question that we can discuss you raise several points which I do not know the answer, perhaps somebody else can get better ones, Some of this is advanced ongoing research.

We here do not even know whether the grandmother was a carrier. Since you say there is a 2/3 chance the mother was a carrier by the same token that makes 4/9 chance the grandmother was, hence a 4/18 = 2/9 chance the sister is. High risk. Only other thing we know is the grandmother had one unaffected son; whatever change that makes to the probability I would not place any bets when the stakes are so high.

One can say that it is the nature of severe disease-causing X chromosome mutations that a large fraction of them are de novo. The affected males do not have progeny so there is high selection pressure eliminating the mutant genes, therefore a high fraction of the mutations must be recent, either de novo, or a very few generations back. This is also the case for dominant mutations in general, but quite different for recessive autosomal mutations whose origins can often be traced or inferred centuries and even millennia back.

It was not obvious to me until the moment of writing this why the novo fraction is not higher than ⅓ , why it is not ½. Suppose you have a population with some fraction of females with a deleterious de novo mutation an X- chromosome gene who each produce two offspring. 1/4 of this offspring are afflicted males who do not produce offspring. An equal number are carrier females. These will in future generations give rise to afflicted males which will count as inherited, not de novo And without so much as summing a series all mutant genes will sooner or later end up this way, so the total number of inherited mutations expressed as disease will be equal to the first generation number. I.e. a 1:1 ratio de novo and inherited X'.

However in that case the whole sub population has halved. Suppose instead their reproduction rate is higher so as to maintain a constant population, i.e. averagely each female instead of two children produces 2x4/3, then they will maintain a constant population and I think I actually produce twice as many as the F1 afflicted males. And an even higher ratioif the population is expanding.

At this point I should say what I think is generally understood by the term 'de novo' mutation.

Definition of de novo mutation - NCI Dictionary of Genetics Terms - National Cancer Institute
https://www.cancer.gov › dictionaries › def

"A genetic alteration that is present for the first time in one family member as a result of a variant (or mutation) in :a germ cell (egg or sperm) of one of the parents, or a variant that arises in the fertilizd egg itself during early embryogenesis. Also called de novo variant, new mutation, and new variant'"l



When I hear the term 'de novo mutation' this mostly means a mutation that has originated not all that recently in a grandparent of the affected male individual. (More likely grandfather than grandmother for reasons explained in the reference below.) It is present in one of the chromosomes of the mother Who is consequently a 'carrier' but was not present in the somatic chromosomes of her parents nor an ancestor further back.The most easily found online sources and of the elementary textbooks as far as I know strongly emphasise these carrier females, and much less how this mutation comes to be present in them. At some point in the differentiation and proliferation of germ cells during development of grandparents the mutation occurs and is present in all the cells that have descended from the mutant one, but not in others. The grandparents are then is said to be mosaics with respect to this type of cell.Then if it is the descendent of one of these mutant cells that happens to end up in the gamete that forms their daughter (the 'mother') she will have homogeneously half her chromosomes bearing the mutation and will be what we call the carrier. That seems to be the classical or textbook story.

However it is also possible that this mother who has not inherited a mutation acquires one in her own germ cells which form a mosaic. So with reduced probability this could be transmitted to a daughter - that is just the process we already described – but also to a son.As far as I could make a note from the difficult publication cited below something in the range 4-10% of the de novo mutations found in males are of this kind of origin. Perhaps in earlier years the mosaicism of mothers would not have been detected, but now the techniques are better.

Finally the publication points out further that this type of disease can come about via what I think used to be called 'somatic mutation' but the publication calls 'postzygotic de novo .mutation'.Here a mutation has occurred not in the germline of the affected individual, but in some of his somatic cells - He is himself a mosaic with respect to the mutation. "close to 7% of seemingly de novo mutations arise as postzygotic events in the offspring". This is a relatively recent biomedical research and discovery theme.

https://genomebiology.biomedcentral.com/articles/10.1186/s13059-016-1110-1

Some of this subject is not new to me, but some of it is and I am not an expert and have described my limitations in a link to my profile.
 
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jim mcnamara

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Thread moved to Biology
 

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