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Given $N=5\times10^3$ and mutation rate is $\mu=10^{-5}$ per site, find the length of a DNA sequence so that the probability of mutation occuring M, is greater or equal than 0.95.

Is there a method or a formula for this type of calculation?

Btzzzz
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  • Welcome to Biology.SE. Homework (or homework-like) questions are off-topic on Biology unless you have shown your attempt at an answer. For more information see our homework policy. – Remi.b Feb 17 '18 at 20:46
  • Hint: Try to compute the probability that a mutation occurs at a single site (in a single generation). Then compute this same probability for 100 sites. And from there you should see what is going on. – Remi.b Feb 17 '18 at 20:48
  • is $n$ the population size? We typically use $N$ for the population size. Also, I would assume that $\mu$ is the per site (per base pair) mutation rate. Finally by probability of mutation, I would assume it is meant probability of 1 or more mutations, just like the formula is suggesting if $M$ means the number of mutations. – Remi.b Feb 17 '18 at 20:48
  • I just realized there are two ways to understand the question. By mutation do you mean mutant allele or do you mean mutational event (see here)? – Remi.b Feb 17 '18 at 21:13
  • I mean by an event , yes – Btzzzz Feb 17 '18 at 21:20
  • So, you don't need any a priori about the sequence. The probability of a mutation to happen at a given site on a given haploid is $10^{-5}$. What is this probability for the whole population (don't forget individuals are probably assumed to be diploid)? You can then compute this probability for 100 sites and you'll understand the pattern. Does it make sense? – Remi.b Feb 17 '18 at 21:22
  • Ok, so for each site: $P(M>0) = 10^{-5} $ meaning that for $N=5000$, we get $P(M>0) = 5000*10^{-5} = 0.05$ for one site? – Btzzzz Feb 17 '18 at 21:31
  • I am really not sure how this method would allow us to find the length to ensure $P(M>0)\ge 0.95$. I know that $\theta = 2 N \mu $, which I think may be used so that $\theta=0.1$ per base. But i don't get how to answer this in terms of the actual question – Btzzzz Feb 17 '18 at 22:07
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    Ok, good! 0.1 is correct. (0.05 would be correct if assuming haploidy). Now for 100 sites? And then just ask for how many sites is the probability 0.95 – Remi.b Feb 17 '18 at 22:10

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The mutation rate per haplotype per site is $\mu = 10^{-5}$. Assuming diploidy and a population size of $N=5000$, the population wide mutation rate per site is $10^{-5} * 5000 * 2 = 0.1$.

$0.1$ is hence the probability that a mutation occurs a at a given site (in the whole population). For 10 sites the probability that a mutation occurs at at least one site is $1 - (1-0.1)^{10} = 0.65$.

The probability we are aiming for is 0.95. So let's write the equation

$$1 - (1-0.1)^{x} = 0.95$$

, where $x$ is the number of sites we are looking for. You just have to solve for $x$ now and round up to the larger integer.

Remi.b
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  • Ok, so this is in terms of the number of sites, so to calculate the length of the sequence in terms of base pairs, would we multiply by $10^3$ say? I am trying to read up on base pairs and this is what i gathered, but i may be mistaken. (Otherwise this answer is perfectly fine, i just see in many literature the lengths are usually given in bp?) – Btzzzz Feb 18 '18 at 00:36
  • A site is a bp. The number of $x$ is the length of the sequence (in sites of in base pairs) that you were looking for. If you want to express the answer in kilo bp, then you can divide the result by $10^3$ but I would not recommend it. $x$, once rounded, is your answer – Remi.b Feb 18 '18 at 00:49
  • Ok! I am comparing this to mitochondrial dna which is generally 16,500 bp in length, so solving the above equation of 29 seems very unusual.. (ref: https://www.ncbi.nlm.nih.gov/m/pubmed/26798311/) – Btzzzz Feb 18 '18 at 00:56
  • The answer depends upon the question. You typically don't attempt to answer everyday the question "what is the sequence length which probability to be hit by a mutation in a given generation is 0.95?". I don't think you should expect from yourself to have much intuition in the answer and therefore there is nothing unusual to be found. I don't understand why the length of the human mtDNA matters here. – Remi.b Feb 18 '18 at 01:09
  • Only because for instance the mutation rate (per site) of mtDNA is also generally $\mu = 10^{-5}$ so i just wanted to compare – Btzzzz Feb 18 '18 at 01:11
  • oh ok. Sure... so yes, a lot of mutations happen on mtDNA every generations (even more so if you consider 7 billions people). – Remi.b Feb 18 '18 at 01:14