# What is the expected number of children that need to be born for every possible point mutation to occur once? [closed]

I'm reading The Perfect Health Diet, and in it the author says that the probability of a point mutation is (175/3*10^9) per new child. He then goes on to write:

In the Paleolithic, with 100000 children per generation, it would have taken 8000 generations, or 160000 years, for each possible mutation to occur once.

Today, with more than a billion children per generation, every possible point mutation now appears about twenty times per generation, or almost yearly.

Using his probability, I get that you need 1 / (175 / (3 * 10^9)) (which is about 17 million children). But he writes that for the Paleolithic era, you need 10000 * 8000 = 80 million children! Can someone help me reason out how he got that number?

EDIT My math was wrong :-) leaving up the og calculation. Here's the reference that he cites:

"Estimate of the Mutation Rate per Nucleotide in Humans" by Michael W. Nachman and Susan L. Crowell In the abstract they write "The average mutation rate was estimated to be 2.5x10^-8 mutations per nucleotide site or 175 mutations per diploid genome per generation."

## closed as primarily opinion-based by fileunderwater, Chris♦, One Face, WYSIWYG, user12874 Feb 18 '15 at 15:43

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

• Does he give a reference for this mutation probability? I remember smaller numbers. – Chris Feb 18 '15 at 9:04
• This is assuming that mutation rates are constant across the genome, which is generally not the case (see e.g. Smith et al. 2002. Deterministic Mutation Rate Variation in the Human Genome). So I think the statement/calculation rests on unfounded assumptions, and therefore vote to close as "primarily opinion-based". – fileunderwater Feb 18 '15 at 10:05
• The authenticity of that text seems doubtful. – WYSIWYG Feb 18 '15 at 13:34
• I added the reference that he cited. – gsastry Feb 18 '15 at 19:41

I don't think the book's reasoning is correct.

There are about 3 billion base pairs in the human genome, but only ca 25.000 protein-coding genes. From the top of my head, I recall that each child has about 30 mutations. If that's in the total genome and if they were all point mutations and there's 4 options for the base pair, then we have $30$ mutations out of 12 billion options in each child. So that each possible point mutation has a probability of $30/12 \text{ billion} = 2.5 \times 10^{-9}$ or $1/(2.5 \times 10^{-9})$ if you like. The book's number is 23 times larger but in the same ballpark range. Lets call this probability $p$.

Now the probability of a mutation occuring in n children at least once is equal to one minus the probability of it never occuring in n children. In other words: $$1 - (1 - p)^n =$$ $$1 - (1 - 2.5 \times 10^{-9})^n =$$ $$1 - (0.9999999975)^n$$

If we want the mutation to occur with probability 95%, we have to solve: $1 - (0.9999999975)^n = 0.9$ This has the solution $n = 1 \times 10^9$

In other words, with 1 billion children, each point mutation has a 95% probability of having occured at least once. Mathematically this is not the same as saying that every mutation will have occured. In fact, this implies that 5% of the mutations will not have occured. But perhaps surprisingly, we only need 4-5 billion to up the chance to 99.999%. That still leaves out thirty thousand point mutations, but is sufficiently close to "everyone".

Compare this to the numbers you mentioned: n = 17 million gives each gene a 4% chance of occuring n = 80 million gives each gene an 18% chance of occurring.

Secondly, a mandatory rant. :-) "The Perfect Health Diet" is written by Paul and Shou-Ching Jaminet, an astrophysicist and molecular biologist respectively. This has the potential to marry the physicist's penchance for oversimplification with the biologist's lack of mathematical prowess. We see both in the passage you ask about. A shoddy calculation about an oversimplification of what mutations and evolution is about. Both the book's dustcover use of a word like "detoxifying" and mention of how diseases spontaneously resolve if you eat right, are both strong crackpot indicators. It's the kind of thinking that killed Steve Jobs because he didn't seek medical help soon enough but tried to eat his way out of his illness.

• There are not 4 chances for each basepair, but only 2. We are diploid, not tetraploid. – Chris Feb 18 '15 at 9:35
• @chris: I was thinking of the 4 bases A, T, C, G. But maybe that should be counted as two too.. – Abulafia Feb 18 '15 at 11:42
• Thanks for busting my 2AM calculation :) nice explanation – gsastry Feb 18 '15 at 19:39