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This and this Quora question ask how many unique human genotypes are possible. The answers take the combinatorial approach of raising the 4 possible base pairs to a power which is some appreciable fraction of the 3 billion base pairs in the human genome, to get absurdly large numbers like $10^\text{many millions}$. But as many of the answers point out, this is a gigantic overestimate, because the vast majority of those gene sequences wouldn't produce anything remotely resembling a human, and even among the tiny fraction that do, there are enormous numbers of sequences that produce phenotypically identical humans (after all, most mutations don't produce any detectable effect). (When I say that two genomes "produce indistinguishable phenotypes," I am holding the environment fixed and only considering the genome's contribution to the phenotype.)

It seems to me that in order to avoid counting non-human genomes or double-counting phenotypically equivalent genomes, the better approach is use genes rather than base pairs as the combinatorial building blocks. The human genome has at most 20,000 protein-coding genes, and apparently most genes have two alleles, although some have only one and others have more than two. (I'm defining an allele as an equivalence class of phenotypically indistinguishable DNA sequences that form a single viable gene. For the purpose of this calculation, we can mentally group together the two chromosomes' versions of the same gene, and increase our allele count to include any interactions between the two chromosomes' sequences.) [Edit: according to the answer below (which I find much more trustworthy than this Quora link), this is a huge underestimate for the typical number of alleles per gene, with a better estimate being ~300. Needless to say, this dramatically changes the final answer.] This would produce a maximum of $2^{20000} \approx 10^{6020}$ possible phenotypically distinguishable genomes. While this is obviously still an astronomically high number, it's an infinitesimally tiny fraction of the estimates produced in the Quora answers.

Is my reasoning correct?

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    $\begingroup$ When you're looking for specific estimates in biology (such as e.g. the number of coding sequences in the human genome), I recommend starting with the website bionumbers rather than Quora or wikipedia. $\endgroup$ – Remi.b Mar 30 '18 at 17:50
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Is my reasoning correct?

I can see at least five major problems in your reasoning.

Do most genes have two alleles?

The claim that most genes have two alleles is wrong. Actually, the entire top answer in the linked quora post is wrong. The person answering confounds simplified cases used to teach intro genetics class with what nature really looks like.

As about one every 100 nucleotides is a SNP (bionumbers) and given that a gene is on average about 30k sites (bionumbers), the average number of alleles is probably closer to 300 although that would be counting synonymous mutations which is a bit unfair. About 20% of substitutions are synonymous (bionumbers), which would lead to an estimate of 240 alleles per gene on average. Even if you consider that 90% of these mutations have no effect on the phenotype (which is maybe an overestimation, bionumbers), you're left with 24 alleles per gene. Of course, this is a very vague estimate but it is, I think, much better than 2.

Non-coding sequences

Also, the majority of phenotypic distinction is caused by non-coding (non - gene) sequences such as regulatory sequences. You cannot ignore them in the calculation.

According to this pie chart (not sure if we can trust this chart though), 5% of the genome is made of regulatory sequences (against 1.5% of the genome that is coding).

One genotype ≠ one phenotype

If you have in mind that one genotype = one phenotype, then it would be very wrong too. There is environmental variance, epigenetic variance and other sources of variance (e.g. developmental noise) that underly phenotypic variance (see this post to understand the sources of phenotypic variance in populations).

indels and CNV

You are only considering substitutions. There are plenty of indels and ohter CNV too.

The existence of this variation does not only complicate the question, it actually renders the question a bit undefined because if you allow the genome to take any size, then number of possibilities is necessarily infinite.

Diploidy

Your calculations assume individuals are haploids. But humans are diplontic.

Gene are not atomic

Genes are not units that cannot be split. Recombination do happen in genes and it would be wrong to take the number of variants at genes (or at any other type of sequences) as basis for your calculation

Attempt at a calculation

So, ignoring many of the problems, here is my attempt at a calculation.

There are about $1.5 \cdot 10^7$ SNPs in the human genome (bionumbers). Based on this pie chart (not sure if we can trust this chart though), I will assume that about 6.5% of them are in coding or regulatory regions and I will assume (based on that paper) that 90% of them have no fitness effect and by extension I will assume it also mean no phenotypic effect (although the actually percentage should been increased). This will result in $1.5 \cdot 10^7 \cdot 0.065 \cdot 0.1 = 97500$ SNPs of interest. It results into $2^{97500} ≈ 10^{32500}$ possible haplotypes of interest. The number of possible genotypes of interest is therefore $\left(10^{32500}\right)^2 = 10^{65000}$. I also assumed absence of epistatic interactions otherwise the number would become so fantastically closer to infinity!

Interest in the calculation

I fail to find much interest in making such calculation because it entirely depends upon the model we want to consider. There is no offense against the OP's question here (and in fact the OP shows that he understands the importance of the considered model), I just want to make sure readers understand that these calculations are based on a very arbitrary model and one cannot make any sense of the given result without understanding the model.

The calculations from the Quora post that OP linked assumes that every SNP could be polymorphic. As long as the intention behind the calculation is not clearly stated, this is a perfectly reasonable model. Here, in my attempt, I assume that the number of SNPs is fixed, but any combination of it is possible. This can or cannot be a fine assumptions. Also, the estimates of the number of SNPs are of course, estimates of the number of SNPs that are at a frequency that is not too low to be detected! So, if we were, for some reason, to consider all the SNPs (by assuming some drift-mutation equilibrium), then the result would differ quite a bit. But again, it all depends upon the specific model one wants to consider.

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    $\begingroup$ Thank you for the thorough answer. I completely agree that the question is ill-defined and any numerical answer is pretty much meaningless. My motivation for asking the question was simply to verify that a naive calculation based solely on the number of base pairs in the genome is even more meaningless and arbitrary, and gives enormous overestimates compared to any possible even vaguely reasonable model. $\endgroup$ – tparker Mar 30 '18 at 19:05
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    $\begingroup$ i actually would also agree with @Remi.b the number of phenotypes would include all the possible physical behaviors, appearances and other properties of a living thing. since most phenotypes are the result of combinations of genotypes in unpredictable and nonlinear ways, with continuous (as opposed to 0/1) sorts of behaviors - the number of possible phenotypes are incalculable. $\endgroup$ – shigeta Mar 30 '18 at 20:05
  • $\begingroup$ @tparker "My motivation for asking the question was simply to verify that a naive calculation based solely on the number of base pairs in the genome is even more meaningless and arbitrary" One additional thing to think about here is the actual number of people, somewhere around 10^10 or 10^11 depending on how many generations you want to consider. In that context, the distinction between 10^65000 and 10^6000 is not really all that important, either, from a sampling perspective. The distinction is going to be more important if you think about more numerous organisms with smaller genomes. $\endgroup$ – Bryan Krause Mar 30 '18 at 20:07

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