# What is the mathematical relationship between selection coefficient and dN/dS

dN/dS is often used as a measure of the intensity of selective pressure on a mutation or gene, but I'm curious about how it can be written as a function of the selection coefficient. I'm specifically interested in the scenario of somatic mutations in human cells; suppose there's a gene, and all non-silent mutations in the gene are equivalent (such as if they're all loss of function mutations). The mutation rate is μ mutations per bp per cell division and once both alleles are (nonsilently) mutated, there's no fitness gain (or loss) from additional mutations. The selection coefficient for mutations in the gene for homozygous mutations is:

w_hom = 1 + s


and for heterozygous:

w_het = 1 + hs


How would one derive the expected dN/dS ratio (where the selection coefficient of a silent mutation is 0)? I realize there are a few questions for which answers must be assumed to begin to answer this question, such as:

1. Is population static, or growing?
2. Is dN/dS only including clonal/fixed mutations, or perhaps all mutations with a frequency in the population greater than or equal to some value?

I'd be interested in what the relationship with dN/dS is even under the simplest assumptions, e.g. h = 1, only considering clonal mutations, and static population. I've found one paper that posits such a formula, citing a finding from another paper, but I don't think either shows all the math.

• Are you suggesting that all non-synonimous mutations have the same fitness? Also, I think homo vs. heterozygocity is not really the issue here, unless we also need to account for recombination. Commented Sep 25, 2021 at 13:37
• All non-synonymous mutations might have the same marginal effect fitness, in that they all just break the copy that gets mutated, but having a cell with one remaining functional copy may be more/less fit than one with no functional copies remaining. Commented Sep 25, 2021 at 18:34
• Now you seem to suggest that these mutations are wekly deleterious. Perhaps, you could add a reference to what you consider standard use if dn/ds - we might be talking about different things. Commented Sep 25, 2021 at 19:03

## 1 Answer

dN/dS is usually calculated for an alignment or a phylogenetic tree as if the gene is haploid. Dioloidy is an interesting complication, since it can render deleterious mutations neutral. Selection coefficient can be calculated, if we make additional assimptions about the population growth, etc., as suggested in the OP. However, ut would limit the applicability of the method to the situations covered by these assumptions.

References
Perhaps this article is relevant, as it associates the fitness edge (i.e. accumulation of multiple beneficial mutations) with dN/dS: Predicting evolution from the shape of genealogical trees

• I think an example of it being used in this kind of context is in this paper: ncbi.nlm.nih.gov/pmc/articles/PMC5720395. They're using dN/dS to assess selective advantage/disadvantage of mutations in cancer. Commented Sep 27, 2021 at 4:23