Drummond:Akashi's Test

Akashi's test on a single gene

Akashi's test is very simple. Suppose you have two aligned codon sequences (a target sequence and an orthologous sequence) and a list of preferred codons. The question we wish to answer: Is there an association between preferred codons and conserved amino acids, controlling for differences between amino acids?

From the aligned codon sequences, build a 2x2 contingency table with entries a, b, c, and d like this:

AA=Ser	Conserved	Variable
Preferred	[math]\displaystyle{ a }[/math]	[math]\displaystyle{ b }[/math]
Unpreferred	[math]\displaystyle{ c }[/math]	[math]\displaystyle{ d }[/math]

for each amino acid. You'll usually have 18 tables; W and M have no synonymous codon alternatives and therefore don't contribute to Akashi's test.

• [math]\displaystyle{ a }[/math] = the number of codons in your target sequence that encode amino acid AA, are PREFERRED, and encode an AA which is unchanged (CONSERVED) in the orthologous sequence
• [math]\displaystyle{ b }[/math] = the number of codons in your target sequence that encode amino acid AA, are PREFERRED and encode an AAwhich is different (VARIABLE) in the orthologous sequence
• [math]\displaystyle{ c }[/math] = the number of codons in your target sequence that encode amino acid AA, are UNPREFERRED and encode an AA which is unchanged (CONSERVED) in the orthologous sequence
• [math]\displaystyle{ d }[/math] = the number of codons in your target sequence that encode amino acid AA, are UNPREFERRED and encode an AA which is different (VARIABLE) in the orthologous sequence

Now the statistics. Assuming no association -- that is, assuming that the probability of a codon being preferred (which we designate [math]\displaystyle{ p }[/math]) is independent of the probability that it encodes a conserved amino acid (which we designate [math]\displaystyle{ q }[/math]) -- we can write down estimates for the expected value and variance of [math]\displaystyle{ a }[/math], [math]\displaystyle{ E(a) }[/math] and [math]\displaystyle{ V(a) }[/math]:

[math]\displaystyle{ n = a + b + c + d }[/math]

[math]\displaystyle{ \hat{p} = \frac{a + b}{n} }[/math]

[math]\displaystyle{ \hat{q} = \frac{a + c}{n} }[/math]

[math]\displaystyle{ \hat{E}(a) = n \hat{p}\hat{q} }[/math]

[math]\displaystyle{ \hat{V}(a) = \frac{1}{n-1} n\hat{p}(1-\hat{p}) n\hat{q}(1-\hat{q}) }[/math]

With the mean and variance, we could write down a [math]\displaystyle{ Z }[/math]-score for one table:

[math]\displaystyle{ \hat{Z} = \frac{a - \hat{E}(a)}{\sqrt{\hat{V}(a)}} }[/math]

And because a [math]\displaystyle{ Z }[/math]-score only gives us a measure of statistical significance, we also want an effect size -- the magnitude of the association between preferred codons and conserved sites -- which we can compute as an odds ratio, the ratio of finding a preferred/conserved association divided by the odds of finding a nonpreferred/variable association. An unbiased estimate of the odds ratio [math]\displaystyle{ \psi }[/math] is given by:

[math]\displaystyle{ \hat{\psi} = \frac{ad}{bc} }[/math].

Akashi's test on multiple genes

Calculating [math]\displaystyle{ Z }[/math] and [math]\displaystyle{ OR }[/math] for a single amino acid in a single gene is perhaps of limited interest. How do we combine tables so that we can ask questions like, "What is the overall association between preferred codons and conserved sites across the genome?" or, "How statistically significant is the preferred/conserved association for alanine compared to glycine?"

To combine tables, we use the basic principle that tables are independent. Expectations add, variances add, and observed values add. That is, indexing tables by [math]\displaystyle{ i }[/math], with table i equal to

AA=X	Conserved	Variable
Preferred	[math]\displaystyle{ a_i }[/math]	[math]\displaystyle{ b_i }[/math]
Unpreferred	[math]\displaystyle{ c_i }[/math]	[math]\displaystyle{ d_i }[/math]

and [math]\displaystyle{ \hat{E}(a_i) }[/math] and [math]\displaystyle{ \hat{V}(a_i) }[/math] computed as before for each table, we have the combined [math]\displaystyle{ Z }[/math]-score

[math]\displaystyle{ \hat{Z} = \frac{\sum_i{a_i} - \sum_i{\hat{E}(a_i)}}{\sqrt{\sum_i{\hat{V}(a_i)}}} }[/math]

and the Mantel-Haenszel estimator for the combined odds ratio,

[math]\displaystyle{ \hat{\psi} = \frac{\sum_i{\frac{a_i d_i}{n_i}}}{\sum_i{\frac{b_i c_i}{n_i}}} }[/math]

With enough tables, we assume that [math]\displaystyle{ \hat{Z} }[/math] follows the standard normal distribution [math]\displaystyle{ \Phi }[/math], so that a [math]\displaystyle{ P }[/math]-value can be computed as [math]\displaystyle{ \Phi(\hat{Z}) }[/math].

Examples

Coming...

References

1. Akashi H. Synonymous codon usage in Drosophila melanogaster: natural selection and translational accuracy. Genetics. 1994 Mar;136(3):927-35. DOI:10.1093/genetics/136.3.927 |