by Michel Heiniger and Patrick Pfeifer

Various research papers published during the past three decades are focusing on this particular problem. One paper by Frank Wright that seems to be widely cited and was published on the 1st of March 1990 in Volume 87, Issue 1, Pages 23–29 of Elsevier's "Gene" Journal is titled "The ‘effective number of codons’ used in a gene". It is available at http://dx.doi.org/10.1016/0378-1119(90)90491-9

The Student's $t$-test can be used to test whether the mean of a normally distributed population has a value specified in a null hypothesis.

"The codon usage table of a gene can be subdivided according to the number of synonymous codons belonging to each aa. Thus for a gene using the 'universal' code, there are 2 aa with only one codon choice, 9 with two, 1 with three, 5 with four, and 3 with six. These represent five SF types, designated SF types 1, 2, 3, 4, and 6 according to their respective number of synonymous codons." (Wright, 1990) (aa = amino-acid)

Then, for each aa, calculate:

\begin{align}\hat{F} &= \frac{n \displaystyle\sum_{i = 1}^{k}{(p_i^2 - 1)} }{n - 1}\end{align}

And for a complete gene, calculate:

\begin{align}\hat{N}_c &= 2 + 9 / \overline{\hat{F}_2} + 1 / \hat{F}_3 + 5 / \overline{\hat{F}_4} + 3 / \overline{\hat{F}_6} \end{align}

"where $\overline{\hat{F_i}}$, is the average homozygosity estimate for SF type i, and $F_i$ for each aa are calculated using Eqn. (1)." (Wright, 1990)

We will (a) carry out $t$- and $\chi^2$-tests for the usage of each codon in respect to the possible codons for Wright's SF2, SF3, SF4 and SF6 families of aa and (b) calculate $\hat{N}_c$ for the whole genome.

Assuming there are $n$ codons coding for a given amino acid $AA$, then the probability for any of those codons $C_x$ to be actually used (i.e. the "mean value" of this codon, if "codon used" is 1 and "codon not used" is 0), would be:

$$P(C_x, AA) = \mu_x = \frac{1}{n}$$

We examine the actual codon usage in E. Coli and S. solfataricus. For a given amino-acid, e.g. Glycin: CGA, GGC, GGG, GGT, we examine if codon-usage satisfies the null-hypothesis.

e.g. $$P(C_x, Glycin) = \mu_x = \frac{1}{4}$$

To test this, assuming the null-hypothesis, we can calculate the probability that out of $m$ identical amino-acids $\mu - \frac{s}{2}$ to $\mu + \frac{s}{2}$ are coded using a specific aa. $s$ is the interval for our chosen confidence level of 95%.