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Most amino acids can be encoded by more than one codon. For example, Serine can be encoded by any one of {UCU, UCC, UCA, UCG, AGU, AGC}.

My question is in two parts:

1) what are the relative frequencies of those codons? This question can be equivalently phrased in probabilistic terms: What is the probability that a given amino acid was encoded by particular codon?

2) Is this probability dependent on the context (the neighboring codons/amino acids)?

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The phenomenon of using different codons with different probabilities is called codon usage bias. How the different codons are used is sometimes pretty different between species and requires some care. This is especially true when you want to overexpress sequences in bacteria to produce protein, which may need codon optimization to ensure an optimal codon usage (otherwise rare tRNAs can slow down the translation process). There is a database (based on the NCBI Genbank) which gives you the codon probabilities for different species:

Additionally, these papers might be interesting:

Regarding your second question, I don't think (and I haven't found any evidence either) that the codon usage is dependent on the neighbours.

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The codon usage biases are species-dependant, so you may have to measure it for the genome or set of genes you are interested in.

This bias is at least in some cases correlated to the genome composition in terms of A, T, G and C (see figure 3 of this paper about cyanobacteria). So in a sense, it is indeed dependent on the context if you include nucleotide composition in what you call "context".

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