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A matrix Lie algebra is a linear space of matrices closed under the operation $ [A, B] = AB-BA $. The Lie closure of a set of matrices is the smallest matrix Lie algebra which contains the set. In the context of Markov chain theory, if a set of rate matrices form a Lie algebra, their corresponding Markov matrices are closed under matrix multiplication; this has been found to be a useful property in phylogenetics. Inspired by previous research involving Lie closures of DNA models, it was hypothesised that finding the Lie closure of a codon model could help to solve the problem of mis-estimation of the non-synonymous/synonymous rate ratio, $ omega $. We propose two different methods of finding a linear space from a model: the first is the emph{linear closure} which is the smallest linear space which contains the model, and the second is the emph{linear version} which changes multiplicative constraints in the model to additive ones. For each of these linear spaces we then find the Lie closures of them. Under both methods, it was found that closed codon models would require thousands of parameters, and that any partial solution to this problem that was of a reasonable size violated stochasticity. Investigation of toy models indicated that finding the Lie closure of matrix linear spaces which deviated only slightly from a simple model resulted in a Lie closure that was close to having the maximum number of parameters possible. Given that Lie closures are not practical, we propose further consideration of the two variants of linearly closed models.
Models of codon evolution are commonly used to identify positive selection. Positive selection is typically a heterogeneous process, i.e., it acts on some branches of the evolutionary tree and not others. Previous work on DNA models showed that when
Empirical substitution matrices represent the average tendencies of substitutions over various protein families by sacrificing gene-level resolution. We develop a codon-based model, in which mutational tendencies of codon, a genetic code, and the str
In this work it is shown that 20 canonical amino acids (AAs) within genetic code appear to be a whole system with strict distinction in Genetic Code Table (GCT) into some different quantums: 20, 23, 61 amino acid molecules. These molecules distinctio
Recent work on mutation-selection models has revealed that, under specific assumptions on the fitness function and the mutation rates, asymptotic estimates for the leading eigenvalue of the mutation-reproduction matrix may be obtained through a low-d
In this work it is shown that 20 canonical amino acids (AAs) within genetic code appear to be a whole system with strict AAs positions; more exactly, with AAs ordinal number in three variants; first variant 00-19, second 00-21 and third 00-20. The or