eCite Digital Repository

The impracticalities of multiplicativelyclosed codon models: a retreat to linear alternatives


Shore, JA and Sumner, JG and Holland, BR, The impracticalities of multiplicativelyclosed codon models: a retreat to linear alternatives, Journal of Mathematical Biology pp. 1-25. ISSN 0303-6812 (2020) [Refereed Article]

Copyright Statement

Copyright 2020 Springer-Verlag GmbH Germany, part of Springer Nature

DOI: doi:10.1007/s00285-020-01519-5


A matrix Lie algebra is a linear space of matrices closed under the operation [A, B] = ABBA. 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, ω. We propose two different methods of finding a linear space from a model: the first is the linear closure which is the smallest linear space which contains the model, and the second is the 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.

Item Details

Item Type:Refereed Article
Keywords:Lie algebra, Markov chain, synonymous/non-synonymous rate ratio
Research Division:Mathematical Sciences
Research Group:Applied mathematics
Research Field:Biological mathematics
Objective Division:Expanding Knowledge
Objective Group:Expanding knowledge
Objective Field:Expanding knowledge in the biological sciences
UTAS Author:Shore, JA (Mrs Julia Shore)
UTAS Author:Sumner, JG (Dr Jeremy Sumner)
UTAS Author:Holland, BR (Professor Barbara Holland)
ID Code:141855
Year Published:2020
Funding Support:Australian Research Council (DP150100088)
Deposited By:Mathematics
Deposited On:2020-11-26
Last Modified:2020-12-09

Repository Staff Only: item control page