Phylogenetic mixtures and linear invariants for equal input models
The reconstruction of phylogenetic trees from molecular sequence data relies on modelling site substitutions by a Markov process, or a mixture of such processes. In general, allowing mixed processes can result in different tree topologies becoming indistinguishable from the data, even for infinitely...
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| Published in: | Journal of mathematical biology Vol. 74; no. 5; pp. 1107 - 1138 |
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| Main Authors: | , |
| Format: | Journal Article Publication |
| Language: | English |
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Springer Berlin Heidelberg
01.04.2017
Springer Nature B.V |
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| ISSN: | 0303-6812, 1432-1416, 1432-1416 |
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| Abstract | The reconstruction of phylogenetic trees from molecular sequence data relies on modelling site substitutions by a Markov process, or a mixture of such processes. In general, allowing mixed processes can result in different tree topologies becoming indistinguishable from the data, even for infinitely long sequences. However, when the underlying Markov process supports linear phylogenetic invariants, then provided these are sufficiently informative, the identifiability of the tree topology can be restored. In this paper, we investigate a class of processes that support linear invariants once the stationary distribution is fixed, the ‘equal input model’. This model generalizes the ‘Felsenstein 1981’ model (and thereby the Jukes–Cantor model) from four states to an arbitrary number of states (finite or infinite), and it can also be described by a ‘random cluster’ process. We describe the structure and dimension of the vector spaces of phylogenetic mixtures and of linear invariants for any fixed phylogenetic tree (and for all trees—the so called ‘model invariants’), on any number
n
of leaves. We also provide a precise description of the space of mixtures and linear invariants for the special case of
n
=
4
leaves. By combining techniques from discrete random processes and (multi-) linear algebra, our results build on a classic result that was first established by James Lake (Mol Biol Evol 4:167–191,
1987
). |
|---|---|
| AbstractList | The reconstruction of phylogenetic trees from molecular sequence data relies on modelling site substitutions by a Markov process, or a mixture of such processes. In general, allowing mixed processes can result in different tree topologies becoming indistinguishable from the data, even for infinitely long sequences. However, when the underlying Markov process supports linear phylogenetic invariants, then provided these are sufficiently informative, the identifiability of the tree topology can be restored. In this paper, we investigate a class of processes that support linear invariants once the stationary distribution is fixed, the 'equal input model'. This model generalizes the 'Felsenstein 1981' model (and thereby the Jukes-Cantor model) from four states to an arbitrary number of states (finite or infinite), and it can also be described by a 'random cluster' process. We describe the structure and dimension of the vector spaces of phylogenetic mixtures and of linear invariants for any fixed phylogenetic tree (and for all trees-the so called 'model invariants'), on any number n of leaves. We also provide a precise description of the space of mixtures and linear invariants for the special case of [Formula: see text] leaves. By combining techniques from discrete random processes and (multi-) linear algebra, our results build on a classic result that was first established by James Lake (Mol Biol Evol 4:167-191, 1987).The reconstruction of phylogenetic trees from molecular sequence data relies on modelling site substitutions by a Markov process, or a mixture of such processes. In general, allowing mixed processes can result in different tree topologies becoming indistinguishable from the data, even for infinitely long sequences. However, when the underlying Markov process supports linear phylogenetic invariants, then provided these are sufficiently informative, the identifiability of the tree topology can be restored. In this paper, we investigate a class of processes that support linear invariants once the stationary distribution is fixed, the 'equal input model'. This model generalizes the 'Felsenstein 1981' model (and thereby the Jukes-Cantor model) from four states to an arbitrary number of states (finite or infinite), and it can also be described by a 'random cluster' process. We describe the structure and dimension of the vector spaces of phylogenetic mixtures and of linear invariants for any fixed phylogenetic tree (and for all trees-the so called 'model invariants'), on any number n of leaves. We also provide a precise description of the space of mixtures and linear invariants for the special case of [Formula: see text] leaves. By combining techniques from discrete random processes and (multi-) linear algebra, our results build on a classic result that was first established by James Lake (Mol Biol Evol 4:167-191, 1987). (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) The reconstruction of phylogenetic trees from molecular sequence data relies on modelling site substitutions by a Markov process, or a mixture of such processes. In general, allowing mixed processes can result in different tree topologies becoming indistinguishable from the data, even for infinitely long sequences. However, when the underlying Markov process supports linear phylogenetic invariants, then provided these are sufficiently informative, the identifiability of the tree topology can be restored. In this paper, we investigate a class of processes that support linear invariants once the stationary distribution is fixed, the 'equal input model'. This model generalizes the 'Felsenstein 1981' model (and thereby the Jukes-Cantor model) from four states to an arbitrary number of states (finite or infinite), and it can also be described by a 'random cluster' process. We describe the structure and dimension of the vector spaces of phylogenetic mixtures and of linear invariants for any fixed phylogenetic tree (and for all trees--the so called 'model invariants'), on any number n of leaves. We also provide a precise description of the space of mixtures and linear invariants for the special case of ... leaves. By combining techniques from discrete random processes and (multi-) linear algebra, our results build on a classic result that was first established by James Lake (Mol Biol Evol 4:167-191, 1987 ). The reconstruction of phylogenetic trees from molecular sequence data relies on modelling site substitutions by a Markov process, or a mixture of such processes. In general, allowing mixed processes can result in different tree topologies becoming indistinguishable from the data, even for infinitely long sequences. However, when the underlying Markov process supports linear phylogenetic invariants, then provided these are sufficiently informative, the identifiability of the tree topology can be restored. In this paper, we investigate a class of processes that support linear invariants once the stationary distribution is fixed, the 'equal input model'. This model generalizes the 'Felsenstein 1981' model (and thereby the Jukes-Cantor model) from four states to an arbitrary number of states (finite or infinite), and it can also be described by a 'random cluster' process. We describe the structure and dimension of the vector spaces of phylogenetic mixtures and of linear invariants for any fixed phylogenetic tree (and for all trees-the so called 'model invariants'), on any number n of leaves. We also provide a precise description of the space of mixtures and linear invariants for the special case of n = 4 leaves. By combining techniques from discrete random processes and (multi-) linear algebra, our results build on a classic result that was first established by James Lake (Mol Biol Evol 4:167-191, 1987). The reconstruction of phylogenetic trees from molecular sequence data relies on modelling site substitutions by a Markov process, or a mixture of such processes. In general, allowing mixed processes can result in different tree topologies becoming indistinguishable from the data, even for infinitely long sequences. However, when the underlying Markov process supports linear phylogenetic invariants, then provided these are sufficiently informative, the identifiability of the tree topology can be restored. In this paper, we investigate a class of processes that support linear invariants once the stationary distribution is fixed, the ‘equal input model’. This model generalizes the ‘Felsenstein 1981’ model (and thereby the Jukes–Cantor model) from four states to an arbitrary number of states (finite or infinite), and it can also be described by a ‘random cluster’ process. We describe the structure and dimension of the vector spaces of phylogenetic mixtures and of linear invariants for any fixed phylogenetic tree (and for all trees—the so called ‘model invariants’), on any number n of leaves. We also provide a precise description of the space of mixtures and linear invariants for the special case of n = 4 leaves. By combining techniques from discrete random processes and (multi-) linear algebra, our results build on a classic result that was first established by James Lake (Mol Biol Evol 4:167–191, 1987 ). The reconstruction of phylogenetic trees from molecular sequence data relies on modelling site substitutions by a Markov process, or a mixture of such processes. In general, allowing mixed processes can result in different tree topologies becoming indistinguishable from the data, even for infinitely long sequences. However, when the underlying Markov process supports linear phylogenetic invariants, then provided these are sufficiently informative, the identifiability of the tree topology can be restored. In this paper, we investigate a class of processes that support linear invariants once the stationary distribution is fixed, the 'equal input model'. This model generalizes the 'Felsenstein 1981' model (and thereby the Jukes-Cantor model) from four states to an arbitrary number of states (finite or infinite), and it can also be described by a 'random cluster' process. We describe the structure and dimension of the vector spaces of phylogenetic mixtures and of linear invariants for any fixed phylogenetic tree (and for all trees-the so called 'model invariants'), on any number n of leaves. We also provide a precise description of the space of mixtures and linear invariants for the special case of [Formula: see text] leaves. By combining techniques from discrete random processes and (multi-) linear algebra, our results build on a classic result that was first established by James Lake (Mol Biol Evol 4:167-191, 1987). The reconstruction of phylogenetic trees from molecular sequence data relies on modelling site substitutions by a Markov process, or a mixture of such processes. In general, allowing mixed processes can result in different tree topologies becoming indistinguishable from the data, even for infinitely long sequences. However, when the underlying Markov process supports linear phylogenetic invariants, then provided these are sufficiently informative, the identifiability of the tree topology can be restored. In this paper, we investigate a class of processes that support linear invariants once the stationary distribution is fixed, the ‘equal input model’. This model generalizes the ‘Felsenstein 1981’ model (and thereby the Jukes–Cantor model) from four states to an arbitrary number of states (finite or infinite), and it can also be described by a ‘random cluster’ process. We describe the structure and dimension of the vector spaces of phylogenetic mixtures and of linear invariants for any fixed phylogenetic tree (and for all trees—the so called ‘model invariants’), on any number n of leaves. We also provide a precise description of the space of mixtures and linear invariants for the special case of n=4 leaves. By combining techniques from discrete random processes and (multi-) linear algebra, our results build on a classic result that was first established by James Lake (Mol Biol Evol 4:167–191, 1987). Peer Reviewed |
| Author | Steel, Mike Casanellas, Marta |
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| BackLink | https://www.ncbi.nlm.nih.gov/pubmed/27604275$$D View this record in MEDLINE/PubMed |
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| CitedBy_id | crossref_primary_10_1093_sysbio_syz054 crossref_primary_10_1089_cmb_2020_0315 crossref_primary_10_1137_23M1605302 |
| Cites_doi | 10.1089/cmb.1995.2.39 10.1016/j.mbs.2003.10.004 10.1089/cmb.1994.1.153 10.1093/sysbio/syv086 10.1007/s11538-007-9293-y 10.1016/0025-5564(91)90083-U 10.1093/molbev/msr259 10.1186/1748-7188-7-33 10.1089/cmb.2005.12.204 10.1006/jtbi.1995.0067 10.1016/S0025-5564(96)00075-2 10.1093/sysbio/syq069 10.1016/j.matpur.2010.11.002 10.1089/cmb.2006.0126 10.1093/sysbio/sys064 10.1093/oso/9780198509424.001.0001 |
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| Contributor | Universitat Politècnica de Catalunya. GEOMVAP - Geometria de Varietats i Aplicacions Universitat Politècnica de Catalunya. Departament de Matemàtiques |
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| Copyright | Springer-Verlag Berlin Heidelberg 2016 Journal of Mathematical Biology is a copyright of Springer, 2017. info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by-nc-nd/3.0/es |
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| Keywords | Markov processes 60J28 Linear invariants 92D15 Phylogenetic tree 05C05 |
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