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## GENERATING MARKOV EVOLUTIONARY MATRICES FOR A GIVEN BRANCH LENGTH

### Citations

2517 |
A simple method for estimating evolutionary rate of base substitutions through comparative studies of nucleotide sequences
- Kimura
- 1980
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Citation Context ...n te so that the substitution matrix (or transition matrix) Pe equals exp(Q · λete). Among them there are the (stationary and time-reversible) models Jukes-Cantor JC69 [10], Kimura two-parameters K80 =-=[13]-=-, Kimura three-parameters K81 [14], HKY [9], and GTR [20]. In this paper we consider a broader class of evolutionary models, the (discretetime) Markov models on phylogenetic trees. These models have b... |

1457 | PAML: a program package for phylogenetic analysis by maximum likelihood
- Yang
- 1997
(Show Context)
Citation Context ...n distribution according to these parameters. There are several programs available for generating data under mostused continuous-time evolutionary models, for example seq-gen [16] and evolver in PAML =-=[21]-=-. Here we deal with the problem of generating data evolving under the more general discrete-time models when the branch lengths of the tree are fixed. FromGENERATING MARKOV EVOLUTIONARY MATRICES 3 wh... |

1087 |
Evolution of protein molecules
- Jukes, Cantor
- 1969
(Show Context)
Citation Context ...tree) that operates at intensity λe and for duration te so that the substitution matrix (or transition matrix) Pe equals exp(Q·λete). Among them there are the time-reversible models Jukes-Cantor JC69 =-=[JC69]-=-, Kimura twoparameters K80 [Kim80], Kimura three-parameters K81 [Kim81], HKY [HKY85], and GTR [Tav86]. In this paper we consider a broader class of evolutionary models, the (discretetime) Markov model... |

1013 |
Dating of the human-ape splitting by a molecular clock of mitochondrial DNA
- Hasegawa, Kishino, et al.
- 1985
(Show Context)
Citation Context ...nsition matrix) Pe equals exp(Q · λete). Among them there are the (stationary and time-reversible) models Jukes-Cantor JC69 [10], Kimura two-parameters K80 [13], Kimura three-parameters K81 [14], HKY =-=[9]-=-, and GTR [20]. In this paper we consider a broader class of evolutionary models, the (discretetime) Markov models on phylogenetic trees. These models have been widely used in phylogenetics, but mostl... |

403 |
Some probabilistic and statistical problems on the analysis of DNA sequences
- Tavare
- 1986
(Show Context)
Citation Context ...x) Pe equals exp(Q · λete). Among them there are the (stationary and time-reversible) models Jukes-Cantor JC69 [10], Kimura two-parameters K80 [13], Kimura three-parameters K81 [14], HKY [9], and GTR =-=[20]-=-. In this paper we consider a broader class of evolutionary models, the (discretetime) Markov models on phylogenetic trees. These models have been widely used in phylogenetics, but mostly in the theor... |

81 | Toric ideals of phylogenetic invariants,
- Sturmfels, Sullivant
- 2005
(Show Context)
Citation Context ..., a − b − c + d, a − b + c − d, a + b − c − d} (in this order). Remark 2.4. The change of variables considered in the Proposition above corresponds to the discrete Fourier transform in the setting of =-=[19]-=-. ⎞ ⎟ ⎠ . 3. Generating JC69 ∗ matrices with given determinant Proposition 3.1. Let K ∈ (0, 1) and let ⎛ ⎞ a b b b A = ⎜ b a b b ⎟ ⎝ b b a b ⎠ , a + 3b = 1, b b b a be a JC69 ∗ matrix. Then A is a str... |

67 | Recovering a tree from the leaf colourations it generates under a Markov model
- Steel
- 1994
(Show Context)
Citation Context ...ies and rows summing to one), this is not an easy task. We solve this problem for the so-called equivariant models JC69 ∗ , K81 ∗ , K80 ∗ and SSM ([8],[5]), and for the general Markov model GMM ([3], =-=[18]-=-, [1]). Models JC69 ∗ , K81 ∗ , K80 ∗ correspond to the discrete-time version of the corresponding continuous-time models, and SSM contains HKY as a submodel. Our results for the first four models (Pr... |

56 |
Phylogenetic invariants for the general Markov model of sequence mutation,
- Allman, Rhodes
- 2003
(Show Context)
Citation Context ...d rows summing to one), this is not an easy task. We solve this problem for the so-called equivariant models JC69 ∗ , K81 ∗ , K80 ∗ and SSM ([8],[5]), and for the general Markov model GMM ([3], [18], =-=[1]-=-). Models JC69 ∗ , K81 ∗ , K80 ∗ correspond to the discrete-time version of the corresponding continuous-time models, and SSM contains HKY as a submodel. Our results for the first four models (Proposi... |

56 |
Statistical analysis of hominoid molecular evolution. Stat. Sci
- Barry, Hartigan
- 1987
(Show Context)
Citation Context ...If a DNA sequence has evolved from another according to a substitution matrix Pe, then the number of substitutions per site that have occurred can be approximated by (1) l(e) = − 1 log det(Pe) 4 (see =-=[3]-=-). This is usually known as the branch length of edge e measured in the expected number of substitutions per site. In the case of stationary and timereversible continuous-time models, the expected num... |

36 |
Singular: a computer algebra system for polynomial computations
- Greuel, Pfister, et al.
- 2001
(Show Context)
Citation Context ...rate matrix fixed for the whole tree in these models, so that they account for what is called nonhomogeneous data: different lineages in the tree are allowed to evolve at different rates. We refer to =-=[GPS03]-=-, [AR04], and [SS03, chapter 8] for a mathematical approach to the evolutionary models used in this paper. If a DNA sequence has evolved from another according to a substitution matrix Pe, then the nu... |

35 |
Asynchronous distance between homologous DNA sequences.
- Barry, Hartigan
- 1987
(Show Context)
Citation Context ... If a DNA sequence has evolved from another according to a substitution matrix Pe, then the number of substitutions per site that have occurred can be approximated by (1) l(e) = −1 4 log det(Pe) (see =-=[BH87]-=-). This is usually known as the branch length of edge emeasured in the expected number of substitutions per site. In the case of stationary continuoustime models, it coincides with −1 4 tr(D(Π)Qλete) ... |

25 | Performance of a new invariants method on homogeneous and nonhomogeneous quartet trees
- Casanellas, Fernandez-Sanchez
(Show Context)
Citation Context ...rforming the basis change F (A) = S−1AS where ⎛ ⎞ 1 0 0 −1 S = ⎜ 0 1 1 0 ⎟ ⎝ 0 1 −1 0 ⎠ 1 0 0 1 . When A is an SSM matrix, A can be viewed as an element in HomG(C 4 , C 4 ) where G =< (AT)(CG) > (see =-=[4]-=-). The change of basis above decomposes C 4 into its isotypic components via the natural linear representation G −→ GL(C 4 ). This change of basis is also known as the generalized Fourier transform (s... |

25 |
Estimation of evolutionary sequences between homologous nucleotide sequences
- Kimura
(Show Context)
Citation Context ...bstitution matrix (or transition matrix) Pe equals exp(Q·λete). Among them there are the time-reversible models Jukes-Cantor JC69 [JC69], Kimura twoparameters K80 [Kim80], Kimura three-parameters K81 =-=[Kim81]-=-, HKY [HKY85], and GTR [Tav86]. In this paper we consider a broader class of evolutionary models, the (discretetime) Markov models on phylogenetic trees. Briefly, the parameters of these models consis... |

23 | On the existence and uniqueness of the real logarithm of a matrix
- Culver
- 1966
(Show Context)
Citation Context .... Remark 4.3. Note that by Lemma 2.3, the matrix A above diagonalizes with eigenvalues 1, α (with multiplicity 2) and β. Therefore in this case the matrix A has a real logarithm. Indeed, according to =-=[7]-=-, a non-singular diagonalizable matrix is the exponential of a real matrix if and only if its negative eigenvalues occur with even multiplicity (above, this is the case when α < 0). ⎞ ⎟ ⎠ ,GENERATING... |

19 | On the ideals of equivariant tree models.
- Draisma, Kuttler
- 2009
(Show Context)
Citation Context ...ry model) with given determinant. As the substitution matrices are stochastic matrices, this is not an easy task. We solve this problem for the so-called equivariant models JC69∗, K81∗, K80∗ and SSM (=-=[DK09]-=-,[CFS11]), and for the general Markov model GMM ([BH87], [Ste94], [AR03]). Models JC69∗, K81∗, K80∗ correspond to the discrete-time version of the corresponding continuous-time models, and SSM contain... |

18 | The strand symmetric model
- Casanellas, Sullivant
- 2005
(Show Context)
Citation Context ...e come from well known evolutionary models: in the stochastic case, GMM is a transition matrix for the general Markov model ([BH87], [Ste94], [AR03]), SSM for the strand symmetric model introduced in =-=[CS05]-=-, K81∗ for the discrete-time version of Kimura three-parameters model [Kim81], K80∗ for the discrete-time version of Kimura two-parameters model [Kim80], and JC69∗ for the discrete-time version of Juk... |

14 | Relevant phylogenetic invariants of evolutionary models
- Casanellas, Fernandez-Sanchez
(Show Context)
Citation Context ...l) with given determinant. As the substitution matrices are stochastic matrices, this is not an easy task. We solve this problem for the so-called equivariant models JC69∗, K81∗, K80∗ and SSM ([DK09],=-=[CFS11]-=-), and for the general Markov model GMM ([BH87], [Ste94], [AR03]). Models JC69∗, K81∗, K80∗ correspond to the discrete-time version of the corresponding continuous-time models, and SSM contains HKY as... |

12 |
Mathematical models in biology, an introduction
- Allman, Rhodes
- 2004
(Show Context)
Citation Context ...ix fixed for the whole tree in these models, so that they account for what is called nonhomogeneous data: different lineages in the tree are allowed to evolve at different rates. We refer to [GPS03], =-=[AR04]-=-, and [SS03, chapter 8] for a mathematical approach to the evolutionary models used in this paper. If a DNA sequence has evolved from another according to a substitution matrix Pe, then the number of ... |

7 | Hetero: a program to simulate the evolution of dna on a four-taxon tree. Appl Bioinformatics 2:159–63 - Jermiin, Ho, et al. - 2003 |

6 |
On the ideals of equivariant tree models’, Mathematische Annalen 344
- Draisma, Kuttler
- 2009
(Show Context)
Citation Context ...on matrices are stochastic matrices (nonnegative entries and rows summing to one), this is not an easy task. We solve this problem for the so-called equivariant models JC69 ∗ , K81 ∗ , K80 ∗ and SSM (=-=[8]-=-,[5]), and for the general Markov model GMM ([3], [18], [1]). Models JC69 ∗ , K81 ∗ , K80 ∗ correspond to the discrete-time version of the corresponding continuous-time models, and SSM contains HKY as... |

5 | SPIn: model selection for phylogenetic mixtures via linear invariants. Molecular Biology and Evolution
- Kedzierska, Drton, et al.
- 2012
(Show Context)
Citation Context ...lgorithm until a matrix whose diagonal entries are the largest in each column is produced (see [11]). The quoted software has been already used to simulate data for testing model selection methods in =-=[12]-=-. 2. Preliminaries Definition 2.1. A 4 × 4 matrix A with real entries and row sums equal to 1, ⎛ ⎞ A = ⎜ ⎝ a1,1 a1,2 a1,3 a1,4 a2,1 a2,2 a2,3 a2,4 a3,1 a3,2 a3,3 a3,4 a4,1 a4,2 a4,3 a4,4 ∑ j ai,j = 1,... |

3 |
The strand symmetric model. In Algebraic statistics for computational biology
- Casanellas, Sullivant
- 2005
(Show Context)
Citation Context ...ices above come from well known evolutionary models: in the stochastic case, GMM is a transition matrix for the general Markov model ([3], [18], [1]), SSM for the strand symmetric model introduced in =-=[6]-=-, K81 ∗ for the discrete-time version of Kimura three-parameters model [14], K80 ∗ for the discrete-time version of Kimura two-parameters model [13], and JC69 ∗ for the discrete-time version of Jukes-... |

1 |
models in biology, an introduction
- Mathematical
- 2004
(Show Context)
Citation Context ...fferent rates. Moreover, there is no stationary distribution implicitly assumed. These are some of the main advantages of these models as opposite to the most used continuous-time models. We refer to =-=[2]-=- and [17, chapter 8] for a mathematical approach to the evolutionary models used in this paper. If a DNA sequence has evolved from another according to a substitution matrix Pe, then the number of sub... |

1 |
phylogenetic invariants of evolutionary models
- Relevant
(Show Context)
Citation Context ...atrices are stochastic matrices (nonnegative entries and rows summing to one), this is not an easy task. We solve this problem for the so-called equivariant models JC69 ∗ , K81 ∗ , K80 ∗ and SSM ([8],=-=[5]-=-), and for the general Markov model GMM ([3], [18], [1]). Models JC69 ∗ , K81 ∗ , K80 ∗ correspond to the discrete-time version of the corresponding continuous-time models, and SSM contains HKY as a s... |