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MDCK-SIAT2,6-UF cells [10, 11], which overexpress influenza virus α2,6-linked sialic acid receptors, were used for the isolation of influenza viruses. The cells were propagated as monolayers at 37 °C and 5% CO2 in Advanced Dulbecco’s Modified Eagle’s Medium (Invitrogen Corp., Carlsbad, CA, USA) supplemented with 2 mM L-Alanyl-L-Glutamine (GlutaMAX, Invitrogen Corp.), antibiotics (50 μg/mL penicillin, 50 μg/mL streptomycin, 100 μg/mL neomycin (Invitrogen Corp.)), and 10% (v/v) low IgG, heat-inactivated gamma-irradiated fetal bovine serum (HyClone, Logan, Utah).
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study
| 100.0 |
Influenza virus A/environment/Gainesville/01/2014(H1N1) (GenBank KJ195790) was isolated from collection media used during air sampling of classroom air (J. Lednicky, unpublished). Viruses A/New York/39/2012 (H3N2), A/Texas/50/2012 (H3N2), and A/Switzerland/9715293/2013 (H3N2) were obtained from the American Type Culture Collection Influenza Reagent Resource (catalog numbers FR-1307, FR-1210, and FR-1368).
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other
| 99.8 |
For primary isolation or passage of stock viruses, aliquots of samples containing influenza virus were inoculated onto newly confluent MDCK-SIAT2,6-UF in serum-free aDMEM otherwise supplemented as described above plus L-1-tosylamido-2-phenylethyl chloromethyl ketone treated mycoplasma-and extraneous virus-free trypsin (Worthington Biochemical Company, Lakewood, NJ) in 5% CO2 at 33 °C. The TPCK-trypsin was used at a final concentration 2 μg/mL. For virus passage, cells were infected at a multiplicity of infection of 0.01 or less. The inoculated cells were monitored daily for influenza virus-specific cytopathic effects (formation of focal enlarged granular cells followed by sloughing in rapid progression); in most cases, about 80% of the virus-infected cells had detached from the growth surface at the end of 2 days of infection. The presence of influenza A virus in the cell culture media was quickly determined using a commercial solid phase ELISA test (QuickVue influenza A and B kit, Quidel Corp., San Diego, CA, USA), and the viral gene sequences determined after RT-PCR and sequencing as described previously .
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study
| 100.0 |
ANOVA and Chi-squared were performed online as referenced in Tables 1 and 2. T-tests were performed on line at http://www.graphpad.com/quickcalcs/ttest1.cfm.Table 1Influenza matrix and subtype gene seasonal nAMP averages in 3 institutions over 3 influenza seasonsInfluenza season2012–2013 H3N2 nAMPs ± SD (n)2013–2014 H1N1 nAMPs ± SD (n)2014–2015 H3N2 nAMPs ± SD (n)PMatrix Gene UF Health121.1 ± 109.5 (91)1 172.6 ± 70.2 (194)2 65.1 ± 37.5 (172)< 0.00014 BayCare74.8 ± 77.8 (19)165.4 ± 76.1 (232)50.2 ± 25 (152)< 0.00014 MemorialND3 160.1 ± 80.6 (475)57.3 ± 61.1 (788)< 0.00014 Subtype Gene UF Health169.9 ± 63.6 (91)240.9 ± 103.4 (194)164.3 ± 40.1 (172)< 0.00014 BayCare136.1 ± 50.2 (19)249.2 ± 81 (239)162.4 ± 45.9 (165)< 0.00014 MemorialND239.4 ± 88.6 (475)144.6 ± 53.0 (711)< 0.00014 1() = number of specimens tested 2Includes 22 patients from 1/21/2013–8/30/2013 who were positive for H1N1 3ND = Not Done 4ANOVA http://vassarstats.net/anova1u.html Table 2Ct adjusted nAMPs and GenMark RVP Matrix Gene Sequence MismatchesInfluenza StrainMatrix Gene nAMPsa Matrix Gene MismatchesForwardReverseCaptureSignal A/NY53.5 ± 32b 1c 03d 0 A/Texas56.6 ± 11.4b 1030 A/Switzerland46 ± 16.9b 1030 H3N2 Synthetic Capture Probe ssDNAe 73 ± 16N/Af N/A3N/A H1N1143.6 ± 57.21020 H1N1 Synthetic Capture Probe ssDNA151 ± 17N/AN/A2N/A aMean ± SD of 4 replicates. All 4 strains were diluted to the same copy number based on Ct using a TaqMan assay (see Methods) bA/NY (p = 0.033), A/Texas (p = 0.025) and A/Switzerland (p = 0.017) vs H1N1 t test. https://www.usablestats.com/calcs/2samplet cThe mismatches were the same in all H3N2 strains, but different from the H1N1 dAll 3 mismatches were the same for the H3N2 strains, but differed from the 2 mismatches in the H1N1 strain essDNA = single stranded DNA fN/A = not applicable
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study
| 100.0 |
Population seasonal nAMP means ± SDs are shown in Table 1 and were consistent between institutions within a given year. For example, in Table 1 in 2014, nAMPs of 172.6 ± 70.2 (N = 194), 165.4 ± 76.1 (N = 232) and 160.1 ± 80.6 (N = 475) were calculated for the Matrix gene assay for all positive H1N1 strains at UFHealth, BayCare Health System, and Memorial Healthcare System respectively. These institutions are 150–300 miles apart and do not share patient populations. In contrast, mean ± SD matrix gene nAMPs for the 3 different Influenza seasons (2012–13, 2013–14, and 2014–15) were highly statistically different, the 2013–14 H1N1 seasonal average being more than twice that of the 2014–15 season (ANOVA, p < 0.0001). A similar pattern was observed for the HA subtype gene assay as well (ANOVA, p < 0.0001). The 2012–2013 H3N2 matrix gene nAMPs appear to be higher than those of the 2014–2015 H3N2 season and further analysis (see Figs. 1, 2, 3) suggests that in fact the average nAMPs in the 2012–2013 H3N2 season at UFHealth may have been bi-modal, whereas the distribution is clearly unimodal for 2013–2014 and 2014–2015. In these Figures, the matrix gene nAMPs are graphed vs the corresponding subtype gene nAMPS, making the subpopulation in the 2012–2013 season easily visualized.Fig. 1Matrix gene nAMPs graphed vs Subtype gene for 2012–2013 H3N2 season at UFHealth Shands hospital. There appears to be a subpopulation with relatively higher Matrix gene nAMPs Fig. 2Matrix gene nAMPs graphed vs Subtype gene for 2013–2014 H1N1 season at UFHealth Shands hospital. There appears to be a relatively uniform ratio of the Matrix gene to Subtype gene nAMPs Fig. 3Matrix gene nAMPs graphed vs Subtype gene for 2014–2015 H3N2 season at UFHealth Shands hospital. In contrast to the 2012–2013 season, except for 3 individuals, there is an essentially uniform ratio of the Matrix gene to Subtype gene nAMPs
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study
| 100.0 |
When adjusted for the number of input RNA copies based on the cycle-threshold (Ct) of the TaqMan matrix gene assay, H3N2 strains representative of those circulating in the 2012–2013 and 2014–2015 seasons had significantly lower average nAMPs than the H1N1 strain from 2013 to 2014 (see Table 2) and these average nAMPS were very close to those of the seasonal population averages. As was the case for the population averages, the average nAMPS of the TaqMan copy number adjusted H3N2 strains were statistically significantly lower than that of the H1N1 strain (p = 0.033 vs A/NY, p = 0.025 vs A/Texas and p = 0.017 vs A/Switzerland, t-test).
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study
| 100.0 |
Matrix gene sequencing of A/New York/39/2012 (H3N2) FR-1307, A/Texas/50/2012 (H3N2) FR-1210, A/Switzerland/9715293/2013 (H3N2) FR-1368 and the Gainesville environmental 2013–2014 isolate A/environment/Gainesville/01/2014(H1N1) KJ195788 showed the same 3 bp mismatches in the capture probe for the H3N2 strains as opposed to only 2 different ones for the H1N1 when compared with the sequence of the proprietary GenMark RVP assay. Discussion of the location of these mismatches with GenMark suggested the 3 mismatches in the H3N2 strains were likely to be more destabilizing than the 2 mismatches in the H1N1 strain. Please see Table 2 for complete details. Since the GenMark assay sequence details are proprietary, specific sequence data as to which mutations were observed is not available. In contrast, the subtype genes of both the H3N2 and H1N1 strains have a perfect sequence match with their respective subtype capture and signal probes and a single bp mismatch in both forward and reverse primers.
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study
| 100.0 |
In order to confirm and better understand the relationship between sequence differences and nAMPs, GenMark Diagnostics provided single-stranded DNA matching the capture probe sequences of the H3N2 and H1N1 strains. When this synthetic DNA was run in the eSensor, equimolar concentrations gave average nAMPs of 150.5 ± 17.2 and 72.9 ± 16.3 nAMPs (N = 4, p = 0.0006, t-test), for the sequences matching the H1N1 strain and H3N2 strains, respectively (see Table 2). A capture probe with a perfect sequence match to the GenMark assay was tested at GenMark and gave the same average nAMPs as the H1N1 sequence, despite the 2 mismatches in the H1N1 sequence (data not shown).
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study
| 100.0 |
The use of respiratory virus quantitation has been discussed primarily in relation to severity of individual patient illness and duration of viral shedding [10–12]. Clearly such quantitative values, whether GenMark nAMPs or TaqMan Ct values from commercial or lab developed assays, vary greatly from patient to patient and even within a patient depending primarily time of sample collection since days 1 and 2 of illness typically have the highest titers [12–14]. Other sources of quantitative difference result from variability in specimen type (e.g. nasopharyngeal vs BAL), variation in collection practices within the same specimen type, as well as patient variables such as immunosuppression, prior vaccination, age, etc. Although the semi-quantitative nAMP readings are not FDA approved for use in patient care, we looked at viral quantitation from the perspective of population seasonal averages, so that patient-related and pre-analytic variability should average out. Our data suggest that this is the case. Seasonal Influenza A population matrix gene nAMPs from the GenMark RVP assay were remarkably consistent between 3 different institutions in Florida for 3 Influenza seasons from 2012 to 2015. The slightly higher matrix gene nAMPs seen at UFHealth in 2012–2013 compared with 2014–2015 seems to be explained by a bi-modal distribution in the 2012–2013 season, suggesting there may have been a mixture of strains in circulation.
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study
| 100.0 |
The 2013–2014 H1N1 season matrix gene nAMPS were markedly higher than the prior and following H3N2 seasons. Since the same matrix gene assay is used for both subtypes, it was possible these differences could have reflected seasonal differences in Influenza illness severity; however, sequence analysis suggested that the differences were better explained by sequence drift in the matrix gene that had occurred by the 2012–2013 season, and was unchanged in the 2014–2015 season. Since the nAMP averages from the synthetic single stranded DNA of the H3N2 and H1N1 capture probe sequences closely matches those of the seasonal averages, sequence variation between the H3N2 and H1N1 viruses offers the best explanation for the seasonal differences.
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study
| 100.0 |
Further study is needed to determine whether population averages could reflect the clinical intensity of an influenza season, once it was determined that no assay-related sequence drift had occurred, on the theory that more severely ill the patients might seek medical care earlier in the course of their illness. Hence higher titers could be a reflection of patients being tested earlier when their titers were higher [12–16]. As noted above, the subtype genes of both the H3N2 and H1N1 strains have a perfect sequence match with their respective subtype capture and signal probes and a single bp mismatch in both forward and reverse primers, but since each subtype assay has different target sequences, average nAMPs cannot be compared.
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study
| 100.0 |
One of the limitations of this study is that we did not save the actual strains from the 3 influenza seasons studied, since the laboratory was no longer performing viral cultures. However, the H3N2 strains analyzed for the matrix gene in the 2012–2013 and 2014–2015 seasons are generally considered representative of the sequence variation observed in the matrix genes, and the H1N1 isolate was an environmental isolate obtained on the University of Florida campus near the peak of the 2013–2014 season. Likewise, sequence analysis of HA sequences available from GenBank for representative H3N2 strains showed no changes in the number and location of mismatches in the GenMark assay between the 2012–2013 and 2014–2015 seasons. Another issue, although not strictly a limitation of the study, is that the GenMark assay is more complex than traditional TaqMan assays, in that after the PCR product is produced, it is captured on a solid phase where a 4th probe that carries a ferrocene label is required to bind to permit detection of an electrical signal. Although a positive assay is reported from 3.0 to >300 nAMPs, in our experience the actual linear range is only about 10–30 fold, which is very narrow compared with the generally observed 5 logs of linearity for a typical TaqMan assay.
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study
| 100.0 |
Perhaps the most important implication of this study is that quantitative population averages (whether done by nAMPs as in the GenMark assay, or by Ct as would be the case in other manufacturers’ assays) appear to be sensitive to Influenza sequence changes due to seasonal drift that result in assay mismatches. In addition, it is possible that plotting the nAMPs of the matrix gene vs the subtype gene may offer a simple method to detect subpopulations with sequence variations affecting the performance of the assay. Although it does not appear that any Influenza strains were actually “missed” in our study because of mismatches between the matrix gene assay and the matrix gene sequence in circulating strains, further sequence drift in circulating strains could render the assay falsely negativee at some point.
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study
| 100.0 |
Seasonal average nAMPs were remarkably consistent between institutions within a given year. The differences in the matrix gene averages between H3N2 and H1N1 seasons were consistent with the number and location of mismatches in the molecular assay. Instrument manufacturers, laboratories and regulatory agencies should work out an approach to capture this type of data electronically on national and regional levels so that seasonal sequence drift affecting the performance of molecular assays can be monitored.
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other
| 99.44 |
The evolutionary histories of species and genes can be discordant , necessitating a distinction between genes trees and species trees. Incomplete Lineage Sorting (ILS), modeled by the multi-species coalescent (MSC) model , is one of the main causes of discordance. A fast approach for estimating the species relationships in the face of such discordances is to first estimate a gene tree for each gene and to summarize the gene trees to build a species tree. The summary method, thus, takes as input a set of gene trees and returns a species tree. A desirable property for a summary method is statistical consistency (a theoretical guarantee that it converges in probability to the correct species tree as the number of error-free genes increases). Many statistically consistent summary methods are available (e.g., ASTRAL [3, 4], BUCKy-population , and MP-EST ), and coalescent-based species tree estimation is a vibrant field of research, with many recent examples of successful biological analyses [7–9] (see [10–14] for criticism of these methods, especially their sensitivity to gene tree error).
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review
| 99.9 |
Inferring trees using pairwise distances is a well-studied general method of phylogenetic reconstruction [15–18], and several summary methods are distance-based. These methods first compute a pairwise distance between species based on input gene trees and then use a distance method (e.g., neighbor joining ) to build the species tree; examples of distance-based summary methods are STAR , GLASS , NJst , and its new implementation, ASTRID .
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review
| 99.75 |
Another powerful general approach to phylogenetic reconstruction is analyzing quartets, which are subsets of four leaves in a tree. Quartet methods first infer a set of quartet trees and then combine them to build a tree on the full dataset [16, 23, 24]. Induced quartet trees have also been used [24–28] to combine a collection of input trees to build a so-called supertree . Quartet-based phylogeny estimation has been revived in recent years [3, 5, 30–32] because of its connections to coalescent-based analyses [33–35]. Under the MSC model, for unrooted species trees with four leaves, the most likely unrooted gene tree is identical to the species tree (but this is not true for larger trees [34, 36]). Furthermore, the length of the internal branch in a quartet species tree (in coalescent units) defines the probabilities of the three possible gene tree quartet topologies . Some recent and statistically consistent quartet-based species tree estimation methods rely on these results. For example, ASTRAL seeks the species tree with the maximum number of quartet trees shared with input gene trees [3, 4].
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review
| 99.7 |
In this paper, we introduce a new coalescent-based summary method, called DISTIQUE (Distance-based Inference of Species Trees from Induced QUartet Elements). Like ASTRAL, DISTIQUE is based on quartets, but instead of directly optimizing a quartet score, it uses quartets to compute pairwise distances, which are then used as input to a distance method. The innovative aspect of DISTIQUE is its method of calculating distances. It chooses two arbitrary “anchor” species and computes the frequency of quartet trees induced by gene trees that include the two anchors as sisters. We show that these frequencies can be transformed into an asymptotically additive distance matrix; using this matrix with a consistent distance-based method (e.g., neighbor joining) gives a statistically consistent summary method. This method would generate a species tree on all species except the two anchors in Θ(n 2 k) (for n species and k gene trees). However, using multiple anchor pairs can increase accuracy and can ensure all species are included in the final tree. Various strategies for choosing anchors and combining their results are introduced, with running times ranging between Θ(n 2 k) and Θ(n 4 k).
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study
| 99.94 |
After describing DISTIQUE, we show that the anchoring approach can be generalized to any tree inference problem. Assume we have a way to compute the topology and the internal branch length for any quartet of leaves. We show that as long as this quartet estimator is consistent, our anchoring mechanism and a certain family of transformations can be used to compute an additive distance matrix, which in turn can be used to infer the correct tree topology but not correct branch lengths. This result is rather surprising because, for any pair of anchors and a pair of other leaves, the quartet internal branch length will often be very different from the distance between non-anchor leaves. Thus, anchoring produces incorrect pairwise distances that are nevertheless additive for the correct tree topology. DISTIQUE uses anchoring because for the MSC-based species tree inference, pairwise species distances are not straightforward to define but inferring quartet trees is easy. We evaluate the accuracy of DISTIQUE on simulated and biological data and show that its accuracy is competitive with the best alternative methods even when used with relatively small subsets of all possible anchors.
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study
| 99.94 |
Notation and background: Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {L}$\end{document}ℒ denote the leaf-set of size n. For an unrooted tree T on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {L}$\end{document}ℒ, the set of quartet trees induced on all possible \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${n \choose 4}$\end{document}n4 quartets of leaves is denoted by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {Q}^{T}$\end{document}QT. We use a b.c d to denote that a and b are sisters in the quartet tree on {a,b,c,d}. A tree T is equivalent to a distance matrix D T, computed by summing lengths of the edges between pairs of leaves, and a distance matrix that corresponds to a tree is called additive . We refer to the unique tree associated with the additive distance matrix D as T D or T. Also, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T|\mathcal {L}'$\end{document}T|ℒ′ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D|\mathcal {L}'$\end{document}D|ℒ′ denote T and D restricted to the leaf-set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {L}'$\end{document}ℒ′.
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other
| 99.6 |
To test for the additivity of a distance matrix D, we can use the four point condition . For a quartet of leaves \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$Q=\{a,b,c,d\}\subset \mathcal {L}$\end{document}Q={a,b,c,d}⊂ℒ, the median and the maximum of the following three values should be the same: {D[a,b]+D[c,d],D[a,c]+D[b,d],D[a,d]+D[b,c]}. When internal branch lengths are assumed positive, as we do throughout this paper, the minimum value is strictly smaller than the median. Assuming w.l.o.g. D[a,b]+D[c,d] is the smallest value, we can infer a b.c d is the topology induced by T D. Let τ(Q)>0 denote the length of the single internal branch in this quartet tree, which we call its “quartet length”; i.e., if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ab.cd \in \mathcal {Q}^{T}$\end{document}ab.cd∈QT, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tau (Q)= \frac {1}{2} (D[a,c] + D[b,d] - D[a,b] - D[c,d])$\end{document}τ(Q)=12(D[a,c]+D[b,d]−D[a,b]−D[c,d]).
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study
| 99.94 |
Given two positive constants α,βand a monotonically increasing function f(x) bounded above by β for positive x (i.e., 0<f(x)<β for x>0), two “anchor” leaves \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u,v\in \mathcal {L}$\end{document}u,v∈ℒ, and a tree T equivalent to distance matrix D with the corresponding quartet length function τ(Q), we define: 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{array}{@{}rcl@{}} D^{\prime}_{uv} [a,b] &=& \left\{ \begin{array}{ll} \beta + \alpha. \tau(\{a,b,u,v\}) & ab.uv \notin \mathcal{Q}^{T} \\ \beta - f(\tau(\{a,b,u,v\})) & ab.uv \in \mathcal{Q}^{T} \end{array}\right. \end{array} $$ \end{document}Duv′[a,b]=β+α.τ({a,b,u,v})ab.uv∉QTβ−f(τ({a,b,u,v}))ab.uv∈QT
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other
| 96.44 |
2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{array}{@{}rcl@{}} D^{\prime}_{v}[a,b] &=& \sum_{u \in \mathcal{L} - \{a,b,v\}} D^{\prime}_{uv}[a,b] \end{array} $$ \end{document}Dv′[a,b]=∑u∈ℒ−{a,b,v}Duv′[a,b]
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other
| 99.8 |
3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{array}{@{}rcl@{}} D^{\prime}[a,b] &=& \sum_{v \in \mathcal{L} - \{a,b\}} \sum_{u \in \mathcal{L} - \{a,b,v\}} D^{\prime}_{uv}[a,b] \end{array} $$ \end{document}D′[a,b]=∑v∈ℒ−{a,b}∑u∈ℒ−{a,b,v}Duv′[a,b]
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other
| 99.8 |
4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{array}{@{}rcl@{}} D^{\prime\prime} [a,b] &=& \max_{u,v \in \mathcal{L} - \{a,b\}} \max\left(0,\frac{D^{\prime}_{uv}[a,b]-\beta}{\alpha}\right). \end{array} $$ \end{document}D′′[a,b]=maxu,v∈ℒ−{a,b}max0,Duv′[a,b]−βα.
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other
| 99.8 |
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\prime }, D^{\prime }_{u}$\end{document}D′,Du′, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\prime }_{uv}$\end{document}Duv′ are distance matrices on leaf-sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {L}, \mathcal {L}\{v\}$\end{document}ℒ,ℒ{v}, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {L}-\{u,v\}$\end{document}ℒ−{u,v}, respectively, and are called “all-pairs anchored”, “single anchored”, and “double anchored”. We say \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\prime }_{uv}$\end{document}Duv′ is induced from D anchored by u,v. D ′′ is called an “all-pairs anchored maximum distance matrix” and is defined on the leaf-set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {L}$\end{document}ℒ.
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other
| 99.6 |
Let D T be an additive distance matrix. A double anchored distance matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\prime }_{uv}$\end{document}Duv′ induced from D T anchored by arbitrary leaves \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u,v\in \mathcal {L}$\end{document}u,v∈ℒ is an additive distance matrix for the leaf-set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {L}^{\prime }=\mathcal {L}-\{u,v\}$\end{document}ℒ′=ℒ−{u,v} and corresponds to a tree that is topologically identical to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T|\mathcal {L}^{\prime }$\end{document}T|ℒ′. Similarly, a single anchored distance matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\prime }_{v}$\end{document}Dv′ induced from D T anchored by an arbitrary leaf and an all-pairs anchored distance matrix D ′ induced from D T are additive distance matrices for the leaf-sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {L}-\{v\}$\end{document}ℒ−{v} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {L}$\end{document}ℒ, respectively, and correspond to trees that are topologically identical to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T|\mathcal {L}-\{v\}$\end{document}T|ℒ−{v} and T, respectively.
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other
| 99.44 |
Both theorems are proved in the appendix. Theorem 2 is similar to a result given by Brodal et al. , and is easy to prove. The basic idea is that for any two non-sister leaves {a,b}, there is a pair of anchors such that in the resulting quartet, a and b are not sisters, and the quartet length is exactly the same as the distance between the two leaves minus their terminal branches. We note that similar to us, Brodal et al. use the concept of anchors, but instead of using anchors to define distances, they use anchors to efficiently build Buneman trees from given distances. Thus, despite some parallels, our anchoring mechanism is novel; In particular, Brodal et al. do not prove our surprising result that a single arbitrarily chosen pair of anchors gives additive distances for the correct topology.
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| 98.9 |
Theorem 1 states anchored distances induced from an additive matrix will correspond to the same topology as the initial matrix (albeit with wrong branch lengths). This result is surprising, but its usefulness might be less clear. Theorem 1 enables new estimators of the tree topology that rely on quartets to compute pairwise distances. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {D}$\end{document}D denote the input data to be used for inferring a phylogeny. Regardless of the nature of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {D}$\end{document}D, we require having a quartet estimator. A quartet estimator is a function that given a quartet of leaves Q, uses \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {D}$\end{document}D to estimate the quartet tree topology and the quartet length τ(Q), and is statistically consistent if, as the size of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {D}$\end{document}D increases, the estimated quartet topology and length both converge in probability to correct values. Statistically consistent quartet estimators can be designed for various models (e.g., sequence evolution and the MSC [33, 34]).
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Given a statistically consistent quartet estimator, a family of statistically consistent tree inference algorithms can be designed (Additional file 1: Algorithm S1). Details and proofs are given in the (Additional file 1: Section 2.4). The basic idea is the following. We can use the quartet estimator to infer a distance matrix that asymptotically can be made arbitrarily close to an additive distance matrix for the true tree topology. Using a method such as neighbor-joining that infers the correct tree for additive distance matrices with a safety radius will give a consistent estimator of the tree .
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| 94.4 |
Problem statement: Given an input dataset \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {G}$\end{document}G of a collection of k unrooted gene trees, we seek to find the unrooted species tree topology, assuming gene trees are generated by the MSC model .
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Next, we first describe anchored distances based on the MSC model used in DISTIQUE. We then describe the algorithmic design of DISTIQUE, including its strategies for selecting anchors, combining results from multiple anchors, and dealing with long branches.
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Let p(a b.u v)denote the true probability of observing the quartet topology a b.u v in gene trees generated according to the MSC model. We define MSC-based double, single, and all-pairs anchored distance matrices \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{*}_{u,v}, D^{*}_{v}$\end{document}Du,v∗,Dv∗, and D ∗, respectively on leaf-sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {L}-\{u,v\}, \mathcal {L}-\{v\}$\end{document}ℒ−{u,v},ℒ−{v} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {L}$\end{document}ℒas: 5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{array}{@{}rcl@{}} D^{*}_{u,v} [a,b] &=& -\ln p(ab.uv) \end{array} $$ \end{document}Du,v∗[a,b]=−lnp(ab.uv)
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6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{array}{@{}rcl@{}} D^{*}_{v}[a,b] &=& \sum_{u \in \mathcal{L} - \{a,b,v\}} -\ln p(ab.uv) \end{array} $$ \end{document}Dv∗[a,b]=∑u∈ℒ−{a,b,v}−lnp(ab.uv)
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7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{array}{@{}rcl@{}} D^{*}[a,b] &=& \sum_{v \in \mathcal{L} - \{a,b\}} \sum_{u \in \mathcal{L} - \{a,b,v\}} -\ln p(ab.uv) \end{array} $$ \end{document}D∗[a,b]=∑v∈ℒ−{a,b}∑u∈ℒ−{a,b,v}−lnp(ab.uv)
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For species tree estimation under the MSC model, Eq. (1) simplifies to Eq. (5) for β= ln3,α=1, and f(x)= ln(3−2e −x). Thus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\prime }_{uv}[a,b] = D^{*}_{uv} [a,b] =-\ln p(ab.uv).$\end{document}Duv′[a,b]=Duv∗[a,b]=−lnp(ab.uv).
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Given true quartet probabilities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p(ab.uv), D^{*}_{uv}, D^{*}_{v}$\end{document}p(ab.uv),Duv∗,Dv∗, and D ∗ become additive distance matrices that correspond to the true species tree topology on leaf-sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {L}-\{u,v\}, \mathcal {L}-\{v\}$\end{document}ℒ−{u,v},ℒ−{v}, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {L}$\end{document}ℒ, respectively.
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| 99.4 |
It may be surprising that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{*}_{uv}$\end{document}Duv∗, which is a special case of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\prime }_{uv}$\end{document}Duv′, depends only on quartet topologies and not branch lengths. To see why, readers should recall that p is the quart frequency in gene trees, and relates to both the quartet topology and the quartet length in the species tree.
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| 99.6 |
True quartet probabilities are not known. Instead, we empirically use \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\overline {p}(ab.uv)=\frac {1}{k}|\{t:\mathcal {G}|ab.uv\in \mathcal {Q}^{t}\}|$\end{document}p¯(ab.uv)=1k|{t:G|ab.uv∈Qt}|. Empirical frequencies inferred from gene trees converge in probability to true values as the number of genes increases; thus, it is easy to show (proof omitted):
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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{*}_{uv}, D^{*}_{v}$\end{document}Duv∗,Dv∗, and D ∗ computed using empirical frequencies in a random sample of error-free gene trees converge in probability to an arbitrarily small radius of an additive matrix identical in topology to the true species tree; a consistent distance method with a safety radius [ 40 ] run on these matrices is a consistent estimator of the species tree topology.
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Computing anchored matrices require Θ(n 2 k),Θ(n 3 k), and Θ(n 4 k) time, respectively for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{*}_{uv}, D^{*}_{v}$\end{document}Duv∗,Dv∗, and D ∗. Among these matrices, only D ∗ includes all species.
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DISTIQUE uses double anchored matrices, which can be each computed in Θ(n 2 k). It uses multiple anchors and combines the trees or matrices produced by different anchors. A careful selection of anchors can ensure the final DISTIQUE tree includes all species, and can control its running time between Θ(n 2 k) and Θ(n 4 k). Before presenting our anchoring strategy, we first need to show how DISTIQUE deals with long branches.
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Smoothing For species tree branches that are even moderately long, expected frequencies of alternative quartet topologies become exceedingly close to zero. For example, a species tree quartet length of 12 in coalescent units results in a 99.6 % chance of observing no discordance among 1000 genes. Thus, our simple empirical frequency estimator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\overline {p}$\end{document}p¯ can easily be equal to zero, resulting in distances of infinity (Eq. 5). To avoid this problem, we use Krichevsky-Trofimov (i.e., add-half estimator), which adds a pseudo-count of 0.5 for each of three possible quartet topologies. This estimator has been shown to reach the min-max cumulative loss for KL divergence asymptotically .
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Consensus Smoothing does not fix the larger problem of distinguishing between long distances. For example, branches of length 12, 24, or 48 are all very likely to result in no gene tree discordance given 1000 genes; thus, even with smoothing, it remains impossible to distinguish between branches with these very different distances. This limitation makes it impossible to compute distances that reflect the true topology from limited data when the species tree includes adjacent long branches (resembling the saturation problem in phylogenetics ). We can construct examples when all gene trees are likely identical, yet our smoothed distances are misleading (Additional file 1: Section 2.2; Figure S7). However, long branches are easy to recover because they appear in most gene trees. A simple majority rule (50 %) consensus of gene trees would return all long branches. Thus, we simply compute the majority consensus and resolve its polytomies using DISTIQUE (Additional file 1: Algorithms S2 and S3). Because the majority consensus is proved not positively misleading under the MSC , our method remains statistically consistent.
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| 100.0 |
To resolve a polytomy, Additional file 1: Algorithm S2 first assigns a cluster label to each branch pendant to it, and then builds a tree using DISTIQUE with the cluster labels as leaves; this tree defines a resolution of the polytomy. Given anchor species u,v from two distinct clusters, we compute distances between other pairs of clusters A and B using Eq. (5), defining the quartet frequencies as: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\overline {p}(uv.AB) = \frac {1}{|A||B|}\sum _{a\in A}\sum _{b\in B}{\overline {p}(uv.ab)}.$\end{document}p¯(uv.AB)=1|A||B|∑a∈A∑b∈Bp¯(uv.ab). When all clusters in the consensus tree are correct (expected asymptotically), p(u v.a b) values are identical; thus, all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\overline {p}(uv.ab)$\end{document}p¯(uv.ab) values are empirical estimates of the same true value, and using their average is justified.
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Additional file 1: Algorithm S4 shows DISTIQUE’s targeted sampling strategy for choosing a subset of all possible anchor pairs. Let d 1…d r denote the degree of polytomies in the consensus tree, indexed arbitrarily. For each polytomy i, we randomly partition its d i clusters into sets of size two; if d i is odd, we randomly choose a cluster and pair it with the remaining cluster. Then, we randomly choose one species from each cluster. This produces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lceil \frac {d_{i}}{2}\rceil $\end{document}⌈di2⌉ pairs of anchors for each polytomy i. The total number of anchors is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$m={\sum _{1}^{r}} \lceil \frac {d_{i}}{2}\rceil =O(n)$\end{document}m=∑1r⌈di2⌉=O(n) (Additional file 1: Lemma S2). Each anchor pair is used to resolve all polytomies on the path between them in the consensus tree. This processes may be repeated several rounds (a user-specified input parameter).
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Polytomies of degree 4 or 5 cannot be handled using the double anchored approach because once two clusters are chosen as anchors, only two or three clusters remain which cannot be resolved as unrooted trees. For these small polytomies, we always use all-pairs distance matrices; thus, we choose all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${4 \choose 2}$\end{document}42 or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${5 \choose 2}$\end{document}52 possible pairs of clusters around the polytomy. We need O(n) anchors in this scenario as well (Additional file 1: Lemma S2).
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| 99.94 |
We first compute m trees, each on n−2 leaves using the double anchored method (Corollary 1) and then combine these m trees using a supertree method. Using a compatibility supertree (i.e., one that given a set of compatible input trees, outputs a tree that refines all input trees) would make the approach statistically consistent (Theorem S2, Additional file 1).
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We also use the following approach to filter out outlier anchors. We compute an initial supertree from m anchored trees, then find the RF distance between m trees and the supertree, remove those with an RF distance at least two standard deviations larger than the mean, and recompute the supertree.
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| 99.9 |
The distance-sum approach creates a summary distance matrix and runs neighbor joining on the summary matrix. The summary distance is simply the average distance of each pair in the set of m double anchored matrices. Note that some of the m double anchored matrices might not have a value for a given pair of leaves; we treat those as missing values and ignore them when averaging values. The presence of missing values jeopardizes our proofs of statistical consistency.
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| 99.3 |
Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{*}_{uv}$\end{document}Duv∗ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{*}_{wz}$\end{document}Dwz∗ be two double anchored matrices produced using two disjoint pairs of anchors. If the two matrices are reduced to the n−4 leaves common between them (i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {L}^{\prime }=\mathcal {L}-\{u,v, w,z\}$\end{document}ℒ′=ℒ−{u,v,w,z}), we get two matrices that asymptotically converge to an additive matrix for the same tree topology (Corollary 1). The sum of two additive distance matrices that correspond to the same tree topology is also additive for the same topology. Thus, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{*}_{uv}|\mathcal {L}^{\prime }+D^{*}_{wz}|\mathcal {L}^{\prime }$\end{document}Duv∗|ℒ′+Dwz∗|ℒ′ is asymptotically additive for the correct species tree. This provides a theoretical justification for our distance-sum approach. However, distances between four anchors and other leaves are missing in one of the matrices, and thus, their correct placement cannot be guaranteed.
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| 99.8 |
If all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${n \choose 2}$\end{document}n2 anchors are used, the distance-sum approach becomes equivalent to the all-pairs approach and is provably statistically consistent (Theorem 3). On the other hand, using only two pairs of anchors makes the placement of anchors dependent on averages of two numbers, one of which is missing, a clearly problematic scenario. Choosing an intermediate number of anchors, while insufficient for giving proofs of consistency, clearly reduces the impact of missing values. For example, assume we have m anchors and each species is included in at most only one of those anchors. The summary distance between each pair of leaves becomes an average of m values, among which at most one may be missing.
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For large enough m, we conjecture that the impact of that single missing value is negligible. In the results section, we provide empirical evidence for this conjecture, but future work should explore theoretical proofs. Due to its superior empirical performance, distance-sum is used by default in the DISTIQUE (see Additional file 1: Algorithm S2 for all details).
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Using all-pairs or all-pairs-max clearly require Θ(n 4 k) time to build the distance matrix and using the default O(n 3) neighbor joining algorithm would result in Θ(n 4 k) total running time. The running times of tree-sum and distance-sum depend on the selection of anchors, and also the exact distance method and supertree method used. Building each double anchored distance matrix requires Θ(n 2 k); thus, building m matrices requires Θ(n 2 m k). Using a fast neighbor joining algorithm (e.g., FNJ , or NINJA ), the running time of distance method can be O(n 2).
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Clearly, any function between Θ(n 2 k) and Θ(n 4 k) can be obtained by adjusting m. DISTIQUE’s default strategy requires O(n) anchors and therefore results in O(n 3 k) total running time. For the tree-sum approach, the running time of the supertree method needs to be also added. MRL, which we use here, doesn’t have running time guarantees, but ML methods tend to have average running time close to O(n 2) .
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We use simulated and real datasets to evaluate the accuracy and scalability of DISTIQUE. We measure species tree accuracy using False Negative (FN) rate, which is equivalent to normalized RF distance here because all estimated species trees are fully resolved.
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For biological analyses, we re-analyzed a dataset of 2022 supergene trees from an avian dataset [7 , 11]. We also use three sets of simulated datasets we used before: a 37-taxon mammalian dataset , a 45-taxon avian dataset , and datasets used for evaluating ASTRAL-II . The first two datasets are based on biological data and have a single species tree topology, whereas the last dataset is simulated using SimPhy and has a different species tree per replicate and has heterogeneous parameters. Avian and mammalian datasets enable us to evaluate performance for relatively small numbers of species, varying ILS and the number of genes. The amount of ILS is changed by multiplying or dividing branch lengths by 2 or 5; shorter branches (0.2X and 0.5X) produce more ILS and longer branches reduce ILS (Additional file 1: Table S1). We create two collections for these datasets, one where we fix the number of genes (200 for mammalian and 1000 for avian) and vary the amount of ILS, and a second collection, where we fix the amount of ILS (to very high or 0.2X for mammalian and default 1X for avian) and vary the number of genes (200 to 3200 for mammalian and 200 to 2000 for avian). The simPhy dataset has two collections, and is simulated to capture the range of reasonable biological datasets. In the simPhy-ILS collection, we fix the number of species to 201 and show three levels of ILS, ranging from moderate (10 million generations) to very high (500K generations), and for each case, we vary the number of genes (50, 200, 1000). For each case, we have 100 replicates, half with a speciation rate of 10−6 and the other half with 10−7. In the simPhy-size, we fix ILS to moderate and speciation rate to 10−6, and change the number of species from 10 to 500, with 50 replicates per dataset.
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We compare various versions of DISTIQUE, described below, against each other, and against ASTRAL-II , which is a quartet-based method, the ASTRID (a new implementation of the NJst algorithm ), which is a distance-based method, and concatenation using RAxML (CA-ML). ASTRAL and NJst are statistically consistent summary methods and, like DISTIQUE, work on unrooted gene trees and species trees (most other approaches such as MP-EST and STAR need rooted input). Also, these two are among the most accurate summary methods [3 , 4 , 21 , 22 , 51].
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| 99.75 |
We explore variants of DISTIQUE, changing the distance matrix (comparing all-pairs, all-pairs-max, tree-sum, and distance-sum; see Additional file 1: Algorithm S1), the number of anchoring rounds (2 to 8), and the use of consensus. To compare to other methods, we use the default distance-sum DISTIQUE (Additional file 1: Algorithm S2), with 2 or 8 rounds of anchoring. DISTIQUE is implemented in python and uses the Dendropy library and uses the FastME as its distance method (but we also tested PhyD* ).
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| 99.94 |
We start by comparing all-pairs and all-pairs-max variants, each applied to either the entire set of species or to polytomies of a 50 % majority rule consensus (default), limiting our study to the 37-taxon and 45-taxon avian and mammalian datasets where Θ(n 4 k) methods could run. On both datasets, a surprising pattern emerges. Without the use of consensus, the error unexpectedly goes up with decreased ILS, a pattern that is more pronounced for all-pairs-max (Additional file 1: Figures S1 and S2). As discussed before, we attribute this pattern to difficulties of estimating long quartet lengths. When consensus is used within DISTIQUE, the accuracy improves with decreased ILS, as expected (Additional file 1: Figures S1 and S2). Depending on the level of ILS, the consensus tree is unresolved for 25 to 95 % of branches, leaving much to DISTIQUE to resolve. Overall, all-pairs methods has better accuracy than all-pairs-max, a result that we do not find surprising. Based on these results, hereafter, we only show results for DISTIQUE applied to a majority consensus, and we omit all-pairs-max.
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We compared the three algorithms, all-pairs, tree-sum, and distance-sum (the last two with eight rounds of anchor sampling), and observed that the distance-sum is competitive with all-pairs and outperforms tree-sum (Table 1). The difference between all-pairs and distance-sum was never more than 1 %. Distance-sum consistently outperformed tree-sum, by as much as 5 % in some cases, despite the fact that tree-sum is provably consistent and distance-sum has not been proved consistent. Thus, we chose to set the default DISTIQUE implementation to distance-sum. Table 1DISTIQUE variants on simulated datasetsDataset#genesAll-pairsTree-sumDistance-sumavian-0.5X1000 0.10 0.110.11avian-1X1000 0.08 0.09 0.08 avian-2X1000 0.05 0.080.06mammalian-0.2X200 0.11 0.13 0.11 mammalian-0.5X200 0.06 0.120.07mammalian-1X200 0.04 0.08 0.04 mammalian-2X200 0.02 0.04 0.02 simphySize-10500.030.030.03simphySize-102000.020.020.02simphySize-1010000.020.020.02simphySize-5050 0.07 0.10 0.07 simphySize-50200 0.04 0.07 0.04 simphySize-5010000.030.040.04simphySize-10050 0.08 0.11 0.08 simphySize-100200 0.05 0.06 0.05 simphySize-1001000 0.03 0.050.04Distance-sum and tree-sum are both based on 8 rounds. For simPhy-size, all-pairs could not finish given two days of running time for more than 100 species. Where there is at least 1 % difference between methods, the best method is shown in bold
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We next evaluated the impact of anchor sampling by changing the number of rounds of targeted sampling between 1 and 8 on the avian and mammalian datasets (Additional file 1: Figures S3 and S4). The distance-sum method had substantial improvements when going from one to two rounds, and generally much smaller improvements after that. We show results for both 2 and 8 rounds when comparing DISTIQUE to other methods.
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Finally, we checked the impact of the exact distance method used inside DISTIQUE (Additional file 1: Figure S5), using a variety of methods implemented inside FastME and PhyD* on both mammalian and avian datasets. There were substantial variations of accuracy among distance methods, especially on the avian dataset. PhyD* tended to have more error, and among methods implemented in FastME, balanced minimum evolution (BME) with SPR moves had the highest accuracy. We chose this option of FastME in the default DISTIQUE.
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simPhy-size: On this simulated dataset, we compare running time and tree accuracy across methods. Generally, all the methods we studied had similar patterns of accuracy on the simPhy-size dataset, and the mean errors of different methods tended to be within the standard error of each other (Fig. 1 a). According to a two-way ANOVA test with FDR correction for multiple testing (n=24; see caption of Additional file 1: Table S3) with α=0.05, the error rate of DISTIQUE-8 was statistically indistinguishable from both ASTRAL and ASTRID (Additional file 1: Table S3). In the few cases where the differences seemed substantial, for example on 500 species and 1000 genes, ASTRAL tended to be the best, followed by both versions of DISTIQUE (but there were exceptions; e.g., 50 species and 1000 genes). Unlike the accuracy, running times of summary methods were quite different (Fig. 2). ASTRID was by far the fastest, followed by DISTIQUE-2 and DISTIQUE-8, and ASTRAL was the slowest. For example, with 500 species and 1000 genes, DISTIQUE-2 and DISTIQUE-8 ran in about 1.1 and 2.2 hours, while ASTRAL took 5 hours, and ASTRID took only 7.5 minutes. Fig. 1DISTIQUE versus other methods on (a) simPhy-size and (b) simPhy-ILS datasets using estimated gene trees. Boxes: (a) number of genes and (b) levels of ILS. The mean and standard error of species tree error are shown over (a) 50 and (b) 100 replicates Fig. 2Running time comparisons on the simPhy-size datasets with 1000 genes (Additinal file 1: Figure S6 has other numbers of genes). Lines show the average running times (50 replicates) in hours
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simPhy-ILS: On the simPhy-ILS dataset where the number of species is fixed to 201, differences between various summary methods were generally small (Fig. 1 b), but overall, ASTRAL was significantly better than DISTIQUE-8 (p<0.001). However, DISTIQUE-8 and ASTRID were indistinguishable (Additional file 1: Table S3). The magnitude of the difference between ASTRAL and DISTIQUE-8 significantly depended on the level of ILS (p=0.001), where with low or medium ILS levels, the two methods had a similar error, but with increased ILS, ASTRAL outperformed DISTIQUE; the differences were more pronounced when we had fewer gene trees (significant: p=0.039; Additional file 1: Table S3).
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Avian On the avian dataset (Fig. 3 a), ASTRID was generally the best method, followed by DISTIQUE-8 (which was significantly worse; p=0.004) and then ASTRAL; CA-ML was the worst. Differences between ASTRAL and DISTIQUE-8 were not statistically significant (Additional file 1: Table S3). The largest difference between DISTIQUE-8 and the best method was for 0.5X ILS, where DISTIQUE-8 had 2.9 % more error than ASTRID. Fig. 3The accuracy of methods on Avian (a) and Mammalian (b) datasets using estimated gene trees. Left: number of genes is fixed (1000 for avian, 200 for mammalian) and ILS levels change. Right: ILS level is fixed (default 1X for avian and very high 0.2X for mammalian). We show average and standard error over 20 replicates, except for 1600 and 3200 genes, which have 10 and 5 replicates, respectively. For the mammalian dataset with 0.2X ILS, due to the large number of gene trees, running concatenation was not feasible
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| 100.0 |
The accuracy of methods on Avian (a) and Mammalian (b) datasets using estimated gene trees. Left: number of genes is fixed (1000 for avian, 200 for mammalian) and ILS levels change. Right: ILS level is fixed (default 1X for avian and very high 0.2X for mammalian). We show average and standard error over 20 replicates, except for 1600 and 3200 genes, which have 10 and 5 replicates, respectively. For the mammalian dataset with 0.2X ILS, due to the large number of gene trees, running concatenation was not feasible
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| 100.0 |
Mammalian On this dataset (Fig. 3 b), overall, ASTRAL was the best method, and was significantly better than DISTIQUE (p=0.025). DISTIQUE and ASTRID were overall statistically indistinguishable (Additional file 1: Table S3). The relative error of concatenation depended on the level of ILS, which was much worse than summary methods for high levels of ILS, but better for low levels of ILS.
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| 100.0 |
On the avian dataset, we ran ASTRAL, ASTRID, and DISTIQUE-8 and used both bootstrapping and local posterior probability (pp) to quantify branch support (Additional file 1: Figures S8 and S9). Bootstrap support was generally high, but the local pp was low for many branches. DISTIQUE and ASTRID differed on three branches. Of these, one, related to the first neoavan split, had high local pp support in ASTRID (0.98) but very low local pp in DISTIQUE; the remaining conflicts had local pp below 0.58 in both trees. ASTRAL and DISTIQUE differed in six branches, and all of these had local pp below 0.58 in DISTIQUE, and all but one also had low local pp (≤ 0.9) in ASTRAL. None of these conflicting relationships have been well resolved in the literature. Interestingly, many of conflicting branches with low local pp had high bootstrap support. It can be argued that conflicts are due to uncertainties resulting from insufficient data, but bootstrapping misleadingly computes high support .
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study
| 100.0 |
We compared three statistically consistent summary methods, ASTRAL, ASTRID, and DISTIQUE; overall, ASTRAL was at least as good as other methods on most datasets, but ASTRID was occasionally the best. DISTIQUE was often as good as and never more than 3 % worse than the best method. The choice of the best method depended on the level of ILS and the number of genes, suggesting when the level of ILS is expected to be very high, ASTRAL might be the best choice. On the other hand, the running time of DISTIQUE grows more slowly with increased numbers of genes; for datasets with large number of species and tens of thousands of genes, DISTIQUE and ASTRID provide fast alternatives to ASTRAL.
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study
| 99.94 |
Despite having strong competition in ASTRAL and ASTRID, we believe DISTIQUE is a promising approach, for several reasons. Because of its speed, DISTIQUE can be used for a very fast estimation of species trees, for example, as a starting point for an extensive hill-climbing search. DISTIQUE can also generate a set of trees instead of a single tree, and we plan to study whether these sets of trees can be utilized for defining the search space of ASTRAL.
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| 99.9 |
DISTIQUE is essentially a method for 1) defining distances based on quartets, and 2) subsampling the space of all Θ(n 4) quartets. The first aspect of DISTIQUE can be replaced by improved ways of defining distances, for example those that better handle gene tree estimation error. Co-estimation of gene trees and the species tree is a computationally challenging problem in general. However, it is reasonable to think that a similar problem defined on quartets, and addressed using distances becomes easier, as some recent theoretical results suggest [32 , 59]. DISTIQUE provides a general way for using anchoring introduced in this paper to implement novel distance-based gene tree species tree co-estimation in a scalable fashion. Simpler approaches of taking into account gene tree uncertainty, for example weighting various quartets according to coalescent expectations, might also result in improvements. Finally, we note that DISTIQUE’s anchoring strategy can be paired with site-based ILS methods such as SVDQuartets , and more broadly for other tree inference problems.
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| 99.94 |
We introduced a general approach for computing tree leaf distances by inferring topologies and internal branch lengths for quartets of leaves. We used our novel anchoring to design DISTIQUE, a new statistically consistent summary method for species tree estimation. DISTIQUE has variants, with several strategies for choosing and combining anchors. The default version of DISTIQUE requires O(n 3 k) running time and is much faster than ASTRAL. In terms of accuracy, DISTIQUE was nearly as accurate as ASTRAL with differences that were rarely substantial.
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study
| 99.0 |
Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{a,b,c,d\}\subset \mathcal {L}$\end{document}{a,b,c,d}⊂ℒ be four arbitrary leaves and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {L}^{\prime }=\mathcal {L}- \{a,b,c,d\}$\end{document}ℒ′=ℒ−{a,b,c,d}. W.l.o.g assume \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ab.cd \in \mathcal {Q}^{T}$\end{document}ab.cd∈QT. We prove that the four point conditions hold for this arbitrarily chosen quartet in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\prime }_{uv},D^{\prime }_{v},$\end{document}Duv′,Dv′, and D ′; we also prove that the four point conditions are true for a tree compatible with the tree T. Proving these conditions for arbitrary quartets completes the proof by results of Buneman .
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other
| 99.4 |
We start with the double-anchored matrix. The four point condition can be written in three ways, but only one of them is compatible with the tree T. Since we assumed w.l.o.g that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ab.cd \in \mathcal {Q}^{T}$\end{document}ab.cd∈QT, the four point condition compatible with T is: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{array}{*{20}l} \overbrace{D^{\prime}_{uv}[a,b]+D^{\prime}_{uv}[c,d]}^{L} &< \overbrace{D^{\prime}_{uv}[a,d] + D^{\prime}_{uv}[b,c]}^{R1} \\ &= \overbrace{D^{\prime}_{uv}[a,c] + D^{\prime}_{uv}[b,d]}^{R2}. \end{array} $$ \end{document}Duv′[a,b]+Duv′[c,d]⏞L<Duv′[a,d]+Duv′[b,c]⏞R1=Duv′[a,c]+Duv′[b,d]⏞R2.
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| 91.4 |
Figure 4 shows all ways of placing anchors {u,v} on the quartet tree a b.c d. Anchors can be sisters, placed on the internal branch (Case 1) or on a tip branch (Case 2; w.l.o.g, we pick the branch pending to d). When anchors are not sisters, they can be both placed on the internal branch (Case 3), or one on the internal branch and the other on a tip branch (Case 4), or they can be both on terminal branches, which can be done in three ways: u and v can be on the same terminal branch (Case 5), on different but adjacent branches (Case 6), or on non-adjacent branches (Case 7).
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other
| 99.8 |
In Table 2, for each of the seven cases, we compute L,R1,R2. We use Eq. (1) to derive \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\prime }_{uv}[x,y]$\end{document}Duv′[x,y] values. Where x y.u v is induced by the tree shown in Fig. 4, we use [β−f(t)] and otherwise we use [β+α t], where t=τ(x,y,u,v) is the length of the internal branch for the quartet tree induced by {x,y,u,v}. For example, for Case 1, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\prime }_{uv}[a,b]=[\beta -f(e_{1}+e_{3})]$\end{document}Duv′[a,b]=[β−f(e1+e3)] because a b.u v is induced by the tree, and the length of the edge on the a b.u v quartet tree is e 1+e 3; in Case 7, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\prime }_{uv}[a,b]=\beta +\alpha e_{1}$\end{document}Duv′[a,b]=β+αe1 because a b.u v is not induced by the tree and τ(a,b,u,v)=e 1.
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study
| 100.0 |
We need to show that L<R1 and R1=R2. We remind the reader that all branches are assumed to be strictly positive and that f is a positive and monotonically increasing function bounded from above by β. In all cases, the equality of R1 and R2 is immediately clear from the Table 2. The inequality L<R1 follows directly from the fact that f(x) is monotonically increasing in Cases 1, 2, and 5. For Case 3, because of positivity of f(x) and α, we have L<2β<R. Similarly, for Case 4, L<2β+α e 1<2β+α e 1+2α e 2=R. Case 6 follows from the positivity of f, and Case 7 is trivially correct for positive branch lengths. Thus, we have shown in all possible relationships between {u,v} and the quartet tree, the four point condition holds for the topology consistent with tree T. Therefore, the proof is complete for the double anchored case. Table 2Proof of four-point condition for double anchors. Four point condition for all 7 cases of adding {u,v} to a quartet tree, as shown in Fig. 4 (left side) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L =D^{\prime }_{uv}[a,b]+D^{\prime }_{uv}[c,d]$\end{document}L=Duv′[a,b]+Duv′[c,d] \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ R1 =D^{\prime }_{uv}[a,d] + D^{\prime }_{uv}[b,c]$\end{document}R1=Duv′[a,d]+Duv′[b,c] \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R2 =D^{\prime }_{uv}[a,c] + D^{\prime }_{uv}[b,d]$\end{document}R2=Duv′[a,c]+Duv′[b,d] Case 1[β−f(e 1+e 3)]+[β−f(e 2+e 3)][β−f(e 3)]+[β−f(e 3)][β−f(e 3)]+[β−f(e 3)]Case 2[β−f(e 1+e 2+e 3)]+[β−f(e 3)][β−f(e 3)]+[β−f(e 1+e 3)][β−f(e 1+e 3)]+[β−f(e 3)]Case 3[β−f(e 1)]+[β−f(e 3)][β+α e 2]+[β+α e 2][β+α e 2]+[β+α e 2]Case 4[β−f(e 3)]+[β+α e 1][β+α(e 1+e 2)]+[β+α e 2][β+α e 2]+[β+α(e 1+e 2)]Case 5[β−f(e 2+e 3)]+[β+α e 1][β−f(e 2)]+[β+α e 1][β−f(e 2)]+[β+α e 1]Case 6[β−f(e 1)]+[β+α(e 2+e 3)][β+α e 3]+[β+α e 2][β+α e 2]+[β+α e 3]Case 7[β+α e 1]+[β+α e 3][β+α(e 1+e 2+e 3)]+[β+α e 2][β+α(e 1+e 2)]+[β+α(e 2+e 3)]
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study
| 100.0 |
Now consider the “single anchored” distance matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{*}_{v}$\end{document}Dv∗ on the leaf-set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {L}-\{v\}$\end{document}ℒ−{v} (for a single \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$v\in \mathcal {L}$\end{document}v∈ℒ). To prove additivity of the single anchored distance matrix, we need to prove the following four point condition: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{array}{@{}rcl@{}} & \sum_{u \notin \{a,b\}} D^{\prime}_{uv}[a,b] + \sum_{u \notin \{c,d\}} D^{\prime}_{uv}[c,d] &< \\ & \sum_{u \notin \{a,d\}} D^{\prime}_{uv}[a,d] + \sum_{u \notin \{a,b\}} D^{\prime}_{uv}[b,c] &= \\ & \sum_{u \notin \{a,c\}} D^{\prime}_{uv}[a,c] + \sum_{u \notin \{b,d\}} D^{\prime}_{uv}[b,d] \end{array} $$ \end{document}∑u∉{a,b}Duv′[a,b]+∑u∉{c,d}Duv′[c,d]<∑u∉{a,d}Duv′[a,d]+∑u∉{a,b}Duv′[b,c]=∑u∉{a,c}Duv′[a,c]+∑u∉{b,d}Duv′[b,d]
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| 98.5 |
We divide each sum to terms with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u \in \mathcal {L}^{\prime }$\end{document}u∈ℒ′ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u \notin \mathcal {L}^{\prime }$\end{document}u∉ℒ′: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{array}{*{20}l} &\sum\limits_{u \in \mathcal{L}^{\prime}} D^{\prime}_{uv}[a,b] + D^{\prime}_{uv}[c,d] + \\ &\underbrace{ D^{\prime}_{cv}[a,b] + D^{\prime}_{dv}[a,b] + D^{\prime}_{av}[c,d] + D^{\prime}_{bv}[c,d]}_{L} < \\ &\sum\limits_{u \in \mathcal{L}^{\prime}} D^{\prime}_{uv}[a,d] + D^{\prime}_{uv}[b,c] +\\ &\underbrace{D^{\prime}_{bv}[a,d] + D^{\prime}_{cv}[a,d] + D^{\prime}_{av}[b,c] + D^{\prime}_{dv}[b,c]}_{R1} =\\ &\sum\limits_{u \in \mathcal{L}^{\prime}} D^{\prime}_{uv}[a,c] + D^{\prime}_{uv}[b,d] +\\ &\underbrace{D^{\prime}_{bv}[a,c] + D^{\prime}_{dv}[a,c] + D^{\prime}_{av}[b,d] + D^{\prime}_{cv}[b,d]}_{R2} \end{array} $$ \end{document}∑u∈ℒ′Duv′[a,b]+Duv′[c,d]+Dcv′[a,b]+Ddv′[a,b]+Dav′[c,d]+Dbv′[c,d]⏟L<∑u∈ℒ′Duv′[a,d]+Duv′[b,c]+Dbv′[a,d]+Dcv′[a,d]+Dav′[b,c]+Ddv′[b,c]⏟R1=∑u∈ℒ′Duv′[a,c]+Duv′[b,d]+Dbv′[a,c]+Ddv′[a,c]+Dav′[b,d]+Dcv′[b,d]⏟R2
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| 99.5 |
For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u \in \mathcal {L}^{\prime }$\end{document}u∈ℒ′ terms, the sums are exactly those we analyzed for double anchored distances; thus, the additivity is already proved. Since the sum of two additive distances is additive, it suffices to prove additivity for u∈{a,b,c,d} cases, marked as L,R1, and R2 above.
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study
| 55.8 |
A single anchor v can be placed (Fig. 4) either on the internal branch (Case 8) or on a terminal branch (Case 9) of a quartet tree. We prove L<R1=R2 for these: In Case 8, we have: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{array}{*{20}l} {}L = & [\beta -f(e_{1})] \!+ [\beta -f(e_{1})]+[\beta -f(e_{2})]+ [\beta -f(e_{2})] \\ < & 4\beta <[\beta +\alpha e_{1}]+[\beta +\alpha e_{1}] + [\beta +\alpha e_{2}] = R1=R2 \end{array} $$ \end{document}L=[β−f(e1)]+[β−f(e1)]+[β−f(e2)]+[β−f(e2)]<4β<[β+αe1]+[β+αe1]+[β+αe2]=R1=R2
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study
| 99.94 |
and in case 9, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${{}\begin{aligned} L & = 2 [\beta +\alpha e_{1}] + [\beta - f(e_{2})] +[\beta -f(e_{1}+e_{2})] <\\ &\quad 4\beta+2\alpha e_{1} - f(e_{1}+e_{2}) <\\ &\quad [\beta \,+\, \alpha e_{2}] \,+\, [ \beta -f(e_{1}) ] +[\beta + \alpha e_{1}] + [\beta +\alpha (e_{1}+e_{2})]\\ & = R1=R2. \end{aligned}} $$ \end{document}L=2[β+αe1]+[β−f(e2)]+[β−f(e1+e2)]<4β+2αe1−f(e1+e2)<[β+αe2]+[β−f(e1)]+[β+αe1]+[β+α(e1+e2)]=R1=R2.
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other
| 99.44 |
We now prove the additivity for the all-pairs matrix. Equation (3) has three types of terms: {u,v}∩{a,b,c,d} may have (I) both anchors, (II) one anchor, or (III) none. The four point condition can be written: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${{}\begin{aligned} \overbrace{2 D^{\prime}_{ab}[c,d]}^{I} + &\overbrace{\sum\limits_{v \in \mathcal{L}^{\prime}} \sum\limits_{u \in \{c,d\}} D^{\prime}_{uv}[a,b] + \sum\limits_{u \in \{a,b\}}D^{\prime}_{uv}[c,d]}^{II} + & \\ &\overbrace{\sum\limits_{u,v \in \mathcal{L}^{\prime}} D^{\prime}_{uv}[a,b] + D^{\prime}_{uv}[c,d]}^{III} &< \\ 2 D^{\prime}_{ad}[b,c] +& \sum\limits_{v \in \mathcal{L}^{\prime}} \sum\limits_{u \in \{b,c\}} D^{\prime}_{uv}[a,d] +\sum\limits_{u \in \{a,d\}} D^{\prime}_{uv}[b,c] +&\\ &\sum\limits_{u,v \in \mathcal{L}^{\prime}} D^{\prime}_{uv}[a,d] + D^{\prime}_{uv}[b,c] &=\\ 2 D^{\prime}_{ac}[b,d] + &\sum\limits_{v \in \mathcal{L}^{\prime}} \sum\limits_{u \in \{b,d\}} D^{\prime}_{uv}[a,c] +\sum\limits_{u \in \{a,c\}} D^{\prime}_{uv}[b,d] +&\\ &\sum\limits_{u,v \in \mathcal{L}^{\prime}} D^{\prime}_{uv}[a,c] + D^{\prime}_{uv}[b,d] \end{aligned}} $$ \end{document}2Dab′[c,d]⏞I+∑v∈ℒ′∑u∈{c,d}Duv′[a,b]+∑u∈{a,b}Duv′[c,d]⏞II+∑u,v∈ℒ′Duv′[a,b]+Duv′[c,d]⏞III<2Dad′[b,c]+∑v∈ℒ′∑u∈{b,c}Duv′[a,d]+∑u∈{a,d}Duv′[b,c]+∑u,v∈ℒ′Duv′[a,d]+Duv′[b,c]=2Dac′[b,d]+∑v∈ℒ′∑u∈{b,d}Duv′[a,c]+∑u∈{a,c}Duv′[b,d]+∑u,v∈ℒ′Duv′[a,c]+Duv′[b,d]
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study
| 92.5 |
For terms of type (III) and (II), the additivity is already proved in double and single anchored cases, respectively. Thus, we need to prove additivity only for terms of type (I), which have no anchors. Let x=τ(a,b,c,d). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{array}{*{20}l} &2D^{\prime}_{ab}[c,d] = 2[\beta-f(x)]< 2\beta<2[\beta+\alpha x] = \\ & 2D^{\prime}_{ad}[b,c] = 2D^{\prime}_{ac}[b,d] \end{array} $$ \end{document}2Dab′[c,d]=2[β−f(x)]<2β<2[β+αx]=2Dad′[b,c]=2Dac′[b,d]
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other
| 94.75 |
We prove that Eq. (4) returns the sum of internal branch lengths on the path from a to b on the tree T (we denote this by D T ab). The theorem immediately follows because a distance matrix compatible with the tree T has to be by definition additive and compatible with it (note that the theorem also claims that D ′′ gives internal branch lengths). For simplicity, we prove with α=1; extension to other values is simple. If a and b are not sisters in T, there exists an anchor pair (u,v) with quartet topology a u.b v and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tau (a,b,u,v)=D^{T}_{ab}$\end{document}τ(a,b,u,v)=DabT; to find such u and v, the following procedure can be followed. Pick u arbitrarily from the sister group of a after rooting T on b and pick v arbitrarily from the sister group of b after rooting T on a. With this choice, it’s easy to see that τ(a,b,u,v) becomes simply the sum of internal branches between a and b; thus, from the first case of Eq. (1), we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\prime }_{uv}[a,b]-\beta =\frac {\tau (a,b,u,v)}{\alpha }+\beta -\beta =D^{T}_{ab}$\end{document}Duv′[a,b]−β=τ(a,b,u,v)α+β−β=DabT. Moreover, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\prime }_{wz}[a,b]-\beta $\end{document}Dwz′[a,b]−β for two other anchors w,z cannot be bigger than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{T}_{ab}$\end{document}DabT. That is because if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ab.wz\in \mathcal {Q}^{T}$\end{document}ab.wz∈QT, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\prime }_{wz}[a,b]<\beta $\end{document}Dwz′[a,b]<β; else, τ(a,b,w,z) will give the length for a subpath from a to b. Thus, the max function in Eq. (4) returns \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{T}_{ab}$\end{document}DabT, as desired. When (a,b) are sisters, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{T}_{ab}=0$\end{document}DabT=0; also \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\prime }_{uv}[a,b]<0$\end{document}Duv′[a,b]<0 for any (u,v), and thus, the max function returns D ′′[a,b]=0; this is what we want, since for sisters, the length of the internal branch length is zero. Thus, as desired, Eq. (4) always returns the length of the internal branches in the T between a and b; this completes the proof. □
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study
| 99.94 |
For x>0, ln(3−2e −x) is clearly positive, monotonic, and bounded from above by ln3, as required by Definition 1. Let Q={a,b,u,v} and let T be the true species tree. To prove that Eq. (1) simplifies to (5), consider two cases. If anchors u,v are not sisters in the species tree quartet on Q (i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ab.uv\notin \mathcal {Q}^{T}$\end{document}ab.uv∉QT), by the MSC model, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p(ab.uv)=\frac {1}{3}e^{-\tau (Q)}$\end{document}p(ab.uv)=13e−τ(Q) and thus, τ(Q)=− ln3p(a b.u v). In the first case in (1), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\prime }_{uv}=\beta + \alpha. \tau (Q) = \ln 3+ \tau (Q)= \ln 3 - \ln 3p(ab.uv)=- \ln p(ab.uv)$\end{document}Duv′=β+α.τ(Q)=ln3+τ(Q)=ln3−ln3p(ab.uv)=−lnp(ab.uv). In the second case, u,v are sisters (i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ab.uv\in \mathcal {Q}^{T}$\end{document}ab.uv∈QT), and by the MSC model, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p(au.bv)=1-\frac {2}{3}e^{-\tau (Q)}$\end{document}p(au.bv)=1−23e−τ(Q); thus, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tau (Q)=-\ln \frac {3}{2}(1-p(au.bv))$\end{document}τ(Q)=−ln32(1−p(au.bv)). In the second case in (1), the distance is β−f(τ(Q))= ln3− ln(3−2e −τ(Q))=− lnp(a b.u v). Thus, in both cases, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\prime }_{uv} = D^{*}_{uv}$\end{document}Duv′=Duv∗. □
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study
| 99.94 |
Epigenetic alterations play a critical role in cancer initiation and progression in addition to genetic alterations. Epigenetics includes changes such as DNA methylation, microRNA and histone modifications, which together make up the epigenome . Here we focused on the histone modifications that play a crucial role in cancer development and progression. Chromatin structure plays an important role in regulating various nuclear functions such as transcription, replication, recombination and DNA repair. Regulation of gene expression is known to be involved in the binding of transcription factors in target gene promoters but it is also dependent on how the epigenetic events, including histone marks, are characterized. Basically, the local modification of chromatin structure by histone modifications can lead to activation or inactivation of gene expression. For example, some histone modifications such as tri-methylated histone H3 at lysine 27 (H3K27me3) are known to inactive gene expression.
|
review
| 96.8 |
Studies of alteration in histone methylation at whole-genome scale bring insight into gene regulation. Global changes in histone H3 are emerging as a new biomarker in malignant transformation [2, 3]. Likewise, histone-modifier enzymes control dynamic transcription of gene expression in normal and cancer cells, enabling key physiopathological processes to take place [4, 5]. Polycomb-group proteins (PcGs) are involved in silencing gene expression, particularly during development and differentiation stages [6, 7], and also play the major role in nuclear reprogramming and chromatin remodeling . PcGs are organized into two main polycomb-repressive complexes (PRCs), PRC1 and PRC2, that control gene silencing through post-translational histone modifications [9, 10].
|
review
| 99.44 |
At specific loci, the histone methyltransferase enhancer of zeste homolog 2 (EZH2), a subunit of PRC2, catalyzes H3K27 trimethylation, leading to chromatin compaction and subsequently silencing of genes in prostate cancer . Abnormal functions of PcGs are one of the main factors involved in the initiation and progression phases in many cancers, including prostate cancer . EZH2 is highly overexpressed in prostate cancer and strongly associated with epigenetic silencing in cancer. EZH2 is so prominently involved in aggressive cell growth, metastasis, drug resistance and stem cell maintenance that it has become an attractive therapeutic target in prostate cancer [13–15]. Previous studies show that EZH2 up-regulation is correlated with H3K27me3 deregulation and poor-prognosis prostate tumor . The H3K27me3 repressive mark has been found on many gene promoters that are silenced , and genome-wide profiling studies of the H3K27me3 mark in metastatic and prostate cancer cells suggest a silencing function of EZH2 in prostate cancer [18, 19].
|
review
| 60.34 |
This study used 34 human prostate biopsies and chromatin immunoprecipitation (ChIP) assays to investigate the interactions between H3K27me3 and gene promoter in prostate cancer. ChIP coupled with promoter microarrays enabled us to determine the entire spectrum of in vivo DNA binding sites of the H3K27me3 repressive mark. Identifying region-specific H3K27me3 patterns also helps address additional questions, such as how these observations may resolve or at least suggest causal relationships between histone methylation and prostate cancer progression.
|
study
| 100.0 |
We demonstrated that average number of H3K27me3-enriched genes was higher in tumor tissues than normal tissues. Then, after factorial discriminant analysis and ANOVA, we characterized the significant interaction of H3K27me3 with ALG5, EXOSC8, CBX1, GRID2, GRIN3B, ING3, MYO1D, NPHP3-AS1, MSH6, FBXO11, SND1, SPATS2, TENM4 and TRA2A in tumor tissues compared to normal tissues. These genes were all more H3K27me3-enriched and were able to discriminate different groups according to Gleason score.
|
study
| 100.0 |
Normal and tumoral prostate biopsies were obtained from 34 patients (Table 1) diagnosed with prostate cancer at Clermont-Ferrand University Hospital (France). All biopsies were kept in nitrogen. A pathologist performed tumor evaluation. Patients did not receive chemotherapy before clinical examination. All subjects gave written informed consent to the study, which was approved by the French Ministry for Higher Education and Research (DC-2008-558).Table 1Clinical and biological characteristics of patientsCasesTTNTTotal cases (n = 34)2113Age at diagnosis (years) < 4900 50–5922 60–69118 > 7083Baseline PSA (ng/mL) < 400 4–1089 10–2084 > 2050Clinical stage T1c7- T28- T36-Gleason Score ≤ 710- > 711- TT tumoral tissues, NT normal tissues, PSA prostate-specific antigen
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study
| 100.0 |
Tissues were fixed for 15 min at room temperature (RT) using 1% formaldehyde in phosphate buffered saline (PBS) containing protease inhibitors. Reversal of crosslinking was performed by incubation with 0.125 M glycine for 5 min at RT. Each pellet was re-suspended with lysis buffer (5 mM PIPES pH 8, 0.85 mM KCL, 0.5% Igepal) supplemented with 1X protease inhibitor cocktail, and sonicated for 30 min in 30 s ON/30 s OFF cycles (Bioruptor, Diagenode). The lysate was centrifuged at 14,000 g for 10 min and the supernatant transferred to a fresh tube. Optimal fragmentation was achieved by testing various sonication conditions on chromatin followed by DNA isolation and gel electrophoresis estimation of sonication efficiency. ChIP was performed using an AutoTrue MicroChIP kit (Diagenode #C01010140) on a SX-8G IP-Star Compact Automated System (Diagenode) as per the manufacturer’s instructions. Immunoprecipitation was performed using 3 μg of anti-H3K27me3 (Diagenode #C15410195) and non-specific IgG (Diagenode). Reverse crosslinking was performed with 5 M NaCl for 4 h at 65 °C. The immunoprecipitated DNA and input samples were purified using MicroChIP DiaPure columns (Diagenode #C03040001) and eluted with TE buffer. After ChIP, the crosslink was reversed and the DNAs were purified. To assess the quality and efficiency of the ChIP procedure, quantitative PCR was performed to assess the enrichment of known target genes. GAPDH, a housekeeping gene, was used as negative control for H3K27me3 ChIP. TSH2B gene, which is present in heterochromatin, was used as positive control for H3K27me3. TSH2B showed strong enrichment of H3K27me3 while GAPDH gene showed weak enrichment. Only samples with an enrichment of H3K27me3 above 5 were selected for ChIP-on-Chip analyses. Quantitative PCR was performed using SYBR Green Mix (Applied Biosystems #4309155) following the manufacturer’s instructions. The samples were amplified using an Applied Biosystems ABI Prism®7900 HT Real-Time PCR System (Applied Biosystems). PCR program was 95 °C for 3 min and 40 cycles of 95 °C for 30 s, 60 °C for 30 s and 72 °C for 30 s. The IP and input DNA were then subjected to microarray hybridization.
|
study
| 100.0 |
After ChIP, the immunoprecipitated DNA was pre-amplified with a whole-genome amplification kit (Sigma #WGA2) following the manufacturer’s protocol, and 2 μg DNA was labeled using a SureTag complete DNA labeling kit (Agilent Technologies #51904240) at 37 °C for 2 h then 65 °C for 10 min. Input DNA was then labeled with cyanine 3 while immunoprecipitated DNA was labeled with cyanine 5. Both samples were purified on columns and eluted in TE buffer. Labeled DNAs were mixed and competitively hybridized to DNA microarrays. Hybridization was carried out on 2X400K Sure Print G3 Human promoter microarrays (Agilent #G4874A) in the presence of human Cot-1 DNA for 40 h at 65 °C and the slides were washed according to Agilent’s procedure. After washing, the slides were scanned using an Agilent microarray scanner, and intensity of fluorescent signals was extracted using Agilent feature extraction 11.2 software.
|
study
| 99.94 |
Each slide contained two identical arrays and each microarray contained 414,043 (60-mer) oligonucleotide probes spaced every 172 bp across promoter regions including −5.5 Kb upstream and +2.5 Kb downstream of identified transcriptional start sites (TSS). The probes covered 21,000 of the best-defined human transcripts represented as RefSeq genes.
|
study
| 99.8 |
For the H3K27me3-enriched gene analysis between normal tissues and tumors, ChIP-on-chip data were processed using RINGO software 1.26.1, then the microarray data was analyzed on R software using several Bioconductor packages (www.bioconductor.org/). Enriched regions were defined from enriched probes using the criteria of at least 3 enriched probes within the region. For each probe on the array, a score was calculated as follows: score = ∑ (enrichment values Probes-Enrichment Threshold). Only genes with a threshold of >1.5 were considered as differently enriched. Genes with a score of least than 1.5 were removed from analysis, as were genes with missing data in more than 30% of the samples.
|
study
| 100.0 |
Gene annotation was carried out using the ENSEMBL annotation system. We generated enrichment profiles for H3K27me3 in tumor samples compared to normal tissues. After determining the enriched regions for H3K27me3 modifications, RefSeq genes were downloaded from the ENSEMBL database.
|
study
| 100.0 |
H3K27me3 sites were defined as differentially enriched if the Enrichment Score was >1.5, and for each H3K27me3 site the mean Enrichment Score level was compared in tumor tissue group versus normal tissue. Factorial discriminant analysis (FDA) and ANOVA were performed to discriminate the three groups. Data was analyzed using R statistics. Thresholds set for statistical significance were *p < 0.05 and **p < 0.01.
|
study
| 100.0 |
In order to grasp the role H3K27me3 marks in prostate cancer progression in 34 patients, we investigated H3K27me3 mark binding to determine whether it correlates with tumor progression. First, to examine the epigenetic signature of H3K27me3 in prostate cancer, we mapped the global promoter occupancy profile of H3K27me3 in prostate cancer compared to normal biopsies using ChIP-on-chip methods.
|
study
| 100.0 |
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