Explicit solution of divide-and-conquer dividing by a half recurrences with polynomial independent term.

Divide-and-conquer dividing by a half recurrences, of the form [Formula: see text] appear in many areas of applied mathematics, from the analysis of algorithms to the optimization of phylogenetic balance indices. These equations are usually "solved" by means of a Master Theorem that provides a bound for the growing order of xn, but not the solution's explicit expression. In this paper we give a finite explicit expression for this solution, in terms of the binary decomposition of n, when the independent term p(n) is a polynomial in ⌈n/2⌉ and ⌊n/2⌋. As an application, we obtain explicit formulas for several sequences of interest in phylogenetics, combinatorics, and computer science, for which no such formulas were known so far: for instance, for the Total Cophenetic index and the rooted Quartet index of the maximally balanced bifurcating phylogenetic trees with n leaves, and the sum of the bitwise AND operator applied to pairs of complementary numbers up to n.

TRACEBACK: Testing of Historical Tubo-Ovarian Cancer Patients for Hereditary Risk Genes as a Cancer Prevention Strategy in Family Members.

PURPOSE: Tubo-ovarian cancer (TOC) is a sentinel cancer for BRCA1 and BRCA2 pathogenic variants (PVs). Identification of a PV in the first member of a family at increased genetic risk (the proband) provides opportunities for cancer prevention in other at-risk family members. Although Australian testing rates are now high, PVs in patients with TOC whose diagnosis predated revised testing guidelines might have been missed. We assessed the feasibility of detecting PVs in this population to enable genetic risk reduction in relatives. PATIENTS AND METHODS: In this pilot study, deceased probands were ascertained from research cohort studies, identification by a relative, and gynecologic oncology clinics. DNA was extracted from archival tissue or stored blood for panel sequencing of 10 risk-associated genes. Testing of deceased probands ascertained through clinic records was performed with a consent waiver. RESULTS: We identified 85 PVs in 84 of 787 (11%) probands. Familial contacts of 39 of 60 (65%) deceased probands with an identified recipient (60 of 84; 71%) have received a written notification of results, with follow-up verbal contact made in 85% (33 of 39). A minority of families (n = 4) were already aware of the PV. For many (29 of 33; 88%), the genetic result provided new information and referral to a genetic service was accepted in most cases (66%; 19 of 29). Those who declined referral (4 of 29) were all male next of kin whose family member had died more than 10 years before. CONCLUSION: We overcame ethical and logistic challenges to demonstrate that retrospective genetic testing to identify PVs in previously untested deceased probands with TOC is feasible. Understanding reasons for a family member's decision to accept or decline a referral will be important for guiding future TRACEBACK projects.

Squaring within the Colless index yields a better balance index.

The Colless index for bifurcating phylogenetic trees, introduced by Colless (1982), is defined as the sum, over all internal nodes v of the tree, of the absolute value of the difference of the sizes of the clades defined by the children of v. It is one of the most popular phylogenetic balance indices, because, in addition to measuring the balance of a tree in a very simple and intuitive way, it turns out to be one of the most powerful and discriminating phylogenetic shape indices. But it has some drawbacks. On the one hand, although its minimum value is reached at the so-called maximally balanced trees, it is almost always reached also at trees that are not maximally balanced. On the other hand, its definition as a sum of absolute values of differences makes it difficult to study analytically its distribution under probabilistic models of bifurcating phylogenetic trees. In this paper we show that if we replace in its definition the absolute values of the differences of clade sizes by the squares of these differences, all these drawbacks are overcome and the resulting index is still more powerful and discriminating than the original Colless index.

Characterization of the immune profile of oral tongue squamous cell carcinomas with advancing disease.

We investigated whether a unique immune response was instigated with the development of oral tongue squamous cell carcinomas (OTSCC), with/without nodal involvement, with/without recurrent metastatic disease, or within tumor involved nodes. One hundred and ten formalin-fixed paraffin-embedded samples were collected from a retrospective cohort of 67 OTSCC patients and 10 non-cancerous tongue samples. Targets including CD4, CD8, FOXP3, PD-L1, and PD-1 were analyzed by immunohistochemistry. The Nanostring PanCancer Immune Profiling Panel was used for gene expression profiling. Data were externally validated in the The Cancer Genome Atlas (TCGA) head and neck (HNSCC), melanoma and lung squamous cell carcinoma (LSCC) cohorts. A 24-immune gene signature was identified that discriminated more aggressive OTSCC cases, and although not prognostic in HNSCC was associated with survival in other TCGA cohorts (improved survival for melanoma, P < .001 and worse survival for LSCC, P = .038). OTSCC exhibited concordant gene and immunohistochemical (IHC) features characterized by a TH-2 biased, proinflammatory profile with upregulated B cell and neutrophil gene activity and increased CD4, FOXP3, and PD-L1 expression (P < .001 for all by IHC). Compared to less advanced disease, nodal involvement and recurrent OTSCC did not induce a different immune response although recurrent disease was characterized by significantly higher PD-L1 expression (P = .004 by SP263, P = .013 by 22C3, P = .004 for gene expression). Identification of a gene signature associated with different prognostic effects in other cancers highlights common pathways of immune dysregulation that are impacted by the tumor origin. The significant immunosuppressive signaling in OTSCC indicates primary failure of immune system to control carcinogenesis emphasizing the need for early, combination therapeutic approaches.

On Sackin's original proposal: the variance of the leaves' depths as a phylogenetic balance index.

BACKGROUND: The Sackin indexS of a rooted phylogenetic tree, defined as the sum of its leaves' depths, is one of the most popular balance indices in phylogenetics, and Sackin's paper (Syst Zool 21:225-6, 1972) is usually cited as the source for this index. However, what Sackin actually proposed in his paper as a measure of the imbalance of a rooted tree was not the sum of its leaves' depths, but their "variation". This proposal was later implemented as the variance of the leaves' depths by Kirkpatrick and Slatkin in (Evolution 47:1171-81, 1993), where they also posed the problem of finding a closed formula for its expected value under the Yule model. Nowadays, Sackin's original proposal seems to have passed into oblivion in the phylogenetics literature, replaced by the index bearing his name, which, in fact, was introduced a decade later by Sokal. RESULTS: In this paper we study the properties of the variance of the leaves' depths, V, as a balance index. Firstly, we prove that the rooted trees with n leaves and maximum V value are exactly the combs with n leaves. But although V achieves its minimum value on every space [Formula: see text] of bifurcating rooted phylogenetic trees with n≤183 leaves at the so-called "maximally balanced trees" with n leaves, this property fails for almost every n≥184. We provide then an algorithm that finds the trees in [Formula: see text] with minimum V value in time O(n log(n)). Secondly, we obtain closed formulas for the expected V value of a bifurcating rooted tree with any number n of leaves under the Yule and the uniform models and, as a by-product of the computations leading to these formulas, we also obtain closed formulas for the variance under the uniform model of the Sackin index and the total cophenetic index (Mir et al., Math Biosci 241:125-36, 2013) of a bifurcating rooted tree, as well as of their covariance, thus filling this gap in the literature. CONCLUSION: The phylogenetics community has been wise in preferring the sum S(T) of the leaves' depths of a phylogenetic tree T over their variance V(T) as a balance index, because the latter does not seem to capture correctly the notion of balance of large bifurcating rooted trees. But it is still a valid and useful shape index.

Diagnosis support of sickle cell anemia by classifying red blood cell shape in peripheral blood images.

Red blood cell (RBC) deformation is the consequence of several diseases, including sickle cell anemia, which causes recurring episodes of pain and severe pronounced anemia. Monitoring patients with these diseases involves the observation of peripheral blood samples under a microscope, a time-consuming procedure. Moreover, a specialist is required to perform this technique, and owing to the subjective nature of the observation of isolated RBCs, the error rate is high. In this paper, we propose an automated method for differentially enumerating RBCs that uses peripheral blood smear image analysis. In this method, the objects of interest in the image are segmented using a Chan-Vese active contour model. An analysis is then performed to classify the RBCs, also called erythrocytes, as normal or elongated or having other deformations, using the basic shape analysis descriptors: circular shape factor (CSF) and elliptical shape factor (ESF). To analyze cells that become partially occluded in a cluster during sample preparation, an elliptical adjustment is performed to allow the analysis of erythrocytes with discoidal and elongated shapes. The images of patient blood samples used in the study were acquired by a clinical laboratory specialist in the Special Hematology Department of the "Dr. Juan Bruno Zayas" General Hospital in Santiago de Cuba. A comparison of the results obtained by the proposed method in our experiments with those obtained by some state-of-the-art methods showed that the proposed method is superior for the diagnosis of sickle cell anemia. This superiority is achieved for evidenced by the obtained F-measure value (0.97 for normal cells and 0.95 for elongated ones) and several overall multiclass performance measures. The results achieved by the proposed method are suitable for the purpose of clinical treatment and diagnostic support of sickle cell anemia. We present a new method to obtain erythrocyte shape classification using peripheral blood smear sample images. The aim of the method is to segment the cells, to separate clusters and classify cells (circulars, elongated and others). We compared our method with state-of the-art. Results showed that our method with is superior for the diagnosis support of sickle cell anemia.

Exploring molecular evolution of Rubisco in C3 and CAM Orchidaceae and Bromeliaceae.

BACKGROUND: The CO2-concentrating mechanism associated to Crassulacean acid metabolism (CAM) alters the catalytic context for Rubisco by increasing CO2 availability and provides an advantage in particular ecological conditions. We hypothesized about the existence of molecular changes linked to these particular adaptations in CAM Rubisco. We investigated molecular evolution of the Rubisco large (L-) subunit in 78 orchids and 144 bromeliads with C3 and CAM photosynthetic pathways. The sequence analyses were complemented with measurements of Rubisco kinetics in some species with contrasting photosynthetic mechanism and differing in the L-subunit sequence. RESULTS: We identified potential positively selected sites and residues with signatures of co-adaptation. The implementation of a decision tree model related Rubisco specific variable sites to the leaf carbon isotopic composition of the species. Differences in the Rubisco catalytic traits found among C3 orchids and between strong CAM and C3 bromeliads suggested Rubisco had evolved in response to differing CO2 concentration. CONCLUSIONS: The results revealed that the variability in the Rubisco L-subunit sequence in orchids and bromeliads is composed of coevolving sites under potential positive adaptive signal. The sequence variability was related to δ13C in orchids and bromeliads, however it could not be linked to the variability found in the kinetic properties of the studied species.

Improved next-generation sequencing pre-capture library yields and sequencing parameters using on-bead PCR.

Tumor DNA sequencing results can have important clinical implications. However, its use is often limited by low DNA input, owing to small tumor biopsy size. To help overcome this limitation we have developed a simple improvement to a commonly used next-generation sequencing (NGS) capture-based library preparation method using formalin-fixed paraffin-embedded-derived tumor DNA. By using on-bead PCR for pre-capture library generation we show that library yields are dramatically increased, resulting in decreased sample failure rates. Improved yields allowed for a reduction in PCR cycles, which translated into improved sequencing parameters without affecting variant calling. This methodology should be applicable to any NGS system in which input DNA is a limiting factor.

A balance index for phylogenetic trees based on rooted quartets.

We define a new balance index for rooted phylogenetic trees based on the symmetry of the evolutive history of every set of 4 leaves. This index makes sense for multifurcating trees and it can be computed in time linear in the number of leaves. We determine its maximum and minimum values for arbitrary and bifurcating trees, and we provide exact formulas for its expected value and variance on bifurcating trees under Ford's [Formula: see text]-model and Aldous' [Formula: see text]-model and on arbitrary trees under the [Formula: see text]-[Formula: see text]-model.

Survival Following Chemotherapy in Ovarian Clear Cell Carcinoma Is Not Associated with Pathological Misclassification of Tumor Histotype.

PURPOSE: Although ovarian clear cell carcinomas (OCCC) are commonly resistant to platinum-based chemotherapy, good clinical outcomes are observed in a subset of patients. The explanation for this is unknown but may be due to misclassification of high-grade serous ovarian cancer (HGSOC) as OCCC or mixed histology. EXPERIMENTAL DESIGN: To discover potential biomarkers of survival benefit following platinum-based chemotherapy, we ascertained a cohort of 68 Japanese and Australian patients in whom progression-free survival (PFS) and overall survival (OS) could be assessed. We performed IHC reclassification of tumors, and targeted sequencing and immunohistochemistry of known driver genes. Exome sequencing was performed in 10 patients who had either unusually long survival (N = 5) or had a very short time to progression (N = 5). RESULTS: The majority of mixed OCCC (N = 6, 85.7%) and a small proportion of pure OCCC (N = 3, 4.9%) were reclassified as likely HGSOC. However, the PFS and OS of patients with misclassified samples were similar to that of patients with pathologically validated OCCC. Absent HNF1B expression was significantly correlated with longer PFS and OS (P = 0.0194 and 0.0395, respectively). Mutations in ARID1A, PIK3CA, PPP2R1A, and TP53 were frequent, but did not explain length of PFS and OS. An exploratory exome analysis of patients with favorable and unfavorable outcomes did not identify novel outcome-associated driver mutations. CONCLUSIONS: Survival benefit following chemotherapy in OCCC was not associated with pathological misclassification of tumor histotype. HNF1B loss may help identify the subset of patients with OCCC with a more favorable outcome.

Sound Colless-like balance indices for multifurcating trees.

The Colless index is one of the most popular and natural balance indices for bifurcating phylogenetic trees, but it makes no sense for multifurcating trees. In this paper we propose a family of Colless-like balance indices [Formula: see text] that generalize the Colless index to multifurcating phylogenetic trees. Each [Formula: see text] is determined by the choice of a dissimilarity D and a weight function [Formula: see text]. A balance index is sound when the most balanced phylogenetic trees according to it are exactly the fully symmetric ones. Unfortunately, not every Colless-like balance index is sound in this sense. We prove then that taking f(n) = ln(n + e) or f(n) = en as weight functions, the resulting index [Formula: see text] is sound for every dissimilarity D. Next, for each one of these two functions f and for three popular dissimilarities D (the variance, the standard deviation, and the mean deviation from the median), we find the most unbalanced phylogenetic trees according to [Formula: see text] with any given number n of leaves. The results show that the growth pace of the function f influences the notion of "balance" measured by the indices it defines. Finally, we introduce our R package "CollessLike," which, among other functionalities, allows the computation of Colless-like indices of trees and their comparison to their distribution under Chen-Ford-Winkel's α-γ-model for multifurcating phylogenetic trees. As an application, we show that the trees in TreeBASE do not seem to follow either the uniform model for multifurcating trees or the α-γ-model, for any values of α and γ.

The Probabilities of Trees and Cladograms under Ford's α-Model.

Ford's α-model is one of the most popular random parametric models of bifurcating phylogenetic tree growth, having as specific instances both the uniform and the Yule models. Its general properties have been used to study the behavior of phylogenetic tree shape indices under the probability distribution it defines. But the explicit formulas provided by Ford for the probabilities of unlabeled trees and phylogenetic trees fail in some cases. In this paper we give correct explicit formulas for these probabilities.

The expected value of the squared cophenetic metric under the Yule and the uniform models.

The cophenetic metrics dφ,p, for p ∈ {0} ∪ [1, ∞), are a recent addition to the kit of available distances for the comparison of phylogenetic trees. Based on a fifty years old idea of Sokal and Rohlf, these metrics compare phylogenetic trees on a same set of taxa by encoding them by means of their vectors of cophenetic values of pairs of taxa and depths of single taxa, and then computing the Lp norm of the difference of the corresponding vectors. In this paper we compute the expected value of the square of dφ,2 on the space of fully resolved rooted phylogenetic trees with n leaves, under the Yule and the uniform probability distributions.

Correction: Positively selected amino acid replacements within the RuBisCO enzyme of oak trees are associated with ecological adaptations.

[This corrects the article DOI: 10.1371/journal.pone.0183970.].

Positively selected amino acid replacements within the RuBisCO enzyme of oak trees are associated with ecological adaptations.

Phylogenetic analysis by maximum likelihood (PAML) has become the standard approach to study positive selection at the molecular level, but other methods may provide complementary ways to identify amino acid replacements associated with particular conditions. Here, we compare results of the decision tree (DT) model method with ones of PAML using the key photosynthetic enzyme RuBisCO as a model system to study molecular adaptation to particular ecological conditions in oaks (Quercus). We sequenced the chloroplast rbcL gene encoding RuBisCO large subunit in 158 Quercus species, covering about a third of the global genus diversity. It has been hypothesized that RuBisCO has evolved differentially depending on the environmental conditions and leaf traits governing internal gas diffusion patterns. Here, we show, using PAML, that amino acid replacements at the residue positions 95, 145, 251, 262 and 328 of the RuBisCO large subunit have been the subject of positive selection along particular Quercus lineages associated with the leaf traits and climate characteristics. In parallel, the DT model identified amino acid replacements at sites 95, 219, 262 and 328 being associated with the leaf traits and climate characteristics, exhibiting partial overlap with the results obtained using PAML.

EIF1AX and NRAS Mutations Co-occur and Cooperate in Low-Grade Serous Ovarian Carcinomas.

Low-grade serous ovarian carcinomas (LGSC) are associated with a poor response to chemotherapy and are molecularly characterized by RAS pathway activation. Using exome and whole genome sequencing, we identified recurrent mutations in the protein translational regulator EIF1AX and in NF1, USP9X, KRAS, BRAF, and NRAS RAS pathway mutations were mutually exclusive; however, we found significant co-occurrence of mutations in NRAS and EIF1AX Missense EIF1AX mutations were clustered at the N-terminus of the protein in a region associated with its role in ensuring translational initiation fidelity. Coexpression of mutant NRAS and EIF1AX proteins promoted proliferation and clonogenic survival in LGSC cells, providing the first example of co-occurring, growth-promoting mutational events in ovarian cancer. Cancer Res; 77(16); 4268-78. ©2017 AACR.

Breast ductal carcinoma in situ carry mutational driver events representative of invasive breast cancer.

The spectrum of genomic alterations in ductal carcinoma in situ (DCIS) is relatively unexplored, but is likely to provide useful insights into its biology, its progression to invasive carcinoma and the risk of recurrence. DCIS (n=20) with a range of phenotypes was assessed by massively parallel sequencing for mutations and copy number alterations and variants validated by Sanger sequencing. PIK3CA mutations were identified in 11/20 (55%), TP53 mutations in 6/20 (30%), and GATA3 mutations in 9/20 (45%). Screening an additional 91 cases for GATA3 mutations identified a final frequency of 27% (30/111), with a high proportion of missense variants (8/30). TP53 mutations were exclusive to high grade DCIS and more frequent in PR-negative tumors compared with PR-positive tumors (P=0.037). TP53 mutant tumors also had a significantly higher fraction of the genome altered by copy number than wild-type tumors (P=0.005), including a significant positive association with amplification or gain of ERBB2 (P<0.05). The association between TP53 mutation and ERBB2 amplification was confirmed in a wider DCIS cohort using p53 immunohistochemistry as a surrogate marker for TP53 mutations (P=0.03). RUNX1 mutations and MAP2K4 copy number loss were novel findings in DCIS. Frequent copy number alterations included gains on 1q, 8q, 17q, and 20q and losses on 8p, 11q, 16q, and 17p. Patterns of genomic alterations observed in DCIS were similar to those previously reported for invasive breast cancers, with all DCIS having at least one bona fide breast cancer driver event. However, an increase in GATA3 mutations and fewer copy number changes were noted in DCIS compared with invasive carcinomas. The role of such alterations as prognostic and predictive biomarkers in DCIS is an avenue for further investigation.

Copy number analysis by low coverage whole genome sequencing using ultra low-input DNA from formalin-fixed paraffin embedded tumor tissue.

Unlocking clinically translatable genomic information, including copy number alterations (CNA), from formalin-fixed paraffin-embedded (FFPE) tissue is challenging due to low yields and degraded DNA. We describe a robust, cost-effective low-coverage whole genome sequencing (LC WGS) method for CNA detection using 5 ng of FFPE-derived DNA. CN profiles using 100 ng or 5 ng input DNA were highly concordant and comparable with molecular inversion probe (MIP) array profiles. LC WGS improved CN profiles of samples that performed poorly using MIP arrays. Our technique enables identification of driver and prognostic CNAs in archival patient samples previously deemed unsuitable for genomic analysis due to DNA limitations.