Equilibrium Points of an AND-OR Tree: under Constraints on Probability

We study a probability distribution d on the truth assignments to a uniform binary AND-OR tree. Liu and Tanaka [2007, Inform. Process. Lett.] showed the following: If d achieves the equilibrium among independent distributions (ID) then d is an independent identical distribution (IID). We show a stronger form of the above result. Given a real number r such that 0 < r < 1, we consider a constraint that the probability of the root node having the value 0 is r. Our main result is the following: When we restrict ourselves to IDs satisfying this constraint, the above result of Liu and Tanaka still holds. The proof employs clever tricks of induction. In particular, we show two fundamental relationships between expected cost and probability in an IID on an OR-AND tree: (1) The ratio of the cost to the probability (of the root having the value 0) is a decreasing function of the probability x of the leaf. (2) The ratio of derivative of the cost to the derivative of the probability is a decreasing function of x, too.Comment: 13 pages, 3 figure

Integrated Multiregional Analysis Proposing a New Model of Colorectal Cancer Evolution.

Understanding intratumor heterogeneity is clinically important because it could cause therapeutic failure by fostering evolutionary adaptation. To this end, we profiled the genome and epigenome in multiple regions within each of nine colorectal tumors. Extensive intertumor heterogeneity is observed, from which we inferred the evolutionary history of the tumors. First, clonally shared alterations appeared, in which C>T transitions at CpG site and CpG island hypermethylation were relatively enriched. Correlation between mutation counts and patients' ages suggests that the early-acquired alterations resulted from aging. In the late phase, a parental clone was branched into numerous subclones. Known driver alterations were observed frequently in the early-acquired alterations, but rarely in the late-acquired alterations. Consistently, our computational simulation of the branching evolution suggests that extensive intratumor heterogeneity could be generated by neutral evolution. Collectively, we propose a new model of colorectal cancer evolution, which is useful for understanding and confronting this heterogeneous disease

Correction: Integrated Multiregional Analysis Proposing a New Model of Colorectal Cancer Evolution.

[This corrects the article DOI: 10.1371/journal.pgen.1005778.]

Genetic insights into ossification of the posterior longitudinal ligament of the spine

Ossification of the posterior longitudinal ligament of the spine (OPLL) is an intractable disease leading to severe neurological deficits. Its etiology and pathogenesis are primarily unknown. The relationship between OPLL and comorbidities, especially type 2 diabetes (T2D) and high body mass index (BMI), has been the focus of attention; however, no trait has been proven to have a causal relationship. We conducted a meta-analysis of genome-wide association studies (GWASs) using 22,016 Japanese individuals and identified 14 significant loci, 8 of which were previously unreported. We then conducted a gene-based association analysis and a transcriptome-wide Mendelian randomization approach and identified three candidate genes for each. Partitioning heritability enrichment analyses observed significant enrichment of the polygenic signals in the active enhancers of the connective/bone cell group, especially H3K27ac in chondrogenic differentiation cells, as well as the immune/hematopoietic cell group. Single-cell RNA sequencing of Achilles tendon cells from a mouse Achilles tendon ossification model confirmed the expression of genes in GWAS and post-GWAS analyses in mesenchymal and immune cells. Genetic correlations with 96 complex traits showed positive correlations with T2D and BMI and a negative correlation with cerebral aneurysm. Mendelian randomization analysis demonstrated a significant causal effect of increased BMI and high bone mineral density on OPLL. We evaluated the clinical images in detail and classified OPLL into cervical, thoracic, and the other types. GWAS subanalyses identified subtype-specific signals. A polygenic risk score for BMI demonstrated that the effect of BMI was particularly strong in thoracic OPLL. Our study provides genetic insight into the etiology and pathogenesis of OPLL and is expected to serve as a basis for future treatment development

Our model of colorectal cancer evolution.

<p>First, founder alterations containing a set of drive alterations are accumulated in the genome and epigenome as a result of aging. After establishment of a parental clone, extensive ITH is generated by neutral evolution, although a few driver alterations are acquired as progressor alterations. Note that this illustration is based on the evolutionary tree of case 3 (<a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1005778#pgen.1005778.g002" target="_blank">Fig 2</a>). However, an actual tumor should harbor numerous subclones, as suggested by the local ITH data (<a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1005778#pgen.1005778.g005" target="_blank">Fig 5D</a>) and simulated single-cell mutation profile (<a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1005778#pgen.1005778.g005" target="_blank">Fig 5B</a>).</p

Analysis of epigenetic ITH.

<p>(<b>A</b>) A heat map of multiregional methylation profiles of the 8 cases. (<b>B</b>) Differential contribution of different categories of probe to intratumor and intertumor variance. According to intratumor and intertumor variance, Probes were ranked to obtain each indicated number of top-ranked probes. Probes were categorized based on their genomic positions and enrichment of each category in the top-ranked probes was measured. (<b>C</b>) Proportion of indicated types of methylation alterations was calculated for founder (F) and progressor (P) methylation alterations in the 8 cases. P-values were calculated by Wilcoxon signed-rank test on the 8 cases.</p

Evolutionary trees of the 9 colorectal tumors.

<p>Evolutionary trees inferred from the multiregional mutation profiles have orange trunks, green branches and variously colored leaves, which correspond to founder, progressor mutations and samples, respectively. The leaves were colored based on the color-coding scheme used in <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1005778#pgen.1005778.g001" target="_blank">Fig 1</a>. Mutation timings of reported driver genes in colorectal cancer were indicated along the trees, and schemas or photos of multiregionally sampled tumors were also provided. Red and blue scales measure tumor size and tree size based on the number of mutations, respectively.</p

Simulation of cancer evolution.

<p>(<b>A</b>) A simulated tumor. Different colors represent different clones. White rectangles labeled with alphabets indicate regions subjected to multiregional sampling. (<b>B</b>) A simulated single-cell mutation profile matrix. Columns represent 500 cells sampled from the simulated tumor, and the top colored bars label each clone. Rows represent mutated genes and driver genes are indicated by left blue bars. (<b>C</b>) A simulated multiregional mutation profile matrix. VAFs of each gene were calculated for cell subpopulations from the 8 regions indicated in (A). (<b>D</b> and <b>E</b>) Distribution of VAFs (D) and Proportion of driver genes (E) in different categories of mutations. The mutations were obtained from 20 multiregional mutation profile matrices generated by independent simulation trials. In (E), the width of each bar is proportional to the count of each category of mutations. Therefore, the area of each bar is proportional to the count of driver genes that belong to each category of mutations.</p