The Complexity of Combinations of Qualitative Constraint Satisfaction Problems

The CSP of a first-order theory $T$ is the problem of deciding for a given finite set $S$ of atomic formulas whether $T \cup S$ is satisfiable. Let $T_1$ and $T_2$ be two theories with countably infinite models and disjoint signatures. Nelson and Oppen presented conditions that imply decidability (or polynomial-time decidability) of $\mathrm{CSP}(T_1 \cup T_2)$ under the assumption that $\mathrm{CSP}(T_1)$ and $\mathrm{CSP}(T_2)$ are decidable (or polynomial-time decidable). We show that for a large class of $\omega$-categorical theories $T_1, T_2$ the Nelson-Oppen conditions are not only sufficient, but also necessary for polynomial-time tractability of $\mathrm{CSP}(T_1 \cup T_2)$ (unless P=NP).Comment: Version 2: stronger main result with better presentation of the proof; multiple improvements in other proofs; new section structure; new example

Molecular overview of the glycan “node” analysis procedure.

<p>For glycans from blood plasma and other biofluids, O-linked glycans are released during permethylation, while N-linked glycans and glycolipids are released during acid hydrolysis. The unique pattern of methylation and acetylation in the final partially methylated alditol acetates (PMAAs) corresponds to the unique “glycan node” in the original glycan polymer and provides the molecular basis for separation and quantification by GC-MS. Figure adapted with permission from Borges CR et al. Anal. Chem. 2013, 85(5):2927–2936. Copyright 2013 American Chemical Society.</p

Distributions and ROC curves for the most highly elevated glycan node markers in former & current UCC patients relative to healthy controls when data were normalized to heavy glucose or heavy GlcNAc.

<p>Patient distributions are shown in (a-d). The Kruskal-Wallis test was performed followed by Dunn’s post hoc test. The letters at the top of the data points show statistically significant differences between the patient groups; groups with same letter do not have a significant difference. (e-h) ROC curves for the different sub-cohorts of UCC patients vs. healthy individuals. Areas under the ROC curves are provided in parenthesis next to the stated patient groups. As explained in the Discussion, despite the promising AUCs and shapes of some of these ROC curves, these data do not indicate that plasma/serum glycan nodes will potentially serve as clinically useful diagnostic markers of UCC.</p

Distributions and ROC curves for the most highly elevated glycan node markers in former & current UCC patients relative to healthy controls when data were normalized to sum of endogenous Hexoses or HexNAcs.

<p>Patient distributions are shown in (a-d). The Kruskal-Wallis test was performed followed by Dunn’s post hoc test. The letters at the top of the data points show statistically significant differences between the patient groups; groups with a common letter do not have a significant difference. (e-h) ROC curves for different groups of bladder cancer patients vs. certifiably healthy individuals. Area under the ROC curves are provided in parenthesis next to the stated patient groups. “NS” next to the area under the ROC curves shows that there is no significant difference between the two groups that are being compared. These data do not indicate that plasma/serum glycan nodes will potentially serve as clinically useful diagnostic markers of UCC.</p

Correlation of CRP and glycan nodes.

<p>Log of CRP concentration vs. (a) α2–6 sialylation; r = 0.34 and (b) β1–6 branching; r = 0.38 are plotted. Both correlations are statistically significant (Pearson correlation; <i>p</i> < 0.001).</p

Statistically significant differences between controls and bladder cancer patient sub-cohorts^a.

<p>Statistically significant differences between controls and bladder cancer patient sub-cohorts<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0201208#t001fn001" target="_blank">^a</a>.</p

Statistically significant differences between controls and bladder cancer patient sub-cohorts with data normalization to the sum of all endogenous hexoses or HexNAcs.

<p>Statistically significant differences between controls and bladder cancer patient sub-cohorts with data normalization to the sum of all endogenous hexoses or HexNAcs.</p

Conceptual overview of the glycan “node” analysis concept.

<p>The procedure consists of applying glycan methylation analysis (i.e., linkage analysis) to whole biofluids. Intact normal and abnormal glycans including O-glycans, N-glycans and glycolipids, are processed and transformed into partially methylated alditol acetates (PMAAs, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0201208#pone.0201208.g001" target="_blank">Fig 1</a>), each of which corresponds to a particular monosaccharide-and-linkage-specific glycan “node” in the original polymer. As illustrated, analytically pooling together the glycan nodes from amongst all the aberrant intact glycan structures provides a more direct surrogate measurement of abnormal glycosyltransferase activity than any individual intact glycan, while simultaneously converting unique glycan features such as “core fucosylation”, “α2–6 sialylation”, “bisecting GlcNAc”, and “β1–6 branching” into single analytical signals. Actual extracted ion chromatograms from 9-μL blood plasma samples are shown. Numbers adjacent to monosaccharide residues in glycan structures indicate the position at which the higher residue is linked to the lower residue. Figure adapted with permission from Borges CR et al. Anal. Chem. 2013, 85(5):2927–2936. Copyright 2013 American Chemical Society.</p

Correlation between age and the most highly elevated glycan node markers in former & current UCC patients relative to healthy controls when data were normalized to heavy glucose or heavy GlcNAc.

<p>Pearson correlation was used to evaluate this correlation. The common age range between all cohorts was 45–67. “NS” next to the r-value indicates that the Pearson correlation was not statistically significant. Distribution of the healthy controls is demonstrated by red dots. Distribution of the different sub-cohorts of UCC patients is demonstrated by black triangles.</p

Stage Dependence, Cell-Origin Independence, and Prognostic Capacity of Serum Glycan Fucosylation, β1–4 Branching, β1–6 Branching, and α2–6 Sialylation in Cancer

Glycans represent a promising but only marginally accessed source of cancer markers. We previously reported the development of a molecularly bottom-up approach to plasma and serum (P/S) glycomics based on glycan linkage analysis that captures features such as α2–6 sialylation, β1–6 branching, and core fucosylation as single analytical signals. Based on the behavior of P/S glycans established to date, we hypothesized that the alteration of P/S glycans observed in cancer would be independent of the tissue in which the tumor originated yet exhibit stage dependence that varied little between cancers classified on the basis of tumor origin. Herein, the diagnostic utility of this bottom-up approach as applied to lung cancer patients (<i>n</i> = 127 stage I; <i>n</i> = 20 stage II; <i>n</i> = 81 stage III; and <i>n</i> = 90 stage IV) as well as prostate (<i>n</i> = 40 stage II), serous ovarian (<i>n</i> = 59 stage III), and pancreatic cancer patients (<i>n</i> = 15 rapid autopsy) compared to certifiably healthy individuals (<i>n</i> = 30), nominally healthy individuals (<i>n</i> = 166), and risk-matched controls (<i>n</i> = 300) is reported. Diagnostic performance in lung cancer was stage-dependent, with markers for terminal (total) fucosylation, α2–6 sialylation, β1–4 branching, β1–6 branching, and outer-arm fucosylation most able to differentiate cases from controls. These markers behaved in a similar stage-dependent manner in other types of cancer as well. Notable differences between certifiably healthy individuals and case-matched controls were observed. These markers were not significantly elevated in liver fibrosis. Using a Cox proportional hazards regression model, the marker for α2–6 sialylation was found to predict both progression and survival in lung cancer patients after adjusting for age, gender, smoking status, and stage. The potential mechanistic role of aberrant P/S glycans in cancer progression is discussed