Aberrant regulation of the Wnt/β-catenin pathway has an important role during the onset and progression of colorectal cancer, with over 90% of cases of sporadic colon cancer featuring mutations in APC or β-catenin. However, it has remained a point of controversy whether these mutations are sufficient to activate the pathway or require additional upstream signals. Here we show that colorectal tumours express elevated levels of Wnt3 and Evi/Wls/GPR177. We found that in colon cancer cells, even in the presence of mutations in APC or β-catenin, downstream signalling remains responsive to Wnt ligands and receptor proximal signalling. Furthermore, we demonstrate that truncated APC proteins bind β-catenin and key components of the destruction complex. These results indicate that cells with mutations in APC or β-catenin depend on Wnt ligands and their secretion for a sufficient level of β-catenin signalling, which potentially opens new avenues for therapeutic interventions by targeting Wnt secretion via Evi/Wls.

Modern psychiatric epidemiology researches complex interactions between multiple variables in large datasets. This creates difficulties for causal inference. We argue for the use of probabilistic models represented by directed acyclic graphs (DAGs). These capture the dependence structure of multiple variables and, used appropriately, allow more robust conclusions about the direction of causation. We analyzed British national survey data to assess putative mediators of the association between bullying victimization and persecutory ideation. We compared results using DAGs and the Karlson–Holm–Breen (KHB) logistic regression commands in STATA. We analyzed data from the 2007 English National Survey of Psychiatric Morbidity, using the equivalent 2000 survey in an instant replication. Additional details of methods and results are provided in the supplementary material. DAG analysis revealed a richer structure of relationships than could be inferred using the KHB logistic regression commands. Thus, bullying had direct effects on worry, persecutory ideation, mood instability, and drug use. Depression, sleep and anxiety lay downstream, and therefore did not mediate the link between bullying and persecutory ideation. Mediation by worry and mood instability could not be definitively ascertained. Bullying led to hallucinations indirectly, via persecutory ideation and depression. DAG analysis of the 2000 dataset suggested the technique generates stable results. While causality cannot be fully determined from cross-sectional data, DAGs indicate the relationships providing the best fit. They thereby advance investigation of the complex interactions seen in psychiatry, including the mechanisms underpinning psychiatric symptoms. It may consequently be used to optimize the choice of intervention targets.

Acyclic digraphs are the underlying representation of Bayesian networks, a widely used class of probabilistic graphical models. Learning the underlying graph from data is a way of gaining insights about the structural properties of a domain. Structure learning forms one of the inference challenges of statistical graphical models.MCMC methods, notably structure MCMC, to sample graphs from the posterior distribution given the data are probably the only viable option for Bayesian model averaging. Score modularity and restrictions on the number of parents of each node allow the graphs to be grouped into larger collections, which can be scored as a whole to improve the chain's convergence. Current examples of algorithms taking advantage of grouping are the biased order MCMC, which acts on the alternative space of permuted triangular matrices, and non ergodic edge reversal moves.Here we propose a novel algorithm, which employs the underlying combinatorial structure of DAGs to define a new grouping. As a result convergence is improved compared to structure MCMC, while still retaining the property of producing an unbiased sample. Finally the method can be combined with edge reversal moves to improve the sampler further.