A Unified Approach to Quasiregular Linearization in the Plane

We generalize the classical K\"onig's and B\"ottcher's Theorems in complex dynamics to certain quasiregular mappings in the plane. Our approach to these results is unified in the sense that it does not depend on the local injectivity, or not, of the map at the fixed point. By passing to the logarithmic transform we obtain a quasiconformal mapping in either case. Certain restrictions on the quasiregular mappings are needed in order for there to be a candidate to linearize to. These are provided by requiring a simple infinitesimal space of the mapping at the fixed point and restricting to the BIP mappings introduced by the authors in prior work

Genus 2 Cantor sets

We construct a geometrically self-similar Cantor set $X$ of genus $2$ in $\mathbb{R}^3$. This construction is the first for which the local genus is shown to be $2$ at every point of $X$. As an application, we construct, also for the first time, a uniformly quasiregular mapping $f:\mathbb{R}^3 \to \mathbb{R}^3$ for which the Julia set $J(f)$ is a genus $2$ Cantor set.Comment: 20 pages, 6 figure

Iteration of quasiregular tangent functions in three dimensions

We define a new quasiregular mapping T in three dimensions that generalizes the tangent function on the complex plane and shares a number of its geometric properties. We investigate the dynamics of the family \lambda T for \lambda>0, establishing results analogous to those of Devaney and Keen for the meromorphic family \lambda tan z, \lambda>0, although the methods used are necessarily original.Comment: 24 pages, 3 figure

Cell proliferation within small intestinal crypts is the principal driving force for cell migration on villi

The functional integrity of the intestinal epithelial barrier relies on tight coordination of cell proliferation and migration, with failure to regulate these processes resulting in disease. It is not known whether cell proliferation is sufficient to drive epithelial cell migration during homoeostatic turnover of the epithelium. Nor is it known precisely how villus cell migration is affected when proliferation is perturbed. Some reports suggest that proliferation and migration may not be related while other studies support a direct relationship. We used established cell-tracking methods based on thymine analog cell labeling and developed tailored mathematical models to quantify cell proliferation and migration under normal conditions and when proliferation is reduced and when it is temporarily halted. We found that epithelial cell migration velocities along the villi are coupled to cell proliferation rates within the crypts in all conditions. Furthermore, halting and resuming proliferation results in the synchronized response of cell migration on the villi. We conclude that cell proliferation within the crypt is the primary force that drives cell migration along the villus. This methodology can be applied to interrogate intestinal epithelial dynamics and characterize situations in which processes involved in cell turnover become uncoupled, including pharmacological treatments and disease models

Coronary Atherosclerotic Plaque Activity and Future Coronary Events

This study was funded by a Wellcome Trust Senior Investigator Award (WT103782AIA). Image analysis was supported by National Institutes for Health (R34HL161195 and 1R01HL135557). The content is solely the responsibility of the authors and does not necessarily represent the official views of the Wellcome Trust or the National Institutes of Health. The British Heart Foundation supports DEN (CH/09/002, RG/16/10/32375, RE/18/5/34216), MRD (FS/SCRF/21/32010), NLM (CH/F/21/90010, RG/20/10/34966, RE/18/5/34216) AJM (AA/18/3/34220) and MCW (FS/ICRF/20/26002) and DD (FS/RTF/20/30009, NH/19/1/34595, PG/18/35/33786, PG/15/88/31780, PG/17/64/33205). MRD is the recipient of the Sir Jules Thorn Award for Biomedical Research 2015 (15/JTA). PJS is supported by outstanding investigator award National Institutes for Health (R35HL161195). JK is supported by the National Science Centre 2021/41/B/NZ5/02630. EvB is supported by SINAPSE (www.sinapse.ac.uk). AB is supported by a Clinical Research Training Fellowships (MR/V007254/1). DD is supported by Chest Heart and Stroke Scotland (19/53), Tenovus Scotland (G.18.01), and Friends of Anchor and Grampian NHS-Endowments. The Edinburgh Clinical Research Facilities, Edinburgh Imaging facility and Edinburgh Clinical Trials Unit are supported by the National Health Service Research Scotland through National Health Service Lothian Health Board. The Leeds Clinical Research Facilities are supported by the UK National Institute for Health Research (NIHR) via its Clinical Research Facility programme. The work at Cedars-Sinai Medical Center (the Los Angeles site) was supported in part by the Dr. Miriam and Sheldon G. Adelson Medical Research Foundation. For the purpose of open access, the author has applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising from this submission. The Chief Investigator and Edinburgh Clinical Trials Unit had full access to all the data in the study and takes responsibility for the integrity of the data and the accuracy of the data analysis.Peer reviewedPostprin

A hierarchical Bayesian model for understanding the spatiotemporal dynamics of the intestinal epithelium

Our work addresses two key challenges, one biological and one methodological. First, we aim to understand how proliferation and cell migration rates in the intestinal epithelium are related under healthy, damaged (Ara-C treated) and recovering conditions, and how these relations can be used to identify mechanisms of repair and regeneration. We analyse new data, presented in more detail in a companion paper, in which BrdU/IdU cell-labelling experiments were performed under these respective conditions. Second, in considering how to more rigorously process these data and interpret them using mathematical models, we use a probabilistic, hierarchical approach. This provides a best-practice approach for systematically modelling and understanding the uncertainties that can otherwise undermine the generation of reliable conclusions-uncertainties in experimental measurement and treatment, difficult-to-compare mathematical models of underlying mechanisms, and unknown or unobserved parameters. Both spatially discrete and continuous mechanistic models are considered and related via hierarchical conditional probability assumptions. We perform model checks on both in-sample and out-of-sample datasets and use them to show how to test possible model improvements and assess the robustness of our conclusions. We conclude, for the present set of experiments, that a primarily proliferation-driven model suffices to predict labelled cell dynamics over most time-scales

Generation in vivo of peptide-specific cytotoxic T cells and presence of regulatory T cells during vaccination with hTERT (class I and II) peptide-pulsed DCs

Optimal techniques for DC generation for immunotherapy in cancer are yet to be established. Study aims were to evaluate: (i) DC activation/maturation milieu (TNF-α +/- IFN-α) and its effects on CD8+ hTERT-specific T cell responses to class I epitopes (p540 or p865), (ii) CD8+ hTERT-specific T cell responses elicited by vaccination with class I alone or both class I and II epitope (p766 and p672)-pulsed DCs, prepared without IFN-α, (iii) association between circulating T regulatory cells (Tregs) and clinical responses

Multiple novel prostate cancer susceptibility signals identified by fine-mapping of known risk loci among Europeans

Genome-wide association studies (GWAS) have identified numerous common prostate cancer (PrCa) susceptibility loci. We have fine-mapped 64 GWAS regions known at the conclusion of the iCOGS study using large-scale genotyping and imputation in 25 723 PrCa cases and 26 274 controls of European ancestry. We detected evidence for multiple independent signals at 16 regions, 12 of which contained additional newly identified significant associations. A single signal comprising a spectrum of correlated variation was observed at 39 regions; 35 of which are now described by a novel more significantly associated lead SNP, while the originally reported variant remained as the lead SNP only in 4 regions. We also confirmed two association signals in Europeans that had been previously reported only in East-Asian GWAS. Based on statistical evidence and linkage disequilibrium (LD) structure, we have curated and narrowed down the list of the most likely candidate causal variants for each region. Functional annotation using data from ENCODE filtered for PrCa cell lines and eQTL analysis demonstrated significant enrichment for overlap with bio-features within this set. By incorporating the novel risk variants identified here alongside the refined data for existing association signals, we estimate that these loci now explain ∼38.9% of the familial relative risk of PrCa, an 8.9% improvement over the previously reported GWAS tag SNPs. This suggests that a significant fraction of the heritability of PrCa may have been hidden during the discovery phase of GWAS, in particular due to the presence of multiple independent signals within the same regio