Enumerating Colorings, Tensions and Flows in Cell Complexes

We study quasipolynomials enumerating proper colorings, nowhere-zero tensions, and nowhere-zero flows in an arbitrary CW-complex $X$, generalizing the chromatic, tension and flow polynomials of a graph. Our colorings, tensions and flows may be either modular (with values in $\mathbb{Z}/k\mathbb{Z}$ for some $k$) or integral (with values in $\{-k+1,\dots,k-1\}$). We obtain deletion-contraction recurrences and closed formulas for the chromatic, tension and flow quasipolynomials, assuming certain unimodularity conditions. We use geometric methods, specifically Ehrhart theory and inside-out polytopes, to obtain reciprocity theorems for all of the aforementioned quasipolynomials, giving combinatorial interpretations of their values at negative integers as well as formulas for the numbers of acyclic and totally cyclic orientations of $X$.Comment: 28 pages, 3 figures. Final version, to appear in J. Combin. Theory Series

Linkage analysis of candidate regions for coeliac disease genes.

A strong HLA association is seen in coeliac disease [specifically to the DQ(alpha1*0501,beta1*0201 heterodimer], but this cannot entirely account for the increased risk seen in relatives of affected cases. One or more genes at HLA-unlinked loci also predispose to coeliac disease and are probably stronger determinants of disease susceptibility than HLA. A recent study has proposed a number of candidate regions on chromosomes 6p23 (distinct from HLA), 6p12, 3q27, 5q33.3, 7q31.3, 11p11, 15q26, 19p13.3, 19q13.1, 19q13.4 and 22cen for the location of a non-HLA linked susceptibility gene. We have examined these regions in 28 coeliac disease families by linkage analysis. There was excess sharing of chromosome 6p markers, but no support for a predisposition locus telomeric to HLA. No significant evidence in favour of linkage to coeliac disease was obtained for chromosomes 3q27, 5q33.3, 7q31.3, 11p11, 19p13.3, 19q13.1, 19q13.4 or 22cen. There was, however, excess sharing close to D15S642. The maximum non-parametric linkage score was 1.99 (P = 0.03). Although the evidence for linkage of coeliac disease to chromosome 15q26 is not strong, the well established association between coeliac disease and insulin dependent diabetes mellitus, together with the mapping of an IDDM susceptibility locus (IDDM3) to chromosome 15q26, provide indirect support for this as a candidate locus conferring susceptibility to coeliac disease in some families

Contribution of the MHC region to the familial risk of coeliac disease

Susceptibility to coeliac disease is genetically determined by possession of specific HLA-DQ alleles, acting in concert with one or more non-HLA linked genes. The pattern of risk seen in sibs and twins in coeliac disease is most parsimonious with a multiplicative model for the interaction between the two classes of genes. Based on a sib recurrence risk for coeliac disease of 10% and a population prevalence of 0.0033, the sib relative risk is 30. To evaluate the contribution of the MHC region to the familial risk of coeliac disease, we have examined haplotype sharing probabilities across this region in 55 coeliac disease families. Based on these probabilities the sib relative risk of coeliac disease associated with the MHC region is 3.7. Combining these results with published data on allele sharing at HLA, the estimated sib relative risk associated with the MHC region is 3.3. Therefore, the MHC genes contribute no more than 40% of the sib familial risk of coeliac disease and the non-HLA linked gene (or genes) are likely to be the stronger determinant of coeliac disease susceptibility.


Keywords: coeliac disease; familial risk; MH