The Legendre Formula in Clifford Analysis

2000 Mathematics Subject Classification: 30A05, 33E05, 30G30, 30G35, 33E20.

Genetics of calcium homeostasis in humans: continuum between monogenic diseases and continuous phenotypes

Extracellular calcium participates in several key physiological functions, such as control of blood coagulation, bone calcification or muscle contraction. Calcium homeostasis in humans is regulated in part by genetic factors, as illustrated by rare monogenic diseases characterized by hypo or hypercalcaemia. Both serum calcium and urinary calcium excretion are heritable continuous traits in humans. Serum calcium levels are tightly regulated by two main hormonal systems, i.e. parathyroid hormone and vitamin D, which are themselves also influenced by genetic factors. Recent technological advances in molecular biology allow for the screening of the human genome at an unprecedented level of detail and using hypothesis-free approaches, such as genome-wide association studies (GWAS). GWAS identified novel loci for calcium-related phenotypes (i.e. serum calcium and 25-OH vitamin D) that shed new light on the biology of calcium in humans. The substantial overlap (i.e. CYP24A1, CASR, GATA3; CYP2R1) between genes involved in rare monogenic diseases and genes located within loci identified in GWAS suggests a genetic and phenotypic continuum between monogenic diseases of calcium homeostasis and slight disturbances of calcium homeostasis in the general population. Future studies using whole-exome and whole-genome sequencing will further advance our understanding of the genetic architecture of calcium homeostasis in humans. These findings will likely provide new insight into the complex mechanisms involved in calcium homeostasis and hopefully lead to novel preventive and therapeutic approaches. Keyword: calcium, monogenic, genome-wide association studies, genetics.

Monogenic polynomials of four variables with binomial expansion

In the recent past one of the main concern of research in the field of Hypercomplex Function Theory in Clifford Algebras was the development of a variety of new tools for a deeper understanding about its true elementary roots in the Function Theory of one Complex Variable. Therefore the study of the space of monogenic (Clifford holomorphic) functions by its stratification via homogeneous monogenic polynomials is a useful tool. In this paper we consider the structure of those polynomials of four real variables with binomial expansion. This allows a complete characterization of sequences of 4D generalized monogenic Appell polynomials by three different types of polynomials. A particularly important case is that of monogenic polynomials which are simply isomorphic to the integer powers of one complex variable and therefore also called pseudo-complex powers.

The Expanded Human Kallikrein (KLK) gene family : genomic organisation, tissue specific expression and potential functions

The tissue kallikreins are serine proteases encoded by highly conserved multigene families. The rodent kallikrein (KLK) families are particularly large, consisting of 13 26 genes clustered in one chromosomal locus. It has been recently recognised that the human KLK gene family is of a similar size (15 genes) with the identification of another 12 related genes (KLK4-KLK15) within and adjacent to the original human KLK locus (KLK1-3) on chromosome 19q13.4. The structural organisation and size of these new genes is similar to that of other KLK genes except for additional exons encoding 5 or 3 untranslated regions. Moreover, many of these genes have multiple mRNA transcripts, a trait not observed with rodent genes. Unlike all other kallikreins, the KLK4-KLK15 encoded proteases are less related (25–44%) and do not contain a conventional kallikrein loop. Clusters of genes exhibit high prostatic (KLK2-4, KLK15) or pancreatic (KLK6-13) expression, suggesting evolutionary conservation of elements conferring tissue specificity. These genes are also expressed, to varying degrees, in a wider range of tissues suggesting a functional involvement of these newer human kallikrein proteases in a diverse range of physiological processes.