On the Foundations of Constructive Mathematics – Especially in Relation to the Theory of Continuous Functions

We discuss the foundations of constructive mathematics, including recursive mathematics and intuitionism, in relation to classical mathematics. There are connections with the foundations of physics, due to the way in which the different branches of mathematics reflect reality. Many different axioms and their interrelationship are discussed. We show that there is a fundamental problem in BISH (Bishop’s school of constructive mathematics) with regard to its current definition of ‘continuous function’. This problem is closely related to the definition in BISH of ‘locally compact’. Possible approaches to this problem are discussed. Topology seems to be a key to understanding many issues. We offer several new simplifying axioms, which can form bridges between the various branches of constructive mathematics and classical mathematics (‘reuniting the antipodes’). We give a simplification of basic intuitionistic theory, especially with regard to so-called ‘bar induction’. We then plead for a limited number of axiomatic systems, which differentiate between the various branches of mathematics. Finally, in the appendix we offer BISH an elegant topological definition of ‘locally compact’, which unlike the current definition is equivalent to the usual classical and/or intuitionistic definition in classical and intuitionistic mathematics, respectively.     

Antioxidant survey to assess antagonism to redox stress using a prokaryotic and an eukaryotic system

Using a prokaryote (Escherichia coli) and a metazoa-resembling eukaryote (Ochromonas danica), we surveyed antioxidants which might overcome redox stress imposed by menadione sodium bisulphite (MD) and buthionine sulphoximine (BSO). BSO oxidant stress was evident only inO. danica; MD oxidant stress was evident in both organisms. Glutathione, its precursors, e.g. cysteine, homocysteine, and 2-oxo-4-thiazolidine carboxylic acid, and red blood cells, emerged as prime antioxidants for relieving BSO and MD oxidant stress. BSO and MD oxidant activity and antioxidant-annulling effect inO. danica were judged comparable to those found in animal cells whereas the resultsE. coli were not entirely equivalent. TheO. danica system emerged as a practical, rapid, and useful system for pinpointing oxidant stressors and antioxidants, and shows promise for studies with mammalian systems.     

Experimental validation of inter-subspecific genetic diversity in rice represented by the differences between the DNA sequences of ‘Nipponbare’ and ‘93-11’

The pedigrees of three sequenced rice cultivars were analyzed to show that a majority of the genetic composition of 'Nipponbare' originates from japonica cultivars while the minority originates from indica cultivars. In contrast, '93-11' is derived mainly from indica cultivars with a smaller contribution from japonica cultivars. All ancestors of 'Guang lu ai 4' appeared to be indica lines. A set of molecular markers (46 InDels and 53 SSRs) polymorphic between 'Nipponbare' and '93-11' were examined in 46 typical indica and 47 typical japonica cultivars selected from 443 accessions according to Cheng's index. All cultivars were divided into indica and japonica groups without overlapping when clustered by Cheng's index, InDels and SSRs. Much higher InDel and SSR diversity between groups than within groups implies that the marker polymorphisms between 'Nipponbare' and '93-11' represent a large proportion of inter-subspecific diversity. About 85% of indica cultivars and more than 90% of japonica cultivars were confirmed to have the same PCR banding patterns as '93-11' and 'Nipponbare', respectively. Some polymorphic loci between 'Nipponbare' and '93-11' cannot be validated in other indica and japonica cultivars, either as subspecies-specific but not predominant alleles, or alleles not specific between the two groups. It was concluded that molecular markers developed from sequence polymorphism between 'Nipponbare' and '93-11' often represent inter-subspecific diversity, although some exceptions were sensitive to either particular marker loci or particular cultivars.     

Enhancements of energetic particles near the heliospheric termination shock

The spacecraft Voyager 1 is at a distance greater than 85 au from the Sun, in the vicinity of the termination shock that marks the abrupt slowing of the supersonic solar wind and the beginning of the extended and unexplored distant heliosphere. This shock is expected to accelerate 'anomalous cosmic rays', as well as to re-accelerate Galactic cosmic rays and low-energy particles from the inner Solar System. Here we report a significant increase in the numbers of energetic ions and electrons that persisted for seven months beginning in mid-2002. This increase differs from any previously observed in that there was a simultaneous increase in Galactic cosmic ray ions and electrons, anomalous cosmic rays and low-energy ions. The low-intensity level and spectral energy distribution of the anomalous cosmic rays, however, indicates that Voyager 1 still has not reached the termination shock. Rather, the observed increase is an expected precursor event. We argue that the radial anisotropy of the cosmic rays is expected to be small in the foreshock region, as is observed.     

Intermediates in the transformation of phosphonates to phosphate by bacteria

Phosphorus is an essential element for all known forms of life. In living systems, phosphorus is an integral component of nucleic acids, carbohydrates and phospholipids, where it is incorporated as a derivative of phosphate. However, most Gram-negative bacteria have the capability to use phosphonates as a nutritional source of phosphorus under conditions of phosphate starvation. In these organisms, methylphosphonate is converted to phosphate and methane. In a formal sense, this transformation is a hydrolytic cleavage of a carbon-phosphorus (C-P) bond, but a general enzymatic mechanism for the activation and conversion of alkylphosphonates to phosphate and an alkane has not been elucidated despite much effort for more than two decades. The actual mechanism for C-P bond cleavage is likely to be a radical-based transformation. In Escherichia coli, the catalytic machinery for the C-P lyase reaction has been localized to the phn gene cluster. This operon consists of the 14 genes phnC, phnD, …, phnP. Genetic and biochemical experiments have demonstrated that the genes phnG, phnH, …, phnM encode proteins that are essential for the conversion of phosphonates to phosphate and that the proteins encoded by the other genes in the operon have auxiliary functions. There are no functional annotations for any of the seven proteins considered essential for C-P bond cleavage. Here we show that methylphosphonate reacts with MgATP to form α-D-ribose-1-methylphosphonate-5-triphosphate (RPnTP) and adenine. The triphosphate moiety of RPnTP is hydrolysed to pyrophosphate and α-D-ribose-1-methylphosphonate-5-phosphate (PRPn). The C-P bond of PRPn is subsequently cleaved in a radical-based reaction producing α-D-ribose-1,2-cyclic-phosphate-5-phosphate and methane in the presence of S-adenosyl-L-methionine. Substantial quantities of phosphonates are produced worldwide for industrial processes, detergents, herbicides and pharmaceuticals. Our elucidation of the chemical steps for the biodegradation of alkylphosphonates shows how these compounds can be metabolized and recycled to phosphate.     

Mutations in embryonic myosin heavy chain (MYH3) cause Freeman-Sheldon syndrome and Sheldon-Hall syndrome

The genetic basis of most conditions characterized by congenital contractures is largely unknown. Here we show that mutations in the embryonic myosin heavy chain (MYH3) gene cause Freeman-Sheldon syndrome (FSS), one of the most severe multiple congenital contracture (that is, arthrogryposis) syndromes, and nearly one-third of all cases of Sheldon-Hall syndrome (SHS), the most common distal arthrogryposis. FSS and SHS mutations affect different myosin residues, demonstrating that MYH3 genotype is predictive of phenotype. A structure-function analysis shows that nearly all of the MYH3 mutations are predicted to interfere with myosin's catalytic activity. These results add to the growing body of evidence showing that congenital contractures are a shared outcome of prenatal defects in myofiber force production. Elucidation of the genetic basis of these syndromes redefines congenital contractures as unique defects of the sarcomere and provides insights about what has heretofore been a poorly understood group of disorders.     

Male-pattern baldness susceptibility locus at 20p11

We conducted a genome-wide association study for androgenic alopecia in 1,125 men and identified a newly associated locus at chromosome 20p11.22, confirmed in three independent cohorts (n = 1,650; OR = 1.60, P = 1.1 x 10(-14) for rs1160312). The one man in seven who harbors risk alleles at both 20p11.22 and AR (encoding the androgen receptor) has a sevenfold-increased odds of androgenic alopecia (OR = 7.12, P = 3.7 x 10(-15)).     

Genetic variation in DPP6 is associated with susceptibility to amyotrophic lateral sclerosis

We identified a SNP in the DPP6 gene that is consistently strongly associated with susceptibility to amyotrophic lateral sclerosis (ALS) in different populations of European ancestry, with an overall P value of 5.04 x 10(-8) in 1,767 cases and 1,916 healthy controls and with an odds ratio of 1.30 (95% confidence interval (CI) of 1.18-1.43). Our finding is the first report of a genome-wide significant association with sporadic ALS and may be a target for future functional studies.