Digitisation, representation, and formalisation - Digital libraries of mathematics

One of the main tasks of the mathematical knowledge management community must surely be to enhance access to mathematics on digital systems. In this paper we present a spectrum of approaches to solving the various problems inherent in this task, arguing that a variety of approaches is both necessary and useful. The main ideas presented are about the differences between digitised mathematics, digitally represented mathematics and formalised mathematics. Each has its part to play in managing mathematical information in a connected world. Digitised material is that which is embodied in a computer file, accessible and displayable locally or globally. Represented material is digital material in which there is some structure (usually syntactic in nature) which maps to the mathematics contained in the digitised information. Formalised material is that in which both the syntax and semantics of the represented material, is automatically accessible. Given the range of mathematical information to which access is desired, and the limited resources available for managing that information, we must ensure that these resources are applied to digitise, form representations of or formalise, existing and new mathematical information in such a way as to extract the most benefit from the least expenditure of resources. We also analyse some of the various social and legal issues which surround the practical tasks.

Presubmergence and green manure affect the transformations of nitrogen-15-labeled urea under lowland soil conditions

The effect of presubmergence and green manuring on various processes involved in [N-15]-urea transformations were studied in a growth chamber after [N-15]-urea application to floodwater. Presubmergence for 14 days increased urea hydrolysis rates and floodwater pH, resulting in higher NH3 volatilization as compared to without presubmergence. Presubmergence also increased nitrification and subsequent denitrification but lower N assimilation by floodwater algae caused higher gaseous losses. Addition of green manure maintained higher NH4+-N concentration in floodwater mainly because of lower nitrification rates but resulted in highest NH3 volatilization losses. Although green manure did not affect the KCl extractable NH4+-N from applied fertilizer, it maintained higher NH4+-N content due to its decomposition and increased mineralization of organic N. After 32 days about 36.9% (T-1), 23.9% (T-2), and 36.4% (T-3) of the applied urea N was incorporated in the pool of soil organic N in treatments. It was evident that the presubmergence has effected the recovery of applied urea N.

13C-2H correlation NMR spectroscopy studies of the in vivo transformations of natural products from Artemisia annua

13C-2H correlation NMR spectroscopy (13C-2H COSY) permits the identification of 13C and 2H nuclei which are connected to one another by a single chemical bond via the sizeable 1JCD coupling constant. The practical development of this technique is described using a 13C-2H COSY pulse sequence which is derived from the classical 13C-1H correlation experiment. An example is given of the application of 13C-2H COSY to the study of the biogenesis of natural products from the anti-malarial plant Artemisia annua, using a doubly-labelled precursor molecule. Although the biogenesis of artemisinin, the anti-malarial principle from this species, has been extensively studied over the past twenty years there is still no consensus as to the true biosynthetic route to this important natural product – indeed, some published experimental results are directly contradictory. One possible reason for this confusion may be the ease with which some of the metabolites from A. annua undergo spontaneous autoxidation, as exemplified by our recent in vitro studies of the spontaneous autoxidation of dihydroartemisinic acid, and the application of 13C-2H COSY to this biosynthetic problem has been important in helping to mitigate against such processes. In this in vivo application of 13C-2H COSY, [15-13C2H3]-dihydroartemisinic acid (the doubly-labelled analogue of the natural product from this species which was obtained through synthesis) was fed to A. annua plants and was shown to be converted into several natural products which have been described previously, including artemisinin. It is proposed that all of these transformations occurred via a tertiary hydroperoxide intermediate, which is derived from dihyroartemisinic acid. This intermediate was observed directly in this feeding experiment by the 13C-2H COSY technique; its observation by more traditional procedures (e.g., chromatographic separation, followed by spectroscopic analysis of the purified product) would have been difficult owing to the instability of the hydroperoxide group (as had been established previously by our in vitro studies of the spontaneous autoxidation of dihydroartemisinic acid). This same hydroperoxide has been reported as the initial product of the spontaneous autoxidation of dihydroartemisinic acid in our previous in vitro studies. Its observation in this feeding experiment by the 13C-2H COSY technique, a procedure which requires the minimum of sample manipulation in order to achieve a reliable identification of metabolites (based on both 13C and 2H chemical shifts at the 15-position), provides the best possible evidence for its status as a genuine biosynthetic intermediate, rather than merely as an artifact of the experimental procedure.

In vivo transformations of dihydroartemisinic acid in Artemisia annua plants

[15-(CH3)-C-13-H-2]-dihydroartemisinic acid (2a) and [15-(CH3)-H-2]-dihydroartemisinic acid (2b) have been fed via the root to intact Artemisia annua plants and their transformations studied in vivo by one-dimensional H-2 NMR spectroscopy and two-dimensional, C-13-H-2 correlation NMR spectroscopy (C-13-(2) H COSY). Labelled dihydroartemisinic acid was transformed into 16 12-carboxy-amorphane and cadinane sesquiterpenes within a few days in the aerial parts of A. annua, although transformations in the root were much slower and more limited. Fifteen of these 16 metabolites have been reported previously as natural products from A. annua. Evidence is presented that the first step in the transformation of dihydroartemisinic acid in vivo is the formation of allylic hydroperoxides by the reaction of molecular oxygen with the Delta(4,5)-double bond in this compound. The origin of all 16 secondary metabolites might then be explained by the known further reactions of such hydroperoxides. The qualitative pattern for the transformations of dihydroartemisinic acid in vivo was essentially unaltered when a comparison was made between plants, which had been kept alive and plants which were allowed to die after feeding of the labelled precursor. This, coupled with the observation that the pattern of transformations of 2 in vivo demonstrated very close parallels with the spontaneous autoxidation chemistry for 2, which we have recently demonstrated in vitro, has lead us to conclude that the main 'metabolic route' for dihydroartemisinic acid in A. annua involves its spontaneous autoxidation and the subsequent spontaneous reactions of allylic hydroperoxides which are derived from 2. There may be no need to invoke the participation of enzymes in any of the later biogenetic steps leading to all 16 of the labelled 11,13-dihydro-amorphane sesquiterpenes which are found in A. annua as natural products. (C) 2003 Elsevier Ltd. All rights reserved.

In vivo transformations of artemisinic acid in Artemisia annua plants

Artemisinic acid labeled with both C-13 and H-2 at the 15-position has been fed to intact plants of Artemisia annua via the cut stem, and its in vivo transformations studied by 1D- and 2D-NMR spectroscopy. Seven labeled metabolites have been isolated, all of which are known as natural products from this species. The transformations of artemisinic acid-as observed both for a group of plants, which was kept alive by hydroponic administration of water and for a group, which was allowed to die by desiccation-closely paralleled those, which have been recently described for its 11,13-dihydro analog, dihydroartemisinic acid. It seems likely therefore that similar mechanisms, involving spontaneous autoxidation of the Delta(4,5) double bond in both artemisinic acid and dihydroartemisinic acid and subsequent rearrangements of the resultant allylic hydroperoxides, may be involved in the biological transformations, which are undergone by both compounds. All of the sesquiterpene metabolites, which were obtained from in vivo transformations of artemisinic acid retained their unsaturation at the 11,13-position, and there was no evidence for conversion into any 11,13-dihydro metabolite, including artemisinin, the antimalarial drug, which is produced by A. annua. This observation led to the proposal of a unified biosynthetic scheme, which accounts for the biogenesis of many of the amorphane and cadinane sesquiterpenes that have been isolated as natural products from A. annua. In this scheme, there is a bifurcation in the biosynthetic pathway starting from amorpha-4,11-diene leading to either artemisinic acid or dihydroartemisinic acid; these two committed precursors are then, respectively, the parents for the two large families of highly oxygenated 11,13-dehydro and 11,13-dihydro sesquiterpene metabolites, which are known from this species. (C) 2007 Elsevier Ltd. All rights reserved.

In vivo transformations of dihydro-epi-deoxyarteannuin B in Artemisia annua plants

[15-(CH3)-C-13-H-2]-dihydro-epi-deoxyarteannuin B (4a) has been fed to intact Artemisia annua plants via the root and three labeled metabolites (17a-19a) have been identified by 1D- and 2D-NMR spectroscopies. The in vivo transformations of 4a in A. annua are proposed to involve enzymatically-mediated processes in addition to possible spontaneous autoxidation. In the hypothetical spontaneous autoxidation pathway, the tri-substituted double bond in 4a appears to have undergone 'ene-type' reaction with oxygen to form an allylic hydroperoxide, which subsequently rearranges to the allylic hydroxyl group in the metabolite 3 alpha-hydroxy-dihydro-epi-deoxyarteannuin B (17a). In the enzymatically-mediated pathways, compound 17a has then been converted to its acetyl derivative, 3 alpha-acetoxy-dihydro-epi-deoxyarteannuin B (18a), while oxidation of 4a at the 'unactivated' 9-position has yielded 9 beta-hydroxy-dihydro-epi-deoxyarteannuin B (19a). Although all of the natural products artemisinin ( 1), arteannuin K ( 7), arteannuin L ( 8), and arteannuin M ( 9) have been suggested previously as hypothetical metabolites from dihydro-epi-deoxyarteannuin B in A. annua, none were isolated in labeled form in this study. It is argued that the nature of the transformations undergone by compound 4a are more consistent with a degradative metabolism, designed to eliminate this compound from the plant, rather than with a role as a late precursor in the biosynthesis of artemisinin or other natural products from A. annua. (C) 2007 Elsevier Ltd. All rights reserved.

The thermal transformations of bicyclopropylidene and methylenespiropentane revisited

The overall and the individual rate constants of the unimolecular thermal isomerization of methylenespiropentane (4) to 1,2- and 1,3-dimethylenecyclobutanes (7 and 8) have been determined to be lg (k(-4)/s(-1)) = (13.78 +/- 0.06) - (49.7 +/- 0.2) kcal mol(-1)/RT.ln 10, lg(k(7)/s(-1)) = (13.03 +/- 0.19) - (48.0 +/- 0.6) kcal mol(-1)/RT.ln 10 and lg(k(8)/s(-1)) = (14.15 +/- 0.19) - (52.4 +/- 0.5) kcal mol(-1)/RT.ln 10, respectively. The activation energies are significantly lower than that for the rearrangement of the parent spiropentane. ((c) Wiley-VCH Verlag GmbH & Co.

Models, mathematics and meaning in interdisciplinary research

Interdisciplinary research presents particular challenges for unambiguous communication. Frequently, the meanings of words differ markedly between disciplines, leading to apparent consensus masking fundamental misunderstandings. Researchers can agree on the need for models, but conceive of models fundamentally differently. While mathematics is frequently seen as an elitist language reinforcing disciplinary distinctions, both mathematics and modelling can also offer scope to bridge disciplinary epistemological divisions and create common ground on which very different disciplines can meet. This paper reflects on the role and scope for mathematics and modelling to present a common epistemological space in interdisciplinary research spanning the social, natural and engineering sciences.