Stieltjes integration and differential geometry: A model for enzyme recognition,discrimination, and catalysis

A model for enzymic catalysis is presented using the mathematical theories of differential geometry and Stieltjes integration. The Stieltjesintegrator is a complex-valued function of bounded variation which represents the curvature and torsion, hence the conformation, of the backbone of an enzyme molecule. Theintegrand is a complex-valued continuous function which describes the shape of the surface of a substrate molecule. We postulate that enzyme-substrate interactions correspond to evaluations of Stieltjes integrals, and that observables of enzymic catalysis correspond to projections. Results from the mathematical theory of the Stieltjes integral are discussed together with their biological interpretations. We contrast the difference between structural and functional proteins, and construct analogues of enzyme cofactors, modifications, and regulation. Various techniques of locating the active site on enzymes are also given. We construct a total variation metric, which is particularly useful for detecting similarities among proteins. An examination on the many different modes of convergence of mathematical functions representing biological molecules leads to a mathematical statement of the fundamental dogma of molecular biology, that ‘structure implies function’. Similar arguments also result in the converse statement ‘function dictates structure’, which is a basic premise of relational biology. Stepped-helical approximations of the backbone space curves of enzymes provide a concrete computational tool with which to calculate the Stieltjes integrals that model enzymic catalysis, by replacing the integral with a finite series. The duality between enzymes and substrates (that they aremeters ‘observing’ one another) is shown to be a consequence of the mathematical duality of Banach spaces. The Stieltjes integrals of enzyme-substrate interactions are hence shown to be bounded bilinear functionals. The mechanism of enzymic catalysis, the transformation from substrate to product, is also formulated in the Stieltjes integration context via the mathematical theory of adjoints. The paper closes with suggestions for generalizations, prospects for future studies, and a review of the correspondence between mathematical and biological concepts.     

Hybrid origin of some ornamentals of Allium subgenus Melanocrommyum verified with GISH and RAPD

 Random amplified polymorphic DNA (RAPD) and genomic in situ hybridization (GISH) methods have been used to verify the hybridogenic origin and to identify the parental species of some ornamental cultivars in the subgenus Melanocrommyum of the genus Allium. The cultivars had been selected from seed obtained after uncontrolled pollination in breeders’ fields. The combination of GISH analysis with RAPD markers is very suitable for testing the hybridogenic origin of plants and to ascertain the parental species of the hybrids in such cases. As suspected, A. macleanii and A. cristophii are the parental species of ‘Globemaster’. The parental species of cultivar ‘Globus’ are A. karataviense and A. stipitatum, and not A. cristophii and A. giganteum as has been assumed on morphological grounds. Cultivars ‘Lucy Ball’ and ‘Gladiator’ are of hybrid origin, though only one of the parental species, A. hollandicum, could be confirmed. The cultivars ‘Purple Sensation’, ‘Mount Everest’, ‘White Giant’, ‘Michael H. Hoog’ and ‘Mars’ are not hybrids since neither GISH nor RAPD suggest the presence of a second genome. ‘Purple Sensation’ belongs to A. hollandicum, ‘Mount Everest’, ‘White Giant’ and ‘Mars’ to A. stipitatum,‘Michael H. Hoog’ to A. rosenorum. Received: 3 July 1997 / Accepted: 9 October 1997     

Visual depth encoding in populations of neurons with localized receptive fields

 Stereopsis is the ability to perceive three-dimensional structure from disparities between the two-dimensional retinal images. Although disparity-sensitive neurons have been proposed as a neural representation of this ability many years ago, it is still difficult to link all qualities of stereopsis to properties of the neural correlate of binocular disparities. The present study wants to support efforts directed at closing the gap between electrophysiology and psychophysics. Populations of disparity-sensitive neurons in V1 were simulated using the energy-neuron model. Responses to different types of stimuli were evaluated with an efficient statistical estimator and related to psychophysical findings. The representation of disparity in simulated population responses appeared to be very robust. Small populations allowed good depth discrimination. Two types of energy neurons (phase- and position-type models) that are discussed as possible neural implementations of disparity-selectivity could be compared to each other. Phase-type coding was more robust and could explain a tendency towards zero disparity in degenerated stimuli and, for high-pass stimuli, exhibited the breakdown of disparity discrimination at a maximum disparity value. Contrast-inverted stereograms led to high variances in disparity representation, which is a possible explanation of the absence of depth percepts in large contrast-inverted stimuli. Our study suggests that nonlocal interactions destroy depth percepts in large contrast-inverted stereograms, although these percepts occur for smaller stimuli of the same class. Received: 21 December 2001 / Accepted: 29 April 2002 RID="*" ID="*" Present address: Bayer AG BTS-PT-MVT-MKM, Geb. K9, 51368 Leverkusen, Germany Acknowledgement. This work was supported by a scholarship from the Studienstiftung des deutschen Volkes to J.L. Correspondence to: J. Lippert (e-mail: joerg.lippert.jl@bayer-ag.de)     

Resistance to Phytophthora fragariae var. fragariae in strawberry: the Rpf2 gene

  Phytophthora fragariae var. fragariae is the causal agent of red stele (red core) root rot in strawberry (Fragaria spp.). The inheritance of resistance to one isolate of this fungus was studied in 12 segregating populations of F.×ananassa derived from crosses between four resistant cultivars (‘Climax’, ‘Redgauntlet’, ‘Siletz’, and ‘Sparkle’) and three susceptible cultivars (‘Blakemore’, ‘Glasa’, and ‘Senga’ Sengana’). The analysis clearly supports the hypothesis of a single segregating dominant resistance gene. It is proposed that this gene be designated Rpf2. Received 12 November 1996 / Accepted: 22 November 1996     

Simulation of ‘hitch-hiking’ genealogies

An ancestral influence graph is derived, an analogue of the coalescent and a composite of Griffiths' (1991) two-locus ancestral graph and Krone and Neuhauser's (1997) ancestral selection graph. This generalizes their use of branching-coalescing random graphs so as to incorporate both selection and recombination into gene genealogies. Qualitative understanding of a ‘hitch-hiking’ effect on genealogies is pursued via diagrammatic representation of the genealogical process in a two-locus, two-allele haploid model. Extending the simulation technique of Griffiths and Tavaré (1996), computational estimation of expected times to the most recent common ancestor of samples of n genes under recombination and selection in two-locus, two-allele haploid and diploid models are presented. Such times are conditional on sample configuration. Monte Carlo simulations show that ‘hitch-hiking’ is a subtle effect that alters the conditional expected depth of the genealogy at the linked neutral locus depending on a mutation-selection-recombination balance. Received: 21 July 2000 / Published online: 5 December 2000     

Localization of the rice stripe disease resistance gene, Stv-bi, by graphical genotyping and linkage analyses with molecular markers

 We used graphical genotyping and linkage analyses with molecular markers to determine the chromosomal location of the rice stripe disease resistance gene, Stv-b ⁱ . The stripe resistance gene from the indica rice (Oryza sativa) cv ‘Modan’ was introgressed into several Japanese rice varieties. We found 4 RFLP markers in ‘Modan’, five susceptible parental rice varieties (‘Norin No. 8’, ‘Sachihikari’, ‘Kanto No. 98’, ‘Hokuriku No.103’ and ‘Koganebare’) and four resistant progeny varieties (‘St. No. 1’, ‘Aichi No. 6’, ‘Aoisora’ and ‘Asanohikari’). Graphical genotyping of the resistant progeny revealed a chromosomal segment ascribable to ‘Modan’ and associated with stripe resistance. The chromosomal segment from ‘Modan’ was located at 35.85 cM on chromosome 11. Linkage analysis using 120 F₂ individuals from a cross between ‘Koshihikari’ (susceptible) and ‘Asanohikari’ (resistant) revealed another 8 RFLP markers in the same chromosome. We performed a bioassay for rice stripe resistance in F₃ lines of the F₂ individuals using infective small brown planthoppers and identified an 1.8-cM segment harboring the rice stripe disease resistance gene, Stv-b ⁱ , between XNpb220 and XNpb257/ XNpb254. Furthermore, Stv-b ⁱ was linked by 0.0 cM to a RFLP marker, ST10, which was developed on the basis of the results of RAPD analysis. These DNA markers near the Stv-b ⁱ locus may be useful in marker-assisted selection and map-based cloning of the Stv-b ⁱ gene. Received: 26 September 1997 / Accepted: 4 November 1997     

Parametric-equation representation of biconcave erythrocytes

The representation of the shape of a biconcave erythrocyte by a set of three parametric equations was achieved by using the expressions that transform the curvilinear coordinates from the disc-cyclide coordinate system [denoted J2R; Moon and Spencer (1988), Field Theory Handbook, Springer-Verlag, Berlin] to Cartesian coordinates. The equations are products of elliptic functions, so the challenge was to relate the three major ’shape-defining’ measurements of the human erythrocyte in Cartesian coordinates to three parameters in the new curvilinear coordinates, to give a realistic representation of the shape of the membrane-surface. The relationships between the coefficients of the Cartesian degree-4 surface that describes the discocyte and the coordinate transformation equations were derived with the aid of Mathematica; and the membrane-surface of the cell was drawn using the ParametricPlot3D function in this ‘package’. By having the erythrocyte shape expressed in its new form it is readily amenable to further transformations that might be used to model those changes in shape that are seen when the cells are immersed in media of various osmolalities, or when they change metabolic ’states’. On the other hand, the relationship between the coefficients of the Cartesian expression for the disc-cyclide surface is relevant to image analysis of erythrocytes, as determined by physical methods that rely on Cartesian imaging ’slices’. These methods include confocal microscopy and various nuclear magnetic resonance microimaging procedures.     

‘Chillum’ jar,a better technique for screening of rhizobia under summer conditions

Summary In the present investigations ‘Chillum’ jar assembly was found to provide more favourable environmental conditions for rhizobia to nodulate leguminous plants particularly under summer conditions than the usual Leonard jar assembly. When thirty pigeon pea rhizobia isolates were tested for their nodulation efficiency in both Leonard jars as well as ‘Chillum’ jars, it was noticed that there was no nodulation in any of the isolates under Leonard jars whereas all isolates were nodulating well under ‘Chillum’ jars conditions. This was probably due to lowering of temperature in ‘Chillum’ jar caused by rapid evaporation from the outer surface of ‘Chillum’ jar assembly. The maximum temperature recorded in ‘Chillum’ jar was 34°C whereas in Leonard jars it was 46.5°C.     

Self-sterility in the mutant ‘Zigui shatian’ pummelo (<Emphasis Type="Italic">Citrus grandis</Emphasis> Osbeck) is due to abnormal post-zygotic embryo development and not self-incompatibility

‘Shatian’ pummelo (Citrus grandis Osbeck), one of the main citrus cultivars in China, is self-incompatible, and its pollen tubes are believed to be arrested in style after self-pollination.We have characterized one ‘Shatian’ pummelo mutant, named ‘Zigui shatian’ pummelo. The mutant pummelo had identical DNA ploidy level, morphology (leaf shape, stoma size and density, pollen shape and size) and developmental progress of pistil and male organs to that of the common ‘Shatian’ pummelo. However, unlike the common ‘Shatian’ pummelo, ‘Zigui shatian’ is self-compatible since its pollen tubes can self-pollinate allowing for successful fertilization. Histological analyses of ‘Shatian’ pummelo further verified abnormal post-zygotic development which led to seed abortion. Simple sequence repeats (SSR) analysis revealed polymorphism in 1 of the 120 primers screened showing that ‘Zigui shatian’ and ‘Shatian’ pummelo are different at the DNA level. Taken together, these data suggested mutant ‘Zigui shatian’ pummelo might be derived from ‘Shatian’ pummelo with self-sterility by self-incongruity after self-fertilization.     

Geometry and mechanics of the morphogenesis of active membranes on the example of plant trichome cells of the genus <Emphasis Type="Italic">Draba</Emphasis> L.

The stages of the early morphogenesis of simple (unbranched) and complex (branched) unicellular trichomes are studied in two species of the genus Draba—D. sibirica (Pall.) Thell. and D. daurica DC. The geometry of morphogenesis is estimated by analyzing intraindividual variation of quantitative morphological characteristics of the developing leaf blade and peduncle trichomes. The surface of all types of trichome cells first acquires a spherical shape, followed by a U-shaped configuration with cylindrical proximal and spherical distal regions. In the development of complex trichomes, the area of the distal zone grows at a higher rate, which leads to separation of its volume into individual spherical regions, the morphogenesis of which repeats the early morphogenetic stages of the overall trichome cell, forming simple (unbranched) or complex (branched) trichome rays. As a rule, the lateral polarity of a trichome cell coincides with the proximodistal polarity of the leaf. Quantitative morphological data make it possible to infer an algorithm of the changes in shape common for all trichome cells, namely, the growth cycle comprising alternation of the phases of increase and decrease in the curvature of the outer cell surface. This surface is an active membrane expanded by the internal pressure and concurrently capable of actively increasing its area by incorporation of new structural elements. A distinctive feature of the proposed model is the geometrical inhomogeneity of the surface movement, changing the radius of curvature and creating internal (active) mechanical stresses in this membrane. A decrease in the ratio of the membrane surface area to the volume deprives the spatially homogeneous shape of its stability; correspondingly, the transition from elastic resistance to internal pressure to active resistance with the help of curvature differentiation becomes more energetically favorable. The source for growth and morphogenesis of the active membrane is alternation of the phases of local curvature leveling, which “charges” the membrane with active mechanical stresses and “discharge” of these stresses, leading to differentiation of the membrane’s local curvatures.     

The wiring-in of neural nets revisited

Earlier and some recent ideas about the possible modes of specification of the wiring-in of nervous systems are reviewed in the light of older and several recent experiments, and some new ideas are suggested. It is argued that certain general principles, notably the postulated ‘principle of alternative matching’ (PALMA) and a suggested and related ‘kaleidoscopic effect’ (KALEF), as well as the notion of an ‘extracellular guidance network’ (ECGN), are in good agreement with recent and older findings concerning axonal guidance during neural wiring-in. It seems possible that by means of genetically programmed processes, neurons become systematically combinatorially labelled to such a degree that possibly all neurons areuniquely specified, as regards the combination oftypes of cell labels they make. Yet, there remains considerable freedom as regards the modes of arrangements of cell labels within cell surface membranes and the KALEF permits to overcome apparent difficulties that confronted earlier versions of the cell labelling hypotheses (cf. Edelman,Science 219, 450–457, 1983, for mention of such difficulties). Apart from label specification, neural development seems to depend on trophic factors, which are also essential for the maintenance of the developed nervous system. The systematic programmes for cell labelling, apart from generating all the required neurons, also produces inappropriate neurons and synaptic connections. These are got rid of by systematic cell death and/or atrophy of inappropriate synapses and/or elimination of inappropriate axon collaterals. The resulting neural net seems then very specifically wired-in for each species, apparently without redundant neurons.     

Genetic analysis of resistance to blackspot (Diplocarpon rosae) in tetraploid roses

  Diplocarpon rosae is the causal agent of rose blackspot, one of the most severe diseases of field-grown roses. The genetics of resistance to this pathogen was investigated in crosses between tetraploid rose genotypes. The hybrid breeding line 91/100-5, which exhibits a broad resistance to all isolates tested so far, was selfed to produce an F₂ population, backcrossed to the susceptible tetraploid variety ‘Caramba’ and crossed to the susceptible varieties ‘Heckenzauber’, ‘Pariser Charme’ and ‘Elina’. Infection experiments resulted in segregation ratios consistent with the presence of a single dominant resistance locus in the duplex configuration in the hybrid 91/100-5. This suggests, together with previous data on the race structure of the fungus, a “gene-for-gene” type of interaction in the pathosystem Diplocarpon/Rosa. We propose to designate this gene Rdr1, which is the first resistance gene described in the genus Rosa. The advantages and limitations of such an interaction type for future rose breeding programmes and for marker-assisted selection strategies are discussed. Received: 29 August 1997 / Accepted: 19 September 1997     

Stability of Hor1-specific YAC-clones and physical mapping of Hor1-loci in barley

 The hordeins are the major class of storage proteins in barley and are encoded by multigene families. Two YAC-clones specific for the C-hordein-coding Hor1-locus of barley (Hordeum vulgare L.) were selected. The clones were constructed with DNA from the cultivars ‘Franka’ and ‘Hockey’ and have insert sizes of 330 kb and 350 kb, respectively. Performing partial digestions and hybridizations with vector-specific probes, a restriction analysis was conducted using restriction enzymes with a 8-bp recognition sequence. Both clones cover the complete region of the Hor1-locus, but exhibit a different pattern of restriction sites reflecting the polymorphic nature of the locus on the scale of long-range restriction mapping. The maximal extent of the regions homologous to the Hor1-specific probe, pBSC5, was 105 kb in the ‘Hockey’-derived YAC and 190 kb in the yeast artificial chromosome constructed with ‘Franka’-DNA. Furthermore the high degree of instability observed with the Hor1-specific YAC-clones is discussed in conjunction with the structure of the Hor1-locus. Received: 19 December 1996 / Accepted: 31 January 1997     

Genetic analysis of the dwarfing gene (Rht8) in wheat. Part I. Molecular mapping of Rht8 on the short arm of chromosome 2D of bread wheat (Triticum aestivum L.)

 Two sets of single chromosome recombinant lines comparing 2D chromosomes from the wheat varieties ‘Ciano 67’ and ‘Mara’ with the common 2D chromosome of ‘Cappelle-Desprez’ in a ‘Cappelle-Desprez’ background were used to detect a diagnostic wheat microsatellite marker for the dwarfing gene Rht8. The genetic linkage maps place the wheat microsatellite marker WMS 261 0.6 cM distal to Rht8 on the short arm of chromosome 2D. By PCR analysis the WMS 261 alleles of ‘Mara’, ‘Cappelle-Desprez’ and ‘Ciano 67’ could be distinguished by different fragment sizes of 192 bp, 174 bp and 165 bp, respectively. A screen of over 100 international varieties of wheat showed that the three allelic variants were all widespread. It also demonstrated that a limited number of varieties carried novel WMS 261 variants of over 200 bp. Following classification of the individual recombinant lines for allelic variants at the WMS 261 locus it was possible to attribute a 7- to 8-cm reduction in plant height with the WMS 261-192-bp allele compared to the WMS 261-174-bp allele in the set of recombinant lines comparing 2D chromosomes of ‘Mara’ and ‘Cappelle-Desprez’. A height reduction of around 3 cm was detected between the WMS 261-174-bp allele and the WMS 261-165-bp allele in the recombinant lines comparing 2D chromosomes of ‘Cappelle-Desprez’ and ‘Ciano 67’. Received: 17 October 1997 / Accepted: 12 November 1997     

Total structure and biological properties of laxaphycins A and B,cyclic lipopeptides from the marine cyanobacterium Lyngbya majuscula

Summary The tropical marine cyanobacteriumLyngbya majuscula produces a series of cytotoxic and antimicrobial cyclic peptides. The total structure of the two major components, laxaphycins A and B, was determined by interpretation of physical data, principally high field NMR, FAB MS and MS/MS, in combination with chemical derivatization and degradation schemes. Absolute stereochemistries of the natural and ‘exotic’ amino acids were determined. The two cyclic peptides exhibited an unusual biological synergism when tested for antifungal or cytotoxic effects.     

Strict paternal inheritance of chloroplast DNA and maternal inheritance of mitochondrial DNA in intraspecific crosses of kiwifruit

 Previous studies have established that chloroplasts are inherited paternally in Actinidia interspecific crosses. However, fertilisation problems in interspecific crosses may affect the transmission of organelles. Six female clones, i.e. ‘Abbott’, ‘Bruno’, ‘Greensill’, ‘Hayward’, ‘Jones’, ‘Monty’, and four male clones were used to identify cpDNA polymorphisms within the cultivated kiwifruit species A. deliciosa. The restriction patterns by HpaII of a chloroplast fragment amplified by PCR with a pair of universal primers revealed a polymorphism at the intraspecific level. The inheritance of cpDNA in 143 seedlings from three intraspecific crosses in kiwifruit (Actinidia deliciosa) was studied. All offspring displayed the restriction pattern of the paternal parent, indicating that maternal inheritance of cpDNA in kiwifruit is rare at best. Strict maternal inheritance of mtDNA was confirmed in the same crosses used to investigate cpDNA transmission. Studies of cytoplasmic inheritance in the Actinidia genus represent to date the best documented report of differential organelle inheritance of cpDNA and mtDNA in angiosperms. Received: 10 November 1998 / Accepted: 14 December 1998     

Use of the multi-allelic self-incompatibility gene in apple to assess homozygocity in shoots obtained through haploid induction

 To obtain homozygous genotypes of apple, we have induced haploid development of either the female or the male gametes by parthenogenesis in situ and anther culture, respectively. Of the shoots obtained, which were mainly of a non-haploid nature, some could be derived from fertilised egg cells or from sporophytic anther tissue. In order to select the shoots having a true haploid origin, and thus homozygotes, we decided to use the single multi-allelic self-incompatibility gene as a molecular marker to discriminate homozygous from heterozygous individuals. The rationale behind this approach was that diploid apple cultivars contain 2 different alleles of the S-gene and therefore the haploid induced shoots obtained from them should have only one of the alleles of the single parent. The parental cultivars used were ‘Idared’ (parthenogenesis in situ) and ‘Braeburn’ (androgenesis), and their S-genotypes were known, except for 1 of the ‘Braeburn’S-alleles. To stimulate parthenogenetic development ‘Idared’ styles were pollinated with irradiated ‘Baskatong’ pollen, the S-alleles of the latter (2n) cultivar were also unknown. The cloning and sequence analysis of these 3 unidentified S-alleles, 1 from ‘Braeburn’ and 2 from ‘Baskatong’ is described, and we show that they correspond to the S ₂₄ -, S ₂₆ - and S ₂₇ -alleles. We have optimised a method for analysis of the S-alleles of ‘Idared/Baskatong’- or ‘Braeburn’-derived in vitro plant tissues and have shown that this approach can be applied for the screening of the in vitro shoots for their haploid origin. Received: 18 August 1997 / Accepted: 10 September 1997     

Expression of an engineered cysteine proteinase inhibitor (Oryzacystatin-IΔD86) for nematode resistance in transgenic rice plants

 We have used a genotype-independent transformation system involving particle gun bombardment of immature embryos to genetically engineer rice as part of a programme to develop resistance to nematodes. Efficient tissue culture, regeneration, DNA delivery and selection methodologies have been established for elite African varieties (‘ITA212’, ‘IDSA6’, ‘LAC23’, ‘WAB56-104’). Twenty-five transformed clones containing genes coding for an engineered cysteine proteinase inhibitor (oryzacystatin-IΔD86, OC-IΔD86), hygromycin resistance (aphIV) and β-glucuronidase (gusA) were recovered from the four varieties. Transformed plants were regenerated from all clones and analysed by PCR, Southern and western blot. Detectable levels of OC-IΔD86 (up to 0.2% total soluble protein) in plant roots were measured in 12 out of 25 transformed rice lines. This level of expression resulted in a significant 55% reduction in egg production by Meloidogyne incognita. Received: 4 August 1997 / Accepted: 22 August 1997     

AFLP analysis of nephthytis (Syngonium podophyllum Schott) selected from somaclonal variants

This study analyzed genetic differences of 19 cultivars selected from somaclonal variants of Syngonium podophyllum Schott along with their parents as well as seven additional Syngonium species and six other aroids using amplified fragment length polymorphism (AFLP) markers generated by 12 primer sets. Among the 19 somaclonal cultivars, ‘Pink Allusion’ was selected from ‘White Butterfly’. Tissue culture of ‘Pink Allusion’ through organogenesis resulted in the development of 13 additional cultivars. Self-pollination of ‘Pink Allusion’ obtained a cultivar, ‘Regina Red Allusion’, and tissue culture propagation of ‘Regina Red Allusion’ led to the release of five other cultivars. The 12 primer sets generated a total of 1,583 scorable fragments from all accessions, of which 1,284 were polymorphic (81.9%). The percentages of polymorphic fragments within ‘White Butterfly’ and ‘Regina Red Allusion’ groups, however, were only 1.2% and 0.4%, respectively. Jaccard's similarity coefficients among somaclonal cultivars derived from ‘White Butterfly’ and ‘Regina Red Allusion’, on average, were 0.98 and 0.99, respectively. Seven out of the 15 cultivars from the ‘White Butterfly’ group and three out of six from the ‘Regina Red Allusion’ group were clearly distinguished by AFLP analysis as unique fragments were associated with respective cultivars. The unsuccessful attempt to distinguish the remaining eight cultivars from the ‘White Butterfly’ group and three from the ‘Regina Red Allusion’ group was not attributed to experimental errors or the number of primer sets used; rather it is hypothesized to be caused by DNA methylation and/or some rare mutations. This study also calls for increased genetic diversity of cultivated Syngonium as they are largely derived from somaclonal variants.