Root Cone Research Articles

Generalized permutohedra in the kinematic space

In this note, we study the permutohedral geometry of the singularities of a certain differential form introduced in recent work of Arkani-Hamed, Bai, He and Yan. There it was observed that the poles of the form determine a family of polyhedra which have the same face lattice as that of the permutohedron. We realize that family explicitly, proving that it in fact fills out the configuration space of a particularly well-behaved family of generalized permutohedra, the zonotopal generalized permutohedra, that are obtained as the Minkowski sums of line segments parallel to the root directions ei − ej.Finally we interpret Mizera’s formula for the biadjoint scalar amplitude m(\U0001d540n, \U0001d540n), restricted to a certain dimension n − 2 subspace of the kinematic space, as a sum over the boundary components of the standard root cone, which is the conical hull of the roots e1 − e2, … , en−2 − en−1.

Narrow root cone angle promotes deeper rooting, cooler canopy temperatures and higher grain yield in a rice (Oryza sativa L.) recombinant inbred line population grown under different water availabilities in aerobic production systems

Rice crops are one of the world’s major consumers of irrigation water. With water becoming more limited, water-saving aerobic production (AP) systems may provide an alternative to traditional flooded culture. Genotypes with narrow root cone angle (RCA) and subsequently deeper rooting systems are hypothesised as key for AP adaptation ensuring water uptake at depth. Using a recombinant inbred line (RIL) population and differential genotypes (narrow and wide RCA groups) derived from a Sherpa/IRAT109 cross, a glasshouse experiment and four field experiments with different water inputs in well-watered (WW, in two seasons), mild water deficit (MWD), and intermittent water deficit (IWD) conditions were carried out to characterise the relationship of RCA with grain yield (GY) and key physiological traits. Significant genotypic variation and group differences were found in root traits, canopy temperature and GY. Expression of genotypes with narrow RCA were found to be consistent across experiments and water availability conditions. Genotypes in the narrow RCA group produced a greater proportion of roots below 20 cm (7.4–17.7%), longer total root length (69.7%), cooler canopies (0.3–1.0 °C) and higher GY (14–62%) compared with wide RCA group. Two promising RILs were identified which expressed high GY in WW (12.0–13.4 t/ha) conditions while maintaining GY in IWD (up to 9.4 t/ha). Higher GY was obtained in genotypes with cooler canopies (r = −0.72) and narrow RCA (r = −0.39) in IWD. Narrower RCA was associated with higher total root length (r = −0.70) and higher stomatal conductance (r = −0.71) in the glasshouse and proportion of deep roots (r = −0.65) and cooler canopies (r = 0.66) in the field. To our knowledge, this was one of the foremost studies exploring and demonstrating such relationships in rice AP systems. Root traits determined in the field were congruent with the other root-related traits driving differences between RCA groups measured in the glasshouse providing confidence with the field phenotyping methods. This work demonstrated the advantage conferred by the development of narrow RCA in aerobic conditions, presented improved physiological understanding of RCA and with genomics assisted breeding, is considered valuable for the development of AP-adapted, sustainably produced rice.

QTL Validation and Development of SNP-Based High Throughput Molecular Markers Targeting a Genomic Region Conferring Narrow Root Cone Angle in Aerobic Rice Production Systems.

Aerobic rice production (AP) provides potential solutions to the global water crisis by consuming less water than traditional permanent water culture. Narrow root cone angle (RCA), development of deeper rooting and associated genomic regions are key for AP adaptation. However, their usefulness depends on validation across genetic backgrounds and development of linked markers. Using three F2 populations derived from IRAT109, qRCA4 was shown to be effective in multiple backgrounds, explaining 9.3–17.3% of the genotypic variation and introgression of the favourable allele resulted in 11.7–15.1° narrower RCA. Novel kompetitive allele specific PCR (KASP) markers were developed targeting narrow RCA and revealed robust quality metrics. Candidate genes related with plant response to abiotic stress and root development were identified along with 178 potential donors across rice subpopulations. This study validated qRCA4’s effect in multiple genetic backgrounds further strengthening its value in rice improvement for AP adaptation. Furthermore, the development of novel KASP markers ensured the opportunity for its seamless introgression across pertinent breeding programs. This work provides the tools and opportunity to accelerate development of genotypes with narrow RCA through marker assisted selection in breeding programs targeting AP, which may ultimately contribute to more sustainable rice production where water availability is limited.

Stable and Novel Quantitative Trait Loci (QTL) Confer Narrow Root Cone Angle in an Aerobic Rice (Oryza sativa L.) Production System

BackgroundAerobic rice production (AP) may be a solution to the looming water crisis by utilising less water compared to traditional flooded culture. As such, development of genotypes with narrow root cone angle (RCA) is considered a key AP adaptation trait as it could lead to deeper rooting and ensure water uptake at depth. Quantitative trait loci (QTL) and genes associated with rooting angle have been identified in rice, but usually in conventional transplanted systems or in upland and drought conditions. This study aimed to identify QTL associated with RCA in AP systems using a recombinant inbred line population derived from IRAT109.ResultsFour experiments conducted in glasshouse and aerobic field conditions revealed significant genotypic variation existed for RCA in the population. Single and multiple QTL models identified the presence of eight QTL distributed in chromosomes 1, 2, 3, 4, and 11. Combined, these QTL explained 36.7–51.2% of the genotypic variance in RCA present in the population. Two QTL, qRCA1.1 and qRCA1.3, were novel and may be new targets for improvement of RCA. Genotypes with higher number of favourable QTL alleles tended to have narrower RCA. qRCA4 was shown to be a major and stable QTL explaining up to 24.3% of the genotypic variation, and the presence of the target allele resulted in as much as 8.6° narrower RCA. Several genes related to abiotic stress stimulus response were found in the qRCA4 region.ConclusionStable and novel genomic regions associated with RCA have been identified. Genotypes which had combinations of these QTL, resulted in a narrower RCA phenotype. Allele mining, gene cloning, and physiological dissection should aid in understanding the molecular function and mechanisms underlying RCA and these QTL. Ultimately, our work provides an opportunity for breeding programs to develop genotypes with narrow RCA and deep roots for improved adaptation in an AP system for sustainable rice production.

Faces of Root Polytopes

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 2 October 2020Accepted: 07 June 2021Published online: 13 September 2021Keywordsroot polytope, root cone, face structureAMS Subject Headings05E99, 52B05Publication DataISSN (print): 0895-4801ISSN (online): 1095-7146Publisher: Society for Industrial and Applied MathematicsCODEN: sjdmec

Genome-wide association mapping for root cone angle in rice

BackgroundPlant root systems play a major role in anchoring and in water and nutrient uptake from the soil. The root cone angle is an important parameter of the root system architecture because, combined with root depth, it helps to determine the volume of soil explored by the plant. Two genes, DRO1 and SOR1, and several QTLs for root cone angle have been discovered in the last 5 years.ResultsTo find other QTLs linked to root cone angle, a genome-wide association mapping study was conducted on two panels of 162 indica and 169 japonica rice accessions genotyped with two sets of SNP markers (genotyping-by-sequencing set with approximately 16,000 markers and high-density-rice-array set with approximately 300,000 markers). The root cone angle of all accessions was measured using a screen protractor on images taken after 1 month of plant growth in the Rhizoscope phenotyping system. The distribution of the root cone angle in the indica panel was Gaussian, but several accessions of the japonica panel (all the bulus from Indonesia and three temperate japonicas from Nepal or India) appeared as outliers with a very wide root cone angle. The data were submitted to association mapping using a mixed model with control of structure and kinship. A total of 15 QTLs for the indica panel and 40 QTLs for the japonica panel were detected. Genes underlying these QTLs (+/−50 kb from the significant markers) were analyzed. We focused our analysis on auxin-related genes, kinases, and genes involved in root developmental processes and identified 8 particularly interesting genes.ConclusionsThe present study identifies new sources of wide root cone angle in rice, proposes ways to bypass some drawbacks of association mapping to further understand the genetics of the trait and identifies candidate genes deserving further investigation.

Accurate modeling of logarithmic spiral bevel gear based on the tooth flank formation and Boolean addition operation

The logarithmic spiral bevel gear is a new type of spiral bevel gear and has received great attention in the industry for its excellent engineering characteristics. The objective of this study is to develop an accurate geometric model for the logarithmic spiral bevel gear. Based on the gear tooth flank formation mechanism, a conical logarithmic spiral curve with a constant spiral angle was constructed as the tooth trace curve. The profiles for the exterior and interior transverse of the tooth were built with an accurate involutes curve, with transition being circular arcs and straight lines. The first tooth model was established by sweeping the tooth profile accurately along the tooth trace curve, and the rest of the teeth were created by an array operation. The accurate three-dimensional model of logarithmic spiral bevel gear was finally obtained by means of a Boolean addition operation among all of the gear teeth and the root cone. An experimental study was carried out for such a gear whose number of teeth was 37, with modules being 4.5 mm, normal pressure angle being 20° and spiral angle being 35°. A DMU 40 monoBLOCK five-axis computer numerical control milling machine tool was used to produce the prototype, and a Zeiss CONTURA G3 three-dimensional coordinate instrument was employed to measure the exterior transverse addendum diameter. The linear error between the theoretical model value and the measured average value was 0.0027 mm, indicating the effectiveness and practicability of this modeling method, which provides an accurate geometric model for design and for subsequent tasks like computer-aided engineering analysis and manufacturing.

On the roots of generalized Wills $\mu$-polynomials

We investigate the roots of a family of geometric polynomials of convex bodies associated to a given measure \mu on the non-negative real line \mathbb R_{\geq 0} , which arise from the so called Wills functional. We study its structure, showing that the set of roots in the upper half-plane is a closed convex cone, containing the non-positive real axis \mathbb R_{\leq0} , and strictly increasing in the dimension, for any measure \mu . Moreover, it is proved that the 'smallest' cone of roots of these \mu -polynomials is the one given by the Steiner polynomial, which provides, for example, additional information about the roots of \mu -polynomials when the dimension is large enough. It will also give necessary geometric conditions for a sequence \{m_i\colon i=0,1,\dots\} to be the moments of a certain measure on \mathbb R_{\geq0} , a question regarding the so called (Stieltjes) moment problem.

Odonata larvae associated with the root system of eichhornia crassipes (pontederiaceae), in the chalons dam, Santiago de Cuba, Cuba

Were studied the Odonata’s larvae associated with the root system of Eichhornia crassipes (Mart. Solms) in two localities of the “Chalons”, dam overflow channel locate in Santiago de Cuba, from March 2005 to March 2006. For the Odonata’s larvae in the root system was determining the taxonomic composition and the trophic structure. A Spatial-temporal species group was estimated by the density of individuals and taxa, calculating the total volume by the method of root cone volume. In the study were collected 19 species belonging to four families. The temporal variation was correlated with the rainfall and the number of rainy days, according to Spearman Correlation. On the period of homogeneity of climatic variables the taxa’s and individuals’ density remained stable; however, the oscillations in the both densities were associated with the heterogeneity in the precipitation patterns. The densities of individual were different in both the sampled localities. The relationship between climatic variables, root volume and density of individuals suggests that the number of rainy days and accumulated rainfall influence in the abundance of certain taxa and that the roots of Eichhornia crassipes can be appraised as an important habitat for dragonflies larvae and a great place to find food for them.

Spiral Bevel Gears with Optimised Tooth-end Geometry

Industrially-manufactured, spiral bevel gears with complementary tooth-end surfaces as a rule have reduced tooth thickness and subsequently reduced strength at the tooth ends. The weakened tooth ends of octoid gearing are generally produced by machining, inevitably resulting in a certain lengthwise dimensional discrepancy and slight profile difference. This in turn very often leads to systematic breakage of teeth. Serious, systematic damage occurs irrespective of the transmission tightness of the outer or inner tooth ends.Gear manufacturers currently use several solutions to avoid such breakages, including artificial inclination near to the teeth or deliberately reducing the back or outer cone angle and increasing the front or root cone angle.This paper outlines an improved design of gear blanks, with optimised tooth ends. Our research concluded that it is possible to optimise rectangular-generated, spiral bevel pinion/gear pairs with constant tooth height and a common pitch cone apex. We can successfully achieve this through recalculation of the gear blanks, without any changes in the flank geometry or tooth-cutting process. Thus the gear pair with optimised tooth ends can be cut without interference to the customary tooth-cutting process. To prove the concept, an example of the recalculation is provided. These improvements result in increased tooth strength, simplification of the gear blank geometry and more suitable geometry for modern machining, as well as a smaller outer diameter of the gear.

STEINER POLYNOMIALS VIA ULTRA-LOGCONCAVE SEQUENCES

We investigate structural properties of the cone of roots of relative Steiner polynomials of convex bodies. We prove that they are closed, monotonous with respect to the dimension, and that they cover the whole upper half-plane, except the positive real axis, when the dimension tends to infinity. In particular, it turns out that relative Steiner polynomials are stable polynomials if and only if the dimension is ≤ 9. Moreover, pairs of convex bodies whose relative Steiner polynomial has a complex root on the boundary of such a cone have to satisfy some Aleksandrov–Fenchel inequality with equality. An essential tool for the proofs of the results is the characterization of Steiner polynomials via ultra-logconcave sequences.

Research on the 3D Modeling Method of Epicycloids’ Bevel Gear Based on Pro/E

In order to obtain 3D solid model of an Oerlikon system’s prolate epicycloidal bevel gear by full generating processing, the solid modeling method for the gear was studied in Pro/E. First, Drawing benchmark circles (base on the circle and reference circle as well as tip circle and root circle)and involute tooth profile of virtual equivalent gears of both large end and small end of the real gear, and then an intercepted segment of prolate epicycloid which has the same trend to the tooth and is controlled by prolate epicycloidal equations is projected to the surface of the gear’s root cone for the sake of getting tooth curve along axial direction. In the end, along the tooth curve and on the root cone conducting a variable cross section solid scanning to tooth shaped cross section, a complete gear tooth will be gained and at last model establishing will be finished. This method provided digital basis for optimization design, reverse design and rapid prototyping of this kind of gear, and so its design efficiency is improved.

Allometry and stilt root structure of the neotropical palmEuterpe precatoria(Arecaceae) across sites and successional stages

The lack of secondary meristems shapes allometric relationships of the palms, influencing species segregation according to their capacity to adjust form and function to spatial and temporal changes in environmental conditions. We examined the stem height vs. diameter allometry of Euterpe precatoria across environmental gradients and measured how terrain inclination and palm size affected stilt root structure at two sites in Costa Rica. We dissected the root cone into eight variables and used principal component analysis to summarize their correlation structure. The fit of the stem diameter-height relationship to the stress, elastic, and geometric similarity models was examined using data from 438 palms. Terrain inclination did not affect stilt roots, whose structure was determined by palm size. Palms under 1 m showed geometric similarity, whereas palms above 1 m had slope values that were one and a half times higher, independent of successional stage, and did not adjust to any mechanical model. Taller palms departed from these models when they were large because they had stilt root support. We conclude that height in E. precatoria is constrained by structural support at the base and that diameter at the base of the stem and stilt roots balances height increments over all the size ranges examined.

The function of stilt roots in the growth strategy of Socratea exorrhiza (Arecaceae) at two neotropical sites

Arboreal palms have developed a variety of structural root modifications and systems to adapt to the harsh abiotic conditions of tropical rain forests. Stilt roots have been proposed to serve a number of functions including the facilitation of rapid vertical growth to the canopy and enhanced mechanical stability. To examine whether stilt roots provide these functions, we compared stilt root characteristics of the neotropical palm tree Socratea exorrhiza on sloped (>20 degrees) and flat locations at two lowland neotropical sites. S. exorrhiza (n=80 trees) did not demonstrate differences in number of roots, vertical stilt root height, root cone circumference, root cone volume, or location of roots as related to slope. However, we found positive relationships between allocation to vertical growth and stilt root architecture including root cone circumference, number of roots, and root cone volume. Accordingly, stilt roots may allow S. exorrhiza to increase height and maintain mechanical stability without having to concurrently invest in increased stem diameter and underground root structure. This strategy likely increases the species ability to rapidly exploit light gaps as compared to non-stilt root palms and may also enhance survival as mature trees approach the theoretical limits of their mechanical stability.

Stilt Root Structure in the Neotropical Palms Iriartea deltoidea and Socratea exorrhiza1

ABSTRACTMany arborescent palms develop a stilt root cone that provides increased mechanical support on steep hillsides, better root aeration under waterlogged conditions, resprouting after mechanical damage, and rapid stem elongation. However, for most species the role of stilt roots is not well understood. We determined how palm size and slope inclination affected the structure of stilt roots in the neotropical palms Iriartea deltoidea and Socratea exorrhiza. We expected palm height to be lower on steep slopes because the effectiveness of root support could decrease as slope inclination increases, and thus, the structure of the root cone would vary mostly with slope. Alternatively, if stilt root development is determined by palm size, their production should match stem height, with taller palms having larger cones. We measured the stilt root cone of 31 Iriartea and 36 Socratea palms in San Ramón Biological Reserve and Golfito Wildlife Refuge, Costa Rica. We divided the cone into five variables (horizontal projections of the cone base and stilt root height up and down the slope, and width of the cone base), from which we extracted the first two principal components and used them to measure the effects of slope and palm size on stilt root structure. We found stilt roots to be determined by palm size, not by slope conditions. Stilt roots matched palm size, with larger palms having taller and larger cones, maintaining stability under diverse slope conditions.

Three-Dimensional Measurement for Straight Bevel Gear.

A method for inspecting the machine settings for straight bevel gear cutting is proposed. The bevel gear is generated by a newly introduced "quasi-complementary crown gear" instead of a conventional complementary crown gear. A tooth bearing of a pair of generated gears is theoretically desirable. The coordinates of many points on a tooth surface are measured at random using a coordinate measuring machine. The measured data should be including various type of information derived from the machine settings. Taking this fact into consideration, each information on the machine settings is extracted from the data using the method of least squares. The extracted items can be corrected, and they are: tool pressure angle, tooth angle, root cone angle, apex to back and tooth thickness. Using the proposed method, it is possible to produce interchangeable bevel gears.

The Anchorage Mechanics of Maize,Zea mays

The anchorage system of mature maize Zea mays was investigated by combining morphological and anatomical study of the root system with mechanical tests on roots and with studies in which plants were pulled over. The root system is dominated by 20–30 adventitious roots which emerge in rings from the stem base pointing radially downwards and outwards, approximately 30° from the vertical. Roots are strengthened near their base by a heavily lignified exodermis which makes them rigid in bending; distally, strength and rigidity both decrease because roots become thinner and less lignified. When plants were pulled over, a maximum anchorage moment of 5–20 Nm was mobilized at angles of 8–10°, larger plants having stronger anchorage. Movement was initially centred on the leeward side of the stem, anchorage being due to the resistance of both windward and leeward roots to axial motion through the soil and to bending. At displacements over 10°, however, leeward roots buckled under combined bending and compression and the centre of rotation shifted to the windward perimeter of the root system; subsequent movement of the cone of roots and soil was resisted only by the bearing strength of the soil beneath it. The differences between anchorage failure in balsam and sunflowers and that in maize probably results from the lower angular spread and the weakness in compression of the maize roots which prevents the leeward side of the root system from bearing large downward loads. The system behaves more like that of wheat; these results suggest that the lodging resistance of both plants may be improved by increasing the bending strength and angle of spread of the adventitious roots.

Variation in the developing root cone angle of the permanent mandibular teeth of modern man and certain fossil hominids

The radiographic appearance of the developing root apex of 461 mandibular teeth was studied using ellipsopantomographs of 77 individuals. The angle formed by the mesial and distal aspects of the developing root cone was measured for each quarter of root development in the permanent mandibular teeth of each subject. Large values for this angle in the more coronal part of the root gradually decreased towards the root apex where values were small in all teeth studied. This gradual decrease in the angle of the developing root cone is concordant with the faster increase in root length but slower rates of dentinogenesis reported for the apical region of human teeth in the literature, as the decrease is likely to reflect the increasing numbers of odontoblasts that become involved in root formation as it proceeds apically. The developing root cone angles of one of the five fossil hominid specimens studied (KMN-ER 820) fall outside the range for the developing root cone angle of the developing premolar and permanent second molar roots in modern Homo sapiens and is evidence of a faster rate of root elongation than is typical for modern man.

Stilt Roots and Growth of Arboreal Palms

Patterns in the of palms support the hypothesis that stilt roots enable arboreal palms to produce an axis early in their development. The height of stilt roots is positively related to axis diameter for Socratea durissima and Iriartea gigantea. Both species achieve greater heights per unit diameter earlier in development than the non-stilt root species Welfia georgii; however, at maturity all three palms exhibit similar relationships of height and diameter. Stilt root production and axis elongation are more rapid for S. durissima than for I. gigantea. Among immature palms, W. georgii has a greater percent of biomass underground than the stilt root species. Comparing the stilt root species, S. durissima produces a lower percent of leaves but a higher percent of stilt roots than I. gigantea. THE GROWTH OF PALM STEMS iS characterized by a lack of secondary growth, a result of the absence of a vascular cambium. Growth is confined primarily to the apical meristem. The adult diameter is produced by a stepwise process, termed establishment growth by Tomlinson and Zimmermann (1966), whereby successive nodes are progressively wider through gradual ontogenetic change. Therefore, the developing axis, devoid of secondary vascular tissue, assumes an obconical shape that is necessarily mechanically unstable. In most arboreal palms (e.g., Welfia and Washingtonia) this instability is obviated by initial development of the axis below ground (Halle et al. 1978). Consequently, arboreal palms spend years locked in the understory as a rosette while developing an axis of sufficient diameter to produce a stem capable of arboreal support (Putz 1983). Our intention is to demonstrate that one group of arboreal Arecaceae, the stilt root or iriarteoid palms (Moore 1973), exhibit a fundamentally different pattern. The obconical axis begins elongation very early in life, often immediately after germination. As a result, the diameter of the axis increases along the height of the stem. Instability would result were it not for the development of adventitious roots that form a cone of stilt roots. Thus, some arboreal palms may achieve more rapid vertical early in development. In order to verify these two different developmental patterns, stilt root versus rosette, we measured parameters on palms of different sizes.