Universal Lax Pair for Generalised Calogero–Moser Models

ABSTRACTThe rice root system consists of three types of roots; main root, L-type and S-type lateral root (LR). These component roots are morphologically and histologically different, which is termed as heterorhizy. Root system hydraulic architecture is related to the unique features of component roots. We hypothesized that each component root contributes in different degrees to water uptake of the whole root system. Rice varieties IRAT 109 and Taichung 65 were grown in pots filled with soil under continuous waterlogged (CWL) and drought (CD) conditions until two weeks after heading. Morphology and histological structures of roots, which may regulate radial water movement, were compared among the three component roots. Moreover, hydraulic conductivity (Lpr) of the root system, which represents the water uptake ability, were measured with a pressure chamber. Based on a model that Lpr of the whole root system is a product of Lpr of each of the component roots and their surface areas, we found that the differences in Lpr between the two varieties and the plants grown under different soil water conditions for any of the component roots did not support the corresponding differences in the measured Lpr of the whole root system. In contrast, a significant and positive correlation was found between Lpr of the whole root system and the percentage of surface area of S-type LR but not for the other component roots. These results indicate S-type LR might have a higher contribution to Lpr of the whole root system than the other component roots.

Parabolic subgroup orbits on finite root systems

Well‐adapted root systems allow plants to grow under resource‐limiting environmental conditions and are important determinants of yield in agricultural systems. Important staple crops such as rice and maize belong to the family of grasses, which develop a complex root system that consists of an embryonic root system that emerges from the seed, and a postembryonic nodal root system that emerges from basal regions of the shoot after germination. While early seedling establishment is dependent on the embryonic root system, the nodal root system, and its associated branches, gains in importance as the plant matures and will ultimately constitute the bulk of below‐ground growth. In this review, we aim to give an overview of the different root types that develop in cereal grass root systems, explore the different physiological roles they play by defining their anatomical features, and outline the genetic networks that control their development. Through this deconstructed view of grass root system function, we provide a parts‐list of elements that function together in an integrated root system to promote survival and crop productivity.

Root systems of woody plants generally display a strong relationship between the cross-sectional area or cross-sectional diameter (CSD) of a root and the dry weight of biomass (DWd) or root volume (Vd) that has grown (i.e., is descendent) from a point. Specification of this relationship allows one to quantify root architectural patterns and estimate the amount of material lost when root systems are extracted from the soil. However, specifications of this relationship generally do not account for the fact that root systems are comprised of multiple types of roots. We assessed whether the relationship between CSD and Vd varies as a function of root type. Additionally, we sought to identify a more accurate and time-efficient method for estimating missing root volume than is currently available. We used a database that described the 3D root architecture of Pinus pinaster root systems (5, 12, or 19 years) from a stand in southwest France. We determined the relationship between CSD and Vd for 10,000 root segments from intact root branches. Models were specified that did and did not account for root type. The relationships were then applied to the diameters of 11,000 broken root ends to estimate the volume of missing roots. CSD was nearly linearly related to the square root of Vd, but the slope of the curve varied greatly as a function of root type. Sinkers and deep roots tapered rapidly, as they were limited by available soil depth. Distal shallow roots tapered gradually, as they were less limited spatially. We estimated that younger trees lost an average of 17% of root volume when excavated, while older trees lost 4%. Missing volumes were smallest in the central parts of root systems and largest in distal shallow roots. The slopes of the curves for each root type are synthetic parameters that account for differentiation due to genetics, soil properties, or mechanical stimuli. Accounting for this differentiation is critical to estimating root loss accurately.

A root system is any collection of vectors that has properties that satisfy the roots of a semi simple Lie algebra. If g is semi simple, then the root system A, (Q) can be described as a system of vectors in a Euclidean vector space that possesses some remarkable symmetries and completely defines the Lie algebra of g. The purpose of this paper is to show the essentiality of the root system on the Lie algebra. In addition, the paper will mention the connection between the root system and Ways chambers. In addition, we will show Dynkin diagrams, which are an integral part of the root system.

The patterns of ring width throughout the root systems and stems of several 32- to 36-year-old, plantation-grown red pine were analyzed. All root systems showed a basic similarity in their radial sequences of ring width. Within a root system, individual sequences ranged from those similar to the radial sequence at the stem base to ones apparently dissimilar. At the stem–root base, the increasing distance to the tree crown with time and changes in mechanical stresses with increasing tree size appeared to play major roles in the development of the basic pattern. The position and type of root, the environment of its apices, and changes in the transportation pathways in the xylem and phloem at branching points may all have influenced the pattern in individual roots. The pattern of deposition of xylem at branching points was visualized as an interaction between the stream of growth regulators and assimilates towards the root tips and the stream of xylem water (and (or) dissolved substances) towards the root base. Discontinuities in the growth layers did not occur in the stem but increased with time in the roots, particularly in the surface horizontal roots. Moisture availability modified the radial sequence patterns and probably influenced the shifts in longitudinal distribution of radial increment from year to year within and between the stem and the root system. The maximum current annual increment in root wood volume occurred before that in the stem.

Let R be an irreducible root system. A Lie Algebra L is called graded by R if L is graded with grading group the root lattice of R such that the only nonzero homogeneous subspaces of L have degree 0 or a root in R , the grading is induced by the adjoint action of a split Cartan subalgebra of a finite-dimensional simple Lie subalgebra of L with root system R , and L is generated by the homogeneous subspaces of nonzero degree. This class of Lie algebras was introduced and studied by S. Berman and R. Moody in Invent. Math. 108 (1992), where, in particular, a classification up to central equivalence is given in the simply-laced case. The doubly-laced cases have recently been classified by G. Benkart and E. Zelmanov. Let R be a 3-graded root system, i.e., R is not of type E_8, F_4 or G_2. In this paper, Lie algebras graded by R are described in a unified way, without case-by-case considerations. Namely, it is shown that a Lie algebra L is 3-graded if and only if L is a central extension of the Tits-Kantor-Koecher algebra of a Jordan pair covered by a grid whose associated 3-graded root system is isomorphic to R . This result is then used to classify Lie algebras graded by R : we give the classification of Jordan pairs covered by a grid and describe their Tits-Kantor-Koecher algebras. One of the advantages of this approach is that it works over rings containing 1/2 and 1/3, and also for infinite root systems. Another application is the description of Slodowy's intersection matrix algebras arising from multiply-affinized Cartan matrices.

In numerous studies dealing with roots of woody plants, a description of the root system architecture is needed. During the twentieth century, several manual measurement methods were used, depending on the objectives of study. Due to the difficulties in accessing the roots and the duration of measurements, the studies generally involved a low number of root systems, were often qualitative and focused only on one specific application. Quantitative methods in plant architecture were largely developed in the last 40 years for aerial architecture. However, root systems have particular features and often need specific procedures. Since the end of the 1990s, new devices and techniques have been available for coarse root architecture measurements including volume location techniques (non-invasive or destructive) and manual or semi-automatic 3D digitising. Full 3D root system architecture dynamics was also reconstructed from partial measurements using modelling procedures. On the one hand, non-invasive and automatic techniques need more development to obtain full 3D architecture, i.e. geometry and topology. On the other hand, both one inexpensive manual and one semi-automatic digitizing procedure are now available to measure precisely and rapidly the full 3D architecture of uprooted and excavated coarse root systems. Specific software and a large number of functions are also available for an in-depth analysis of root architecture and have already been used in a dozen of research papers including a fairly large sample of mature trees. A comprehensive analysis of root architecture can be achieved by classifying individual roots in several root types through architectural analysis. The objective of this paper is both to give a detailed overview of the state of the art techniques for 3D root system architecture measurement and analysis and to give examples of applications in this field. Practical details are also given so that this paper can be used as a sort of manual for people who want to improve their practice or to enter this quite new research field.

Root architecture can be very important in plant productivity. The importance of studies on root morphology and development is discussed to improve seedling growth. Root systems of dicotyledonous species are reviewed, with emphasis on differences between growth of basal and lateral roots. The presence of different types of roots in plant species suggests possible differences in function as well. The architecture of a root system related to its functions is considered. Classical methods for studying root systems comprise excavation of root system, direct observation, and indirect analyses. While the first method is destructive and the third is effective in understanding root architecture only on a relatively gross scale, observation methods allow the scientist a complete a nondestructive architectural study of a root system. The three groups are reviewed related to their potential to give valuable information related to the root architecture and development of the seedling, with emphasis on the availability of a medium-transparent plant-growing system, enabling nondestructive daily observations and plant measurements under controlled environmental conditions. Effects of CO 2 enrichment on seedling growth is reviewed, emphasizing the effects of CO 2 on root growth.

The heavy metal cadmium (Cd) affects root system development and quiescent center (QC)-definition in Arabidopsis root-apices. The brassinosteroids-(BRs)-mediated tolerance to heavy metals has been reported to occur by a modulation of nitric oxide (NO) and root auxin-localization. However, how BRs counteract Cd-action in different root types is unknown. This research aimed to find correlations between BRs and NO in response to Cd in Arabidopsis’s root system, monitoring their effects on QC-definition and auxin localization in root-apices. To this aim, root system developmental changes induced by low levels of 24-epibrassinolide (eBL) or by the BR-biosynthesis inhibitor brassinazole (Brz), combined or not with CdSO4, and/or with the NO-donor nitroprusside (SNP), were investigated using morpho-anatomical and NO-epifluorescence analyses, and monitoring auxin-localization by the DR5::GUS system. Results show that eBL, alone or combined with Cd, enhances lateral (LR) and adventitious (AR) root formation and counteracts QC-disruption and auxin-delocalization caused by Cd in primary root/LR/AR apices. Exogenous NO enhances LR and AR formation in Cd-presence, without synergism with eBL. The NO-signal is positively affected by eBL, but not in Cd-presence, and BR-biosynthesis inhibition does not change the low NO-signal caused by Cd. Collectively, results show that BRs ameliorate Cd-effects on all root types acting independently from NO.

The perfect storm. That is the term used by Gerald C. Nelson to describe the triple challenge of increasing food production while adapting to climate change and reducing the environmental impact of agricultural systems. Nowadays, conventional farming systems are showing some limitations, such as low resources use efficiency and poor ecosystems services that appear to be associated to the loss of plant diversity and perenniality in crop rotations. In addition, water, the most important yield limiting factor worldwide, will increasingly restrict food production in the future due to rainfall shortage and increase in human consumption. In such context, perennial crops, with denser and deeper root system could use resources in deep soil layers that are logically inaccessible to crops with shallower root system. The goal of this thesis was therefore to investigate the root growth and water uptake capacity of intermediate wheatgrass (Kernza®) and alfalfa, two deep rooted perennial crops, under field conditions and at great soil depth (i.e. 1.0-2.5 m).
\n\tMaintaining hydraulic continuity along the soil-plant-atmosphere continuum is a prerequisite for deep water uptake. At the plant level, hydraulic conductivity depends on complex anatomical and physiological processes among which the root system constitutes the second largest resistance to water flow. Therefore, in depth characterisation of root and xylem anatomy was done to understand the hydraulic properties of the crop root systems, with a focus on their evolution with soil depth. Crops were grown in the field, rhizoboxes, mesocosms and solution culture to take into account the variability of root type and soil depth as well as growing environment. For both crops, axial hydraulic conductance decreased with soil depth and along individual root segment. Alfalfa roots had greater axial hydraulic conductance in comparison to intermediate wheatgrass roots, especially at depth. Root and xylem anatomy were highly variable across crop species, root types and growing environments. In parallel, a combination of imaging and sensor technology, stable isotope techniques and a modelling approach was used to study root growth and water uptake under field conditions during the 2018-2019 seasons. Both crops presented roots down to 2.0 m soil depth that were active in terms of water uptake. Alfalfa had greater root length at depth and absorbed twice as much water below 1 m soil depth, than intermediate wheatgrass. For both crops, model simulations predicted that water uptake in deep soil layers (i.e. 1.5 – 2.0 m) increase (i.e.>30%) under dry conditions.
\n\tThis thesis brings insights into the understudied field of root growth and water uptake at great soil depth. Particular efforts were put in understanding the environmental and agricultural contexts in which deep root growth, deep water uptake and the development of perennial cropping systems would be possible and favourable.

A plant's ability to maintain or improve its yield under limiting conditions, such as nutrient deficiency or drought, can be strongly influenced by root system architecture (RSA), the three-dimensional distribution of the different root types in the soil. The ability to image, track and quantify these root system attributes in a dynamic fashion is a useful tool in assessing desirable genetic and physiological root traits. Recent advances in imaging technology and phenotyping software have resulted in substantive progress in describing and quantifying RSA. We have designed a hydroponic growth system which retains the three-dimensional RSA of the plant root system, while allowing for aeration, solution replenishment and the imposition of nutrient treatments, as well as high-quality imaging of the root system. The simplicity and flexibility of the system allows for modifications tailored to the RSA of different crop species and improved throughput. This paper details the recent improvements and innovations in our root growth and imaging system which allows for greater image sensitivity (detection of fine roots and other root details), higher efficiency, and a broad array of growing conditions for plants that more closely mimic those found under field conditions.

BackgroundPlant roots release a variety of organic compounds into the soil which alter the physical, chemical and biological properties of the rhizosphere. Root exudates are technically challenging to measure in soil because roots are difficult to access and exudates can be bound by minerals or consumed by microorganisms. Exudates are easier to measure with hydroponically-grown plants but, even here, simple compounds such as sugars and organic acids can be rapidly assimilated by microorganisms. Sterile hydroponic systems avoid this shortcoming but it is very difficult to maintain sterility for long periods especially for larger crop species. As a consequence, studies often use small model species such as Arabidopsis to measure exudates or use seedlings of crop plants which only have immature roots systems.ResultsWe developed a simple hydroponic system for cultivating large crop plants in sterile conditions for more than 30 days. Using this system wheat (Triticum aestivum L.) and barley (Hordeum vulgare L.) plants were grown in sterile conditions for 30 days by which time they had reached the six-leaf stage and developed mature root systems with seminal, nodal and lateral roots. To demonstrate the utility of this system we characterized the aluminium-activated exudation of malate from the major types of wheat roots for the first time. We found that all root types measured released malate but the amounts were two-fold greater from the seminal and nodal axile roots compared with the lateral roots. Additionally, we showed that this sterile growth system could be used to collect exudates from intact whole root systems of barley.ConclusionsWe developed a simple hydroponic system that enables cereal plants to be grown in sterile conditions for longer periods than previously recorded. Using this system we measured, for the first time, the aluminium-activated efflux of malate from the major types of wheat roots. We showed the system can also be used for collecting exudates from intact root systems of 30-day-old barley plants. This hydroponic system can be modified for various purposes. Importantly it enables the study of exudates from crop species with mature root systems.

To address the issues of a great variety and a high degree of morphological diversity for a plant root system, this paper proposes an editable modeling method to simulate root growth. This method extends the traditional L-system to describe the root structure and topology with a semantic feature rule, and achieves a natural growth simulation according to a continuous equation of root stem radius and single root length. The global morphology of the root system is controlled by guiding vectors, and a variety of measures are provided to edit the local shape of the root stem to generate a natural-looking root model, such as stem skeleton, surface shape, irregular surface, etc. In the experiment, this paper simulates six species of plants in nature, including four different types of tap roots, fibrous roots, storage roots and plate roots. The average similarity between the simulated root system and the real picture is up to 79.16%. The experimental results show that the method has the ability to model many types of root systems, and it is possible to build root models with special morphology by setting the shape guidance curve or adjusting some parameters in a realistic way.

The root system is responsible for water and nutrients uptake from the soil, and therefore, its extension is basic for an efficient acquisition. The maize root system is formed by different types of roots, and the lateral root branching substantially increases the surface for nutrient uptake. Therefore, the regulation of lateral root formation is fundamental in the development of root functions. Root architecture is basically controlled by auxin and cytokinins, which antagonize in the formation of lateral roots (LR) along the primary root axis, with auxin, a stimulator, and cytokinins inhibitors of LR development. This interaction has been analyzed in several zones along the primary root where LRs in different developmental stages were located. The root has been divided into several zones, such as meristem, elongation zone, and mature zone, according to the developmental processes occurring in each one. As Arabidopsis root elongated more slowly than maize root, these zones are shorter, and its delimitation is more difficult. However, these zones have previously been delimitated clearly in maize, and therefore, they analyze the effect of exogenous hormones in several LR developmental stages. The inhibitory effect of cytokinin on lateral root formation was observed in already elongated primary root zones in which initial events to form new lateral roots are taking place. Contrarily, auxin increased LR formation in the primary root segments elongated in the presence of the hormone. The inhibitory effect of cytokinin was reversed by auxin in a concentration-dependent manner when both hormones were combined. However, auxin is unable to recover LR development in primary root zones that have been previously elongated only in the presence of cytokinin. This antagonistic auxin-cytokinin effect on LR development depended on the balance between both hormones, which controls the root system architecture and determines the formation of LR during the process of initiation.