Cluster Roots Research Articles

Interspecific facilitation of micronutrient uptake between cluster-root-bearing trees and non-cluster rooted-shrubs in a Banksia woodland

Background and aimsBelowground interspecific plant facilitation is supposed to play a key role in enabling species co-existence in hyperdiverse ecosystems in extremely nutrient-poor, semi-arid habitats, such as Banksia woodlands in southwestern-Australia. Manganese (Mn) is readily mobilised by Banksia cluster root activity in most soils and accumulates in mature leaves of native Australian plant species without significant remobilisation during leaf senescence. We hypothesised that neighbouring shrubs are facilitated in terms of Mn uptake depending on distance to surrounding cluster root-forming Banksia trees.MethodsWe mapped all Banksia trees and selected neighbouring shrubs within a study site in Western Australia. Soil samples were collected and analysed for physical properties and nutrient concentrations. To assesses the effect of Banksia tree proximity on leaf Mn concentrations [Mn] of non-cluster-rooted woody shrubs, samples of similarly aged leaves were taken. We used multiple linear models to test for factors affecting shrub leaf [Mn].ResultsNone of the assessed soil parameters showed a significant correlation with shrub leaf Mn concentrations. However, we observed a significant positive effect of very close Banksia trees (2 m) on leaf [Mn] in one of the understorey shrubs. We found additional effects of elevation and shrub size.ConclusionsLeaf micronutrient concentrations of understorey shrubs were enhanced when growing within 2 m of tall Banksia trees. Our model predictions also indicate that belowground facilitation of Mn uptake was shrub size-dependent. We discuss this result in the light of plant water relations and shrub root system architecture.

Broadcasting‐induced colorings of preferential attachment trees

AbstractWe consider random two‐colorings of random linear preferential attachment trees, which includes recursive trees, plane‐oriented recursive trees, binary search trees, and a class of d‐ary trees. The random coloring is defined by assigning the root the color red or blue with equal probability, and all other vertices are assigned the color of their parent with probability and the other color otherwise. These colorings have been previously studied in other contexts, including Ising models and broadcasting, and can be considered as generalizations of bond percolation. With the help of Pólya urns, we prove limiting distributions, after proper rescalings, for the number of vertices, monochromatic subtrees, and leaves of each color, as well as the number of fringe subtrees with two‐colorings. Using methods from analytic combinatorics, we also provide precise descriptions of the limiting distribution after proper rescaling of the size of the root cluster; the largest monochromatic subtree containing the root.

Facilitation of phosphorus acquisition by Banksia attenuata allows Adenanthos cygnorum (Proteaceae) to extend its range into severely phosphorus-impoverished habitats

Abstract Background and aims In extremely low-phosphorus (P) environments, most Proteaceae exude carboxylates from cluster roots. These carboxylates mobilise inorganic P which leads to a relatively high leaf manganese concentration ([Mn]). However, we found that Adenanthos cygnorum (Proteaceae) in a low-P habitat did not invariably have a high leaf [Mn] in south-western Australia. We aimed to explore how A. cygnorum acquires P in severely P-impoverished habitats. Methods We determined soil P concentrations and leaf [Mn] of A. cygnorum growing within 1 m and more than 10 m away from other large Proteaceae. We also grew plants in a glasshouse to determine its root carboxylate exudation and rhizosheath phosphatase activity. Results Adenanthos cygnorum did not produce functional cluster roots. It depended on carboxylates released by a P-mobilising neighbour, Banksia attenuata (Proteaceae), to acquire P when growing in severely P-impoverished soil (< 8 mg P kg− 1 dry soil). In slightly less P-impoverished soil (> 11 mg P kg− 1 dry soil), phosphatases released by A. cygnorum hydrolysed sufficient organic P that was relatively mobile. Conclusion The reliance on facilitation of P acquisition in A cygnorum depended strongly on location. We demonstrated the exudation of phosphatases, which mobilise inorganic P; this P was adequate for growth when there was sufficient organic P in soil. Facilitation of P acquisition by B. attenuata allowed A. cygnorum to extend its range into severely P-impoverished habitats where it cannot exist without facilitation. This knowledge provides a better understanding of the diversity of P-acquisition strategies in severely P-impoverished environments.

Improving phosphorus acquisition efficiency through modification of root growth responses to phosphate starvation in legumes.

Phosphorus (P) is one of the essential macronutrients for plant growth and development, and it is an integral part of the major organic components, including nucleic acids, proteins and phospholipids. Although total P is abundant in most soils, a large amount of P is not easily absorbed by plants. Inorganic phosphate (Pi) is the plant-available P, which is generally immobile and of low availability in soils. Hence, Pi starvation is a major constraint limiting plant growth and productivity. Enhancing plant P efficiency can be achieved by improving P acquisition efficiency (PAE) through modification of morpho-physiological and biochemical alteration in root traits that enable greater acquisition of external Pi from soils. Major advances have been made to dissect the mechanisms underlying plant adaptation to P deficiency, especially for legumes, which are considered important dietary sources for humans and livestock. This review aims to describe how legume root growth responds to Pi starvation, such as changes in the growth of primary root, lateral roots, root hairs and cluster roots. In particular, it summarizes the various strategies of legumes to confront P deficiency by regulating root traits that contribute towards improving PAE. Within these complex responses, a large number of Pi starvation-induced (PSI) genes and regulators involved in the developmental and biochemical alteration of root traits are highlighted. The involvement of key functional genes and regulators in remodeling root traits provides new opportunities for developing legume varieties with maximum PAE needed for regenerative agriculture.

Invasiveness, Monitoring and Control of Hakea sericea: A Systematic Review.

Solutions for ecological and economic problems posed by Hakea sericea invasions rely on scientific knowledge. We conducted a systematic review to analyze and synthesize the past and current scientific knowledge concerning H. sericea invasion processes and mechanisms, as well as monitoring and control techniques. We used ISI Web of Science, Scopus, and CAPES Periodicals to look for publications on the ecological and environmental factors involved in H. sericea establishment (question 1); responses of H. sericea to fire in native and invaded ecosystems (question 2); and H. sericea monitoring and control methods (question 3). We identified 207 publications, 47.4% of which related to question 1, mainly from Australia and South Africa, with an increasing trend in the number of publications on monitoring and modeling. The traits identified in our systematic review, such as adaptations to dystrophic environments, drought resistance, sclerophylly, low transpiration rates, high nutrient use efficiency, stomatal conductance and photosynthetic rates, strong serotiny, proteoid roots and high post-fire seed survival and seedling recruitment, highlighted that H. sericea is a successful invader species due to its long adaptive history mediated by an arsenal of ecophysiological mechanisms that place it at a superior competitive level, especially in fire-prone ecosystems. Integrated cost-effective control methods in selected areas and the incorporation of information on the temporal invasion dynamics can significantly improve invasion control and mitigate H. sericea impacts while maintaining the supply of ecosystem services in invaded areas.

CO2 concentration and water availability alter the organic acid composition of root exudates in native Australian species

PurposeRoot exudation of organic acids (OAs) facilitates plant P uptake from soil, playing a key role in rhizosphere nutrient availability. However, OA exudation responses to CO2 concentrations and water availability remain largely untested.MethodsWe examined the effects of CO2 and water on OA exudates in three Australian woodland species: Eucalyptus tereticornis, Hakea sericea and Microlaena stipoides. Seedlings were grown in a glasshouse in low P soil, exposed to CO2 (400 ppm [aCO2] or 540 ppm [eCO2]) and water treatments (100% water holding capacity [high-watered] or 25–50% water holding capacity [low-watered]). After six weeks, we collected OAs from rhizosphere soil (OArhizo) and trap solutions in which washed roots were immersed (OAexuded).ResultsFor E. tereticornis, the treatments changed OArhizo composition, driven by increased malic acid in plants exposed to eCO2 and increased oxalic acid in low-watered plants. For H. sericea, low-watered plants had higher OAexuded per plant (+ 116%) and lower OArhizo per unit root mass (–77%) associated with larger root mass but fewer cluster roots. For M. stipoides, eCO2 increased OAexuded per plant (+ 107%) and per unit root mass (+ 160%), while low-watered plants had higher citric and lower malic acids for OArhizo and OAexuded: changes in OA amounts and composition driven by malic acid were positively associated with soil P availability under eCO2.ConclusionWe conclude that eCO2 and altered water availability shifted OAs in root exudates, modifying plant–soil interactions and the associated carbon and nutrient economy.

Slice-Relation-Clustering Framework via Horizontal Angle Information for 3-D Tree Roots Reconstruction

Tree root system 3D reconstruction and spatial distribution analysis are prevalent aspects of tree root investigation using ground penetrating radar (GPR). Precedent 3D reconstruction methods are found to be effective in mapping simple, smooth root structures. However, repetitive and dense B-scans are needed, otherwise, the retrieved roots’ spatial distribution and growth extension trend accuracy would deteriorate with the increase in the root systems’ complexity. To address these issues, this paper, for the first time, explores the possibility of integrating the horizontal angle information of the tree roots and a slice-relation-clustering (SRC) algorithm to reconstruct the complex tree root systems in a 3D manner. The proposed framework, which takes the roots’ horizontal angle as an analyzing condition instead of biological properties that are similar among neighboring branches used in existing methods, clusters pre-processed and focused 2D reflection patterns from the same single root together. The whole roots system is the combination of every single root cluster. Real measurement results show that our proposed method achieves a high efficiency in accurate root system reconstruction.

Clustering of the West African Starchy Roots and Tubers using Nutritive Value

The objective of this study was to sub-group the West African starchy roots and tubers’ food sources according to their nutritive values. The k means algorithm was deployed on a dataset consisting of starchy roots and tubers food items extracted from the West African Food Composition Table. Various measures of evaluating clustering validity were employed and the results showed that the clustering was valid. Three clusters were extracted, each consisting of food sources that are very similar in nutritive value. Findings prove that in terms of nutritive value, some kinds of yam and cassava could be classified together, while some kinds of sweet potato and cocoyam could also be classified together, and so on. The clusters can be explored by nutritionists, dieticians, food processing enterprises, and food scientists to find alternative food sources when their original choices are unavailable. Though clustering validation showed that food sources within the same cluster are significantly similar on a general note, a future study is required to research on the extent to which food sources within the same cluster are similar to each other at a granular level.

Response of root morphogenesis of medicinal plants to low phosphorus stress and molecular mechanism

The content of available phosphorus in soil is generally low worldwide. Phosphorus, one of the necessary macroelements for plant growth and development, plays an important role in cell structure, material composition and energy metabolism, and signal transduction in plants. Phosphate transporter(PHT) genes are important for plant growth and development, root morphogenesis, secondary metabolism, hormone response, and phosphorus balance. Most of the active components in medicinal plants are secondary metabolites. Thus, it is essential to reveal the relationship between the regulation of phosphorus and the accumulation of active components in medicinal plants, especially the effect of phosphorus starvation on root morphogenesis of root medicinal materials and its coupling with hormone response. With advancement of molecular biology, scholars gradually emphasize the mechanism of PHT regulating the secondary metabolism of medicinal plants. This study summarized the strategies of plants to adapt to low phosphorus environment, such as changing root morphogenesis, inhibiting taproot growth, forming cluster root and changing physiological metabolism, PHT, its regulatory network, phenotypic biological characteristics and key genes in medicinal plants related to phosphorus starvation, and the response mechanism. The findings are expected to lay a basis for the cultivation of medicinal plants with high quality, excellent shape, and high price.

Soilborne bacterium Klebsiella pneumoniae promotes cluster root formation in white lupin through ethylene mediation.

Cluster roots of white lupin are induced by low phosphorus (LP) to efficiently access unavailable P, but how soilborne microbes are associated with cluster root formation (CRF) is unclear. We investigated the roles of soilborne bacteria in CRF response to LP by high-throughput sequencing and root-bacteria interactions. Cluster root number was significantly decreased in plants grown in sterilized soil compared with nonsterilized soil. Proteobacteria was enriched in CR, as shown by microbiome analysis of soil (bulk, rhizosphere, and rhizosheath) and roots (main, lateral, and CR). Large-scale gene expression level implicated ethylene mediation in CRF. Klebsiella pneumoniae (P7), a soilborne bacterium belonging to Proteobacteria, was isolated from CR. Among 11 isolated strains, P7 exhibited the highest 1-aminocyclopropane-1-carboxylate deaminase (ACCD) activity; this enzyme inhibits the biosynthesis of ethylene in plants by the cleavage of the ethylene precursor 1-aminocyclopropane-1-carboxylic acid and promotes CRF under LP. We constructed an ACCD-deficit mutant accd in the P7 genetic background. The loss-of-function mutation failed to promote CRF under LP conditions. Also, auxin responses may be involved in K. pneumoniae-ethylene-mediated CRF. Overall, we propose that the soilborne bacterium K. pneumoniae promotes CRF of white lupin in response to LP by ethylene mediation.

The vacuolar transporter LaMTP8.1 detoxifies manganese in leaves of Lupinus albus.

Manganese (Mn) is an essential microelement, but overaccumulation is harmful to many plant species. Most plants have similar minimal Mn requirements, but the tolerance to elevated Mn varies considerably. Mobilization of phosphate (P) by plant roots leads to increased Mn uptake, and shoot Mn levels have been reported to serve as an indicator for P mobilization efficiency in the presence of P deficiency. White lupin (Lupinus albus L.) mobilizes P and Mn with outstanding efficiency due to the formation of determinate cluster roots that release carboxylates. The high Mn tolerance of L. albus goes along with shoot Mn accumulation, but the molecular basis of this detoxification mechanism has been unknown. In this study, we identify LaMTP8.1 as the transporter mediating vacuolar sequestration of Mn in the shoot of white lupin. The function of Mn transport was demonstrated by yeast complementation analysis, in which LaMTP8.1 detoxified Mn in pmr1∆ mutant cells upon elevated Mn supply. In addition, LaMTP8.1 also functioned as an iron (Fe) transporter in yeast assays. The expression of LaMTP8.1 was particularly high in old leaves under high Mn stress. However, low P availability per se did not result in transcriptional upregulation of LaMTP8.1. Moreover, LaMTP8.1 expression was strongly upregulated under Fe deficiency, where it was accompanied by Mn accumulation, indicating a role in the interaction of these micronutrients in L. albus. In conclusion, the tonoplast-localized Mn transporter LaMTP8.1 mediates Mn detoxification in leaf vacuoles, providing a mechanistic explanation for the high Mn accumulation and Mn tolerance in this species.

Percolation probability and critical exponents for site percolation on the UIPT

AbstractWe derive three critical exponents for Bernoulli site percolation on the uniform infinite planar triangulation (UIPT). First, we compute explicitly the probability that the root cluster is infinite. As a consequence, we show that the off-critical exponent for site percolation on the UIPT is $\beta = 1/2$ . Then we establish an integral formula for the generating function of the number of vertices in the root cluster. We use this formula to prove that, at criticality, the probability that the root cluster has at least n vertices decays like $n^{-1/7}$ . Finally, we also derive an expression for the law of the perimeter of the root cluster and use it to establish that, at criticality, the probability that the perimeter of the root cluster is equal to n decays like $n^{-4/3}$ . Among these three exponents, only the last one was previously known. Our main tools are the so-called gasket decomposition of percolation clusters, generic properties of random Boltzmann maps, and analytic combinatorics.

Auxin triggers pectin modification during rootlet emergence in white lupin.

Emergence of secondary roots through parental tissue is a highly controlled developmental process. Although the model plant Arabidopsis has been useful to uncover the predominant role of auxin in this process, its simple root structure is not representative of how emergence takes place in most plants, which display more complex root anatomy. White lupin is a legume crop producing structures called cluster roots, where closely spaced rootlets emerge synchronously. Rootlet primordia push their way through several cortical cell layers while maintaining the parent root integrity, reflecting more generally the lateral root emergence process in most multilayered species. In this study, we showed that lupin rootlet emergence is associated with an upregulation of cell wall pectin modifying and degrading genes under the active control of auxin. Among them, we identified LaPG3, a polygalacturonase gene typically expressed in cells surrounding the rootlet primordium and we showed that its downregulation delays emergence. Immunolabeling of pectin epitopes and their quantification uncovered a gradual pectin demethylesterification in the emergence zone, which was further enhanced by auxin treatment, revealing a direct hormonal control of cell wall properties. We also report rhamnogalacturonan-I modifications affecting cortical cells that undergo separation as a consequence of primordium outgrowth. In conclusion, we describe a model of how external tissues in front of rootlet primordia display cell wall modifications to allow for the passage of newly formed rootlets.

Evolutionary adaptation of plants to phosphorus deficiency: the multifaceted role of cluster roots

Evolutionary adaptation of plants to phosphorus deficiency: the multifaceted role of cluster roots

Magnesium stable isotopes as a potential geochemical tool in agronomy – Constraints and opportunities

A sustainable use of soil resources is urgently required to cope with the increasing demand for agricultural products during climate change. To inspire farmers on new soil cultivation methods like subsoil management requires not only yield measurements but also nutrient use efficiency measurements for which analytical tools are still missing. Stable isotopes of the macronutrient magnesium (Mg) are a potential novel subsoil management evaluation tool in agronomy and soil/plant sciences because its isotope ratios shift considerably during Mg uptake by crops. The feasibility of Mg stable isotopes was first demonstrated conceptually by simulating subsoil management on soils with low, middle, and high inventories of bioavailable Mg and crop plants typically cultivated in Germany. This simulation showed that the magnitude of Mg isotope shifts among crops and the exchangeable fraction of Mg in soil is resolvable from the long-term external precision of Mg isotope analyses only if three conditions are met. First, the crop uptake-related Mg isotope fractionation factor should be at the upper end of hitherto published fractionation factors. Second, a high Mg uptake flux of crop plants (e.g., sugar beets) is matched by a low Mg supply from the exchangeable fraction in soil (e.g., sandy soils). Third, subsoil management causes a considerable deepening of the rooting system (e.g., flipping the topsoil root cluster below 30 cm depth). If these conditions are met, Mg stable isotopes can be used in a qualitative manner to identify the main Mg uptake depth, and in a quantitative manner by calculating the Mg use efficiency, defined here as the ratio of Mg uptake versus Mg supply, solely from Mg isotope ratios. This concept was tested for Alfisols on field trials by conducting deep loosening with and without the incorporation of compost. Magnesium isotope shifts in crops and the exchangeable fraction of Mg in soil were mostly unresolvable from the long-term external precision of Mg isotope analyses, which positively tested the Mg isotope concept for well nurtured soils. However, systematic Mg isotope shifts among bulk crops cultivated on and beside a melioration strip were found and attributed to the uplift of isotopically distinct compost-derived Mg on the melioration strip and root restricting layers beside the melioration strip. Given that the Mg isotope composition of the exchangeable fraction barely varies with depth, field-based crop uptake-related ‘apparent’ Mg isotope fractionation factors of winter wheat and spring barley could be determined, which differed from one another (Δ26Mgwheat-rem. exch. = 0.63 ± 0.05‰, Δ26Mgbarley-rem. exch. = 0.37 ± 0.12‰). Nonetheless, the quantitative approach of Mg isotopes was violated when calcareous fertilizer was applied to the field as differences in the isotope-derived Mg use efficiency could be attributed to the uneven distribution of lime-derived Mg with depth. Using Mg stable isotopes as a new geochemical routine for agronomy and soil/plant sciences requires future work focussing on isotope fractionation factors related to crop uptake and intra-plant translocation of Mg – which may depend on species, environmental conditions, and nutrient status – to allow minimally invasive sampling of the soil-plant system and to reduce sample sets.

Optimal Derivative-Free One-Point Algorithms for Computing Multiple Zeros of Nonlinear Equations

In this paper, we describe iterative derivative-free algorithms for multiple roots of a nonlinear equation. Many researchers have evaluated the multiple roots of a nonlinear equation using the first- or second-order derivative of functions. However, calculating the function’s derivative at each iteration is laborious. So, taking this as motivation, we develop second-order algorithms without using the derivatives. The convergence analysis is first carried out for particular values of multiple roots before coming to a general conclusion. According to the Kung–Traub hypothesis, the new algorithms will have optimal convergence since only two functions need to be evaluated at every step. The order of convergence is investigated using Taylor’s series expansion. Moreover, the applicability and comparisons with existing methods are demonstrated on three real-life problems (e.g., Kepler’s, Van der Waals, and continuous-stirred tank reactor problems) and three standard academic problems that contain the root clustering and complex root problems. Finally, we see from the computational outcomes that our approaches use the least amount of processing time compared with the ones already in use. This effectively displays the theoretical conclusions of this study.

Global Identification of White Lupin lncRNAs Reveals Their Role in Cluster Roots under Phosphorus Deficiency.

Phosphorus (P) deficiency heterogeneously affected plant nutritional status and physiological performance, ultimately leading to a severe yield reduction. A few putative long non-coding RNAs (lncRNAs) responding to P-starvation in the model crops Arabidopsis thaliana and Oryza sativa have been characterized. White lupin (Lupinus albus) is of prime importance, and is a legume with increasing agronomic value as a protein crop as it exhibits extreme tolerance to nutrient deficiency, particularly P deficiency. Despite its adapted nature to P deficiency, nothing is known about low P-induced lncRNAs in white lupin roots. To address this issue, we identified 39,840 mRNA and 2028 lncRNAs in the eight developmental stages of white lupin root (S0–S7 and lateral root, LR) grown under P deficiency. From these 2028 lncRNAs, 1564 were intergenic and 464 natural antisense intergenic transcript (NAT) lncRNAs. We further predicted six potential targets of miRNAs with twelve lncRNAs, which may regulate P-deficiency-related processes. Moreover, the weighted gene co-expression network analysis (WGCNA) revealed seven modules that were correlated with the expression pattern of lncRNAs. Gene Ontology (GO) and Kyoto Encyclopedia of Genes and Genomes (KEGG) analysis revealed 606 GO terms and 27 different pathways including signal transduction, energy synthesis, detoxification, and Pi transport. In addition, we screened 13 putative lncRNAs that showed a distinct expression pattern in each root, indicating their role in the P deficiency regulatory network. Therefore, white lupin may be a reference legume to characterize P-deficiency-responsive novel lncRNAs, which would highlight the role of lncRNAs in the regulation of plant responses to P deficiency.

Relationship between soil phosphorus dynamics and low-phosphorus responses at specific root locations of white lupine

ABSTRACT Phosphorus (P) is an essential element for crop production; however, availability tends to be low due to slow diffusion and high fixation in soils. To cope with phosphate deficiency, white lupines (Lupinus albus L.) form unique root structures called cluster roots. The objective of this study was to elucidate detailed spatial differences of the mechanisms under low-P condition in the root system using rhizoboxes. We cultivated plants in rhizoboxes with P-deficient soil for 33 days. We then harvested roots and the adjacent soil from 128 compartments by dividing 2 × 2 cm squares in the rhizoboxes. We investigated relative expression levels of several genes that encode proteins assumed to be involved in P solubilization or translocation. Additionally, we analyzed fractionated P, soluble metal cations (Fe, Al, and Mn), and enzyme activities in the soil of each compartment. We observed a significant positive correlation between one of the MATE (multidrug and toxic compound extrusion/detoxification) genes, LaMATE6, and soluble metal cations, suggesting the secretion of citric acid into the rhizosphere via MATE proteins increased soluble metal cation concentrations. Furthermore, we demonstrated that roots in the same developmental stages were likely to have different influences on the mobilization of fractionated P in the rhizosphere soil. Our findings highlight the importance of investigating the relationship between soil chemical properties and root functions at a high spatial resolution to elucidate the detailed mechanisms of P mobilization by plants.

Effects of Root Temperature and Cluster Position on Fruit Quality of Two Cocktail Tomato Cultivars

Managing root temperature can strongly influence plant growth in various species. Our previous work has shown that a positive effect of root cooling on the nutritional quality (sugars, vitamin C, and carotenoids levels) of two cocktail tomato cultivars, while the yield and mineral content was moderately reduced in two seasons. But few studies have investigated which cluster quality is more influenced by root temperature. The main purpose of this study was to evaluate the effects of root temperature and different cluster positions (the first to the fourth clusters) on fruit quality parameters of two cocktail tomato cultivars (‘Amoroso’ and ‘Delioso’). Two root temperatures, cool (10 °C) and control (18–22 °C), were applied to the roots of hydroponically cultivated tomato plants after inflorescence until the final harvest in the greenhouse. The results showed that root cooling has no influence on the biomass of fruits (both dry weight and fresh weight) of all clusters, but it increased the sugar (6.1–8.4%) and vitamin C (9.1–12.5%) concentration of the second cluster of ‘Amoroso’ and the third cluster of ‘Delioso’ compared to the control. In most cases, significant positive changes (8–23.8%) in the levels of lycopene were observed in response to low root temperature treatment, but the effect was dependent on different cluster positions and different cultivars. Most mineral element levels showed no response to the root temperature. Significant reductions (10% and 17.6%) along with the root cooling was only found in the level of zinc and potassium of the first and the second cluster of ‘Delioso’, respectively. It was concluded that the fruits from the second or third clusters containing more sugars, vitamin C, and lycopene, were more influenced by root cooling. In addition, the effect of cluster position on fruit quality was different between two root temperature groups. Root cooling reduced the heterogeneity in fruit quality parameters among the clusters of ‘Amoroso’ and increased the differences in ‘Delioso’.

Derivative-Free Iterative Schemes for Multiple Roots of Nonlinear Functions

The construction of derivative-free iterative methods for approximating multiple roots of a nonlinear equation is a relatively new line of research. This paper presents a novel family of one-parameter second-order techniques. Our schemes are free from derivatives and have been designed to find multiple roots (m≥2). The new techniques involve the weight function approach. The convergence analysis for the new family is presented in the main theorem. In addition, some special cases of the new class are discussed. We also illustrate the applicability of our methods on van der Waals, Planck’s radiation, root clustering, and eigenvalue problems. We also contrast them with the known methods. Finally, the dynamical study of iterative schemes also provides a good overview of their stability.