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Proximal root diameter as predictor of total root size for fractal branching models. I.Theory
Abstract
In a fractal branching pattern the same rules govern branching at each subsequent level. The initial size (diameter) and the essential branching rules thus contain the information required to construct the whole pattern. If root branching patterns have fractal characteristics, measurement of the proximal root diameter at the stem base and the branching rules as observed anywhere in the root system, would be enough to predict total root length, root diameter distribution and root length per unit dry weight (specific root length).
A ‘pipe stem’ model is used to derive algebraic relations between total root size and proximal root diameter for two classes of branching patterns, determinate and proportionate. To predict total root length from the proximal root diameter, at least information is needed on the minimum root diameter, the average length of internal and external links (segments) and the proportionality factor between total cross sectional areas before and after branching. For the length of the longest root or the specific root length further information on the branching rules is needed, as it is highest for determinate and proportionate branching rules, respectively.
... In most instances, the area ratio is close to 1, with most values ranging between 0.9 and 1.2, which implies the conservation of area across a bifurcation. This agrees with the pipe stem theory, which contends that this hydraulic architecture naturally evolves to preserve crosssectional areas for e cient water and nutrient transport [9,34]. In addition to natural variability, there can be deviations in the hydraulic architecture from the external diameter owing to the thickness of nonconducting material encasing the root [6, 17,19]. ...
... Full description of root architecture requires the elongation rate (or length), branching interval, branching angle, growth tortuosity, and gravitropism for each order (or generation) of root [35]. As an alternative to the above geometric descriptor-based approach, a fractal model that de nes the architecture based on approximate rules of proportionality, such as allocation rules at branch points and scaling of lengths and branching angles, can be used in particular cases (e.g., [34]). This fractal model is particularly well-suited to capture the architecture of roots with self-repeating patterns. ...
... L-system modeling is a powerful formulation to store or represent a larger or more complicated structure in a compact manner through the use of reproduction rules [36]. It is a kind of fractal model [34] that also facilitates the use of different root types or stages with corresponding patterns and distributions in the case of stochastic modeling. One modeling method might look like the following: a parent root extends for some branch length sampled from the measured distributions and then divides, at which the child root diameters are sampled from the measured distribution of local thickness allocation, and the directions are sampled from the distributions of branch angle and azimuthal angle. ...
Background
Statistical analysis of root architectural parameters is necessary for development and exploration of root structure representations and their resulting anchorage properties. Three-dimensional (3D) models of orchard tree root systems, Lovell (from seed, prunus persica), Marianna (from cutting, prunus cerasifera), Myrobalan (from cutting, also prunus cerasifera), that were extracted from the ground by vertical pullout are reconstructed through photogrammetry, and then skeletonized as nodes and root branch segments. Combined analyses of the 3D models and skeletonized models enable detailed examination of basic bulk properties and quantification of architectural parameters divided into simple root segment classifications— trunk root, main lateral root, and remaining roots.
Results
The patterns in branching and diameter distributions show significant difference between the trunk and main laterals versus the remaining lateral roots. In general, the branching angle decreases with branching order. The main lateral roots near the trunk show significant spreading while the lateral roots near the end tips grow roughly parallel to the parent root. For branch length, the roots branch more frequently near the trunk than further from the trunk. The root diameter decays at a higher rate near the trunk than in the remaining lateral roots, while the total cross-sectional area across a bifurcation node remains mostly conserved. The histograms of branching angle, and branch length and thickness gradient can be described using lognormal and exponential distributions, respectively.
Conclusions
Statistical measurements of root system architecture upon hierarchy provide a basis for representation and exploration of root system structure. This unique study presents data to characterize mechanically important structural roots, which will help link root architecture to the mechanical behaviors of root structures.
... In most instances, the area ratio is close to 1, with most values ranging between 0.9 and 1.2, which implies the conservation of area across a bifurcation. This agrees with the pipe stem theory, which contends that this hydraulic architecture naturally evolves to preserve crosssectional areas for e cient water and nutrient transport [9,34]. In addition to natural variability, there can be deviations in the hydraulic architecture from the external diameter owing to the thickness of nonconducting material encasing the root [6, 17,19]. ...
... Full description of root architecture requires the elongation rate (or length), branching interval, branching angle, growth tortuosity, and gravitropism for each order (or generation) of root [35]. As an alternative to the above geometric descriptor-based approach, a fractal model that de nes the architecture based on approximate rules of proportionality, such as allocation rules at branch points and scaling of lengths and branching angles, can be used in particular cases (e.g., [34]). This fractal model is particularly well-suited to capture the architecture of roots with self-repeating patterns. ...
... L-system modeling is a powerful formulation to store or represent a larger or more complicated structure in a compact manner through the use of reproduction rules [36]. It is a kind of fractal model [34] that also facilitates the use of different root types or stages with corresponding patterns and distributions in the case of stochastic modeling. One modeling method might look like the following: a parent root extends for some branch length sampled from the measured distributions and then divides, at which the child root diameters are sampled from the measured distribution of local thickness allocation, and the directions are sampled from the distributions of branch angle and azimuthal angle. ...
Aims
This study explores structural root architectures of orchard trees to understand the interplays between the mechanical behavior of roots and the root architecture.
Methods
Full three-dimensional (3D) models of natural tree root systems, Lovell, Marianna, Myrobalan, that were extracted from the ground by vertical pullout are reconstructed through photogrammetry, and later skeletonized as nodes and root branch segments. Combined analyses of the full 3D models and skeletonized models enable detailed examination of basic bulk properties and quantification of architectural parameters. The segments from the skeletonized models are divided into three categories — trunk roots, main lateral roots, and remaining roots.
Results
The patterns in branching and diameter distributions show significant difference between the trunk and main laterals versus the remaining lateral roots. In general, the branching angle decreases over the course of successive bifurcations. The main lateral roots near the trunk show significant spreading while the lateral roots near the end tips grow roughly parallel to the parent root. For branch length, the roots bifurcate more frequently near the trunk than further from the trunk. The local thickness analysis confirms that the root diameter decays at a higher rate near the trunk than in the remaining lateral roots, while the total cross-sectional area across a bifurcation node remains mostly conserved. The histograms of branching angle, and branch length and thickness gradient can be described using lognormal and exponential distributions, respectively.
Conclusions
This unique study presents data to characterize mechanically important structural roots, which will help link root architecture to the mechanical behaviors of root structures.
... For instance, Wang et al. (2006) found that the R b of the second-order roots was critical for roots' uptake of nutrients and water. Trubat et al. (2012) reported that fewer nitrogen and phosphorus elements reduced the topological index of a root system, which tended toward a herringbone branching pattern. Recently, Yildirim et al. (2018) pointed out that incorporating the lateral root link length can help to build a more efficient root system for water uptake in plants adapting to drought conditions. ...
... We calculated the topological index (TI, Equation 1) (Fitter, 1994), revised topological index (q a and q b , Equations 2 and 3) (Oppelt et al., 2001), the cross-sectional area of the root branching (Equation 4) (van Noordwijk et al., 1994), and the root branching rate (R b and R i :R i+I , Equation 5) (Wang et al., 2006), using the equations in Table 2. ...
... Searching the literature, we found several studies similar to ours. For instance, Trubat et al. (2012) reported that with lower nitrogen and phosphorus, the root system topological index was reduced and tended toward a herringbone branching pattern. Later, Correa et al. (2019) summarized that soil compaction was significantly and positively correlated with the topological index; i.e., the dichotomous branching pattern of RSA was a way for plants to adapt to the higher strength of soil compaction. ...
The root system architecture (RSA), being a key characteristic of the root economic spectrum, describes the spatial arrangement and positioning of roots that determines the plant's exploration of water and nutrients in the soil. Still, it remains poorly understood how the RSA of woody plants responds to the demand for water and nutrients in different soil environments and how the uptake of these resources is optimized. Here we selected single-species plantations of Cupressus funebris and determined their topological index (TI), revised topological index (qa and qb), root link length (RLL), root branching rate (Rb and Ri:Ri+1), and in situ soil physicochemical properties to assess which root foraging strategies adopt in different soil environments among Guang'an City (GA), Suining City (SN), Mianyang City (MY), and Deyang City (DY) in China. We also tested the potential effects of different nutrients upon RSA according to its plastic phenotype. Principal component analysis (PCA) showed that levels of soil nutrients were the highest at DY, followed by MY and SN, and lower at GA. A dichotomous branching pattern was observed for GA, SN, and MY, but a herringbone branching pattern for DY. The RLL was ranked as GA, > SN, > MY > DY. The Rb of GA, SN, and MY was significantly lower than that of DY (p < 0.05). Among the different city regions, values of R1/R2 were the largest in different regions and those of R4/R5 the smallest. The cross-sectional area of the root system did not differ between any two connected branch orders. The TI, qa, and RLL were significantly and negatively correlated with soil's water content, porosity, total nitrogen, total potassium, available nitrogen, and available phosphorus (p < 0.05), whereas they all had significant, positive relationships with soil temperature (p < 0.05). The Rb was significantly and positively correlated with total potassium in soil (p < 0.05). Redundancy analysis showed that total potassium was the main factor driving variation in RSA. Our results emphasize that the RSA is capable of corresponding plastic alterations by changing its number of internal or external links and the root link length of fine roots vis-à-vis a heterogeneous environment, thereby optimizing the rates of water capture and space utilization.
... where α d is a proportionality factor introduced to make the resulting cross-sectional area varying with depth (Oppelt et al., 2001;Spek and van Noordwijk, 1994;van Noordwijk et al., 1994). In a more general formulation, as in Arnone et al. (2016), it is assumed that from each node more than two branches can emerge, so to have n daughter branches; additionally, cross sectional area can vary with an exponent different than 2, according to generic exponent Δ: ...
Leonardo's rule (Lrule) applied to below-ground systems defines a simple topological scheme that describes how the branches of root architectures develop within the soil. The approach does not consider the soil-climate-root interactions. From another hand, eco-hydrological approaches exploit physically-based formulations to derive the dynamic evolution of root profile based on soil and climate characteristics. In homogenous soil and simplified hydrological conditions, analytical solutions can be derived, as demonstrated by Laio's model, who proposed a simple exponential formulation to derive the Root Area (AR) profile. Apart from Laio's model, more generalized functions, i.e. derived by two and three parameters gamma distribution or others, can be efficiently used to derive the AR profile.
This communication proposes a combination of the Lrule and eco-hydrological approaches to derive the AR profile, at given soil and climate conditions, allowing to identify a physical and theoretical meaning of the Lrule's parameters. A comprehensive root dataset from field measurements carried out in the region of Tuscany (Italy) is used. Results demonstrate that values of Lrule's parameters derived throughout the proposed mathematical relationships tend to constant values in case of exponential function, which is valid for homogenous soils. Moreover, in a realistic vegetated soil, where top-soil is different than deep-soil, functions derived from a two and three parameters gamma distribution may reproduce better root data observations.
... For DM determination, subsamples were oven dried at 105 0 C for 24 h or up to when no further changes in weight occurred. The biomass of unexcavated roots was determined by allometric equation relating the biomass of excavated root segment to their proximal and distal diameters (van Noordwijk et al., 1994;.. ...
The proceedings cover results of project on Development of Drought Tolerant Trees for Adaptation to Climate Change in Drylands of Kenya. The project worked on Acacia tortilis and Melia volkensii
... Many structural features of plant root architectures have been studied, including root length and root depth [13], the distribution of root hairs [14][15][16], the size and number of lateral branches [17][18][19][20], and biomass allocation to fine roots [21]. More global properties of root shapes have also been analyzed [22], including allometric and fractal scaling [23][24][25], root foraging precision [26,27], and topological morphology via persistent homology methods [28]. These properties can affect numerous biological functions performed by root architectures, including anchorage, carbon sequestration, and search for water and nutrients in the soil [29][30][31][32][33]. Identifying the molecular mechanisms that drive the formation of different shapes can aid in uncovering genotype-to-phenotype relationships [34] and in breeding specific traits of interest in crops [35][36][37][38][39]. ...
Numerous types of biological branching networks, with varying shapes and sizes, are used to acquire and distribute resources. Here, we show that plant root and shoot architectures share a fundamental design property. We studied the spatial density function of plant architectures, which specifies the probability of finding a branch at each location in the 3-dimensional volume occupied by the plant. We analyzed 1645 root architectures from four species and discovered that the spatial density functions of all architectures are population-similar. This means that despite their apparent visual diversity, all of the roots studied share the same basic shape, aside from stretching and compression along orthogonal directions. Moreover, the spatial density of all architectures can be described as variations on a single underlying function: a Gaussian density truncated at a boundary of roughly three standard deviations. Thus, the root density of any architecture requires only four parameters to specify: the total mass of the architecture and the standard deviations of the Gaussian in the three x , y , z growth directions. Plant shoot architectures also follow this design form, suggesting that two basic plant transport systems may use similar growth strategies.
... Root architecture and adaptation to soil water deficit are determined by various important characteristics (Meister et al. 2014). For example, the maximum root length contributes the baseline for water acquisition; root number, root diameter and root length density, which are strongly correlated with root dry weight, root volume and root surface area (Noordwijk et al. 1994;Courtois et al. 2009;Meister et al. 2014;Colombi and Walter 2017), affect the intensity of fixation in the soil profile and the ability to absorb the soil solution within the baseline (Courtois et al. 2009;Wasson et al. 2012). Wasson et al. (2012) suggested an approach for developing new varieties that make better use of deepstored soil water, via the identification and employment of genetic diversity for superior wheat root traits (e.g., deep and highly branched roots). ...
Synthetic hexaploid wheat (SHW), possesses numerous genes for drought that can help breeding for drought-tolerant wheat varieties. We evaluated 10 root traits at seedling stage in 111 F9 recombinant inbred lines derived from a F2 population of a SHW line (SHW-L1) and a common wheat line, under normal (NC) and polyethylene glycol-simulated drought stress conditions (DC). We mapped quantitative trait loci (QTLs) for root traits using an enriched high-density genetic map containing 120370 single nucleotide polymorphisms (SNPs), 733 diversity arrays technology markers (DArT) and 119 simple sequence repeats (SSRs). With four replicates per treatment, we identified 19 QTLs for root traits under NC and DC, and 12 of them could be consistently detected with three or four replicates. Two novel QTLs for root fresh weight and root diameter under NC explained 9 and 15.7% of the phenotypic variation respectively, and six novel QTLs for root fresh weight, the ratio of root water loss, total root surface area, number of root tips, and number of root forks under DC explained 8.5–14% of the phenotypic variation. Here seven of eight novel QTLs could be consistently detected with more than three replicates. Results provide essential information for fine-mapping QTLs related to drought tolerance that will facilitate breeding drought-tolerant wheat cultivars.
... Other models were based on a discrete and/or random formalism that simulates root architecture, growth, branching and death processes. Some of them were based on mathematical processes such as L-systems (Leitner et al., 2010) or fractals (van Noordwijk et al., 1994;Ozier-Lafontaine et al., 1999), or using combinations of the two (Shibusawa, 1994), others were derived from direct measurements performed on real plants taking into account root-soil interaction functions (Diggle, 1988;Jourdan and Rey, 1997a,b;Lynch et al., 1997;Pagès et al., 2004;Dunbabin et al., 2013;Pagès et al., 2014;Barczi et al., 2018), with water and nutrient uptake (Wu et al., 2007;Javaux et al., 2008;Leitner et al., 2010;Postma et al., 2017). Most of these models, with the exception of some of them (Jourdan and Rey, 1997a, b;Pagès et al., 2014;Barczi et al., 2018), are mainly designed for annual and herbaceous plants, with a short lifespan and a limited rooting depth. ...
While the number of studies dealing with fine root dynamics in deep soils layers (depth > 1 m) has increased sharply recently, the phenology, the morphology, the anatomy and the role of deep fine roots are still poorly known in forest ecosystems. This review summarizes the current knowledge on fine root production, mortality and longevity in deep soil layers, mycorrhizal association with deep roots, and the role of deep fine roots on carbon, water and nutrient cycling in forest ecosystems. Plant species are known to be more deeply rooted in tropical ecosystems than in temperate and boreal ecosystems, but deep-rooted species are common in a wide range of climates. Deep fine roots are highly plastic in response to changes in environmental conditions and soil resources. Recent studies show that functional traits can be different for deep and shallow roots, with a possible functional specialization of deep fine roots to take up nutrients. With higher vessel diameter and larger tracheid, the anatomy of deep fine roots is also oriented toward water acquisition and transport by increasing the hydraulic conductivity. Deep fine roots can have a great impact on the biogeochemical cycles in many forests (in particular in tropical areas where highly weathered soils are commonly very deep), making it possible to take up water and nutrients over dry periods and contributing to store carbon in the soil. The biogeochemical models in forest ecosystems need to consider the specificity of deep root functioning to better predict carbon, water and nutrient cycling as well as net ecosystem productivity.
In the context of a recent massive increase in research on plant root functions and their impact on the environment, root ecologists currently face many important challenges to keep on generating cutting‐edge, meaningful and integrated knowledge. Consideration of the below‐ground components in plant and ecosystem studies has been consistently called for in recent decades, but methodology is disparate and sometimes inappropriate. This handbook, based on the collective effort of a large team of experts, will improve trait comparisons across studies and integration of information across databases by providing standardised methods and controlled vocabularies. It is meant to be used not only as starting point by students and scientists who desire working on below‐ground ecosystems, but also by experts for consolidating and broadening their views on multiple aspects of root ecology. Beyond the classical compilation of measurement protocols, we have synthesised recommendations from the literature to provide key background knowledge useful for: (1) defining below‐ground plant entities and giving keys for their meaningful dissection, classification and naming beyond the classical fine‐root vs coarse‐root approach; (2) considering the specificity of root research to produce sound laboratory and field data; (3) describing typical, but overlooked steps for studying roots (e.g. root handling, cleaning and storage); and (4) gathering metadata necessary for the interpretation of results and their reuse. Most importantly, all root traits have been introduced with some degree of ecological context that will be a foundation for understanding their ecological meaning, their typical use and uncertainties, and some methodological and conceptual perspectives for future research. Considering all of this, we urge readers not to solely extract protocol recommendations for trait measurements from this work, but to take a moment to read and reflect on the extensive information contained in this broader guide to root ecology, including sections I–VII and the many introductions to each section and root trait description. Finally, it is critical to understand that a major aim of this guide is to help break down barriers between the many subdisciplines of root ecology and ecophysiology, broaden researchers’ views on the multiple aspects of root study and create favourable conditions for the inception of comprehensive experiments on the role of roots in plant and ecosystem functioning.
Plants use roots to access soil resources, so differences in root traits and their ecological consequences could be a mechanism of species coexistence and niche divergence. Current views of the evolution of root diversity are informed by large‐scale evolutionary analyses based on taxonomically coarse sampling and led to the ‘root trait phylogenetic conservatism hypothesis’. Here we test this hypothesised conservatism among closely related species, and whether root variation plays an ecological role.
We collected root architectural traits for the species‐rich Cape rushes (Restionaceae) in the field and from herbaria. We used machine learning to interpolate missing data. Using model‐based clustering we classified root syndromes. We modelled the proportion of the syndromes along environmental gradients using assemblages and environmental data of 735 plots. We fitted trait evolutionary models to test for the conservatism hypothesis.
We recognised five root syndromes. Responses to environmental gradients are syndrome specific and thus these represent ecomorphs. Trait evolutionary models reveal an evolutionary lability in these ecomorphs.
This could present the mechanistic underpinning of the taxonomic radiation of this group which has been linked to repeated habitat shifts. Our results challenge the perspective of strong phylogenetic conservatism and root trait evolution may more generally drive diversification.
This paper contains a strategy for estimating total aboveground biomass of tropical forests. We developed regression equations to estimate aboveground biomass of individual trees as a function of diameter at breast height, total height, wood density, and Holdridge life zone (sensu Holdridge 1967). The regressions are applied to some 5,300 trees from 43 independent sample plots, and 101 stand tables from large-scale forest inventories in four countries, to estimate commercial and total aboveground biomass per unit area by forest type, and to estimate expansion factors defined as the ratio of aboveground to commercial biomass. The quadratic stand diameter (QSD, i.e., the diameter of a tree of average basal area) in a given forest stand influences the magnitude of the expansion factor. Stands of small trees have large expansion factors (up to 6.4), and as QSD increases, the expansion factor decreases to a constant value (about 1.75). For undisturbed forests in moist, moist transition to dry, and dry life zones respectively, the expansion factors for total aboveground biomass were 1.74, 1.95, and 1.57 respectively. For undisturbed, logged, and nonproductive forest categories used by the FAO to report global commercial wood volume data, we estimated expansion factors of 1.75, 1.90, and 2.00 respectively. Applying these factors to FAO data results in a 28 to 47% increase in previous volume-derived estimates of tropical forest biomass. However, estimates of tropical forest biomass based on small destructive samples continue to be high relative to estimates based on volume data. For. Sci. 35(4):881-902.
Root research in agricultural systems is aimed at increasing the efficiency of using external inputs. Nutrient uptake efficiency is defined as the ratio between uptake during a growing season and the size of the available pool in the soil. Part of the available pool is “unrestrictedly available”, i.e. it can be taken up at the rate required for maximum crop growth; the remaining part is “restrictedly available” and its rate of transport to the root determines the uptake rate possible and hence crop growth. Root characteristics determine for a given soil-nutrient combination which part of the available pool is unrestrictedly available and how fast the restrictedly available pool can be depleted. A mathematical formulation of these effects is now available.
1. The fractal dimensions of seedling root systems of two grasses and two dicots growing in situ against glass sheets have been measured using the dividers method. Values were mostly below 1.5. 2. D was little affected by nutrient supply in Trifolium pratense, but differed significantly between four other species. 3. Values of D increased with age, slowly at first, then more rapidly and finally levelled off or declined slightly. This behaviour appears to correspond to the initial phase of radical branching pattern, which is then maintained. 4. D was negatively correlated with exterior link length and positively with topological index. 5. The fractal dimension appears to be a valuable additional index of the topology and geometry of root systems.
Extreme topologies for root systems can be viewed either as minimizing cost or maximizing efficiency of soil exploration. Plants of red clover Trifolium pratense L., lettuce Lactuca sativa L., four species of Festuca and three of Geranium were grown in a range of nutrient supply conditions, to test this hypothesis. Topology was generally insensitive to changes in the supply of N and P but large differences between species were found: slow-growing perennials had efficient, herring-bone systems, whereas annuals were at the opposite extreme. This accords with earlier predictions. Lengths of external links were reduced by N and of internal links by P. There were large differences between species in mean link lengths. These data confirm the value of topological analyses in understanding the functional architecture of root systems.
Recent collections of Sawdonia ornata have been made from the Strathmore Group which crops out in the Midland Valley of Scotland. The sediments are Lower Devonian, probably Emsian, in age. New morphological features discovered include planar branching of vegetative and fertile axes, abaxial emergences close to branch points and H branching. Well preserved cuticles have permitted reinterpretation of the structure of stornata, rosettes (hair bases), papillae and spines, and striated cuticle is identified as S. ornata. The morphology of sporangia and in situ spores is described. Pyritised axes have revealed details of xylem shape, maturation, changes in the vicinity of branching, tracheidal structure and the arrangement of xylem and cortex in deformed axes.
Drepanophycus spinaeformis Göppert is a long ranging Devonian plant. Although reported from numerous localities, it is far from well understood. Observations on specimens from the Emsian Strathmore Group of Scotland have revealed variation in forms of branching including root-like axes on branches interpreted as rhizomes. The anatomy of the xylem is described from uncompressed pyrite permineralisations. Cuticle characteristics are also described and discussed. The systematic position of the plant is debated since, although the morphology of the leaves, leaf traces, roots and xylem indicate an affinity with the lycopods, there is some doubt as to whether sporangial position permits inclusion in this group.
The morphology of root systems of crop plants was analysed by fractal geometry using an image processing system. Results indicate that these root systems have a fractal structure (D; 1.48 ~ 1.58) within a scale range measuring (0.28 ~ 21 mm), implying that the intricacy of shape of the root systems is characterized by D, called fractal dimension. This suggests the possible application of the fractal analysis to the quantification of root system morphology.