Modelling the root system architecture of Poaceae. Can we simulate integrated traits from morphological parameters of growth and branching?

Authors


Summary

  • Our objective was to calibrate a model of the root system architecture on several Poaceae species and to assess its value to simulate several ‘integrated’ traits measured at the root system level: specific root length (SRL), maximum root depth and root mass.
  • We used the model ArchiSimple, made up of sub-models that represent and combine the basic developmental processes, and an experiment on 13 perennial grassland Poaceae species grown in 1.5-m-deep containers and sampled at two different dates after planting (80 and 120 d).
  • Model parameters were estimated almost independently using small samples of the root systems taken at both dates. The relationships obtained for calibration validated the sub-models, and showed species effects on the parameter values. The simulations of integrated traits were relatively correct for SRL and were good for root depth and root mass at the two dates. We obtained some systematic discrepancies that were related to the slight decline of root growth in the last period of the experiment.
  • Because the model allowed correct predictions on a large set of Poaceae species without global fitting, we consider that it is a suitable tool for linking root traits at different organisation levels.

Introduction

Poaceae species are present in many ecosystems and contribute greatly to biodiversity and ecosystem services, as well as to a large number of agricultural products. Most of these grasses have a fasciculate secondary root system made up of a large number of adventitious roots (Cannon, 1949; Kutschera, 1960) which usually give rise to high root length densities in the upper layers of soil (Jackson et al., 1997). Understanding better the functional attributes of these species requires studies on their particular structure at different levels.

In many ecological studies, the functions of particular plant species in ecosystems are often evaluated through ‘integrated’ traits that are measured at the plant level or even at the community level (Suding et al., 2003; Roumet et al., 2006). Examples of such traits are leaf area, leaf mass, specific leaf area (SLA), specific root length (SRL), root mass, root length density and root depth. These integrated traits are valuable because they can give synthetic information about interactions between the plant organisms and their environment, and about important functions in the ecosystem, such as carbon acquisition, water and mineral fluxes, avoidance of water stress and carbon allocation to the soil.

Although valuable in ecosystem analysis, the use of such traits also raises a number of difficulties. From a practical point of view, they may be difficult to estimate, especially those concerning the root systems, because the data they require are very difficult to obtain. For example root mass and root depth have a high acquisition cost, especially on big plants and on heavy or stony soils. Sub-sampling is generally used to estimate some of them, such as SLA and SRL (Cornelissen et al., 2003), adding some uncertainty or bias to their estimate (Birouste et al., 2014). Moreover, several traits exhibit a clear dependence on time, in relation to developmental processes and ageing (Picon-Cochard et al., 2012). For example, important root traits, such as root length, root mass or root depth, are usually growth stage-dependent. Analysing the basic components and the dynamic construction of these integrated traits can be a way to improve their measurement and to interpret their variations more accurately.

Ecophysiologists and geneticists used to consider these integrated traits as targets. In their mechanistic approaches focused on plants (generally over a narrower range of genetic diversity and environmental conditions), they used to evaluate more ‘analytical’ traits, by measurements at lower organisation levels (e.g. typically from the tissue to organ and plant). The developmental variations that are determined by genotype and environment are usually relevant to study at these finer levels, but scaling up the results stays a challenge.

Therefore, developing tools dedicated to linking organisation levels (or scales) is a very important task, both to enable dialogue between these different scientists and to bridge the knowledge between organisation levels, from the organ to the plant and community, and from genotype–environment interaction to ecosystem behaviour. For example, in the aggregative (bottom-up) approach, it is necessary to define and to combine interesting developmental characteristics to create new ideotypes. Conversely, in the analytical (top-down) approach, it is necessary to identify the basic characters of organs and developmental processes which confer interesting features at more integrated levels. The value of architectural models to make this bridge has been shown in several cases, either focusing on the whole plant (Soussana et al., 2012; Maire et al., 2013) or on the root system (Dunbabin, 2007; Pagès, 2011). In the latter work, this question of bridging the scales was addressed from a theoretical viewpoint to understand better how foraging performance could be related to specific developmental parameters or their combinations. Our aim is to present additional modelling work closely associated with experiments in order to evaluate further the capacity of such models to predict, for several species, various aspects of root system dimensions and distribution in the soil.

The success of such an approach will depend a priori on the quality of the model that is used (de Dorlodot et al., 2007). Saltelli & Funtowicz (2014) reminded us that ‘all models are wrong but some can be useful’ if they satisfy some conditions, which depend on the utilisation domain. For our purpose of linking scales in plant studies, the model first should be generic in order to be able to represent several genotypes or species with about the same level of quality. The basic assumptions should be correctly verified for all genotypes. Second, the so-called ‘plant parameters’ that are included in the model should represent mostly the genetic diversity, at least in the target environments. Their inter-species variations should be large in comparison to other variations and uncertainties. Finally, the model should be simple, with a small number of parameters, each with a concrete meaning and able to be obtained independently.

In this paper, we want to test whether a particular model of the root system architecture fulfils these requirements. We will use the ArchiSimple model, presented in a recent paper (Pagès et al., 2013a). It was claimed to be a generic model, because it is based on the synthesis of several particular plant models, and it has a very small number of parameters. The calibration procedure for the basic processes (sub-models) was shown. The whole model is made of several sub-models which could be rather well validated against various sets of data from different species. However, the model was not tested regarding integrated traits, at the upper organization level. Thus, the questions that we want to address concern: our capacity to independently estimate the parameters for a set of 13 different Poaceae species; and the ability of the model to predict the dynamics of integrated traits (SRL, root depth, root mass) without any global fitting procedure.

Materials and Methods

Model

The model was presented in detail with its literature justifications in Pagès et al. (2013a). Here we just recall the main points which are used in the present study.

Structure and general aspects

The ArchiSimple model is a dynamic architectural model in which the root system is represented as a set of segments (some millimetres long) and meristems. Time is divided into 1-d steps during which the virtual root system is modified by the application of rules that formalize the main developmental processes: emission of new adventitious roots from the shoot, elongation of existing roots, acropetal branching, and self-pruning following root decay. The soil is assumed to be homogeneous in the present study. A file containing the list of simulated segments with their attributes (connection information, length, diameter, 3D coordinates and age) is saved at the end of the simulation and used to calculate the integrated traits.

In the following, each developmental process is described. The meanings of parameters, abbreviations and units are summarized in Table 1.

Table 1. Abbreviations, meanings and units of the calibrated parameters
Parameter abbreviationMeaningUnit
D min Minimal diameter (quantile 2%)mm
D max Maximal diametermm
EMEmission rate of adventitious rootsd−1
ELElongation rate vs diametermm d−1 mm−1
IPDInter-branch distancemm
RDMSlope of the regression of lateral diameter vs mother diameter
CVDDCoefficient of variation of the diameter of lateral roots for a given mother diameter
GDsGrowth duration coefficientd mm−2
RTDRoot tissue densityg cm−3

Emission

After the seminal radicle, adventitious roots are emitted. Their number is assumed to be a linear function of time (slope EM, in d−1). Each new emitted root is given a tip diameter drawn at random from a uniform distribution between Dmax/2 and Dmax (parameter Dmax, mm).

Elongation

The elongation of any root starts after a lag time from its initiation, representing the primordium stage, fixed at 5 d. From this time onwards, the root tip becomes mature and the root can elongate until it reaches its growth duration GD. From its tip diameter (D) and age (Age), the elongation rate (ER) of any root is calculated using the following equation:

display math(Eqn 1)

Three parameters are used: EL, the slope of elongation rate against diameter (mm mm−1 d−1); GD, the growth duration of the root (day) calculated from another parameter GDs (see later); and Dmin (mm), a threshold minimal diameter below which there is no possible elongation (diameter of the finest roots).

Growth duration (GD) is also related to apical diameter (D), according to the following equation:

display math(Eqn 2)

(GDs (d mm−2), the growth duration coefficient.) Roots are assumed to stop branching when they stop growing.

Because of soil obstacles and tropisms, the root trajectory usually does not follow a straight line. At each time step, the new growth direction is calculated using three components: its initial direction, a random perturbation representing the effect of mechanical constraints, and a vertical downward component whose strength is given by the gravitropism coefficient (parameter G).

Branching

Lateral roots are initiated as primordia in an acropetal sequence near the tip of the mother root. They are regularly spaced (parameter IPD specifies the inter-primordium distance, in millimetres).

Each new lateral root emerges with a diameter (Dl) related to that of its mother (Dm). The average value Dl is calculated from the following equation:

display math(Eqn 3)

It involves the parameter RDM which quantifies the hierarchy between diameters of mother and lateral roots. Moreover, a standard deviation SD around the mean is calculated as the product Dl × CVDD, CVDD being a parameter (coefficient of variation) which quantifies the variability of the diameters of lateral roots. Then, the diameter value of any lateral (Dl) is drawn at random from a Gaussian distribution with a mean Dl and standard deviation SD. This lateral can actually emerge from the mother root if its diameter is above Dmin and if the mother root is still growing.

Decay and abscission

This process was not activated in the present simulations. We assumed that root decay was rather low in these conditions, because the growing period (4 months) was not very long. This point will be discussed later.

Experimentation

Plant material and growing conditions

We used tillers of 13 different species of Poaceae (see Table 2) found in upland grasslands near Clermont-Ferrand (Massif Central, Auvergne Region, France). We grew these plants in containers, which were foam-insulated PVC tubes (150 cm long, 10 cm diameter). Plastic sleeves were placed inside the tubes in order to easily extract the whole root system at sampling. These were filled with a sieved (below 5 mm) grassland soil (granitic brown soil). This soil was mixed with a controlled release fertilizer (3.5 kg m−3, NPK 14-7-14 Multicote 12, Haifa Israel), and manually packed into the pots. We used 12.5 kg of dry soil per pot (inner volume: 11.4 dm3) giving a mean bulk density of 1.1 g cm−3. The set up was left outside under the natural climate of Clermont-Ferrand (latitude: 45°77′; longitude: 3°14′; altitude: 339 m). During the whole experiment, pots were rainfed and irrigated. We used two containers per species, and five plants were planted in each, transplanted from individual tillers, on 21 March 2013. Plants were sampled (one tube per species) at two dates: mid-June (called date 1 hereafter, c. 80 d after planting, before flowering for all species) and mid-July 2013 (called date 2 hereafter, c. 120 d after planting, when all species had flowered). Each harvest lasted one full week, harvesting two species per day.

Table 2. Poaceae species used in the study and corresponding abbreviations for the figures
SpeciesAbbreviation
Alopecurus pratense AlPr
Antoxanthum odoratum AnOd
Arrhenaterum elatius ArEl
Dactylis glomerata DaGl
Elymus repens ElRe
Festuca arundinacea FeAr
Festuca rubra FeRu
Holcus lanatus HoLa
Lolium perenne LoPe
Phleum pratense PhPr
Poa pratense PoPr
Poa trivialis PoTr
Trisetum flavescens TrFl

Root system excavation and measurements

The tubes were leant on a long sloping tray on which intact extracted root systems attached to the shoots were gently washed with running tap water. Using this protocol, because the soil was loose and root decay very limited at these stages, we did not observe any root loss during washing.

The maximal depth was measured at this stage, and the bases of the plants were excised to count the number of tillers and adventitious roots. The shoots and the bases of the root systems of the plants could be separated to count the tillers and adventitious roots, but the entire root systems could not be disentangled. Therefore, root mass was measured for the five plants together. We could also measure individually the approximate length of three of the long adventitious roots as well as their apical diameter. Then, samples containing young branched parts of roots were taken near the base and in the deepest part of the pot for subsequent scanning. These samples were chosen from ‘internal’ roots, that is roots which did not grow along the tube wall. However, we did not notice differences between the roots which had grown along the tube wall or inside. Root samples were carefully separated and spread in a layer of water several millimetres deep, in a glass tray, using mounted needles. They were scanned with a flatbed scanner (EPSON perfection V700; Seiko Epson Corp., Japan) at a resolution of 1200–3200 dots per inch, using the transparent mode. At least eight images were recorded for each species and date, thus in total 105 and 110 images for date 1 and date 2, respectively. The resolution was adjusted for each species in order to get at least six pixels transversally to the finest roots to measure them with sufficient precision. Previous tests had shown that this adjustment did not introduce any bias because we obtained the same values (on average) when measuring the same roots at these various resolutions.

All diameters and inter-branch distances were measured manually (by eye and mouse click) from the images on the computer screen using the ImageJ software v.1.48 (http://rsbweb.nih.gov/ij/). On the images, we identified a number of branched substructures (with one mother and some daughter roots) and measured on each the diameter of the mother root, the diameter of the laterals and the distance from each lateral to its proximal closest neighbour. Depending on the architectural position of the mother root, the substructures studied had 1–15 laterals. For each species, we measured 373–805 lateral roots on the different scanned samples, for a total number of 5711 roots.

All diameters (also called ‘apical diameter’ hereafter) were measured on the young part of the root, close to the tip, at a location where it was nearly cylindrical. The distance from this position to the very tip was typically between 5 and 50 mm on the thickest roots, and from 2 to 20 mm on the finest. The youngest (most distal) lateral roots, < 2 mm long, were discarded from these measurements. Zones of local thickening could be locally observed, but they were systematically discarded for diameter measurements.

We measured root dry matter content (RDMC), root tissue density (RTD) and specific root length (SRL) on date 2. Fresh scanned roots were weighed and then oven-dried (48 h, 60°C) to calculate RDMC (g g−1) as the ratio of dry mass to fresh mass. Corresponding images were analysed with WinRhizo (Pro 2012b; Regent Instruments, Sainte-Foy, CA, USA) to estimate root length and root volume (with automatic configuration), allowing calculation of SRL (m g−1) and RTD (g cm−3).

Integrated traits for comparison

The comparison of observed and simulated variables was based on three integrated traits (root mass, maximal root depth and SRL) that we chose because they could be measured accurately in pots (for a valid comparison with simulated values), and they have functional significance. Root mass integrates the plant allocation to its roots, and eventually to the soil (Ryser, 2006). Rooting depth is often used to evaluate the water and nutrient reserve available for the plant, and is known to be correlated with drought resistance (Comas et al., 2013). SRL is often used as a synthetic indicator of the efficiency of producing root length with a given root mass (e.g. in relation to acquisition or conservation strategies, Ryser, 2006).

Data analyses

All further data treatments, plots and analyses were done with the R software (R Development Core Team, 2013; http://www.r-project.org/). Linear models were estimated with the ‘lm’ function in order to carry out analysis of variance and covariance and to obtain parameters (with P-values). These analyses allowed us to test the overall effect of the species on parameter values.

Simulations were done using a home-made program that is described in detail in Pagès et al. (2013a).

Results

Model calibration

Parameters were estimated using data from the two sampling dates pooled. Parameter values are given in Table 3.

Table 3. Parameter estimates for each species
Species D min D max EMELIPDRDMCVDDGDsRTD
  1. Units are specified in Table 1. Numbers in parenthesis are standard errors when they could be calculated (on slopes or means).

Alopecurus pratense 0.0780.560.879 (0.11)22.2 (2.4)1.65 (0.072)0.254 (0.0052)0.128 (0.0085)4650.185
Antoxanthum odoratum 0.0700.640.968 (0.088)18.8 (2.4)1.43 (0.072)0.212 (0.0035)0.130 (0.0061)7740.143
Arrhenaterum elatius 0.0980.950.598 (0.050)17.6 (1.2)3.78 (0.070)0.309 (0.0054)0.148 (0.0066)3020.156
Dactylis glomerata 0.0751.10.309 (0.040)14.7 (1.3)2.43 (0.071)0.235 (0.0042)0.170 (0.0089)8030.107
Elymus repens 0.0970.690.324 (0.059)18.8 (4.8)2.43 (0.073)0.238 (0.0042)0.129 (0.0071)2900.118
Festuca arundinacea 0.0960.960.336 (0.041)12.8 (0.35)3.93 (0.050)0.219 (0.0031)0.173 (0.0070)4380.102
Festuca rubra 0.0690.440.557 (0.075)17.8 (0.67)2.02 (0.068)0.157 (0.0030)0.126 (0.0061)8530.131
Holcus lanatus 0.0690.821.13 (0.10)22.4 (1.7)1.69 (0.068)0.233 (0.0058)0.240 (0.018)6760.0961
Lolium perenne 0.0580.480.738 (0.042)16.6 (1.7)2.44 (0.069)0.292 (0.0061)0.169 (0.011)15330.204
Phleum pratense 0.0710.511.01 (0.11)21.0 (1.6)1.46 (0.064)0.248 (0.0040)0.125 (0.0080)6590.145
Poa pratense 0.0560.480.540 (0.044)17.8 (3.9)1.73 (0.069)0.228 (0.0044)0.170 (0.0095)15980.159
Poa trivialis 0.0510.381.08 (0.099)25.8 (1.1)1.09 (0.067)0.255 (0.0043)0.119 (0.0054)14070.164
Trisetum flavescens 0.0510.5700.512 (0.044)16.3 (2.6)1.09 (0.065)0.214 (0.0031)0.125 (0.0074)23030.195

Extreme diameters (Dmin and Dmax) were estimated from the quantiles (probability 0.02 and 1.0, respectively) of the population of mother and daughter roots. During sampling, we took care to include the thickest and the finest roots that we could see on the root systems. The thickest roots were found in rather distal (deep) positions, and were easily visible because the root density was low at this depth. The finest ones were found among the tertiary and quaternary roots in proximal (surface) samples. They were so numerous that it was easy to measure a sufficient number of them. We used the quantile 0.02 of the diameter instead of the strict minimum to smooth the relative uncertainty in their measurement. In Table 3, we can see rather large variations from one species to another, both for Dmin (from 0.051 to 0.097 mm) and Dmax (from 0.38 to 1.1 mm). For most species, we could notice that the apical diameter of the longest roots was lower at the second date than at the first (data not shown).

We estimated the emission rate of adventitious roots (EM parameter) from the slope of the regression line of the number of adventitious roots against growth duration (i.e. from the planting day to the sampling day). For the different regression coefficients, we obtained P-values between 3.7 × 10−4 and 2.7 × 10−8. We noticed large interspecific variations of the emission rate, from 0.31 to 1.13 d−1. Moreover, individual plants also had highly variable numbers of adventitious roots within the same species. Coefficients of variation were c. 32% on average at each date. This rate parameter was expected to be sensitive in the simulation of root mass.

Elongation rate vs diameter (EL parameter) were estimated from the slope of the regression of the mean elongation rate (length divided by duration from planting) against apical diameter. We used the data obtained on the adventitious roots sampled in distal positions, and we assumed that it held for all roots, whatever their branching order (Pagès et al., 2013a,b). The regression line was constrained to pass through the origin (i.e. simple proportionality, see Eqn 1). A covariance analysis on the elongation rate, using the diameter as the covariable, confirmed the significant effect of the species on the slope. P-values on these slopes varied within 7.8 × 10−2 and 2.6 × 10−9. Again, for this parameter, estimates varied greatly, in the range 13–26 mm d−1 mm−1. We will make a sensitivity analysis on this parameter, to evaluate its possible influence in root mass simulation (see later).

We estimated IPD from the average distance between successive branches that we measured on our root samples, for each lateral root. With an ANOVA made on the logarithm of this variable, we checked that the species had a highly significant effect on this trait. In this analysis we used the logarithms instead of the natural values of the variable because the distributions of the inter-branch distances were much skewed for all species. The variations of estimated IPD were remarkably high, ranging from 1.1 to 3.9 mm.

Using Eqn 3, we estimated RDM as the slope of the regression line (without intercept) of the lateral diameters (minus Dmin) against mother diameters (minus Dmin). The species effect was significant in the covariance analysis, and all slope coefficients were highly significant (all P-values were below 10−15). Estimated values were between 0.16 and 0.31.

We estimated CVDD on the residuals of these regressions (Dl observed – Dl estimated) grouped in classes of Dl values. The class intervals were adjusted in order to have 10 or 11 lateral roots per class. The slope of the regression (without intercept) of the standard deviation of these residuals vs their mean value was used as the CVDD estimate. Here again, all regression coefficients were highly significant (all P-values were below 10−15) and substantially variable from one species to another. They ranged between 0.12 and 0.24.

The gravitropism coefficient (G parameter) was not estimated. It was attributed a high value, common to all species, to constrain the root system to fit into the growing containers.

GDs, the growth duration coefficient, was not estimated on the different species, but it was simply adjusted so that the finest roots could reach a length of 5 mm, using a formula combining Eqns 1 and 2: math formula.

RTD was obtained directly from the measurements made on the root samples. It ranged from 0.096 to 0.204 g cm−3. We checked that it was correlated to RDMC (r2 = 0.65).

Simulations, calculation of integrated traits and tests

We simulated 20 sets (replicates) of five plants for the 13 species at the two different dates corresponding to the number of days elapsed from the planting day to the sampling day. The ArchiSimple model was run with the parameters presented in Table 3. From the simulated structures, we calculated the maximal root depth, total root mass and SRL (total root length/total root mass), with one value of these three variables for each set of five plants. These values were compared to the corresponding measurements of the experiment. On several graphs (Figs 1-5), we show the range, which represented a simulated confidence interval, at the 5% level.

Figure 1.

Relationship between the specific root length (SRL) simulated at two dates after planting (c. 80 and 120 d) for the 13 species. Each double segment (vertical and horizontal) represents one individual species with its simulated interval for 20 simulations.

Figure 2.

Comparison between observed and simulated specific root length (SRL) at the second sampling date (c. 120 d after planting). The triangles represent the extreme values obtained for 20 different simulations. Abbreviations are specified in Table 2.

Figure 3.

Comparison between observed and simulated maximal root depth at two dates after planting (a, c. 80 d after planting; b, c. 120 d after planting). The triangles represent the extreme values obtained for 20 different simulations. Abbreviations are specified in Table 2.

Figure 4.

Comparison between observed and simulated root mass at two dates after planting (a, c. 80 d after planting; b, c. 120 d after planting). The triangles represent the extreme values obtained for 20 different simulations. Abbreviations are specified in Table 2.

Figure 5.

Comparison of observed and simulated root mass at the first sampling date (c. 80 d after planting). The points ending the vertical segments represent the average simulated values obtained in a sensitivity analysis when the elongation rate was varied according to its uncertainty (see the text). The position of the label corresponds to the simulations presented in Fig. 4(a). Abbreviations are specified in Table 2.

Regarding the simulation of SRL (Fig. 1), we obtained very similar values for the two dates (r2 = 0.99). On average, SRL increased slightly from date 1 to date 2. The simulated confidence intervals at both dates were very small for all species. On Fig. 2, we present the comparison between the measurements (made on the second date) and the simulations at this same date (r = 0.78). The wide range of SRL was well described, as well as the general variations. However, some relative differences were high for several species (e.g. Festuca rubra) and could not be explained by the expected variability from the model. We also observed a slight overestimation trend.

Observed and simulated root depths at both dates are presented in Fig. 3. The correspondence was rather good, because the points were close to the bisecting line, and most segments intercepted the bisecting line on date 1 (9 out of 13). The correlations were high and consistent: for date 1, = 0.86 (P =1.4 × 10−4); for date 2, = 0.85 (= 5.9 × 10−4). However, we could detect several types of discrepancies. First, there was a general trend with an overall underestimation of rooting depth at date 1 and an overestimation at date 2. This was related to the decrease in growth rate that we could detect on the longest adventitious roots of most species. These roots had a lower apical diameter and a lower elongation rate during the second period. Therefore, it is logical that the overall estimation of EL induced an underestimation at the beginning which was compensated for by an overestimation afterwards. Second, there was a significant correlation between the relative differences (simulated – observed/observed) at date 1 and the relative differences at date 2 (= 0.012). Other differences seemed to be randomly distributed.

The quality of the correspondence between observed and simulated root masses was good (Fig. 4). The correlations were rather high: for date 1, = 0.88 (= 7.1 × 10−5); for date 2, = 0.91 (= 1.9 × 10−5). We also detected a trend of overestimation at date 2. Unlike root depth, there was no significant correlation between the relative differences of root mass at date 1 and the relative differences at date 2 (= 0.71). However, we detected a significant correlation between the relative differences of root mass at date 2 and the relative differences of root depth at date 2 (= 0.012).

Sensitivity analysis on the elongation rate

We thought that our estimates of the elongation rate parameter (EL) were not very accurate because of the small size of our sample and observed variations. Moreover, we saw that discrepancies in root mass could be related to those of root elongation. Therefore, we made a sensitivity analysis on this EL parameter, to analyse the consequences on simulated root mass of a given uncertainty level. We assumed that 10% was the magnitude of our uncertainty on these estimates. Hence, we made simulations with values of 0.9×EL and 1.1×EL, EL being the original estimate presented in Table 3.

The results are presented in Fig. 5. Simulated intervals are represented by vertical segments ended by the average values of the simulations made with the extreme values of EL. They confirm the large effect of these plausible uncertainties of EL on the simulated root masses (21% for their mean variation around the average).

Discussion

The model could be calibrated on all 13 species

Most parameters of this simulation model could be estimated from a rather simple experimental design, given a sufficient number of measurements and the use of high-resolution images from a scanner. For example, the dependency of some variables (elongation rate on diameter, lateral root diameter on mother root diameter), which justify the use of regression slope parameters such as EL or RDM, were well confirmed on all these species. Thus, this work contributed to validate the basic rules of this model regarding the emission of adventitious roots, their elongation rate depending on apical diameter, and the diameter relationships at branching. From this experiment, we could not estimate the growth duration coefficient or the life duration parameter. For GDs, we found a rough approximation which must be checked further. A more precise estimation of these parameters (as well as the elongation rate parameter EL) would require dynamic data, coming from the monitoring of growth and decay of the fine roots. Rhizotrons or minirhizotrons (Garré et al., 2012) would be necessary.

The relative independence of parameter estimates was ensured by the use of a specific protocol which avoided fitting a set of parameter values from a single output variable. The roots that were measured were sampled from different locations (within the whole root systems) at two different dates and they were measured from different viewpoints to increase this independence. Even on the simple model equation defining the diameter relationship in the branching process (Eqn 3) it was possible to estimate RDM and Dmin separately. Branching density (parameter IPD) was also estimated independently using the measurement of inter-branch distance. Conversely, the estimate of the coefficient of variation (CVDD) could depend slightly on that of RDM, because the same data were used, although from another viewpoint. We think that the general problem of independence in parameter estimation is an important one in modelling such systems, and deserves attention (see e.g. Saltelli & Funtowicz, 2014). The subsequent possibility of comparing species or environmental conditions for the same species depends on this independence.

Parameters exhibited large inter-species variations

A species effect was evident and confirmed statistically on all the estimated parameters, except for those on which we could not make statistical tests (diameters Dmax and Dmin). However, these parameters exhibited the same magnitude of interspecific variations. Because all species had grown in a common environment, this point validates the genetic determinism of parameter values in these conditions. Moreover, interspecific differences could be very high for these parameters, because we typically obtained variations of 2–4-fold between extreme values. Morphological and physiological trait variations were presented on the same species but in other conditions by Picon-Cochard et al. (2012). The comparison is not obvious because the measured traits were generally different. However, our results in containers appear to be consistent with their results in grasslands regarding the diameters. The ranking of species was about the same. Regarding root tissue density, our values were often lower.

It would be very interesting to repeat these observations and model calibration in a very different environment to assess the environmental variation, and to find out which parameters are strictly controlled and which exhibit a plastic response. For example, soil strength variations observed in field conditions are likely to modify significantly the root elongation rates and their relationship to root diameter as shown by Bécel et al. (2012) on peach trees. Moreover, the various species might exhibit different responses, as shown on other Poaceae species (Bingham & Bengough, 2003). The model that we used was designed to potentially include such knowledge, and to integrate it from the local root to the whole root system. However, this capacity was not activated in the present experiment because the soil was rather loose and uniform, and because a systematic interspecific calibration is still to be done.

Our calibration work demonstrated that the model can be a guide and a valuable tool for phenotyping root systems in the Poaceae family, because all species conformed to the sub-models and exhibited large quantitative differences for various criteria. This phenotyping exercise should be tested among the genotypes of some given species, in order to assess the magnitude of intra-specific variations of the parameters (Chen et al., 2011).

The simulations gave correct estimates of the integrated traits

As we did not fit on these target variables, we were rather satisfied by the approximations of the integrated traits given by the model. With a small number of parameters, this model made it possible to describe the dynamics of several key aspects of root development for a set of species, and thus to achieve part of the objective of linking scales. This was checked for three different integrated traits (SRL, root depth, root mass) measured on two different dates (for root depth and root mass). The range of variation of these integrated traits was high (from 2- to 8-fold depending on the traits), and it was quite well reproduced.

From such a comparison we cannot expect a perfect quantitative correspondence, because both data and simulated values exhibited variations. We have illustrated this point by simulating virtual replicates, using the stochastic aspect of the model. In most cases, the observed plants came within the simulated intervals, except for the SRL. Regarding this particular trait, we used the RTD measurements which can be also flawed by large uncertainty (see for example the discussions of Birouste et al., 2014) and have a direct impact on mass simulations. Moreover, large variations of RTD can be observed within the plants (Drouet et al., 2004) which must be very seriously considered during the sampling procedures (Cornelissen et al., 2003). The emerging variations that we obtained by simulation are probably underestimated, at least because we considered a fixed root tissue density. The growth duration coefficient (GDs), which has a big effect on the final length and branching of the finest roots, may also have a large impact on SRL. Although difficult, a more accurate view of this phenomenon of growth and branching cessation would be necessary to define more precisely the overall fineness of roots. Further knowledge is also necessary to model intra-plant variations and environmental responses of the density of root tissues (Drouet et al., 2004).

A common approach to improve the agreement between simulated and observed data is to fit a number of parameters against the output variable simultaneously. In our case, it is likely that the simultaneous optimization of three parameters (adventitious emission rate, root tissue density, elongation rate) would have improved the agreement. But applying such a procedure would induce artefact correlations between these parameters, jeopardising their use for subsequent species comparisons. It seemed more important to take a critical look at the discrepancies.

How to explain the observed discrepancies?

In our comparisons of time-dependent traits, we saw three types of discrepancies: an overall time trend, species-dependent discrepancies and ‘random’ discrepancies.

We can interpret the overall trend by the fact that the model simulated a constant elongation rate of the main roots, which was not exactly in agreement with some of our observations. Most species exhibited a decline in their main root elongation rates associated with a reduction of their apical diameter during the second period of growth. Because individual plants were limited in their lateral extension by the container and the neighbours, this fact was probably due to an increase in competition for light, and a consequent limitation of carbohydrate provision to the root system. Both associated symptoms (elongation rate and diameter decrease) might reflect this limitation, according to Thaler & Pagès (1996). The trend could be reinforced by the fact that we did not simulate root decay and self-pruning, while those phenomena began to occur during the second period for the finest roots. The model helped in quantifying the trend as an average difference, and it allowed us to quantify the links between elongation and mass reduction at the root system scale. These competition phenomena occur also in the field. Therefore, the calibration of the maximal biomass allocated over time (a feature that exists in the model and that we did not use in the present case) would be useful to make representations of these conditions of competition for carbohydrates. This would require estimating the radiation intercepted by the individual plant and its conversion into root biomass.

The species-dependent discrepancies were indicated by the correlations that we observed in the differences at the two dates. Because we used a common set of parameters estimated for each species, they could originate from common calibration errors which had common effects at both dates. This was likely to happen. In this sense, it would demonstrate consistency in the prediction errors. They could also mean that some particular features, existing in particular species, were not accounted for by the model. Calibrating the model at each date separately would help to discriminate between these hypotheses, but we are worried that the split datasets would become too small to guarantee correct estimates for each period.

Regarding the random discrepancies, the simulation of confidence intervals helped us to separate the ‘normal’ hazard differences from those which deserve particular interpretations. This was possible because the model was stochastic; it simulated both expected central values and variations.

Future prospects

Beyond some discrepancies that we found in this particular study, we think that the calibrated model is a useful tool which can serve other purposes, particularly in the general arena of root–soil modelling. For example, it should help to better evaluate the consequences of the interspecific variations of several developmental parameters that we have observed on the overall dynamics of soil colonization. To complete this calibration, in order to make more use of the spatial capacities of the model, it will be necessary to calibrate the gravitropism parameter. This could be done for several species using the drawings made by Kutschera (1960). In the near future, new imaging methods (e.g. X-ray CT, MRI) will become available and should help with the validation of the 3D aspects in more detail.

The simplicity and generality of this model contrast with many others (Dunbabin et al., 2013) which should be an advantage for many uses.

Acknowledgements

We thank Sandrine Revaillot, Patrick Pichon and Maurice Crocombette for their technical assistance and Valérie Tourel for her meticulous measurements on the root images. This work has been granted by Agropolis fondation (‘Rhizopolis’ project) and INRA (EA Department) in the frame of a ‘Pari scientifique’ project.

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