Biomass-Based Leaf Curvilinear Model for Rapeseed (Brassica napus L.)

: Leaf is one of the most important photosynthetic organs of rapeseed ( Brassica napus L.). To quantifyrelationships between the leaf curve and the corresponding leaf biomassfor rapeseed on main stem, this paper presents a biomass-based leaf curvilinear model for rapeseed. Various model variables, includingleaf length, bowstring length, tangential angle,and bowstring angle, were parameterized based on data derived from the field experiments withvarieties, fertilizer, andtransplanting densities during 2011 to 2012, and 2012 to 2013growingseasons. And then weanalysed the biological significance of curvilinear equation for straight leaves,constructed the straight leaf probabilistic modelon main stem,quantified the relationship between leaf curvature and the corresponding leaf biomass, and constructed the leaf curvilinear model based on the assumption and verification of the curvilinear equation form for curvingleaf. The probability of straight leaf can be quantified with piecewise function according tothe different trend in the normalized leaf ranks ((0, 0.4],and (0.4, 1]). The leaf curvature decreased with the increasing of leaf biomass, and can be described with reciprocal function. The curve of straight leaf and the curving leafcan besimulated by linear equation and thequadratic function, respectively. Our models were validated withthe independent dataset from the field experiment, and the results indicatedthat the model could effectively predict thestraight leaf probability and leaf curvature, which would be useful for linking the rapeseed growth model with the rapeseed morphological model, and set the stagefor the development of functional-structural rapeseed models.


Introduction
Rapeseed is the world's important oil crops [1] with harvest area of 25.3 to 30.9 million ha and total yield of 46.5~72.5 million tons during 2004~2013 [2].At the same time, it is the main oilseed crop in China [3], whose harvest area is about 5.6~7.5 million ha, and the total yieldis about 10.6~14.4 million tons [2] in general.Also, it is one of the main raw material of biodiesel [4].Therefore, it isvery important for ensure food and ecological securitythatpromotthe development of rapeseed production.
Light distribution characteristics in crop canopiesdirectly affectthe light energy utilization efficiency for photosynthesis, dry matter accumulation, and yield formation.Allmost all the growth models predicted the crop canopy light distribution through the Beer's law [5][6][7][8], in that the two key factors for light distribution simulation process, the extinction coefficient and the layered leaf area index, are closely related with the leaf curving characteristics [9].Therefore, quantitatively modeling of the leaf curve could provide a mechanistic way for precisely simulating the crop canopy structure, light distribution, and photosynthesis, and lay a foundation for the predicting of light energy utilization efficiency and yield formation.
At present, there aremany studies on leaf curve modeling.In the study of mathematical characterization of maize canopies [10], leaf curve was described asa general quadratic equation expressed by the initial leaf angle, the coordinates of the leaf tipand theleaf'smaximum height.The general quadratic equation was also used to simluate the leaf curve of maize [11][12][13], rice [14], and other crops by many researchers.
Leaf curve was also fitted into a quadratic function for rice [15,16] and winter wheat [17], or a Gaussian function for spring barley [18] and rice [19].Furthermore, Espana et al. [20] decomposed leaf curvature into two parts, the ascending part was described as a parabolic curve, and the descending part, when existing, was characterized by a portion of an ellipse, and then applied the leaf curvature model to maize canopy 3D architecture and reflectance simulation.Watanabe et al. [21]found that leaf curves could be fitted using Hermite functions though analyzing three angles related to the basal, mid, and tip of leaf.Shi et al. [22] characterized the rice leaf curve by a second order differential equation,including the synthetic effect of leaf blade length, width,specific leaf weight,initial leaf angle, and the deformation coefficient on leaf space shape, using force analyzing on rice leaf.Zheng et al. [23] obtain leaf midrib coordinate points by cubic spatial B-spline interpolation, and characterized the leaf curves as the connecting line of these points.
It is difficult to measure leaf curve because of the intricate leaf shapefor rapeseed.Therefore, the objectives of this researchwere to develop straight leaf probabilistic model, straight leaf curvilinear model, and biomass-based leaf curvilinear model by linking leaf morphological parameters with the corresponding leaf biomass, to validate the hypothesis that the curvilinear function for curving leaves could be fitted as quadratic function, and to provide a reference for linking morphological parameters with corresponding organ biomass, and for the establishment of the FSPMs.
The plots of Exp. 2 and Exp. 3 arranged random with 0.42 m row spacing in 3.99 by 3.5 m area, and the plant spacing was calculated by row spacing and transplanting density.Fertilizer contained 90kgP 2 O 5 ha -1 and90kgK 2 Oha -1 for N1 plots, and180kgP 2 O 5 ha -1 and 180 kgK 2 Oha -1 for N2 plots, and 15 kgboron ha -1 was used as foliage spray for both N1 and N2 plots after bolting.

Measurements
The leafrank on the main stem of 50 randomly selected seedlings for each plot were marked using a red number stampbefore transplanting.Leaf morphological parameters including leaf length (the distance between leaf basal and leaf tip in the straight state, including the leaf blade and petiole, if it exists.LL,forshort), leaf tangentialangle (the angle between the tangential direction of leaf basalandthe main stem.TA, for short), leaf bowstring angle (the angle between the straight line from leaf basal to leaf tip in natural stateand the main stem.BA, for short), and leaf bowstring length (the distance between leaf basal and leaf tip in natural state.LBL, for short) were measured usingstraightedge andprotractordirectly (Fig. 1).

Data Analysis
We writed a C# Program to solve the approximate solution of leaf curve equation parameters through a step of 10 -5 cm.Leaf rank data were normalized to (0, 1] interval, in order to eliminate the apparent differences between treatments and replications.The data from Exp. 1 was used for model development, data from Exp. 2, and Exp. 3 were used for validation.

Model Validation
We validated the models developed in this paper by calculating the correlation (r), the root mean square error (RMSE), the average absolute difference (d a ), and the ratio of d a to the average observation (d ap ) [24], and 1:1 line of simulated and observed properties.Some statistical indiceswere defined as follows: whereiis sample number, n is total number of measurements,n-1=n when n≥30, S i is simulated value, and O i is observedvalue.

Probability and CurvilinearModel for Straight Leaves
In order to represent the extension state of leafbetter, we set the leaf in the Cartesian coordinate system with the leaf basal point as the origin and growth direction of main stem as y-axis, regardless of leaf distorting.According to observations in the Exp. 1, some rapeseed leaves could be considered as straight leaves with small difference between leaf length and leaf bowstring length, as well as leaf tangentialangle and leaf bowstring angle.Therefore, we treated the leaves asstraight leaves if difference between the leaf length and the leaf bowstring length was less than 1 cm or difference between the leaf tangentialangle and the leaf bowstring angle is less than 10°.The curvilinearequation (f(x))of straight leaves could be expressed asa linear functionwhich passes through the originand with the cotangent value of the leaf tangentialangle as the slope, and the function could be described byEq.(1).
where TA can be simulatedby our previous model [25].
where NLRs is normalized leaf rank; A 1 ,A     2.

b-ac b-ac=cot(TA)
From Table 2, wesawthat only quadratic function, sine function, and Hoerl-like function had specific biological significance for all the parameters: for quadratic functionax 2 +bx, -aexpresses the leaf curvature, and bexpresses the cotangentvalue of leaf tangentialangle; for sine functionasin(bx), aexpresses the ordinate value of leaf curve peak point, π/2b abscissa value of expresses theleaf curve peak point, and abexpresses the cotangent value of leaf tangentialangle; for Hoerl-like functionaxb x , aexpresses the cotangent value of leaf tangentialangle, and -bexpresses the leaf curvature.So that we addressedthem and validated the assumptions.

3.1.2.2Solution of the Leaf Curvilinear Equation
L t is leaf tip, x t is the abscissa of L t , y t is the ordinate of L t .

Fig. 3.The geometrical properties of leaf curve
As shown in Fig. 3, the leaf tip could be expressed as(sin (BA)•LBL, cos(BA)•LBL) by solving△OL t y t .We substitutedcoordinatesof origin and L t into curvilinear function for curving leaves to get equation set as Eq. 3.

{ 𝑓(sin(𝐵𝐴) • 𝐿𝐵𝐿) = cos(𝐵𝐴) • 𝐿𝐵𝐿 𝑓
The equation sets and their solutions corresponding to the three curvilinear functions for curving leaves we supposed above were shown in Table 3.

Table 3Three curvilinear functionsfor curving leaves and the corresponding equation setsand their solutions f(x)
Equation set Solution As shown in Table .3, quadratic function,and Hoerl-like function could be solved directly, but there was no analytical solution for sine function, and we writed a C# program to calculatethe approximate solutionwith a step of 10 -5 cm.

3.1.2.3Validation for Leaf Curvilinear Functions
We used the observed value of tangentialangle, bowstring angle, and bowstring length for solvingequation set, except leaf length.Meanwhile, the leaf length also could becalculated as the arc length between leaf basal and leaf tip by the formula of arc length.Therefore, we can validate theleaf curvilinear functionsthrough comparing theobserved leaf length with the calculatedarc length.The three leaf curvilinear functions were validated by 1:1 line of observed andcalculatedleaf length (Fig. 4), and the statistical parameterswere shownin Table 4.The results showed that the bestfittedone (with the highestr and lowestda,dap, andRMSE,Table 4) with specific biological significance (Table 2)for leaf curve was quadratic function, so that the curvilinear model for curving leaves could be described as aquadratic function asEq.(4).
where, Lc i is the leaf curvature on the ith day after emergence.
whereDWLB i is the dry weighton the ith day after emergence; Lc a and Lc b are model parameters whose values and testing datashown in Table 5.

Validation
The models developed above were validatedwerevalidated with the independent datasets fromExp. 2 and Exp. 3, and the results showed that the correlation (r) of simulation and observationprobability for straight leavesand curvature for curving leavesall had significant level at P<0.001, and that the average absolute difference (d a ), the ratio of d a to the average observation (d ap ), and the root mean square error (RMSE) all weresmaller (Table 6).Fig. 6 and Fig. 7 indicated that the observed and simulated probability for straight leavesand curvature for curving leaves were all close to the 1:1 line.Functional-structural plant model with better mechanisticembodies the interaction between plant morphogenesis andcultivarsand environmental factors byintegrated the functionsof growth model with the structuresof morphological model [26].So far, there werelot of research on modeling the leaf curve [10][11][12][13][14][15][16][17][18][19][20][21]whichaimed at 3D reconstruction of crop canopy with a high precision.Theywell explainedthe effects of structures on functions by combiningtechnical instrumentalities like geometrical ray trace [27,28], but could not change the structuresas a responseof the changed functions.Shi et al. [22]linked leaf curve with correspondingdry weightthroughforce analyzing on rice leaf, but the leaf curvilinear equation which waslimited by a form of second order differential equation in this research wasdifficult to use.
Groer et al. [29]established a dynamic 3D model of rapeseed using the modellinglanguage XL [30]and made the morphological model response of different nitrogen levels using a sinks and sourcessystem like GREENLAB and theLEAFC3-N model [31]for photosynthesis at different N-regimes.But the time from sowingto the rosette stage and the conditions except nitrogen was neglected in this model.Jullienet al. [32] constructed a FSPM by characterizing the interactions between architecture and source-sinkrelationships in winter oilseed rapeusing the GREENLABmodel.However, it mainly consideredthe relationship betweenbiomass and leaf area [33],and the description of blade shape was relatively simple.Cao et al. [34] and Zhang et al. [25]establishedmodels to meticulouspredict leaf morphological parameters like leaf length, width, and angles, apart from leaf curve.We developedbiomass-based leaf curvilinearmodel for rapeseed by linking the leaf curvature with the corresponding leaf biomass, which could realize combination of structures with functions,explainedeffects of environmental conditions on leaf morphogenesis, and set the stage for the development of functional-structural rapeseed models.

4.2The Research ProvidedAMechanistic and Universal Method for Leaf Curve Modeling
As to facilitate observation and simulation, we set the leaf intoa Cartesian coordinate system with the leaf basal point as the origin and growth direction of main stem as y-axis.To make the model more precisely, two cases of curving characteristic (straight or curving) were analyzed.In order to determiningthe form V2 of leaf curvilinear equation to be difficult to measure directly, we compared various functions based on their biological significance, and calculated and observed leaf length.For eliminating the apparent differences between treatments and replications, we normalized the leaf rank into the interval of (0, 1].
All these practices madethe model to havebetter mechanistic and universal.

4.3Models Developed In This PaperNeeds To Be Improved
The biomass-basedrapeseed leaf curvilinearmodel developed in this paper for the stage from leaf fully expandeduntil senescence, but the processesof leaf extension,senescence, and distorting is neglected.
Thus, it needs to be studied further.A similar method could also be applied in other crops likemaize andrice, and it could help to the regulation and selection for ideal plant typein the future.

Fig. 2 .
Fig. 2. Changes in the probability of straight leaves with the normalized leaf ranks

Fig. 4 .
Fig. 4. Comparision of the observed and the stimulated leaf length on main stem in 2011-2012

Fig. 5 .
Fig. 5. Changes in the leaf curvature with the leaf dry weight in 2011-2012

5Conclusions
This paper presents a biomass-based leaf curvilinear model for rapeseed designed to explain effects of cultivars and environmental conditions on leaf curve.Various model variables, including leaf biomass, length, and angles were parameterized based on datasets derived from the experiments with rapeseed cv.Ningyou 18, and Ningza 19.With the help of our descriptive model, it will be easy tofulfill calculation of leaf curve via biomass, canopy structure via leaf curve,light distribution and the photosynthesis via canopy structure, and biomass via photosynthesis.It should be possible to connectionmorphological model with physiological model via biomass, and to development the FSRM.
2 ,B 1 ,B 2 , and C 1 are model parameters whose values and testing datashown in Table 1.

Table 1Significance
test of the straight leaf probabilistic model and its parameters *** , ** , and * denote significance at P<0.001, P<0.01, and P<0.05, respectively.The same as below.
Table 2The various functions of leaf curve, their derivative function and the derivative value at the origin, and the biological significance of the parameters

Table 4Comparison
of statistical parameters of the observed and the stimulated leaf blade length on main stem in

Table 5 .
Significance test of the leaf curvature model and its parameter