**4. Multi-element Balances in plant nutrition**

#### **4.1. Sequential binary partition**

The sensitivity, specificity, PPV and NPV criteria are presented in Table 3. We expect perform‐ ance criteria to be at least 80%. Low specificity indicates that some interactions with K leading to high yield, possibly involving Ca and Mg, have been ignored. Apparently, the *ceteris paribus* assumption did not apply to this study. The fact that the balance allows to adjust the K to other

(a)

(b)

**Figure 10.** Cate-Nelson partitioning of the relationship between guava fresh fruit yield. critical values were (a) 1.2

mmolc K dm-3. TN = 2; FN = 0; TP = 11; FP = 1 and (b) -2.07. TN = 4; FN = 0; TP = 9; FP = 1.

cationic species may account for failure to meet the *ceteris paribus* assumption.

98 Soil Fertility

Plant nutrients are classified as essential macronutrients measured in % (N, S, P, Mg, Ca, K, Cl), essential micronutrients measured in mg kg-1 (Mn, Cu, Zn, Mo, B) and beneficial nu‐ trients generally measured in mg or µg kg-1 but occasionally in % (Si, Na, Co, Ni, Se, Al, I, V) [64, 65, 15]. The plant ionome is defined as elemental tissue composition as related to the genome [66]. A subcomposition of plant ionome could be defined by the following simplex for conducting statistical analysis:

$$\mathbf{S}^{\,\,D} = \mathcal{C}\{\mathbf{C}, \,\mathbf{N}, \,\mathbf{P}, \,\mathbf{K}, \,\mathbf{C}a, \,\mathbf{M}\mathbf{g}, \,\mathbf{B}, \,\mathbf{S}, \,\mathbf{C}l, \,\mathbf{C}u, \,\mathbf{Z}u, \,\mathbf{M}u, \,\mathbf{F}e, \,\mathbf{M}u, \,\mathbf{F}\_v\} \tag{9}$$

Where *Fv* is the filling value between 1000 g kg-1 and the sum of analytical data and *D* = 15, the total number of components including *Fv*. An SBP scheme can be elaborated based on well documented roles and stoichiometric rules provided by [17, 14, 12], who reported a large number of dual and multiple nutrient interactions in plants such as:


#### **4.2. Datasets**

The tissue composition can be altered by environmental and seasonal factors. A dataset of 1909 potato (*Solanum tuberosum* L. cv. 'Superior') yields and ionomes was collected at five developmental stages between 1987 and 2002 in Quebec, Canada. The first mature leaf from top was sampled at 20-cm height (n = 502), bud stage (n = 544), beginning of flowering (n = 587), full bloom (n = 213) and fast tuber growth (n = 63) and analyzed for N, P, K, Ca, and Mg. The plant nutrient signatures at each developmental stage were compared using box‐ plots and discriminant analysis.

opmental to the other. The *ilrs* can thus be described by trend equations and sample com‐ position be detrended toward a specific developmental stage for diagnostic purposes. The seasonally increasing N/P ratio may indicate possible N or P imbalance at some point in time assuming a stationary N:P stoichiometric rule. However, the N/P ratio was found to vary widely between plant species during plant development, depending on relative growth rates [38]. The Redfield N/P ratio in eukaryotic microbes is a balance between two fundamental processes, protein and rRNA synthesis, resulting in a stable biochemical at‐ tractor toward a given protein: to RNA ratio [68]. The N/P ratio of plant biomass is used as indicator of N or P limitation but critical N/P ratios change with age and function of tissues [38]. Immature leaves of young plants assimilate and grow simultaneously and their demand for N and P follows the stoichiometryic rules of basic biochemical process‐ es such as photosynthesis, respiration, protein synthesis, DNA duplication and transcrip‐ tion; growth becomes restricted to active meristems such as young leaves, shoot tips and inflorescences when plants get older [38]. Mature leaves are still photosynthetically active but no longer grow, which greatly reduces the P requirements for RNA and increases the N/P ratio. Nucleic-acid P can be mobilized from older leaves and transferred to younger leaves, leading to higher N/P ratios in older leaves [69], such as the first mature leaves of

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(a) (b)

**Figure 11.** The (a) boxplots and (b) coda dendrogram of the four balances for five development stages.

potatoes used as diagnostic tissue [67].

A critical hyper-ellipsoid can be viewed as a particular zone of the nutrient balance space where the probability to obtain high yield is high enough to satisfy the practitioner. The points lying inside the hyper- ellipsoid would be qualified as "balanced", and those lying outside the multi-dimensional construct, as "imbalanced". The practitioner might delineate intermediate zones if needed. Fertilizer trials were conducted to monitor balance change to‐ ward optimum nutrient conditions defined by the critical ellipses. In a P trial, P treatments applied to a P deficient soil were 0, 33, 66, 98 and 131 kg P ha-1. In a K trial, K treatments of 0, 50, 100 and 150 kg K ha-1 were applied to a K deficient soil. The diagnostic leaf of potato was sampled at the beginning of flowering [67].

#### **4.3. Seasonal change in nutrient compositions**

The boxplots and the CoDa dendrogram illustrate the center and dispersion of nutrient bal‐ ances per development stage (Figure 11). The [N, P, K | Ca, Mg] balance tended to decrease markedly during the season while the [N | P] and [Ca | Mg] balances tended to increase, and the [N,P | K] balance tended to decrease. The fast decrease in [N,P,K | Ca, Mg] balance is attributable to more N, P and K than Ca and Mg being transferred toward growing leaves during exponential growth and toward tubers during maturation. The K was more affected than N and P.

The discriminant scores (dots) and eigenvectors, as well as confidence regions at 95% lev‐ el delineated the distributions of populations (large grey ellipses) and means (small color colored ellipses) across stages of plant development (Figure 12). The first axis, dominated by the Redfield [N | P] balance followed by the [N, P, K | Ca, Mg] balance, captured 92% of total inertia. It is noteworthy that the nutrient balance changed orderly from one devel‐ opmental to the other. The *ilrs* can thus be described by trend equations and sample com‐ position be detrended toward a specific developmental stage for diagnostic purposes. The seasonally increasing N/P ratio may indicate possible N or P imbalance at some point in time assuming a stationary N:P stoichiometric rule. However, the N/P ratio was found to vary widely between plant species during plant development, depending on relative growth rates [38]. The Redfield N/P ratio in eukaryotic microbes is a balance between two fundamental processes, protein and rRNA synthesis, resulting in a stable biochemical at‐ tractor toward a given protein: to RNA ratio [68]. The N/P ratio of plant biomass is used as indicator of N or P limitation but critical N/P ratios change with age and function of tissues [38]. Immature leaves of young plants assimilate and grow simultaneously and their demand for N and P follows the stoichiometryic rules of basic biochemical process‐ es such as photosynthesis, respiration, protein synthesis, DNA duplication and transcrip‐ tion; growth becomes restricted to active meristems such as young leaves, shoot tips and inflorescences when plants get older [38]. Mature leaves are still photosynthetically active but no longer grow, which greatly reduces the P requirements for RNA and increases the N/P ratio. Nucleic-acid P can be mobilized from older leaves and transferred to younger leaves, leading to higher N/P ratios in older leaves [69], such as the first mature leaves of potatoes used as diagnostic tissue [67].

**•** Mg with N, P, B, Fe, Mn, Mo, Na, and Si;

**•** Fe with N, P, Ca, Mg, Cu, Mn, Co, and Zn;

**•** Zn with N, P, K, Ca, Mg, S, Na, Zn, Fe, and Mn;

The tissue composition can be altered by environmental and seasonal factors. A dataset of 1909 potato (*Solanum tuberosum* L. cv. 'Superior') yields and ionomes was collected at five developmental stages between 1987 and 2002 in Quebec, Canada. The first mature leaf from top was sampled at 20-cm height (n = 502), bud stage (n = 544), beginning of flowering (n = 587), full bloom (n = 213) and fast tuber growth (n = 63) and analyzed for N, P, K, Ca, and Mg. The plant nutrient signatures at each developmental stage were compared using box‐

A critical hyper-ellipsoid can be viewed as a particular zone of the nutrient balance space where the probability to obtain high yield is high enough to satisfy the practitioner. The points lying inside the hyper- ellipsoid would be qualified as "balanced", and those lying outside the multi-dimensional construct, as "imbalanced". The practitioner might delineate intermediate zones if needed. Fertilizer trials were conducted to monitor balance change to‐ ward optimum nutrient conditions defined by the critical ellipses. In a P trial, P treatments applied to a P deficient soil were 0, 33, 66, 98 and 131 kg P ha-1. In a K trial, K treatments of 0, 50, 100 and 150 kg K ha-1 were applied to a K deficient soil. The diagnostic leaf of potato was

The boxplots and the CoDa dendrogram illustrate the center and dispersion of nutrient bal‐ ances per development stage (Figure 11). The [N, P, K | Ca, Mg] balance tended to decrease markedly during the season while the [N | P] and [Ca | Mg] balances tended to increase, and the [N,P | K] balance tended to decrease. The fast decrease in [N,P,K | Ca, Mg] balance is attributable to more N, P and K than Ca and Mg being transferred toward growing leaves during exponential growth and toward tubers during maturation. The K was more affected

The discriminant scores (dots) and eigenvectors, as well as confidence regions at 95% lev‐ el delineated the distributions of populations (large grey ellipses) and means (small color colored ellipses) across stages of plant development (Figure 12). The first axis, dominated by the Redfield [N | P] balance followed by the [N, P, K | Ca, Mg] balance, captured 92% of total inertia. It is noteworthy that the nutrient balance changed orderly from one devel‐

**•** Mn with N, P, K, Ca, Mg, B, Mo, Ni, and Zn;

**•** Cu with N, P, K, Ca, Fe, Mn, and Zn;

**•** Mo with N, P, K, S, Fe, and Mn.

plots and discriminant analysis.

sampled at the beginning of flowering [67].

**4.3. Seasonal change in nutrient compositions**

**4.2. Datasets**

100 Soil Fertility

than N and P.

**•** B with N, P, K, and Ca;

**Figure 11.** The (a) boxplots and (b) coda dendrogram of the four balances for five development stages.

The green and red points in Figure 13 represent specimens showing balanced and imbal‐ anced nutrition, respectively. The fertilization of the potato should move nutrient signature toward the hyper-ellipsoid center. Added P perturbed the internal nutrient balance of cv. 'Superior' growing on a P deficient soil (Figure 14). The P trial showed that an addition of 98

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(a) (b)

**Figure 14.** The P treatments applied to a P-deficient soil at rates of 0, 33, 66, 98 and 131 kg P ha-1 (rates increasing as red circles enlarged) perturbed the potato ionome that moved toward the critical ellipse. The ternary diagram scaled on proportions on the left (A) was centered and rescaled on the right (B) for better appreciation of the shape of the

In Figure 15, it can be observed that added K also perturbed the nutrient balance: the potato ionome moved toward the critical ellipse. The 2nd K rate moved the K deficient plant ionome closer to the critical ellipse, but Ca shortage maintained the crop outside the critical ellipse. From the second application rate up, the perturbation was small. In this case, the Ca was

The perturbation on 5 nutrients can be illustrated by a matrix of ternary diagrams (Figure 16). These diagrams show 2 nutrients and an asterisk (\*) representing the sum of the 3 other components. The central dot is the mean of high yielders surrounded by its 95% confidence

likely to be the most limiting nutrient as shown on the ternary diagram.

ellipse and the trend of the fertilization effect.

region represented by a black line.

kg P ha-1 allowed the balance to penetrate into the critical ellipse.

**Figure 12.** Discriminant analysis shows that nutrient balance changes orderly between the early and late develop‐ mental stages.

#### **4.4. Defining reference balances for diagnostic purposes**

The confidence region of optimum nutrition was defined by a 4-dimensional hyper-ellipsoid (Figure 13).

**Figure 13.** The ellipses define the optimum conditions of potato nutrition as found in surveyed high-yield potato stands (95% confidence regions about mean) illustrated as (a) balances and (b) scaled and centered concentrations.

The green and red points in Figure 13 represent specimens showing balanced and imbal‐ anced nutrition, respectively. The fertilization of the potato should move nutrient signature toward the hyper-ellipsoid center. Added P perturbed the internal nutrient balance of cv. 'Superior' growing on a P deficient soil (Figure 14). The P trial showed that an addition of 98 kg P ha-1 allowed the balance to penetrate into the critical ellipse.

**Figure 12.** Discriminant analysis shows that nutrient balance changes orderly between the early and late develop‐

The confidence region of optimum nutrition was defined by a 4-dimensional hyper-ellipsoid

(a) (b)

**Figure 13.** The ellipses define the optimum conditions of potato nutrition as found in surveyed high-yield potato stands (95% confidence regions about mean) illustrated as (a) balances and (b) scaled and centered concentrations.

**4.4. Defining reference balances for diagnostic purposes**

mental stages.

102 Soil Fertility

(Figure 13).

**Figure 14.** The P treatments applied to a P-deficient soil at rates of 0, 33, 66, 98 and 131 kg P ha-1 (rates increasing as red circles enlarged) perturbed the potato ionome that moved toward the critical ellipse. The ternary diagram scaled on proportions on the left (A) was centered and rescaled on the right (B) for better appreciation of the shape of the ellipse and the trend of the fertilization effect.

In Figure 15, it can be observed that added K also perturbed the nutrient balance: the potato ionome moved toward the critical ellipse. The 2nd K rate moved the K deficient plant ionome closer to the critical ellipse, but Ca shortage maintained the crop outside the critical ellipse. From the second application rate up, the perturbation was small. In this case, the Ca was likely to be the most limiting nutrient as shown on the ternary diagram.

The perturbation on 5 nutrients can be illustrated by a matrix of ternary diagrams (Figure 16). These diagrams show 2 nutrients and an asterisk (\*) representing the sum of the 3 other components. The central dot is the mean of high yielders surrounded by its 95% confidence region represented by a black line.

**5. Compositional modeling of C mineralization of organic residues in**

The carbon, nitrogen, phosphorus and sulfur cycles are interconnected in agroecosystems and often expressed using stoichiometric rules [4]. The ratio between total C and total N is the most simplified rule used in C mineralization studies but the Corganic/Norganic and lignin/ Norganic ratios are also common. However, several biochemical components of organic matter are omitted in most studies, resulting in loss of information on the system. There are few studies on the relationship between labile or recalcitrant C and the biochemical composition of organic residues added to soil. [70] analyzed ash and N contents as well as four C frac‐ tions in organic residues representing pools of increasing resistance to decomposition. In this section, we related labile C in organic residues to this 6-part compositional vector of or‐ ganic residues. The components were expressed as fractions on dry weight basis to compute a biological stability index using multiple linear regression models. The compositional vec‐

Where SOL = soluble matter, HEM = hemicellulose, CEL = cellulose, L IG= lignin, and N =

Because scale dependency induces spurious correlations [71, 72, 73] and linear regression models are solved based on correlations between variables, the interpretation of regression coefficients is scale-dependent. To illustrate the problem of spurious correlations, chemical fractions were scaled on organic mass basis and analyzed using multiple linear regression. The balance scheme reflected the C/N ratio and the order of decomposability of biochemical components (Figure 17). The SOL fraction was isolated from other biochemically labile frac‐ tions because its composition is complex, possibly including sugars, amino-sugars, aminoacids, and polypeptides as well as more recalcitrant or bacteriostatic easily solubilized polyphenols such as fulvic acids, tannic substances, resins, intermediate products, etc. The

*Ilr balance* **SOL HEM CEL LIG Total N Ash r s** [SOL,HEM,CEL,LIG,N | Ash] 1 1 1 1 1 -1 5 1 [SOL,HEM,CEL,LIG | N] 1 1 1 1 -1 0 4 1 [SOL,HEM,CEL | LIG] 1 1 1 -1 0 0 3 1 [SOL | HEM,CEL] 1 -1 -1 0 0 0 1 2 [HEM | CEL] 0 1 -1 0 0 0 1 1

**Table 4.** Sequential binary partition of the biochemical composition of organic residues in Thuriès et al. (2002)

balance scheme was formalized by SBP as shown in Table 4.

*S* <sup>6</sup> =(*SOL* , *HEM* , *CEL* , *LIG*, *N* , *Ash* ) (10)

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**soils**

tor was defined as follows:

total nitrogen.

**Figure 15.** Ternary diagrams showing K treatments applied to a K-deficient soil at rates of 0, 50, 100 and 150 kg K ha-1 (rates increasing as red circles enlarged (a) zoomed on proportions and (b) centered and scaled.

**Figure 16.** General view of plant ionomes in P and K deficient soils moving toward the reference ellpse for highly pro‐ ductive agroecosystems as P (blue) and K (red) fertilizers are added.
