Introduction

Red tomato (Solanum lycopersicum L.) is one of the main vegetables grown in protected agriculture (PA) in Mexico. In 2021, Sinaloa contributed 16.8 % of the total national production under PA (^{Servicio de Información Agroalimentaria y Pesquera [SIAP], 2021}). In PA, intensive vegetable production requires nutrient solutions (NSs) to provide essential nutrients to plants. NSs are generally formed by the mixture of two types of macronutrients, classified as anions (NO₃ ^-, H₂PO₄ ^- and SO₄ ^2-) and cations (K⁺, Ca²⁺ and Mg²⁺), and micronutrients (Fe, Mn, Zn, Mo, B, Cu, Cl and Ni), although producers usually do not apply Cl and Ni (^{Parra-Terraza, 2020}).

The supply of ions to the NS is done through chemical fertilizers (which are electrically neutral compounds), and when dissolved in water they condition the ionic or charge balance (^{Skoog et al., 2001}), which is characterized by having the same number of anion and cation equivalents. This can be represented as: [K⁺] + [Ca²⁺] + [Mg²⁺] = [NO₃ ^-] + [H₂PO₄ ^-] + [SO₄ ^2-]. In other words, a given ion should not be supplied without adding another one with the opposite electrical charge, so the change in the concentration of a cation is carried out with the equivalent change in the accompanying anion. The sum of ions in the NS determines the total concentration in equivalents.

^{Schrevens and Cornell (1993)} and ^{de Rijck and Schrevens (1998)} indicate that the ionic balance defines NSs as systems of ion mixtures, which allows experiments to be designed and analyzed with the mixture methodology and multivariate analysis. In this methodology, the total ionic concentration is constant and only the proportions of the components vary (^{Cornell, 2002}), so the effect of the mixtures on any plant growth or yield variable depends on the proportions of the ions. Multivariate analysis, on the other hand, allows evaluating the effect of one component of the mixture in combination with the effects of the other components. ^{Cornell (2002)} notes that, in experiments with mixtures, the “x_i” proportions of each of the “q” components meet the following restrictions: x_i ≥ 0 (i = 1, 2, 3..., q) and ∑i=1qxi=x1+ x2+…xq=1. These constraints determine the geometry of the experimental region as a dimensional simplex (q - 1); thus, in a three-component system, the geometric space of the mixtures is distributed in an equilateral triangle.

Mixture design has many applications in the areas of food, beverages and health (^{Galvan et al., 2021}). The mixture experiment methodology was proposed by ^{Scheffé (1958)} and implemented in plant nutrition under hydroponics by ^{Schrevens and Cornel (1993)} and ^{de Rijck and Schrevens (1995)} to optimize the chemical composition of the NS. Likewise, with this methodology, a statistical model can be generated that describes the relationship between the proportions of the mixture and the response variable considered, and that allows predicting, within the explored region, the value of the response variable as a function of the regressors.

In the agricultural area, experiments related to mixture design at a global level are scarce, and no reports on this subject were found in scientific journals published in Mexico. ^{De Rijck and Schrevens (1998)} reported the effect of NO₃ ^-, H₂PO₄ ^-, SO₄ ^2- and Cl^- anions in the NS on chicory plants (Cichorium intybus L.). ^{Colla et al. (2008)} optimized the composition of K, Ca and Mg in the NS to improve tomato yield. ^{Valdez-Aguilar and Reed (2010)} evaluated the growth of bean (Phaseolus vulgaris L.) plants in response to NH₄ ⁺, K⁺ and Na⁺ mixtures. ^{Marín et al. (2010)} modeled lily (Lilium cv. Navona) growth with N, K and Ca mixtures. ^{De Rijck and Schrevens (1999)} used the mixture design methodology to optimize the ion composition in the NS and the Ca content in English ryegrass (Lollium perenne).

Considering the above, the aim of this study was to apply the mixture design methodology and multifactorial analysis to optimize the proportions of K⁺, Ca²⁺ and Mg²⁺ in the NS and to obtain the maximum yield of hydroponically-grown tomato using peat as substrate.

Materials and methods

The experiment was conducted from December 15, 2021 to May 6, 2022 in a multi-tunnel greenhouse, with fixed overhead ventilation and anti-aphid mesh on the side walls. The average temperature and relative humidity were 23 °C and 65 %, respectively. Ball-type tomato (Solanum lycopersicum cv. Legionario) plants of indeterminate growth were used. Seedlings were transplanted into a closed-loop hydroponic system consisting of 80 plastic containers with a capacity of 20 L each; 40 containers contained 15 L of brown Sphagnum peat (Kekkila Professional) as substrate plus the tomato plants, and the remaining 40 contained 10 L of NS from the corresponding treatments (Table 1). The substrate containers had a 12 mm diameter hole located 4 cm above the base. A rubber band, an initial irrigation connector, 50 cm of low-density polyethylene irrigation hose and a mini water passage valve were placed in each hole, whose function was to manually drain the excess NS after each irrigation. The drainage was recovered in the container with the corresponding NS. The containers with substrate were placed on six wooden benches 50 cm high and 4.7 m long. The spacing between benches was 1.25 m and the distance between plants on the benches was 0.35 m, giving a density of 2.3 plants∙m^-2.

NSs were designed according to the mixture experiment methodology with Design Expert version 11 (^{Stat-Ease, 2018}). Three components (K⁺, Ca²⁺ and Mg²⁺) were evaluated, where each was expressed as a fraction, and the fractions summed to unity, or 100 % if the components are expressed as a percentage (Table 1).

The cation mixtures in the NSs were established under an axial simplex-lattice design {3,2} (Figure 1), whose geometric space is an equilateral triangle, and each point in the simplex represents a certain mixture of K⁺, Ca²⁺ and Mg²⁺. The number four, placed on the left side of each point, is the number of replications. Due to the constraints 0.25 ≤ K ≤ 0.4, 0.35 ≤ Ca ≤ 0.5 and 0.10 ≤ Mg ≤ 0.25, the evaluated points were represented in the interior region of the inverted simplex triangle. Such constraints limit the size and alter the shape of the experimental region; therefore, transformation of the original components to pseudocomponents based on lower (L-pseudocomponent), upper (U-pseudocomponent), or both lower and upper (U-pseudocomponent) bounds is required. U-pseudocomponents facilitate model fitting and analysis of experimental mixture designs (^{Gorman, 1970}; ^{Crosier, 1984}; ^{Cornell, 2002}). These are calculated with the formula: xi' = Ui-xiU-1, where i = 1, 2..., q, x _i are the actual values of K⁺, Ca²⁺ and Mg²⁺ of the mixture to be transformed to a pseudocomponent, U _i is the upper value of the proportion of the original component to be transformed, U is the sum of the upper values of the proportions of the original components (U=∑i=1qUi) (^{Cornell, 2002}) and xi' is the value of the pseudocomponent, equivalent to the original value of the component.

Figure 1 Geometric arrangement of the K⁺:Ca²⁺:Mg²⁺ mixtures in a nutrient solution in an axial simplex-lattice system {3,2} using the values of the U-pseudocomponents. 

The experimental design used was completely randomized with 10 treatments and four replications, giving a total of 40 experimental units. Each experimental unit consisted of a container with substrate plus a tomato plant and a container with NS. Irrigation was applied daily to the substrate with the plant, and the evapotranspirated water was replenished daily by filling with irrigation water, without adjusting the pH of the NSs, which were renewed every 14 days until the end of the work. The concentration of the NSs was 40 meq∙L^-1 of ions (20 and 20 meq∙L^-1 of cations and anions, respectively).

Cation concentrations (Table 1) were obtained by multiplying the proportion of each cation by the total cation concentration (20 meq∙L^-1), while anion concentrations had the same proportion in all NSs: 0.65, 0.05 and 0.35, equivalent to 12, 1 and 7 meq∙L^-1 of NO₃ ^-, H₂PO₄ ^- and SO₄ ^2-, respectively. Micronutrient concentrations (mg∙L^-1), added to each NS, were: 2, 1.1, 1.2, 0.1, 1.6 and 0.04, of Fe, Mn, B, Cu, Zn and Mo, respectively. Harvesting began 91 days after transplanting (dat) and ended 142 dat. In each cut, fruit number and weight (equivalent to fruit yield [FY]) were recorded. Mean fruit weight (MFW) was obtained from the quotient between fruit weight and fruit number (FN).

Data obtained were subjected to an analysis of variance and Tukey's range test using SAS version 9.4 software (^{SAS Institute, 2013}). In both cases, 0.05 was used as the significance level. The estimated regression models and Piepel’s trace plot to study fruit yield were obtained with Design Expert version 11.

Results and discussion

The analysis of variance in Table 2 shows a significant regression effect of the variables FY (P ˂ 0.0001) and FN (P ˂ 0.0014) on the regressors considering different mixtures of K⁺, Ca²⁺ and Mg²⁺ cations in the NS. The variable mean fruit weight (MFW) had no such effect (P ˂ 0.0817).

The highest values in FY and FN (135.8 t∙ha^-1 and 27, respectively) were obtained with the proportions 0.375 K⁺, 0.400 Ca²⁺ and 0.225 Mg²⁺ in the NS, equivalent to the U-pseudocomponents 0.167x1', 0.667 x2' and 0.167 x3' (Table 3). The lowest FY (84.5 t∙ha^-1) was observed with 0.325 K⁺, 0.475 Ca²⁺ and 0.500 Mg²⁺, corresponding to pseudocomponents 0.167 x1', 0.167 x2' and 0.667 x3' (Table 3).

^{Fageria (2001)} notes that K⁺, Ca²⁺ and Mg²⁺ have similar chemical properties and compete for adsorption sites, absorption, transport and functions in roots and plant tissues; however, under conditions of imbalance of these ions, absorption and yield are affected. Therefore, the highest FY was obtained with the NS that had a better balance of K⁺, Ca²⁺ and Mg²⁺ (0.375, 0.400 and 0.225, respectively), compared to the lowest FY, which was observed with proportions of 0.325 K⁺, 0.475 Ca²⁺ and 0.500 Mg²⁺.

Table 4 shows the multiple linear regression models estimated by Design Expert to predict FN, MFW and FY as a function of the proportions of K⁺, Ca²⁺ and Mg²⁺ in the NSs. These models are of the fourth degree, of the special quartic polynomial type, and their estimated equation is of the form: Ŷ = β₁x₁ β₂x₂ + β₃x₃ + β₁₂x₁x₂ + β₁₃x₁x₃ + β₂₃x₂x₃ + β₁₁₂₃x₁ ²x₂x₃ + β₁₂₂₃x₁x₂ ²x₃ + β₁₂₃₃x₁x₂x₃ ², where Ŷ is the prediction of the response variable based on the multiple regression model estimated for FN, MFW and FY, and the coefficients β_i (β₁, β₂…, β₁₂₃₃) are the estimated values based on the experimental data obtained with the mixtures (x₁ = K, x₂ = Ca and x₃ = Mg) in the NSs. The linear model for FY presented a coefficient of determination (R²) equal to 0.81 (Table 4), which indicates that the model explains 81 % of the variability in the yield data.

The magnitude of the estimated regression effects associated with the linear components of the regressors is: x3' > x2' > x1'; thus, Mg produced the highest FY, followed by K and Ca. The components x1'x2' (KCa) and x2'x3' (CaMg) had a synergistic nonlinear effect, since the yield of each of these mixtures was greater than the simple average yield of the two components: (104.3 + 93.8) / 2 = 99.05 t∙ha^-1 and (93.8 + 109.3) / 2 = 101.6 t∙ha^-1, respectively (Table 3). On the other hand, x1'x3' (KMg) presented a nonlinear effect of antagonism, because the yield of this mixture (94.3 t∙ha^-1) was lower than the simple average of these two components (104.3 + 109.3) / 2 = 106.8 t∙ha^-1, respectively.

^{Xie et al. (2021)} indicate that antagonism between K and Mg cations is common in agricultural production, although such antagonism depends on the plant species, leaf age, and sink- and source organs involved. ^{Tränkner et al. (2018)} note that high K concentrations in the NS inhibit Mg uptake and induce deficiency of this element in plants, which affects photosynthesis, photoassimilate translocation and photoprotection.

With the ternary mixture x1'x2'2x3' (KCa²Mg) equal to 1/6, 2/3, 1/6, equivalent to the proportion 0.375 K⁺, 0.40 Ca²⁺ and 0.225 Mg²⁺ in the NS, the highest yield was obtained (Figure 2). In contrast, the mixtures x1'2x2'x3' (K²CaMg) equal to 2/3, 1/6, 1/6 and x1'x2'x3'2 (KCaMg²) equal to 1/6, 1/6, 2/3 were antagonistic in reducing yield, which is reflected by the terms -510.2 x1'2x2'x3' and -2223.3 x1'x2'x3'2. In this regard, ^{Colla et al. (2008)} obtained the highest yield of hydroponically-grown tomato with proportions of 0.42:0.42:16 of K⁺:Ca²⁺:Mg²⁺ in the NS and a total ion concentration of 42 meq∙L^-1.

Figure 2 Proportions of K⁺, Ca²⁺ and Mg²⁺ in the ten nutrient solutions. 

Based on the regression model for yield (Table 4), it was determined that the optimal proportions of K⁺, Ca²⁺ and Mg^2-, which maximize this variable, were 0.364 K⁺, 0.413 Ca²⁺ and 0.223 Mg²⁺, to obtain an estimated yield of 140.22 t∙ha^-1 and a partial desirability (di) of 0.964. The Design Expert program obtains the yield’s di by converting this response into a function ranging from 0 to 1, where 0 is considered unacceptable and 1 is ideal (^{Derringer & Suich, 1980}).

Piepel's trace plot (Figure 3) (^{Cornell, 2002}; ^{Piepel et al., 2002}) shows the individual effects of the mixture components, where the predicted response for each component is compared to the reference mixture (the centroid, in this case, of the experimental region, with coordinates 0.333:0.333:0.333, K⁺=A:Ca²⁺=B:Mg²⁺=C). This Figure shows the nonlinear effect of the individual components K⁺, Ca²⁺ and Mg²⁺ on the yield.

Figure 3 Piepel’s plot as a function of the individual proportions of K⁺ (A), Ca²⁺ (B) and Mg²⁺(C) in nutrient solution on tomato yield. Predictions are made with respect to the reference mixture (0.333 K⁺, 0.333 Ca²⁺ and 0.333 Mg²⁺). 

Due to the coding with U-pseudocomponents, if their values (represented on the abscissa axis in Figure 3) are to the right of the value of the reference mixture, then the values of the mixture components are lower, and vice versa. For example, by individually increasing the proportion of Ca, Mg and K (in U-pseudocomponents) in the reference mixture by 0.20, the predicted yields are 137.8, 88.2 and 102.8 t∙ha^-1, respectively. On the contrary, by individually reducing said proportion of the mixture by 0.20, the predicted yields are 80.4, 127.6 and 114.4 t∙ha^-1, respectively.

Conclusions

A mixture design with multifactor components was applied to formulate and optimize K⁺, Ca²⁺ and Mg²⁺ in nutrient solutions for greenhouse-grown tomato. A significant effect (P ˂ 0.0015) of the relative proportions of the cations on fruit yield and number was found, which was deduced from the analysis of the estimated special quartic polynomial regression model. The mixture and multiple regression model considered allows optimizing the proportions of K⁺, Ca²⁺ and Mg²⁺ in the nutrient solution.