**3. Models of biological responses to nutrients**

The first studies on the limiting factors in the plants growth were developed by Carl Sprengel in 1826 and 1828, and by Liebig in 1840, leading to the rejection of humus theory and formulation of Law of minimum (van der Ploeg et al., 1999). The Law of minimum or Law of response is associated with the absence of nutrient replacement, linear response in production by increase in the quantity of the limiting factor and a maximum plateau of response, in which the plants do not respond satisfactorily anymore to the limiting nutrient.

A posterior mark was the Law of diminishing return of Mitscherlich (1909). The convex exponential equation of Mitscherlich, with a model that includes the maximum asymptotic yield, allows calculating the optimum economic level of fertilization, based in the benefitcost ratio.

The Michaelis-Menten model (Michaelis & Menten, 1913) was developed to describe the enzymatic kinetic in the beginning of the 20 century. The Lineweaver-Burk model (Lineweaver & Burk, 1934), an equation of the linear regression of the reciprocal of Y (enzymatic activity) as a function of the reciprocal of X (concentration of substrate), was used to obtain the kinetic constants of the Michaelis-Menten model: ks (the amount of substrate needed to reach half of maximum enzymatic activity) and kmax (maximum enzymatic activity).

Later, researchers verified that the microbial growth rate was dependent of substrate concentration and both were related to the saturation kinetic typical of enzymatic systems (Monod, 1949; Russell, 1984).

Although the use of saturation kinetic model to explain the nutrients responses by the superior forms of life is not being adopted (Morgan et al., 1975), the Michaelis-Menten model allows to explain the curvilinear relationship of plants and animals to the nutrients and the model of Lineweaver-Burk allows to obtain the kinetic constants, ks (the amount of substrate needed to reach half theoretical maximum response in rate of growth or production of milk, wool, eggs, among others) and kmax (theoretical maximum response in rate of growth or production), according to Lana et al. (2005).

The responses of plants and animals to nutrients as saturation phenomena have important implications in addition to calculation of the rate of decreasing economical return and estimates of nutrients recommendations, such as the consciousness about the excessive use of non renewable natural resources; soil, water, and air pollution; and global warming.

The knowledge about the efficiency of utilization of fertilizers in agriculture will play an important role in the political decisions about the rational use of non renewable natural resources in the future. The natural fertilizer sources have to be used with maximum efficiency and with minimum negative effects in the environment.

### **4. Marginal response or Law of diminishing return in plants**

Recommendations of fertilization are mostly based in the method of calculation of nutrients requirements of a culture and the mineral contribution of the soil. The fertilizers are then calculated to supply the deficiencies. This method allows recommendation of the lower level that maximize the production. However, the method does not indicate changes in the recommendation based on changes in the costs of nutrients and grains. Also, it does not give direct information of the effect of application of other level than the recommended one (Makowski et al., 1999).

It has being utilized a variety of empirical models to predict the responses to nutrients and to calculate the optimum levels of nutrients. Among then, it is included the model of Mitscherlich, square root (Mombiela et al., 1981; Sain & Jauregui, 1993), exponential, linearplus-plateau, linear-plus-hyperbola, quadratic and quadratic-plus-plateau (Cerrato & Blackmer, 1990; Bullock & Bullock, 1994; Makowski et al., 1999, 2001).

The use of saturation kinetics to explain the nutritional responses to nutrients by superior life beings are rarely employed (Morgan et al., 1975). The model of Michaelis-Menten has not being evaluated to make recommendations of fertilization. This model has a great potential in recommendation of use of nutrients in agriculture, by considering the efficiency of use of nutrients and the Law of diminishing return, as observed by Mitscherlich (1909). This model can aggregate important concepts such as responses to different levels of nutrients, benefit-cost ratio, efficiency of use of nutrients, rationality of use of non renewable natural resources and consciousness about environmental pollution.

Linear regressions of reciprocal of plants responses as a function of reciprocal of nutrients supply, methodology known as data transformation of Lineweaver-Burk (Lineweaver & Burk, 1934; Champe & Harvey, 1994), were proposed by Lana et al. (2005) as follow:

$$1/\Upsilon = \mathbf{a} + \mathbf{b} \ast (1/\chi)$$

where:

270 Recent Trends for Enhancing the Diversity and Quality of Soybean Products

Fig. 3. Theoretical population growth (cumulative and growth rate) as function of time.

Malthus, Hubbert and Club of Roma.

cost ratio.

enzymatic activity).

(Monod, 1949; Russell, 1984).

**3. Models of biological responses to nutrients** 

Studies of these phenomena lead us to understand the need for rational use of non renewable natural resources. The kinetic saturation models are important tools generated by science in order to evaluate the efficiency and allow the rational use of non renewable natural resources (Lana, 2005; Lana et al., 2005; Lana, 2007a,b; Lana et al., 2007a,b; Lana, 2008). As a result, they can avoid the complete depletion of the resources and the collapse in food and energy supply, with dramatic consequences for our civilization, as predicted by

The first studies on the limiting factors in the plants growth were developed by Carl Sprengel in 1826 and 1828, and by Liebig in 1840, leading to the rejection of humus theory and formulation of Law of minimum (van der Ploeg et al., 1999). The Law of minimum or Law of response is associated with the absence of nutrient replacement, linear response in production by increase in the quantity of the limiting factor and a maximum plateau of response, in which the plants do not respond satisfactorily anymore to the limiting nutrient. A posterior mark was the Law of diminishing return of Mitscherlich (1909). The convex exponential equation of Mitscherlich, with a model that includes the maximum asymptotic yield, allows calculating the optimum economic level of fertilization, based in the benefit-

The Michaelis-Menten model (Michaelis & Menten, 1913) was developed to describe the enzymatic kinetic in the beginning of the 20 century. The Lineweaver-Burk model (Lineweaver & Burk, 1934), an equation of the linear regression of the reciprocal of Y (enzymatic activity) as a function of the reciprocal of X (concentration of substrate), was used to obtain the kinetic constants of the Michaelis-Menten model: ks (the amount of substrate needed to reach half of maximum enzymatic activity) and kmax (maximum

Later, researchers verified that the microbial growth rate was dependent of substrate concentration and both were related to the saturation kinetic typical of enzymatic systems

Y = responses of plants (grain yield, x 1,000 kg/ha),

a = intercept,

Rationality in the Use of Non Renewable Natural Resources in Agriculture 273

A

B

C

0 100 200 300 400 Fertilizer (kg/ha)

0 0.01 0.02 0.03 0.04 1/fertilizer (kg/ha)

> 0 100 200 300 400 500 Fertilizer (kg/ha)

Fig. 4. Biological responses to nutrients as a function of a second limiting nutrient (A) control (O), decrease in ks (), increase in kmax () and decrease in ks and increase in kmax

Lineweaver-Burk (B); and effect of a second limiting nutrient in the efficiency of use of the

(); reciprocal of production as a function of reciprocal of fertilizer level – plot of

0.0

0

0

4

8

12

kg of fertilizer)

Efficiency (kg of crop/ .

first one (C)

16

20

1

2

1/yield (x 1,000 kg/ha) .

3

4

0.5

1.0

Yield (x 1,000 kg/ha) .

1.5

2.0

b = coefficient of linear regression,

X = amount of nutrient (kg/ha/year).

The theoretical maximum grain production (kmax) is obtained by the reciprocal of intercept (1/a). The amount of nutrient (X) needed to reach half of theoretical maximum response (ks) is obtained by the model presented above, replacing Y by 1/a x 50(%) x 0.01, or dividing the coefficient of the linear regression by the intercept (b/a).

The efficiency of use of fertilizers is calculated dividing the accretion in grain production (Y2 - Y1) by the accretion in fertilization (X2 - X1), from a specific level of fertilizer in relation to the previous level.

Simulations of biological responses to nutrients in the absence or presence of a second limiting nutrient are presented in Table 1 and Figure 4, in which are expected changes in the maximum yield (kmax) and ks of the first limiting nutrient (increase, no effect or decrease). The Figure 4A illustrates four kind of responses in production and models of doublereciprocal are presented in Table 1 and Figure 4B, demonstrating the combination of two values of kmax by two of ks.

The best effects that a second limiting nutrient can cause are by increasing kmax, decreasing ks, or both changes that is even better. However, the most common kind of response is by increasing both kmax and ks. Increase in kmax by increase in productivity with a second nutrient lead to increase the efficiency of use of the first limiting nutrient (Figure 4C), but this benefit decreases sharply by increase in the amount of the first limiting nutrient, especially when ks is low.


Table 1. Constants of linear regression of reciprocal of grain production (x1,000 kg/ha) as a function of reciprocal of amount of fertilizers (kg/ha/year) in hypothetic situations of high or low values of the saturation constants ks (kg of fertilizer/ha) and kmax (x1,000 kg/ha) – see Figure 4B

The plants responses to fertilization depend on soil fertilization, in which high responses occur when soil fertility is low (Figure 5A) and in low level of fertilization, that is the main factor that affects the efficiency of use of fertilizers (Figure 5B).

Equations of data transformation of Lineweaver-Burk were used to explain the effect of fertilization and the effect of a second factor in the yield, ks, kmax and efficiency of use of fertilizers in soybean, bean, wheat and cotton production (Tables 2, 3 and 4).

When limestone was the second factor, there was change in ks and kmax in 34 and 85%; -75 and -10%; and 33 and 22% for soybean fertilized with P2O5 (Table 2). Limestone as a second factor changed ks and kmax, respectively, in -55 and -12% for wheat fertilized with P2O5, and in 9 to 87% in cotton fertilized with K2O.

As seen above, increase or decrease in kmax is associated with the same effect in ks, but increase in kmax associated with exaggerated increase in ks is not desirable because it requires more fertilizer to reach the plateau. In other words, the greater values of ks present greater response to the use of fertilizers in high level of fertilization, but it cannot be advantageous due to the increase in the cost of fertilization.

The theoretical maximum grain production (kmax) is obtained by the reciprocal of intercept (1/a). The amount of nutrient (X) needed to reach half of theoretical maximum response (ks) is obtained by the model presented above, replacing Y by 1/a x 50(%) x 0.01, or dividing the

The efficiency of use of fertilizers is calculated dividing the accretion in grain production (Y2 - Y1) by the accretion in fertilization (X2 - X1), from a specific level of fertilizer in relation

Simulations of biological responses to nutrients in the absence or presence of a second limiting nutrient are presented in Table 1 and Figure 4, in which are expected changes in the maximum yield (kmax) and ks of the first limiting nutrient (increase, no effect or decrease). The Figure 4A illustrates four kind of responses in production and models of doublereciprocal are presented in Table 1 and Figure 4B, demonstrating the combination of two

The best effects that a second limiting nutrient can cause are by increasing kmax, decreasing ks, or both changes that is even better. However, the most common kind of response is by increasing both kmax and ks. Increase in kmax by increase in productivity with a second nutrient lead to increase the efficiency of use of the first limiting nutrient (Figure 4C), but this benefit decreases sharply by increase in the amount of the first limiting nutrient,

Equation Symbol Intercept (a) Coefficient (b) r2 ks kmax 1 O 0.8163 79.789 1.00 98 1.2 2 0.9195 39.591 1.00 43 1.1 3 0.4082 39.894 1.00 98 2.4 4 0.4768 19.483 1.00 41 2.1 Table 1. Constants of linear regression of reciprocal of grain production (x1,000 kg/ha) as a function of reciprocal of amount of fertilizers (kg/ha/year) in hypothetic situations of high or low values of the saturation constants ks (kg of fertilizer/ha) and kmax (x1,000 kg/ha) –

The plants responses to fertilization depend on soil fertilization, in which high responses occur when soil fertility is low (Figure 5A) and in low level of fertilization, that is the main

Equations of data transformation of Lineweaver-Burk were used to explain the effect of fertilization and the effect of a second factor in the yield, ks, kmax and efficiency of use of

When limestone was the second factor, there was change in ks and kmax in 34 and 85%; -75 and -10%; and 33 and 22% for soybean fertilized with P2O5 (Table 2). Limestone as a second factor changed ks and kmax, respectively, in -55 and -12% for wheat fertilized with P2O5, and

As seen above, increase or decrease in kmax is associated with the same effect in ks, but increase in kmax associated with exaggerated increase in ks is not desirable because it requires more fertilizer to reach the plateau. In other words, the greater values of ks present greater response to the use of fertilizers in high level of fertilization, but it cannot be

factor that affects the efficiency of use of fertilizers (Figure 5B).

advantageous due to the increase in the cost of fertilization.

in 9 to 87% in cotton fertilized with K2O.

fertilizers in soybean, bean, wheat and cotton production (Tables 2, 3 and 4).

b = coefficient of linear regression, X = amount of nutrient (kg/ha/year).

to the previous level.

values of kmax by two of ks.

especially when ks is low.

see Figure 4B

coefficient of the linear regression by the intercept (b/a).

Fig. 4. Biological responses to nutrients as a function of a second limiting nutrient (A) control (O), decrease in ks (), increase in kmax () and decrease in ks and increase in kmax (); reciprocal of production as a function of reciprocal of fertilizer level – plot of Lineweaver-Burk (B); and effect of a second limiting nutrient in the efficiency of use of the first one (C)

Rationality in the Use of Non Renewable Natural Resources in Agriculture 275

Coefficient

Efficiency of use of fertilizers (kg of grains/kg of fertilizer)5 50 2 100 150 200 250 300

(b) r2 ks

2 kmax 3

Source of data

4

Intercept (a)

Soybean P2O5 - 0.7536 57.766 1.00 77 1.3 1 + 0.4096 42.198 1.00 103 2.4 1 Soybean P2O5 - 0.3502 30.524 0.98 87 2.9 2 + 0.3801 8.2987 0.99 22 2.6 2 Soybean P2O5 - 0.3103 3.6726 0.68 12 3.2 2 + 0.2535 3.9962 0.53 16 3.9 2 Wheat P2O5 - 0.4169 174.48 1.00 419 2.4 1 + 0.4781 91.00 1.00 190 2.1 1 Cotton K2O - 0.622 4.6865 0.91 7.5 1.6 1 + 0.3284 2.7052 0.97 8.2 3.0 1

Soybean P2O5 - Lim 1 10.0 4.5 2.5 1.6 1.1 0.8 + Lim 16.0 8.1 4.9 3.3 2.4 1.8 Soybean P2O5 - Lim 15.5 9.7 5.6 3.7 2.6 1.9 + Lim 15.4 6.6 2.7 1.5 1.0 0.7 Soybean P2O5 - Lim 28.0 5.5 2.1 1.1 0.7 0.5 + Lim 37.0 8.2 3.2 1.7 1.1 0.7 Wheat P2O5 - Lim 5.0 4.1 3.4 2.9 2.4 2.1 + Lim 8.3 5.7 4.0 3.0 2.3 1.8 Cotton K2O - Lim 10.5 2.0 0.7 0.4 0.2 0.2 + Lim 20.8 4.0 1.5 0.8 0.5 0.3 1 Limestone: without (-) or with (+) 4,000 to 7,000 kg/ha; 2 Kg of fertilizer/ha - P2O5 or K2O; 3 x1,000 kg/ha of grain; 4 1 = Malavolta (1989), p.61, 275 and 283; 2 = Oliveira et al. (1982), p.36; 5 Considering US\$1.208/kg of P2O5 and US\$0.178/kg of soybean, is necessary 6.8 kg soybean to pay one kg of fertilizer. Efficiency lower than 6.8 kg of soybean/kg of P2O5 is not viable. These calculations can be

Table 2. Changes in the constants of linear regression of the reciprocal of grain production (x1,000 kg/ha) as a function of the reciprocal of amount of fertilizer (kg/ha/year), by the second factor, and the respective efficiency of use of fertilizers (kg of grains/kg of fertilizer)

The weight gain in growing bovines in pasture in the dry season is curvilinear as a function of supplement supply, based on corn and soybean meal, in which the supplement conversion (kg of supplement/kg of accretion in weight gain) becomes worse with increase

The milk production by supplemented cows in pasture or in feedlot is also curvilinear as a function of increase in the concentrate supply, based on corn and soybean meal (Figure 7A), in which the marginal increase in milk production per kg of concentrate decreases with increase in the amount of concentrate (Bargo et al., 2003; Pimentel et al., 2006a; Sairanen et al., 2006; Lana et al., 2007a,b), as shown in Figure 7B, and in some studies the milk response to concentrate was satisfactory only up to 2-4 kg of concentrate/animal/day (Fulkerson et

in the supplementation (Lana et al., 2005; Keane et al., 2006; Lana, 2007b) (Figure 6).

Product Fertilizer

(kg/ha/year)

Fertilizer (kg/ha)

used to choose the level of fertilization.

**5. Marginal response in bovines** 

al., 2006).

Second factor 1

Fig. 5. Plants responses to fertilizers in low (O) and high () soil fertility (A); and the effect of soil fertility in the efficiency of use of fertilizer (B)

In the case of soybean (Table 2), considering US\$1.208/kg of P2O5 and US\$0.178/kg of soybean, it is necessary 6.8 kg of soybean to pay 1 kg of fertilizer. Therefore, it is viable to use 50 to 100 kg of P2O5 in the absence of limestone and 100 kg of P2O5 in the presence of limestone. Above 150 kg of P2O5, although in some cases there was still response to fertilizer, especially in high values of ks, the response is not viable economically.

In bean production (Table 3), the second factor (P2O5) increased the kmax of nitrogen from 0.1-0.5 to 1.0-1.6 x 1,000 kg/ha of bean, but also increased the ks (1 to 13 and 17 to 29 kg/ha of nitrogen). When the second factor was nitrogen, this increased the kmax of P2O5 from 0.7- 0.8 to 1.5 x 1,000 kg/ha of bean, but also increased the ks (5 to 15 and 136 to 199 kg/ha of P2O5). In the second case, the high values of ks for P2O5 caused low improvement in the efficiency of use of this fertilizer in low level of fertilization (Table 4). The extra production in this case cannot be enough to pay the extra amount of fertilizers.

A

B

0 100 200 300 400 Fertilizer (kg/ha)

> 0 100 200 300 Fertilizer (kg/ha)

Fig. 5. Plants responses to fertilizers in low (O) and high () soil fertility (A); and the effect

In the case of soybean (Table 2), considering US\$1.208/kg of P2O5 and US\$0.178/kg of soybean, it is necessary 6.8 kg of soybean to pay 1 kg of fertilizer. Therefore, it is viable to use 50 to 100 kg of P2O5 in the absence of limestone and 100 kg of P2O5 in the presence of limestone. Above 150 kg of P2O5, although in some cases there was still response to

In bean production (Table 3), the second factor (P2O5) increased the kmax of nitrogen from 0.1-0.5 to 1.0-1.6 x 1,000 kg/ha of bean, but also increased the ks (1 to 13 and 17 to 29 kg/ha of nitrogen). When the second factor was nitrogen, this increased the kmax of P2O5 from 0.7- 0.8 to 1.5 x 1,000 kg/ha of bean, but also increased the ks (5 to 15 and 136 to 199 kg/ha of P2O5). In the second case, the high values of ks for P2O5 caused low improvement in the efficiency of use of this fertilizer in low level of fertilization (Table 4). The extra production

fertilizer, especially in high values of ks, the response is not viable economically.

in this case cannot be enough to pay the extra amount of fertilizers.

0

0

6

12 18

24

Efficiency (kg of crop/ ..

kg of fertilizer) .

of soil fertility in the efficiency of use of fertilizer (B)

30

36

1

2

Yield (x 1,000 kg/ha) .

3

4


1 Limestone: without (-) or with (+) 4,000 to 7,000 kg/ha; 2 Kg of fertilizer/ha - P2O5 or K2O; 3 x1,000 kg/ha of grain; 4 1 = Malavolta (1989), p.61, 275 and 283; 2 = Oliveira et al. (1982), p.36; 5 Considering US\$1.208/kg of P2O5 and US\$0.178/kg of soybean, is necessary 6.8 kg soybean to pay one kg of fertilizer. Efficiency lower than 6.8 kg of soybean/kg of P2O5 is not viable. These calculations can be used to choose the level of fertilization.

Table 2. Changes in the constants of linear regression of the reciprocal of grain production (x1,000 kg/ha) as a function of the reciprocal of amount of fertilizer (kg/ha/year), by the second factor, and the respective efficiency of use of fertilizers (kg of grains/kg of fertilizer)
