This article was downloaded by: [Duke University Libraries]On: 07 September 2012, At: 16:44Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Journal of Plant NutritionPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lpla20

A SIMPLE MATHEMATICAL MODEL FOR DIAGNOSIS OFNUTRIENT CONTENT AND DRY MATTER PRODUCTION INWHEATL. Sánchez de la Puente a & Rosa M. Beldaa Instituto de Recursos Naturales y Agrobiología, CSIC, Salamanca, 37071, Spain

Version of record first published: 16 Aug 2006

To cite this article: L. Sánchez de la Puente & Rosa M. Belda (2001): A SIMPLE MATHEMATICAL MODEL FOR DIAGNOSIS OFNUTRIENT CONTENT AND DRY MATTER PRODUCTION IN WHEAT, Journal of Plant Nutrition, 24:4-5, 651-660

To link to this article: http://dx.doi.org/10.1081/PLN-100103659

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form toanyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses shouldbe independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims,proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly inconnection with or arising out of the use of this material.

A SIMPLE MATHEMATICAL MODEL FORDIAGNOSIS OF NUTRIENT CONTENT

AND DRY MATTER PRODUCTIONIN WHEAT

L. SaÂnchez de la Puente1 and Rosa M. Belda2

1Instituto de Recursos Naturales y AgrobiologõÂa, CSIC,

37071 Salamanca, Spain2Dep. BiologõÂa Vegetal, Universidad PoliteÂcnica,

46022 Valencia, Spain

ABSTRACT

The equation y � a� bxa � cx2a proposed as a general function

of nutrition to describe the relationship between the concentra-

tion of nutrients in a plant and the production of dry matter.

The function has a maximum if b> 0 and c< 0, which corre-

sponds to the optimal nutritional value; depending on the value

of the parameter a it may have a point of in¯exion which can

occur before or after the maximum and at varying distances

from it. The parameter a can be set to zero if necessary in

which case the function passes through the origin. Hence its

parametric form is ¯exible and suitable for describing the inci-

dence of nutrients in dry matter production. Furthermore, it is

very simple and easy to use. Its use in describing leaves of

®eld-grown wheat at different stages of growth seems to indicate

that it has some applicability compared with other commonly

used functions, and given its ¯exibility it can offer advances in

the understanding of mineral nutrition in plants. Although

JORUNAL OF PLANT NUTRITION, 24(4&5), 651±660 (2001)

651

Copyright # 2001 by Marcel Dekker, Inc. www.dekker.com

Dow

nloa

ded

by [

Duk

e U

nive

rsity

Lib

rari

es]

at 1

6:44

07

Sept

embe

r 20

12

further assays are necessary, so that the physiological stage of

the plant sample can be accurately de®ned, it seems that the

value of a may de®ne the physiological state of the plant, the

particular circumstances of cultivation de®ning the parameters b

and c. Thus, for example, in 1987, in experimental trials of

wheat close to ¯owering, the value of a for nitrogen in the ¯ag

leaf was 1.75, for the second leaf 1.25, and 1.00 for the third.

The corresponding stages of growth, although not very precisely

de®ned, were close to the maximum for the ®rst leaf, slightly

declining in the second, but more so for the third. Post-anthesis,

nitrogen in the ®rst leaf had an a-value of 0.75; its physiological

state now corresponding to a sharp decline in foliar weight.

Other situations would give characteristic values of a, whose

calibration over several carefully conducted trials would be used

in plant nutrition diagnosis.

INTRODUCTION

Mineral nutrition of plants is, without doubt, one of the mainstays of plant

physiology, a science that has grown most dramatically in the last few decades,

and has, in turn, led to signi®cant increases in plant production. The massive use

of fertilisers, initially applied to produce increased yields was later mediated by

economics, and more recently by conservation considerations. In all this, plant

nutrition has remained a constant theme, never suf®ciently understood.

There is still a need to strengthen the quantitative description of nutrients in

plants. Much work has been done with many species in very different conditions,

yet no general description or universal methodology has emerged from it. It is

important to seek general laws of nutrition which can express the behaviour of

nutrients in plant growth quantitatively (1, 2).

It frequently happens that a set of experimental data can be equally well

®tted by quite different functions. In these conditions which are usually

characterized by reduced-sized trials it is often not possible to have a single law.

To rectify this it is necessary to increase the observation range both in mineral

content and dry matter production.

It is generally thought that the non-existence of a relationship suggests

other problems not related to plant nutrition, or that account must be taken of

functions of the nutrients rather than their total content, thereby demonstrating

even more the importance of a general nutritional equation.

Several functions were tried in order to ®nd one which shaped the nutrient

content-dry matter production curve (2). The conclusion was that square root and

652 SAÂ NCHEZ DE LA PUENTE AND BELDA

Dow

nloa

ded

by [

Duk

e U

nive

rsity

Lib

rari

es]

at 1

6:44

07

Sept

embe

r 20

12

quadratic could approach the general equation for the relationship between

nutrient content and dry matter production (3).

The objective of the present work is to examine a general equation that can

synthesize both the quadratic and square root functions to explain the relationship

between the mineral nutrient composition of a plant and its production.

THEORY

We have studied and examined the `nutrition curve', using many different

functions (1, 2), leading to the choice of two of them for best ®t: the quadratic and

the square-root equation (3).

If we consider the inverted form of both curves (b > 0 and c < 0), the

quadratic is characterized by its horizontal symmetry. The values of x which

correspond to the intersection of the curve with the x-axis are simply calculated as

the roots of equation:

a� bx� cx2 � 0 �1�which are

x1; x2 �ÿb�

������������������b2 ÿ 4cap

2c� ÿb

2c�

������������������b2 ÿ 4cap

2c�2�

(Equation [2] has two distinct roots only if the discriminant b2 ÿ 4ac > 0 and a

single root if b2 ÿ 4ac � 0, otherwise there are no real roots since b2 ÿ 4ac < 0).

The ®rst term of the right-hand side of equation 2 is the x-value at the maximum,

to which the same quantity is added or subtracted for the roots, hence the

symmetry. In the case of the square-root equation,

y � a� b���xp � cx �3�

whose intersections with the x-axis can be determined from the roots of the

equation:

a� b���xp � cx � 0 �4�

Setting v � px changes equation [4] into:

a� bv� cv2 � 0 �5�the roots of which are:

x1; x2 �ÿb�

������������������b2 ÿ 4cap

2c� ÿ b

2c�

������������������b2 ÿ 4cap

2c�6�

MATHEMATICAL MODEL FOR DIAGNOSIS 653

Dow

nloa

ded

by [

Duk

e U

nive

rsity

Lib

rari

es]

at 1

6:44

07

Sept

embe

r 20

12

Resubstituting v by x, and squaring the right hand side gives

x1; x2 �ÿb

2c�

������������������b2 ÿ 4cap

2c

!2

�7�

or

x1; x2 �b2 ÿ 2ca

2c2� b

������������������b2 ÿ 4cap

2c2�8�

Assuming that b2 ÿ 4ac > 0 (the assumption for real roots), the ®rst term is now

displaced from the maximum of x �x � b2=4c2� and the two roots are

equidistantly spaced about it. The curve is asymmetric about the vertex and the

decline of the curve to the right of the vertex is markedly less than its increase

from the left axis.

Both functions correspond to the more general form:

y � a� bxa � cx2a �9�which can be regarded as a `generalized quadratic' in xa, where a is a non-linear

parameter with values greater than 0.

The form that interests us will pass through the origin �a � 0� and have

b > 0 and c < 0 thereby demonstrating the essential status of nutrients at low

concentration but progressing towards a maximum. The x-value at the maximum

of this function is:

xmax �ÿb

2c

� �1a

�10�

and the value of x at the point of in¯exion is

xinf l �ÿb

2c� aÿ 1

2aÿ 1

� �1a

�11�

obtained from setting the second derivative to zero. Equation [11] can be re-

written as:

xinf l �ÿb

2c

� �1a

� aÿ 1

2aÿ 1

� �1a

�12�

in which the ®rst term of the right-hand side is identical to [10] and represents the

x-value corresponding to the maximum. The second factor has the following

properties relative to parameter a: 1) greater than unity if 0 < a < 0:5;2) indeterminate for a � 0:5; 3) imaginary for 0:5 < a < 1:0; 4) zero for

a � 1, and 5) less than unity if a > 1:0.

654 SAÂ NCHEZ DE LA PUENTE AND BELDA

Dow

nloa

ded

by [

Duk

e U

nive

rsity

Lib

rari

es]

at 1

6:44

07

Sept

embe

r 20

12

There is, therefore, a point of in¯exion beyond the maximum in case 1, i.e.,

0 < a < 0:5, but no point of in¯exion is present for cases 2, 3 and 4. For a > 1:0(case 5) the point of in¯exion is before the maximum, but approaches the

maximum as a increases.

The x-value corresponding to the point of in¯exion is therefore determined

by multiplying the maximum by the second term of [12], which varies with a.

The equation with a greater than unity has a point of in¯exion so that the section

of the curve beyond the maximum has considerable ¯exibility and is readily

adaptable to numerous situations without being restricted to the symmetry of, for

example, the logistic equation. It is similar to Hagin's (4) concept of `a production

curve', which for experiments on very de®cient soils shows an almost complete

response with the data following a sigmoidal path. It also has resonance with

Richard's (5) general form for curves of the action of an element that has an

ascending part, which declines after passing through a maximum. The point of

in¯exion can be considered as corresponding to the nutrient content at which

detrimental factors begin to dominate bene®cial ones. The curve is very simple

and easy to calculate and manipulate.

MATERIALS AND METHODS

The data come from ®ve experimental ®elds [for soil analysis results, see

Belda and SaÂnchez de la Puente (6)], each planted with the winter wheat variety

Astral and subjected to seven experimental treatments in four replicate blocks.

Treatments consisted of a control, a combination of two rates of nitrogen (45 and

90 kg ha71 N, N1, and N2, respectively) and two rates of calcium (2,000 and

4,000 kg ha71 Ca, Ca1, and Ca2, respectively). The treatment combinations were:

0, N1, N2, N1Ca1, N1Ca2, N2Ca1, and N2Ca2. All ®elds were given the same

fertilizer regime and received the same cultural treatments. Samples were taken

both at anthesis and post-anthesis. The number of samples at anthesis was

complete (140) but only 112 were taken at post-anthesis (representing four of

the ®ve ®elds). For each sample nitrogen, phosphorus, potassium, calcium,

magnesium, iron, and manganese content and dry weight were determined [for

the analytical procedures, see Belda and SaÂnchez de la Puente (7)]. The sampled

population was large and the range of experimental observations extensive as can

be seen from Table 1.

The aim of the current work is to apply the generalized equation

y � a� bxa � cx2a to the above results with y as dry matter content and x as

nutrient content. The equation parameters have been determined for values of abetween 0.25 and 3.00 at intervals of 0.25 using the program `Lotus 1-2-3 v. 3.1'.

Parameters have been chosen on the basis of statistical signi®cance not only of

the correlation coef®cients, but also of the parameter estimates. Setting a � 3:00

MATHEMATICAL MODEL FOR DIAGNOSIS 655

Dow

nloa

ded

by [

Duk

e U

nive

rsity

Lib

rari

es]

at 1

6:44

07

Sept

embe

r 20

12

gives a value of 0.737 for the second term of [12] pulling the point of in¯exion

towards the maximum. However, higher values of a are not discounted.

RESULTS AND DISCUSSION

The results described here correspond to samples of ®rst, second and third

leaves of wheat enumerated from the top of the plant at anthesis and of the ®rst

leaf post-anthesis. The best-®t equations are given in Table 2 and the

representations of the relationships between the nitrogen content of each leaf at

anthesis and the leaf dry weight are given in Figure 1.

The applicability of the equation y � a� bxa � cx2a is quite extensive in

these experiments. For the ®rst leaf at anthesis it ®ts all seven nutrients studied,

and ®ve for the second and third leaves; post-anthesis, four of the nutrients are

®tted for the ®rst leaf. The level of signi®cance for all ®ts is good. Further,

it provides a more satisfactory ®t than our earlier published equations (1, 2, 3).

Nitrogen gives the best results, having the highest correlation coef®cients,

most particularly at anthesis. The range of nitrogen content is wide (Table 1),

especially in the ®rst leaf, although less in the second leaf and less still in the

third. Figure 1 shows the corresponding equations. Post-anthesis the range is

quite small. The value of a declines and the absolute values of b and c increase

with time (Table 2) which suggests physiological differences between the organs.

The concentrations that give maximum production values occur within the

experimental range, except for the post-anthesis sample, which gives a value

higher than the experimental maximum. This could imply a nitrogen de®cit at that

time. Maximum production values are similar for leaves 1 and 2, but decline for

leaf 3 before increasing again in relation to the leaf at post-anthesis.

Table 1. Experimental Range for Leaf Dry Weight (ldw, g=m2) and for Nutrient

Concentration (N: Nitrogen, %; P: Phosphorus, %; K: Potassium, %; Ca: Calcium, %;

Mg: Magnesium, %; Fe: Iron, ppm; Mn: Manganese, ppm) in Leaves at Anthesis and

Post-anthesis of Wheat

ldw N P K Ca Mg Fe Mn

Anthesis

Leaf 1 7.8±69.8 0.77±4.33 0.19±0.58 1.47±3.46 0.22±0.92 0.11±0.42 74±182 39±413

Leaf 2 7.4±67.5 0.69±3.30 0.19±0.55 1.75±3.79 0.19±0.77 0.05±0.35 43±230 31±432

Leaf 3 4.6±36.3 0.44±3.01 0.09±0.50 1.17±3.20 0.15±1.04 0.03±0.35 44±174 23±546

Post-anthesis

Leaf 1 7.5±58.5 0.44±1.96 0.05±0.32 0.60±1.78 0.48±1.22 0.05±0.64 82±162 39±661

656 SAÂ NCHEZ DE LA PUENTE AND BELDA

Dow

nloa

ded

by [

Duk

e U

nive

rsity

Lib

rari

es]

at 1

6:44

07

Sept

embe

r 20

12

For leaves 1 and 2 the point of in¯exion also occurs within the

experimental range; for the other samples there is no evidence of in¯exion.

In¯exion indicates a physiological changeÐit is the time at which the effect of

negative growth factors starts to increase, slowing down in plant growth until

the maximum is reached, and declining thereafter. In the second leaf it occurs

earlier than in the ®rst, closer to the lowest values, whereas in the third leaf it

does not occur at all, because in its ascending part the function always has a

declining slope.

The different values of nutrient concentrations corresponding to the

maxima, the points of in¯exion, as well as the experimental limits, illustrate

Table 2. Values of the Correlation Coef®cient R � P < 0:01�, the Parameters b and c, and

their Corresponding t-Values, � P < 0:01�, and the Maximum and Point of In¯exion of the

Equation for Different Values of a

a Nutrient R b t(b) c t(c) xmax xin¯

Anthesis ± First Leaf

1.75 N 0.513 8.93 20.1 ÿ0.49 ÿ8.49 3.52 1.77

1.25 P 0.265 272. 16.0 ÿ516. ÿ9.62 0.34 0.08

1.50 K 0.220 18.2 17.6 ÿ2.27 ÿ9.07 2.52 1.00

0.75 Ca 0.218 103. 12.5 ÿ75.0 ÿ5.72 0.61 ±

0.75 Mg 0.233 231. 14.0 ÿ373. ÿ8.56 0.21 ±

1.00 Fe 0.184* 0.587 15.0 ÿ2.4E-03 ÿ8.10 120. 0.00

0.50 Mn 0.242 6.57 17.1 ÿ0.296 ÿ10.5 123. ±

Anthesis ± Second Leaf

1.25 N 0.687 19.2 19.3 ÿ2.33 ÿ6.93 3.11 0.74

1.50 P 0.319 304. 19.2 ÿ713. ÿ10.2 0.36 0.14

2.00 K 0.516 4.67 11.9 ÿ0.112 ÿ2.69 4.57 2.64

0.75 Ca 0.234 85.1 8.31 ÿ55.0 ÿ3.13 0.71 ±

0.50 Mg 0.287 184. 12.8 ÿ259. ÿ8.27 0.13 ±

Anthesis ± Third Leaf

1.00 N 0.650 25.5 30.6 ÿ6.66 ÿ14.1 1.91 0.00

0.75 P 0.338 111. 19.1 ÿ149. ÿ9.22 0.27 ±

1.50 K 0.366 7.44 11.3 ÿ0.553 ÿ3.37 3.56 1.41

0.25 Mg 0.279 84.1 11.2 ÿ83.3 ÿ7.41 0.07 0.33

0.25 Fe 0.178* 18.8 5.80 ÿ3.99 ÿ3.99 31. 155.

Post-anthesis ± First Leaf

0.75 N 0.468 45.1 13.8 ÿ13.1 ÿ4.61 2.05 ±

1.50 Ca 0.298 65.9 12.4 ÿ34.0 ÿ6.02 0.98 0.39

0.75 Mg 0.316 129. 12.7 ÿ130. ÿ7.15 0.39 ±

1.50 Fe 0.239* 0.043 11.4 ÿ1.5E-05 ÿ5.31 129. 51.

*Only at the signi®cant level � P < 0:05�.

MATHEMATICAL MODEL FOR DIAGNOSIS 657

Dow

nloa

ded

by [

Duk

e U

nive

rsity

Lib

rari

es]

at 1

6:44

07

Sept

embe

r 20

12

Figure 1. Relationships between nitrogen content and dry weight of the 1st, 2nd and 3rd

leaf at anthesis. Each regression consists of 140 samples.

658 SAÂ NCHEZ DE LA PUENTE AND BELDA

Dow

nloa

ded

by [

Duk

e U

nive

rsity

Lib

rari

es]

at 1

6:44

07

Sept

embe

r 20

12

the existence and importance of mineral nutrition to the plant. In general terms,

the leaf acts as a sink prior to its maximum development, thereafter providing a

source for younger structures of the plant. These others, in their turn, perform the

same function as far as their weight and mineral content allow. The third leaf

requires less nutrients as it does not need to export as do the ®rst and second.

Thus, for any assay, the nutrient status of each mineral element would be

automatically diagnosed by its position on the curve and the physiological stage

of the leaf sampled. Each curve is uniquely de®ned by its parameters and the

statistical goodness of ®t.

The parameters and signi®cance of the other nutrient elements can be

obtained in similar way. So, for example, potassium is characterised by its high

coef®cient (R� 0.516) in the second leaf, showing greater signi®cance than the

third leaf. Post-anthesis the lack of signi®cance suggests no nutrient problems.

The value of a � 2:0 (for the second leaf ) indicates a more sinuous curve than

for the other leaves. The best-®t equation also has lower values of b and c,

a maximum concentration greater than the experimental values indicating

de®ciency, and a point of in¯exion close to the maximum but within the

observation range. Calcium gives a relatively higher correlation coef®cient in the

leaf post-anthesis, an optimal value within the experimental range and a point of

in¯exion.

Future studies need to de®ne the parameters and shape of the curve,

particularly if the experiments are designed to isolate and study the effects of

individual elements. The applicability of the equation can also be examined using

already published data with a suf®ciently wide experimental range.

CONCLUSIONS

The equation y � a� bxa � cx2a, with a � 0; b > 0; c < 0 and values

of a between 0.25 and 3.00, has been shown to represent the relationship

between the concentration of plant nutrients and plant dry weight. It,

therefore, offers a useful method for the determination of the nutrient status of

plants.

The wide applicability of the equation has been shown by its use on 140

samples from ®eld experiments conducted in 1987 in which leaves from different

positions on the stem and in different phases of growth were used. The hypothesis

is also advanced that a speci®c value of a corresponds to a leaf type of varying

age. The other parameters, b and c vary according to the circumstances of growth

with parameter a usually set to 0.

We would like to suggest the name `Delapuente' equation for the function

y � a� bxa � cx2a. It is a valuable model for the relationship between plant

nutrient content and dry matter production.

MATHEMATICAL MODEL FOR DIAGNOSIS 659

Dow

nloa

ded

by [

Duk

e U

nive

rsity

Lib

rari

es]

at 1

6:44

07

Sept

embe

r 20

12

ACKNOWLEDGMENTS

We thank John Fenlon for his assistance in statistics and in the English

translation of the mathematical terminology, and Dionisio GonzaÂlez for his

technical help.

REFERENCES

1. SaÂnchez de la Puente, L.; Belda, R.M. Analysis of Nine Mathematical

Functions as Models for the Relationships Between the Chemical

Composition and the Dry Weight of Leaves, Shoots and Ears in Wheat.

J. Plant Nutr. 1994, 17, 963±977.

2. SaÂnchez de la Puente, L.; Belda, R.M. Analysis of Nine Mathematical

Functions as Models for Leaf Diagnosis in Wheat Grown in Fields. J. Plant

Nutr. 1995, 18, 2347±2363.

3. SaÂnchez de la Puente, L.; Belda, R.M. Square Root and Quadratic Equations

for the Study of Leaf Diagnosis in Wheat. J. Plant Nutr. 1999, 22 (9), 1469±

1479.

4. Hagin, J. On the Shape of the Yield Curve. Plant Soil 1960, 12, 285±296.

5. Richard, L. Les EÂ tudes de Nutrition MineÂrale Chez les VeÂgeÂtaux:

Contribution aÁ Leur Methodologie; Inst. Rech. Coton et Tex. Exotiques:

Paris, France, 1963; TheÂsis.

6. Belda, R.M.; SaÂnchez de la Puente, L. Mineral Nutrition of Wheat. I. Organ

Stage Relationships. J. Plant Nutr. 1992, 15, 359±369.

7. Belda, R.M.; SaÂnchez de la Puente, L. Mineral Nutrition of Wheat. II.

Importance of Leaves Depending on Their Development and Position on

the Stem. J. Plant Nutr. 1992, 15, 371±384.

660 SAÂ NCHEZ DE LA PUENTE AND BELDA

Dow

nloa

ded

by [

Duk

e U

nive

rsity

Lib

rari

es]

at 1

6:44

07

Sept

embe

r 20

12