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**|**Ann Bot**|**v.102(6); 2008 December**|**PMC2712405

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Ann Bot. 2008 December; 102(6): 967–977.

Published online 2008 October 7. doi: 10.1093/aob/mcn188

PMCID: PMC2712405

Duncan J. Greenwood,^{1,}^{*} Tatiana V. Karpinets,^{2} Kefeng Zhang,^{1} Angela Bosh-Serra,^{3} Arianna Boldrini,^{4} and Lyudmila Karawulova^{5}

Received 2008 July 16; Revised 2008 August 12; Accepted 2008 August 19.

Copyright © The Author 2008. Published by Oxford University Press on behalf of the Annals of Botany Company. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

This article has been cited by other articles in PMC.

Numerous estimates have been made of the concentrations of N and P required for good growth of crop species but they have not been defined by any unifying model. The aim of the present study was to develop such a model for the dependence of the N : P ratio on crop mass, to test its validity and to use it to identify elements of similarity between different crop species and wild plants.

A model was derived between plant N : P ratio (*R*_{w}) and its dry biomass per unit area (*W*) during growth with near optimum nutrition by considering that plants consist of growth-related tissue and storage-related tissue with N : P ratios *R*_{g} and *R*_{s}, respectively. Testing and calibration against experimental data on different crop species led to a simple equation between *R*_{w} and *W* which was tested against independent experimental data.

The validity of the model and equation was supported by 365 measurements of *R*_{w} in 38 field experiments on crops. *R*_{g} and *R*_{s} remained approximately constant throughout growth, with average values of 11·8 and 5·8 by mass. The model also approximately predicted the relationships between leaf N and P concentrations in 124 advisory estimates on immature tissues and in 385 wild species from published global surveys.

The N : P ratio of the biomass of very different crops, during growth with near optimum levels of nutrients, is defined entirely in terms of crop biomass, an average N : P ratio of the storage/structure-related tissue of the crop and an average N : P ratio of the growth-related tissue. The latter is similar to that found in leaves of many wild plant species, and even micro-organisms and terrestrial and freshwater autotrophs.

Elucidating changes in N : P ratios during plant growth and developing mechanistic models for them could help solve major problems in crop, environmental and ecological sciences (Sterner and Elser, 2002; Sadras, 2006). For example, in the context of crop production, although numerous measurements have been made on the N and P concentrations in leaves as an aid to diagnosing deficiencies, it is not clear how far the differences in estimated ‘optimum’ concentrations result from experimental error, differences in plant age, plant mass, leaf position or differences between species. In addition, uncertainty about the changes in the N : P ratio of crop biomass during growth has proved to be a serious problem in simulating the interactive effects of N and P on crop growth (Zhang *et al.*, 2007). A model with a strong theoretical background is needed to explain some of the uncertainties in the estimated optimum nutrient concentrations and thus contribute to a more effective use of added nutrients. Such a model might also be useful in detecting atmospheric N deposition damage to natural vegetation (Güsewell, 2005), and in establishing the effects of imbalance of nutrients on natural vegetation (Güsewell, 2004).

The possibility that such a model could be devised is suggested by the discovery of important principles and relationships that emphasize the similarities in the N and P dynamics in different organisms. The specific growth rates (SGRs) of heterotrophs are generally strongly and positively related to their contents of RNA and thus to their P : N ratios (Sterner and Elser, 2002). The ‘growth rate hypothesis’ was proposed to encompass this relationship (Elser *et al.*, 1996, 2000*b*, Sterner and Elser, 2002) and it applies not only across different micro-organisms, but also at the level of the individual species (Karpinets *et al.*, 2006). gren (2004) has further developed the theory for algae and tree seedlings. On the basis of biochemical considerations, he predicts that their SGR may increase or decrease with an increase in the P : N ratio, depending on whether the SGR is high or low. Measurements on leaves of wild herbaceous species backed up by theory indicate that the SGR increased with an increase in the P : N ratio (Niklas, 2006). Strong correlations were also found at the species level between shoot concentrations of P and N in 117 angiosperm species grown in hydroponic culture with a constant nutrient regime (Broadley *et al.*, 2004).

Lack of knowledge about the changes in N : P ratios during the growth of the entire biomass of both crop and natural vegetation is particularly conspicuous. Thus in one survey of natural vegetation, Kerkhoff and Enquist (2006) found that the N : P ratios were approximately the same, about 11, over a wide range of plant weights, whereas in another study it was found that the N : P ratios were greater in the leaves than in the stems and roots of herbage crops, which implies that the N : P ratio decreased with an increase in plant mass (Kerkhoff *et al.*, 2006). Agricultural studies are sparse, but a recent review (Sadras, 2006) indicates that N : P ratios of cereals varied between 1 and 20. A major cause of the variability lies in variations in the supply of nutrients to crops and, in particular, the tendency of crops to absorb far more P than is needed to meet the immediate needs and to store it (Bollons and Barraclough, 1999). However, this cannot explain the variability in the N : P ratio when there is an optimal supply of nutrients. The present work was therefore initially focused on the crop biomass N : P ratio during growth of very different crop species receiving near optimum levels of nutrients.

Models considering plants to consist of at least two types of tissues have been developed for the decline of N concentration of the whole plant with increasing plant mass (Caloin and Yu, 1982, 1984; Hardwick, 1987; Lemaire and Gastal, 1997). Caloin and Yu (1984) developed a model that predicted a linear increase of plant N concentration with SGR. The prediction was confirmed by a controlled-environment experiment and later by field experiments of others (Greenwood *et al.*, 1991). These studies suggest that an approach based on the relative rates of growth of different plant tissues could lead to a unifying mechanistic insight into relationships between N : P concentrations and plant biomass.

The objective of the present work was to develop a mechanistic model along these lines for the changes in the N : P ratio of whole crop biomass during growth with near optimum nutrition and to test its validity against a wide range of experimental data. It was established that processes governing N : P ratios are defined by the same equations for different species. On this basis, a model for the changes in N : P ratio with the increase in crop mass has been developed. Its validity has been tested and it has been calibrated with experimental data obtained with near optimum nutrients. This led to a simple equation which was tested against data that were entirely independent of those used for developing the model. It enables predictions to be made about the N : P ratios of plant biomass and of leaves. It is believed that the proposed model and the equation and associated principles could be useful in advancing crop, environmental and ecological sciences.

Definitions of the symbols and the units are given in the Appendix. The model is concerned with changes in the N : P ratio of whole crop dry weights, excluding fibrous roots, that occur during growth with optimum levels of nutrients before the onset of senescence. It is considered that most of the changes arise because of a decline in the tissues related to growth relative to those that are related to storage and structure. More specifically, it is considered that growth-related tissues predominate in leaves, and the storage- and structure-related tissues, which will be referred to herein as the storage-related tissue, in the remaining parts of the plant. According to this classification, Kerkhoff *et al.* (2006) found that in natural herbage the N : P ratio of the storage-related tissue is lower than in the growth-related tissue. Thus, as plants get larger, the proportion of storage-related tissue to growth-related tissue would be expected to increase and the N : P ratio to decline. It is proposed that at the beginning of growth, plants are mainly comprised of the proliferating cells organized in metabolically active, mainly growth plant tissues, which will be further referred to as growth-related tissue (*W*_{g}). The N : P ratio of the plant organism at this stage is considered to be similar to those of exponentially growing unicellular organisms. As the plant gets bigger, the proportion of *W*_{g} in the plant biomass decreases with a concomitant decrease in SGR; at the same time there is an increase in the proportion of storage-related tissue (*W*_{s}). On this basis, a simple relationship is derived between the N : P ratio of the whole crop (R_{w}) and crop dry biomass per unit area (*W*) with two constants that represent an average N : P ratio in the growth-related tissue (*R*_{g}) and an average N : P ratio in the storage-related tissue (*R*_{s}). The relationship is based on the following three assumptions.

(1) The growth rate of the crop is always proportional to the weight of growth-related tissue (*W*_{g})

1

where *W* is in Mg ha^{−1}, *t* is in days and α is a proportionality constant.

(2) Growth rate with the optimum levels of nutrients is defined by

2

where *K*_{2} (Mg ha^{−1} d^{−1}) is the growth rate coefficient and *K*_{1} (Mg ha^{−1}) is a constant. The last coefficient represents the value of *W* when the growth rate is half the maximum. The equation was derived by considering that interception of radiation by the crop and thus d*W/*d*t* increases asymptotically with *W* until the onset of senescence (defined as when growth rate starts to decline), and it has been validated by experiments on field vegetables during the main UK growing season (Greenwood *et al.*, 1977).

By considering the previous assumption (assumption 1), d*W/*d*t* can be eliminated from eqns (1) and (2), which gives

3

(3) At the beginning of growth the crop biomass (*W*) is comprised only of growth-related tissues, i.e. as *W* → 0, *W* = *W*_{g}.

This assumption about the initial growth condition is important as it permits the estimation of the coefficient ‘α’ in eqn (3). Indeed, if crop biomass is small, i.e. *W* → 0, *K*_{2}*W*/(*K*_{1} + *W*) can be replaced with *WK*_{2}*/K*_{1}. However, according to assumption 3, crop biomass at the beginning of growth is comprised only of growth-related tissue or *W* = *W*_{g}. Therefore, α*W*_{g} = *W*_{g}*K*_{2}/*K*_{1} or α = *K*_{2}/*K*_{1}. As α remains constant throughout growth (assumption 1), *K*_{2}/*K*_{1} = α may be substituted in eqn (3) to give

4

and

5

Let *P*_{g} and *P*_{s} be the total weights of P, and *N*_{g} and *N*_{s} the total weights of N in the growth- and the storage-related tissues, respectively, then the ratio of N : P in the whole plant (*R*_{w}) is given by

6

Multiplying throughout by (*P*_{g} + *P*_{s}) and then dividing by *P*_{g} gives

which on rearranging and substituting the N : P ratios of the growth- and storage-related tissues, respectively, namely *R*_{g} = *N*_{g}/*P*_{g} and *R*_{s} = *N*_{s}/*P*_{s}, gives

7

Let *C*_{Pg} and *C*_{Ps} be the concentrations of P in the growth- and storage-related tissues then *P*_{s} = *W*_{s}*C*_{Ps} and *P*_{g} = *W*_{g}*C*_{Pg} which, when substituted in eqn (7), gives

Measured differences in P concentration between different plant organs usually differ by a factor of <2 (e.g. Ingestad and Lund, 1979; Greenwood *et al.*, 1980*b*; de Groot *et al.*, 2003; Kerkhoff *et al.*, 2006). The effect of varying *C*_{Ps}/*C*_{Pg} on *R*_{w} was calculated with *W* varying from 0·1 to 20 Mg ha^{−1}, with *W*_{s} and *W*_{g} calculated from *W* by eqns (4) and (5) and with *R*_{g} and *R*_{s} having the average values found for the different species of field vegetables. This sensitivity analysis given in Supplementary Table S1, available online, indicated that *R*_{w} was almost unaffected by *C*_{Pg}*/C*_{Ps} when *W* was either small or large. Even over the entire range, a change in *C*_{Pg}*/C*_{Ps} by ± 50 % on average only resulted in a <10 % change in *R*_{w}. Thus although *C*_{Pg}*/C*_{Ps} deviates considerably from unity, setting *C*_{Pg}*/C*_{Ps} = 1 will result in little error. We therefore consider that *W*_{g}/*W*_{s} can be taken as equal to *P*_{g}/*P*_{s} which when substituted in eqn (7) gives

8

and as *W* = *W*_{g} + *W*_{s}, eqn (8) can be rewritten as

9

The last equation is a linear relationship between the crop N : P ratio and the dry biomass-related function *K*_{1}/(*K*_{1} + *W*) with two constants (*R*_{g} – *R*_{s}) and *R*_{s}.

Details of all three data sets are given in Supplementary Table S2, available online.

Most measurements in the data set are from 24 experiments carried out at Wellesbourne, UK. In these experiments, 14 different species of field vegetables were grown in the same field with near optimum levels of N, P and K fertilizers (Greenwood *et al.*, 1977, 1980*a*). In each experiment there were between five and 21 harvests with measurements of *W*, and of N and P contents, made at intervals during the growing period until the commercial harvest date which is before the onset of senescence. The near optimum levels of fertilizer had, depending on the crop, N : P ratios of from 0·7 to 5·4 and gave average yields that were 92 % of the maximum achieved with any combination of levels of fertilizer–nutrients in an adjacent multilevel NPK fertilizer experiment on the same crop. The remaining experiment in this data set was at Lerida, NE Spain (Bosch Serra, 1999) on a Xerofluvent aquic, fine silty, mixed (calcareous) messic soil (Soil Survey Staff, 1975). Onions were drilled in March, and at 50 % emergence were thinned to 80 plants m^{−2}. They were drip irrigated to maintain the water potential higher than −18 kPa and were fertigated at 334 kg N ha ^{−1}, 62 kg P ha^{−1} and 403 kg K ha^{−1}. Pest disease and weed control were effective.

Altogether there were 287 measurements on 15 different species (Table 1). A special feature of these experiments is that there is a sufficient proportion of measurements (43 %) with small values of *W* (*W* < 2 t h^{−1}) to provide a sensitive test of the model.

Data set A was also used to test the validity of the integrated form of eqn (2), namely

11

where *C* is the constant of integration and where *W* is crop dry weight (excluding fibrous roots in Mg ha^{−1} at time *t* in days, and *K*_{1} is equal to 1 Mg ha^{−1} for field vegetables as this had been found to give good fits to past field experimental data (Greenwood *et al.*, 1977, 1985). *K*_{2} is a growth rate coefficient which is generally constant throughout growth until senescence; in effect it corrects growth for differences in plant mass. *K*_{1}ln*W* + *W* was regressed against *t* for each experiment to provide estimates of *K*_{2}; on average the regressions had an *r*^{2} of 0·96 (Table 1).

This data set was obtained by searching the literature for measurements at intervals, during growth with near optimal nutrition, of *W*, %N and %P that were not included in data set A. It consists of 78 sets of measurements from 13 experiments in five countries; there were six different crops of which five were not in data set A. The data were also different from those in data set A, because they contained few measurements where *W* was <2 Mg ha ^{−1} and, thus, did not allow satisfactory estimates of *R*_{g}. They were used to test the predicted relationship between the N : P ratio and *W* for *W* between 0·1 and 27 Mg ha^{−1}. It was also used to confirm the finding from data set A that the time course dry crop weight increase followed eqn (11) (Supplementary Table S2).

The measured average N : P ratio is 7·9 in data set A and 7·2 in data set B compared with an average of 7·2 in the particulate matter in the sea which is the same as the average ratio in sea water (Sterner and Elser, 2002, p. 29).

Plant analyses are often used to guide fertilizer practice. Many experiments have been carried out to find the ‘optimum or adequate concentrations’ at different stages of growth of many crop species. Tcierling (1978) deduced such concentrations from cereal experiments in Russia, and Bergman (1992) deduced them for numerous crop species from experiments in a wide range of countries. Data set C includes measurements taken directly from Tcierling's review (1978). Bergman (1992) gives the upper and lower values of the adequate range of %N and %P; the mean values of each of these two values are also given in data set C, and these were compared with the predictions as described below. Tcierling's data consisted of values of %N and %P (for the above-ground parts of the plant) at different growth stages of cereals.

Neither the Tcierling nor the Bergman data included values of *W*, which is an essential input into eqn (10) after calibration to form eqn (15), as will be described in the Results section. Apparent values of *W* in the data set were calculated from inputs of %N by

12

(Greeenwood and Draycott, 1989) with β = 3, which had been found to be satisfactory for many field vegetables and has been used widely (Zhang *et al.*, 2007). For the Tcierling data eqn (12) was used to produce a set of corresponding values of %N, and *W. R*_{w} was calculated by inserting these calibrated values of *W* into eqn (15). The Bergman (1992) data covered a considerable range of crop species and of %P and %N. These mean values for each species were compared with a predicted relationship between %P and %N. To generate the prediction, eqn (12) was first used to calculate %N for *W* = 2 t ha ^{−1} to 20 t ha ^{−1} in steps of 1 t ha ^{−1}. The same values of W were then used to calculate *R*_{w} by inserting into eqn (15) and to predict %P multiplying 1/*R*_{w} by %N. These values of %P were regressed against those of %N to give a predicted linear relationship that was compared with Bergman's data in Fig. 5. The same procedure but with a different range of values of *W* was used to calculate a predicted relationship between %P and %N for comparison with experimental data on grasses reported by Salette (1990) given in Fig. 6. Thus, these predicted linear relationships were independent of the data with which they were compared.

The predicted relationship (continuous line) between optimum %P and %N (%P = 0·1152 × %N + 0·0788, *r*^{2} = 0·99) as described in data set C and 124 advisory estimates (symbols) from the literature for leaves and whole plants **...**

Whenever both variables were subject to error, regressions were by the reduced major axis (RMA) procedure (Niklas, 2006) with software by Bohonak (2004). If only one variable was subject to error, regression was by the standard procedure.

The validity of eqn (10) was tested by regression of the plant N : P ratio (*R*_{w}) against 1/(1 + *W*), as *K*_{1} = 1 Mg ha^{−1}, for the measurements in data set A. Figure 1 gives regressions for those experiments with most measurements and illustrates that the points were generally randomly scattered about the best fitting line; Table 1 gives values of *r*^{2} averaged over the experiments for each crop. They show that measurements of *W* and the N : P ratio during growth fit eqn (10) closely. Thus this theoretical equation gives good predictions for all crops of the linear increase in the ratio as a function of 1/(1 + *W*). Therefore, the ratio decreases with an increase in *W*.

Strong evidence that, for a given experiment, eqn (2) with the same value of *K*_{2} held throughout growth is provided by the excellent fits (average *r*^{2} for each experiment = 0·96) to eqn (11), the integral of eqn (2) for data set A (Table 1). Similar results were obtained for data set B (Supplementary Table S2). From eqn (2)

13

Assuming all of the parameters except *R*_{w} and SGR are constant as is indicated by Table 1, it follows that during a given experiment *R*_{w} decreases with a decrease in SGR.

According to eqn (10) the calculated regression coefficients (intercept and slope) allow estimates to be made of the N : P ratio in the growth-related tissue (*R*_{g}) and in the storage-related tissue (*R*_{g}) for each crop (Table 1). Because there were two independent experiments on most crops, an analysis of variance of the values of *R*_{g} and of *R*_{s} was carried out. The standard errors for the differences between crop means, *R*_{g} and *R*_{s}, are given at the bottom of Table 1. Both *R*_{g} and *R*_{s} did not differ significantly between most crops. Given this similarity, the average values of *R*_{g} and *R*_{s}, which are 11·83 and 5·57, respectively, were used as constants in the theoretically derived eqn (10), with *K*_{1} = 1 t ha ^{−1} to give

15

This calibrated equation was further verified by considering the N : P ratio in different crops at different stages of growth using experimental data in data sets B and C.

Agricultural experiments (data set B) that are entirely independent of those used for developing the model give a relationship between N : P ratio and *W* that is similar to that calculated from eqn (15) (Fig. 2). The data set B includes measurements made throughout the growth period on the C_{4} crop maize and on five C_{3} crops. The values of *W* range from 0·1 to 27 Mg ha^{−1} and yet are in reasonably good agreement with eqn (15).

Literature measurements (data set B) of the ratio %N/%P and *W* in Mg ha^{−1} during growth of crops grown with near optimum levels of nutrients and predicted dependence of the ratio on *W* by eqn (15). Details and sources of data are in Table S2 in **...**

As N : P ratios are particularly sensitive to changes in *W*, when *W* is small, the validity of the model was tested against measurements made in the early stages of crop growth. In data set A, the average N : P ratio of crop seedlings, for values of *W* < 0·2 Mg ha^{−1}, is 9·8 compared with 11·16 obtained by substituting *W* = 0·1 Mg ha^{−1} in eqn (15). The average ratio of seedlings in data set B with *W* < 2 Mg ha^{−1} was 9·3, which is close to the predicted value of 8·65 obtained by substituting *W* = 1 Mg ha^{−1} in eqn (15) (Supplementary Tables S3 and S4, available online).

According to the model, the average N : P ratio of the growth-related tissue, *R*_{g}, is an important plant characteristics that does not vary much during growth (Fig. 1, Table 1). As a result, changes in crop N : P ratio during growth are affected by changes in the biomass of the growth-related tissue, but not by its N : P ratio. Because leaves consist mainly of growth-related tissue, the average values of the N : P ratio in leaves should be close to the *R*_{g} found in the study. Ecological measurements have been studied on leaves to check this proposition, namely to confirm that leaf %P is proportional to leaf %N with a proportionality constant of 1/11·83 = 0·0843. Figure 3A gives leaf %P plotted against leaf %N on logarithmic axes for the 206 sets of data on the leaves of herbage species (Kerkhoff *et al.*, 2006), and Fig. 3B gives a similar plot for 177 sets of data on the leaves of evergreen trees of deciduous shrubs (Wright *et al.*, 2004). The lines are calculated by assuming the P : N ratio is always equal to that of the growth-related tissue, namely 0·0843, and appear to give a good representation of the average values of the data. Leaf photosynthesis usually ceases when the leaf nitrogen concentrations are low (Nátr, 1975) so such leaves could not be regarded as growth-related tissue. For this reason, formal statistical analyses on leaf measurements have been confined to leaves with a %N >1. The details of these analyses are given in Table S5 in the Supplementary information (available online). RMA regressions of log_{10}%P against log_{10}%N were carried out separately on the Kerkhoff and the Wright data. The means sums of squares of the deviations between measured and calculated values (from the regressions) of log_{10}%P and vice versa were determined. They were also determined when the values were calculated from P/N = 0·0843. Although the gradients of the RMA fits were >1, indicating a curvature in the relationships, the degree of agreement between measured and calculated values was not significantly different whether calculations were from the RMA fitted equations or from P/N = 0·0843. The ratio of the mean sum of squares of deviations for calculations with the latter equation to that with the former equations was 1·14 which is not significant at *P* = 0·05. Thus P/N = 0·0843 gives a good description of the data. This is supported further by the finding that the average P : N ratios of the Kerkhoff and of the Wright leaves were not significantly different from 0·0843. Therefore, the P : N ratio of wild leaves having a %N >1 is similar to that inferred from the present model (eqn 10) and experiments on field vegetables.

Relatively few experiments (given in Fig. 2) have been published where sequential measurements were made of *W*, %N and %P on crops grown with near optimum levels of nutrients. However, a considerable amount of other data exists on the optimal %N and %P that does not include *W* for different crops and conditions. It was therefore important to find how far they could be predicted with eqn (15). The methods were as previously described under data set C. The data include N : P ratios (data set C) at each of four growth stages of four different C_{3} cereals, grown with ‘adequate nutrition’, that were derived from Tcierling's (1978) review of nationwide Russian experiments. Comparisons between measured N : P ratios and those calculated with eqn (15) are given in Fig. 4. An upper limit for the standard error of the measurements is 0·7 (see legend to Fig. 4), indicating that the least significant difference is about 1·4. As in 12 out of the 15 comparisons the difference between measurement and calculated N : P ratio is <1·4, it follows that agreement between measurement and calculation is reasonably good. Bergman (1992) in his review of world-wide literature gives the ranges of %N and %P of immature tissues of numerous species required for good growth. Altogether the data set includes 124 records (data set C), the average of the range of %P of each record was plotted against the average %N. They are linearly related and are plotted in Fig. 5 which also gives the linear relationship predicted with eqn (15). The slopes of the two relationships are almost identical, but the intercept of the predicted relationship is about 0·1 % higher that that for the measurements. Measurements of %P and %N of the above-ground biomass made at intervals during the growth of tall fescue and rye grass grown with near optimum conditions (Salette, 1990) are also in a good agreement with that obtained with eqn (15) (Fig. 6).

The present study indicates that the N : P ratios in the growth-related tissues of many crops are about 11·83, and are similar to those of the leaves of many naturally grown species (Fig. 3). This value is in the range of that found for micro-organisms (Sterner and Elser, 2002) and is close to the averages of 12·65 and 13·64 for terrestrial and freshwater autotrophs (Elser *et al.*, 2000*a*). These similarities imply that the same cellular processes, such as changes in the transcribed rRNAs (Elser *et al.*, 2000*b*) and in RNA:protein ratios (Karpinets *et al.*, 2006), may underlie the relationships between the N : P ratios of growing cells and their growth rate whether in plants or in micro-organisms. The nutrient requirements of this conserved molecular machinery could dominate the N and P requirements for cellular growth and thus explain their close balance and synergy in natural ecosystems (Elser *et al.*, 2007).

An alternative to the proposed mechanistic modelling of the crop N : P ratio is to fit an empirical model to the experimental data. One widely used model in crop nutrition (Lemaire, 1997) considers the relative changes in %N and %P during growth of the whole crop in terms of simple scaling relationships with *W*; these are %N = a*W*^{b}; %P = c*W*^{d} and N : P = (a/c)*W*^{(b–d)} where a, b, c and d are coefficients that are fitted separately for each experiment. The model gave good fits to each of five sets of the test data (Supplementary Table S6, available online), an example of which is given in Fig. 7. As both the coefficients b and d were always negative and b was less than d, the relative decline in %N was always greater than that of %P. The model also indicates that N : P will never remain constant unless b = d. The present mechanistic model also gave a good description of the dependence of N : P on *W*. In addition, the calibrating coefficients have a clear biological sense, representing the N : P ratios in each of the main plant tissues. One of these parameters appears to be constant, as mentioned above, not only across plants, but also across microbial organisms, which demonstrates that the proposed mechanistic model quantifies global regularity in biological plant species.

The calculated relationship between N : P and *W* (continuous curve) and the measured values of N : P for the turnip 72 experiment. Calculations were made with N : P = (a/c)*W*^{(b–d)} in which the coefficients a and b were obtained by fitting %N = a **...**

In the Introduction, it was emphasized that in accordance with the ‘growth rate hypothesis’ the specific growth rate of heterotrophic organisms increases with a decrease in the N : P ratio (Elser *et al.*, 1996, 2000*b*; Sterner and Elser, 2002). It was also pointed out that a similar relationship had been found for leaves (Niklas, 2006). In fact limited support was found for the relationship as the N : P ratio of the growth-related tissue, which is thought to consist mainly of leaves, for the experiments in data set A was negatively, although weakly, correlated with the growth rate *K*_{2}. Nevertheless, in the present studies, the N : P ratio of the whole plant declined as plants grew and their SGR fell, which is in accordance with gren's predictions (gren, 2004) for whole plants with SGRs (<0·1 d^{−1}) that are similar to the SGRs of the present data. According to the model described here, this difference occurs because whole plants, unlike unicellular organisms and individual parts of plants, consist of different tissues, each with different functions and thus different demands for N and P. During growth, the proportions of the tissues change, leading to changes in the total requirement of the whole plant for N relative to that for P. The present model is a simplification of the various processes, but its validity is supported by much experimental information covering widely different species grown in different ways. It illustrates how strongly changes in the relative proportions of the growth- and storage-related tissues dominate the N : P ratio of the whole crop. For example, according to eqn (4), the weight of growth-related tissue as a fraction of total plant weight, *W*, is equal to 1/(1 + *W*) which indicates that the fraction declines sharply with small increases in *W*; in fact the fraction drops from 1 to 0·5 with an increase in *W* from 0 to 1 Mg ha^{−1}. As the average N : P ratio of this tissue is 11·83 and that of storage-related tissue is 5·58, it follows that there is a very sharp fall in the N : P ratio of the whole plant with small increases in *W*, as found experimentally (Fig. 2). It thus supports the view that the N : P ratio of crop biomass grown with optimum concentrations of nutrient is greatly affected by the partition of assimilates to different tissues during growth.

The low N : P ratio of field vegetables (*R*_{w}) compared with that of much standing mass of natural communities deserves comment. The low N : P ratio is dominated by the low N : P ratio of the storage-related tissues (*R*_{s}) which is subsequently used for reproduction and thus requires a high %P. As the average weight of the standing biomass of many natural communities is much greater than that of crops, most of it could have consisted of wood which contains only very small amounts of nutrients (Altman and Dittmer, 1964). In consequence, the N : P ratio could be largely determined by that of the growth-related tissue and be approx. 11·83 irrespective of the total weight of biomass. Indeed, a survey of such communities indicated that they had an N : P ratio of approx. 13 irrespective of the total weight of biomass (Kerkhoff *et al.*, 2005).

Ecological studies emphasize the similarities in the relationships between %N and %P in the dry matter of leaves grown on soils of differing fertility (Niklas *et al.*, 2005), whereas crop studies emphasize the dependence of crop %N and of %P on soil nutrient levels (Lemaire, 1997; Bollons and Barraclough, 1999). In this study it is shown that the average N : P ratio of naturally grown leaves in a large data set is on average similar to the N : P ratio of the growth-related tissue of crops grown with near optimum levels of nutrients. As growth-related tissue is thought to consist mainly of leaves, the average N : P ratio of natural leaves appears to be that expected with optimal plant nutrition. Presumably the leaves of natural vegetation have adapted over long periods to the stable but low soil nutrient levels, by various feedback mechanisms in such a way that the different nutrients limit growth simultaneously (Bloom *et al.*, 1985; Gleeson and Tilman, 1992; Knecht and Göransson, 2004) and that when this occurs the N : P ratios are similar to those obtained for plants grown with the optimum fertilizer levels. Arable crops are in the ground for much shorter periods and have much less time to adapt to the nutrient regimes, so homeostasis is less marked and plant nutrient concentrations are more dependent on soil nutrient levels. They have also been bred for rapid growth and high yields, possibly at the expense of their root systems and their ability to withstand soil nutrient stresses. Thus inter-site variation in growth of crops could be much more dependent on soil nutrient levels than that of natural vegetation. Also it is possible that the N : P ratio of the growth tissue is approximately the same over a wide range of natural and crop species and the soil nutrient effects on plants are entirely accommodated by differences in the N : P ratios of the storage-related tissues.

The present model could improve crop nutrition by providing general diagnostic criteria for nutrient imbalance and by its incorporation into dynamic models for crop response to fertilizers, as described in the Introduction. Nevertheless some characteristics of the experimental data used for developing eqn (15) impose limits on its application. It may not apply when *W* is >20 Mg ha^{−1} as most values of *W* in the experiments were less than this. It may not apply after the onset of appreciable leaf senescence as the field vegetables were harvested before the onset of senescence. An obvious major uncertainty in these studies is the extent to which deviations from optimum applications of N and P fertilizers have affected the measured N : P ratios. Less obvious is the error that may have resulted from setting *K*_{1} = 1 Mg ha^{−1} for all crops. It is the value of *W* when the growth rate is half the maximum. Such a low value seems to be appropriate for most of the field vegetables grown in the field experiments as these were grown at high plant densities, and complete crop cover and maximum growth rates were attained whilst *W* was small. Setting *K*_{1} = 1 Mg ha^{−1} for all crops was based on the excellent fits that were obtained with this value of *K*_{1} in fitting eqn (11) (Table 1). However, *K*_{1} and *K*_{2} are highly correlated in such fits, and equally good fits can be obtained with different combinations of values of each of these coefficients, and it may be that *K*_{1} should be >1 for widely spaced crops. The same uncertainty exists for eqn (15) governing the dependence of the N : P ratio on *W*. Increasing *K*_{1} in eqn (10) and also in eqn (15) results in the N : P ratio declining less rapidly with an increase in *W*. This could explain why *R*_{w} of Brussels sprouts declined less with an increase in *W* as they were grown at a far wider spacing than the other crops. Wide plant spacing depressed the decline of %N of lucerne with an increase in plant mass compared with close spacing (Lemaire and Gastal, 1997), which could be explained in terms of the interception of light (Hirose and Werger, 1987; Lynch and Gonzalez, 1993). Spacing might therefore be expected to affect the dependence of N : P ratio on *W*.

It is concluded that the same physiologically derived equation with two constants gives a good prediction of the changes in the N : P ratio in the biomass of a wide range of crops grown, in different ways, with near optimal nutrition. The model also gave good estimates of the relationship between %P and %N in the leaves of natural vegetation provided the leaf %N was >1. The N : P ratio of crop biomass declines sharply with increase in plant mass per unit area because of an increase in the proportion of storage-related tissue which has a low N : P ratio relative to growth-related tissue which has a high N : P ratio.

Supplementary information is available online at www.aob.oxfordjournals.org/ and consists of the following tables. Table S1, effect of variation in the ratio *C*_{Ps}/*C*_{Pg} on *R*_{w}; Table S2, details of data sets A, B and C with references to the sources of data; Table S3, N : P ratios for values of *W* < 0·2 Mg ha^{−1} from data set A; Table S4, N : P ratios of *W* < 2 Mg ha^{−1} from data set B with references to the sources of data; Table S5, statistical analyses of predictions of P : N ratios of wild leaves; and Table S6, empirical model fits measurements of %N and %P during crop growth.

We thank Professor D. Kerkhoff of Kenyon College, Gambiier, Ohio, USA for providing us with his original data on herbage plants and permitting us to publish them. Funding for this work was partially provided by the UK Department for the Environment, Food and Rural Affairs through project HH3507SFV.

Symbol | Definition | Unit |
---|---|---|

α | Proportionality constant between dW/dt and W_{g} | d^{−1} |

β | Coefficient for dependence of optimum %N on W | Dimensionless |

C_{Pg} | Concentration of P in W_{g} | Dimensionless |

C_{Ps} | Concentration of P in W_{s} | Dimensionless |

K_{1} | Coefficient in a growth rate equation | Mg ha^{−1} |

K_{2} | Coefficient in a growth rate equation | Mg ha^{−1} d^{−1} |

N_{g} | Weight of N in W_{g} | Mg ha^{−1} |

N_{s} | Weight of N in W_{s} | Mg ha^{−1} |

P_{g} | Weight of P in W_{g} | Mg ha^{−1} |

P_{s} | Weight of P in W_{s} | Mg ha^{−1} |

R_{g} | N : P ratio of growth-related tissue | Dimensionless |

R_{s} | N : P ratio of storage-related tissue | Dimensionless |

RMA | Reduced major axis regression | Dimensionless |

R_{w} | N : P ratio of W | Dimensionless |

SGR | Specific growth rate | d^{−1} |

t | Time | Days |

W | Dry weight of biomass excluding fibrous roots | Mg ha^{−1} |

W_{g} | Dry weight of growth-related tissue | Mg ha^{−1} |

W_{s} | Dry weight of storage-related tissue | Mg ha^{−1} |

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