AOBPreview originally published online on December 17, 2003
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Annals of Botany 93: 127-139, 2004
© 2004 Annals of Botany Company
Equilibrium and Balanced Growth of a Vegetative Crop
1 Civil and Environmental Engineering, Technion, Haifa 32000, Israel
* For correspondence. E-mail: segineri{at}tx.technion.ac.il
Received: 11 June 2003; Returned for revision: 16 September 2003; Accepted: 22 October 2003 Published electronically: 17 December 2003
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMODELEQUILIBRIUM COMPOSITION AND...EQUILIBRIUM GROWTH EXAMPLEDISCUSSIONCONCLUSIONLITERATURE CITED |
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• Model A previously developed dynamic model, NICOLET, designed to predict growth and nitrate content of a lettuce crop, is subjected to (virtual) constant environmental conditions. For every combination of shoot and root environment, the cell sap, here assumed to reside in the ‘vacuole’ compartment, equilibrates at a certain nitrate concentration level. This, in turn, defines the composition of the crop in terms of carbon and nitrogen content in each of the three compartments of the model. Growth under constant environmental conditions is defined as ‘equilibrium’ growth (EG). If, in addition, the source strengths of carbon and nitrogen balance each other, as well as the sink strength of the growing crop, the growth is said to be ‘balanced’ (BG).
• Results It is shown that the range of BG approximately coincides with the range of ‘mild’ nitrogen stress, where reduction in nitrogen availability results in a mild reduction of relative growth rate (RGR). Beyond a certain low nitrate concentration in the cell sap, the N-stress becomes ‘severe’ and the loss of growth increases considerably.
• Conclusions The model is able to mimic the five central observations of many constant-environment growth-chamber experiments, namely (1) the initial exponential growth and later decline of the RGR, (2) the constant chemical composition, (3) the equality of the RGR and the relative nutrient supply rate (RNR), (4) the proportionality between the N : C ratio and the RNR, and (5) the proportionality between the water content and the reduced N content. Guidelines for the optimal combination of the shoot and root environments are suggested.
Key words: Lettuce, Lactuca sativa L., vegetative crop, equilibrium growth, balanced growth, crop composition, relative growth rate, N-stress, NICOLET.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMODELEQUILIBRIUM COMPOSITION AND...EQUILIBRIUM GROWTH EXAMPLEDISCUSSIONCONCLUSIONLITERATURE CITED |
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When seedlings are exposed to a constant environment, they initially grow at an exponential rate while maintaining a constant chemical composition (constant constituent proportions). ‘Constancy’, in this context, may be defined on various time scales, but normally means a repeated daily cycle, such as in growth chamber experiments and, in particular, in nutrition experiments of the Ingestad-type (Ingestad and Ågren, 1992).
Ingestad and Ågren (1988), and many others, refer to growth during this initial stage as ‘steady-state’. Thornley (1997), on the other hand, refers to it as ‘balanced exponential growth’, reserving the expression ‘steady-state’ for the ecological ‘climax’ situation, where net growth (change of state) is zero. The focus in the present paper is on the ‘constant composition’ stage, termed here the ‘equilibrium’ stage because the (constant) internal composition of the plants seems to be in equilibrium with the (constant) environment.
The environmental conditions may be stressful to the plants, meaning that an essential plant process cannot proceed at the rate that matches the other concurrent processes. An example is an insufficient supply of water or nutrients relative to the demand of the crop. Another example is a low temperature level (low sink strength) relative to the prevailing light level (source strength). Under such conditions plants may respond by ‘morphological adjustment’, aimed at reaching a ‘functional equilibrium’ (Brouwer, 1983). Such ‘plasticity’ is best illustrated by the increased root : shoot ratio (RSR) when availability of nutrients or water is the growth-limiting factor. If the adjustment is complete, the various processes become ‘balanced’; otherwise, some processes may have to be ‘inhibited’ such that all ‘match’ each other. In any case, under constant environmental conditions, plants seem to settle into a state of ‘equilibrium’ with the environment. From this point of view, ‘balanced growth’ (BG), where no process is inhibited, is a special case of ‘equilibrium growth’ (EG).
Models of intensive horticultural crops (e.g. tomato: Jones et al., 1991; Bertin and Heuvelink, 1993; lettuce: Sweeney et al., 1981; Seginer, 2003) often ignore water stress and nutrient stress conditions, assuming these inputs to be plentiful. As a result, they usually do not include a root compartment and, hence, cannot posses an explicit root–shoot balancing mechanism. Whether an explicit root compartment is required in models designed to make predictions under nutrient stress conditions depends, however, on the specific predictions that are needed. For example, a recent study (R. Linker and C. J. Rutzke, unpub. res.) shows that adding a root compartment to a lettuce model hardly improves the prediction of growth and chemical composition of the shoots. A root compartment is, of course, essential if prediction of the RSR is intended.
A model to predict the main qualitative observations of ‘equilibrium’ experiments, should be able to mimic the following.
1. Relative growth rate (RGR) of young isolated plants is constant (growth is exponential).
2. Plant chemical composition is constant.
3. If nitrogen supply is limiting growth, RGR is essentially equal to the relative N addition rate (RNR).
4. When N is limiting, the total-N : total-C composition ratio is proportional to the RNR.
5. Plant water content is proportional to reduced N content.
Experimental support for the first four points may be found in the work of Ingestad and his co-workers (e.g. Oscarson et al., 1989; Ingestad and Ågren, 1992) and support for point 5 may be found in Seginer (2003). This list will be referred to below as the ‘list of observations’ or LO.
The recently developed single-organ (shoot) model NICOLET (Seginer, 2003) was designed to predict growth rates and nitrate content of lettuce, mainly under controlled glasshouse conditions. It has the elements necessary to evaluate the effect of a constant environment on the composition and relative growth rate (RGR) of a young vegetative crop. The objective of this paper is to map the equilibrium composition and the RGR of NICOLET over a wide range of (constant) environmental conditions and to compare the results, qualitatively, with the five observations mentioned above. The analysis and mapping are carried out in terms of normalized variables, to produce a compact description of the results.
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MODEL |
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TOPABSTRACTINTRODUCTIONMODELEQUILIBRIUM COMPOSITION AND...EQUILIBRIUM GROWTH EXAMPLEDISCUSSIONCONCLUSIONLITERATURE CITED |
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Only the essentials of the NICOLET model are presented here. The original paper (Seginer, 2003) may be consulted for further detail, explanation and justification.
Carbon and nitrogen balances
The model has three compartments: (1) ‘vacuole’, where the soluble non-structural (raw) material is stored, and the nitrogen : carbon ratio may vary as needed to maintain a constant osmotic potential; (2) ‘structure’, a metabolically active compartment with fixed chemical composition; and (3) an ‘excess-carbon’ (excess-C) compartment, which serves as a long-term storage of ‘waterless’ carbohydrates. The model is described schematically in Fig. 1. Note that in the NICOLET model, ‘growth’ is used to describe structural growth, which is perceived, implicitly, as ‘leaf expansion’. The excess-C compartment is perceived as contributing to dry matter by (mass) thickening of the (implicit) leaves. The excess-C compartment contributes very little to the fresh mass of the crop, since it does not contain water. Hence ‘growth’ describes roughly the growth of fresh mass. During EG all constituents grow at the same RGR.
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The three carbon balances are
vacuole :
structure :
excess-C :
where MC is the molar mass of carbon (subscript C) per unit ground area, and the subscripts v, s and e refer to the vacuole, structure and excess-C compartments, respectively. FC is the flux of carbon. The fluxes with two lower-case subscripts are between compartments (from the first to the second subscript) and the other fluxes, namely FCp, FCg and FCm, are the photosynthesis, growth respiration and maintenance respiration fluxes (all between the ‘vacuole’ and the environment).
The two nitrogen balances are
vacuole :
structure :
where the subscripts N and u denote nitrogen and uptake from the nutrient solution, respectively. Vacuolar nitrogen is considered to be in nitrate form and structural nitrogen is considered to be in reduced N form.
Two constant compositional ratios are assumed: N : C in the structure, and water to structural carbon
MNs = rMCs(6)
V = 
MCs(7)
where V is the volume of water per unit ground area, and r and are the two proportionality coefficients. Dividing eqn (7) by eqn (6), yields
where c is a constant. Equation 8 satisfies point 5 of the LO (list of observations in the Introduction).
Constant proportions also apply to two of the fluxes. First, in view of eqns (2) and (5), eqn (6) implies
FNvs = rFCvs(9)
Second, growth respiration is assumed to be a constant fraction (
) of growth, namely
FCg = FCvs(10)
A central element of the NICOLET model is the osmotica balance
ßCMCv + ßNMNv = 
V(11)
where ßC and ßN are the osmotic contributions of one unit of vacuolar C or N, and is the total osmotic potential of the cell sap, assumed to be constant. Differentiating eqn (11) with respect to time, and substituting from eqns (1), (2), (4), (7), (9) and (10), results in the flux form
GFCvs – ßNFNu = ßC(FCp – FCm – FCve + FCev)(12)
where
G = + ßNr + ßC(1 + )(13)
is a collection of parameters. It is assumed that the four fluxes on the right of eqn (12) depend only on the shoot environment and on the state of the crop, while the two fluxes on the left may also be affected by the availability of nutrients in the nutrient solution, and hence may have to be formulated differently for different nutritional situations (abundant or limited nutrition). Given either FCvs or FNu, the other flux can be determined from eqn (12). Which flux is unknown depends on the nutritional situation (to be considered further below).
The overall model description is now complete. It involves six constituents, MCs, MCv, MCe, MNs, MNv and V, constrained by three relationships, eqns (6), (7) and (11). Hence, only three out of the five differential eqns (1)–(5) are required for a complete description of the model dynamics. In this study, eqns (1)–(3) were chosen to serve as the state equations, so that the three carbon masses MCs, MCv and MCe are the three state variables of the model.
Nutritional situations and carbon partitioning
When nutrients (here nitrogen) are abundant, the model crop takes up as much of them as is required to match the carbon growth (sink strength), denoted by FACvs (A for ‘abundant’). In such a case, the osmotica balance (eqn 12) becomes
which may be used to determine the rate of nitrogen uptake, FANu. If the supply of nitrogen is limited, growth must adjust to the limited supply, FLNu (L for ‘limited’), namely
At any point in simulation time it is possible to decide which one of eqns (14) and (15) is valid, by replacing the actual fluxes FCvs and FNu, on the left of eqn (12), by the ‘abundant’ and ‘limited’ fluxes, FACvs and FLNu. If
is true, the instantaneous growth is N-limited and the crop is N-stressed (demand for nitrogen is larger than its supply). If it is false, growth is not N limited.
Carbon fluxes
Defining the normalized concentration, 
of the vacuolar carbon by
the carbon fluxes on the right of eqn (12) are formulated as
and the potential growth flux (when nutrients are abundant) is formulated as
where the expressions on the right are short-hand notations for convenience. The five fluxes are functions of the shoot environment, namely of I (light), CC (CO2 concentration), Ta (temperature), and of the state of the crop, via MCs and . The functions p, e, g, f and A are referred to as the ‘photosynthesis’, ‘respiration’, ‘growth’, ‘light interception’ and ‘attenuation’ functions, respectively.
Typically, p increases asymptotically with light and CO2 concentration, e increases (for ‘low’ temperatures) exponentially with temperature, g is proportional to e
and f approaches 1 asymptotically, as a function of the size of the crop
where the approximation is valid when the plants are young.
The attenuation functions are
where the limitation functions hp and hg are formulated as
The constants in eqns (25) to (30) are bounded as follows: 0 < 
<1, 0 < bp < 1, 0 < bg < 1, sp > 0 and sg > 0. The limitation function hp protects the vacuole from carbon over-spilling, and hg protects it from carbon over-draining. The inflow limitation, hp, approaches zero as the vacuole approaches saturation with carbon compounds ( 
1), and increases asymptotically to 1 (no limitation) when the vacuole becomes depleted of carbon. The outflow limitation function hg is qualitatively a mirror image of hp (Fig. 2). The flow in and out of the excess-C compartment is restricted to extreme situations, where photosynthesis or growth are inhibited by a carbon-saturated or carbon-depleted vacuole, respectively. The parameters bg, bp and 
are indicated in Fig. 2. The first two define the locations, along the abscissa, where the limiting functions have a value of 0·5 (mid-transition). The parameter is the fraction of the difference between the photosynthetic potential and hp, which is diverted to the excess-C compartment.
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The range where neither photosynthesis nor growth is inhibited is labelled in Fig. 2 as the ‘balanced’ growth (BG) range. If the environment is constant and the system settles within this range, the carbon source and sink strengths are matched, the uptake of nitrogen matches them both, and the excess-C compartment is inactive. In other words, equilibrium growth is maintained without invoking inhibition (attenuation) and hence is regarded as BG. Note, however, that the balanced growth situation is not unique. First, by increasing simultaneously all source and sink strengths, the absolute growth rate increases. Secondly, (nearly) balanced growth (Fig. 2) covers a wide range of vacuolar carbon concentration (
), which affects both chemical composition and growth rate.
The three distinct ranges along the axis, roughly 0 to bg, bg to bp, and bp to 1, are separated by the sloping portions of hg and hp. The separation is sharp when sg and sp are large, as is the case in the numerical example described below.
Nitrogen uptake
The potential (maximum) nitrogen uptake rate of a given root system in a particular environment is formulated as
where CN is the nitrogen concentration of the nutrient solution and Tn is root temperature. The ‘uptake’ function u is typically an asymptotically increasing function of CN and an exponential function of Tn. The actual uptake is
where FLNu and FANu are given by eqns (31) and (14).
Particular forms of p, e and u are proposed in Seginer (2003), but are not required here, since the equilibrium solutions to be developed below will be presented in the two-dimensional normalized environmental space p/g and u/g, rather than in the five-dimensional physical environmental space: I, CC, CN, Ta and Tn. Note that in view of the definitions of p, u and g (Appendix B: notation), p/g and u/g, which represent the shoot and root environments, respectively, may also be viewed as normalized potential photosynthesis and N availability (or N uptake, depending on definition; Fig. 3). Another way to look at p/g is as a (potential) source : sink ratio of carbon, which normally is of order 1. An analogous nitrogen source : sink ratio would be u/rg, which also is of order 1. Whatever the point of view, the arguments of the functions p/g and u/g are purely environmental.
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EQUILIBRIUM COMPOSITION AND RELATIVE GROWTH RATE |
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TOPABSTRACTINTRODUCTIONMODELEQUILIBRIUM COMPOSITION AND...EQUILIBRIUM GROWTH EXAMPLEDISCUSSIONCONCLUSIONLITERATURE CITED |
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General composition ratios
The equilibrium composition ratios are determined from the differential balance-equations by assuming that for any combination of substance and compartment, the incremental ratio is equal to the existing (constant) ratio, namely
where I, J indicate any two substances and i and j are any two compartments. Rearranging eqn (33) to read
shows that, if eqn (33) holds, the relative growth rate (RGR) of all the compartments and constituents is the same (but may vary with time). In the NICOLET model, eqn (33) holds, since all fluxes, eqns (18)–(22) and (31), are proportional to the same function of size (age, time), namely to f{MCs}.
The model has six constituents and therefore five composition ratios. Three of those are obtained from the first four of the five differential eqns (1)–(5) (plus the two auxillary eqns 9 and 10), as indicated by the numbers on the left of the following expressions:
The two remaining ratios, repeated here for completeness, are independent of the environmental conditions and will be used later only for substitutions.
Specific composition ratios
The fluxes of eqns (35)–(37) may be specified further by substitution from the appropriate flux equations. Two cases are to be distinguished: abundant N and limited N. Again, the numbers on the left indicate the equations used to derive the new equations.
Abundant N:
Limited N:
Note that the condition for an empty excess-C compartment [eqn (41) or (44)] is
When the excess compartment is empty, is switched to zero.
Solution
In view of eqn (17), the definition of , each one of eqns (40) and (43) may be written as
since the environment, expressed in terms of p/g and u/g, is constant and hp and hg are functions just of (all other symbols represent constants). The equality on the right is true by derivation. Assuming that the equality on the left [= eqn (33)] also holds, the left and right terms form an equation which can be solved for in terms of the environment, namely
Since the environment is constant, , the composition of the vacuole, is also constant and hence dMCv/dMCs is constant, leading to a constant MCv/MCs, as postulated. Once the equilibrium value of is known, eqns (41), (42), (44) and (45) can be solved for the corresponding (constant) compositional ratios (point 2 of the LO).
In the abundant N case, the solution for p/g in terms of [eqns (40) and (17)] is explicit and unique, namely
When growth is N limited, there are many combinations of p/g and u/g which result in the same vacuolar composition. Utilizing eqn (17) as before, eqn (43) can be used to obtain
which, for a given , is a linear relationship between p/g and u/g. Equations (49) and (50) can be used to tabulate the dependence of on p/g and u/g. Given , p/g and u/g, all the composition ratios and the RGR of eqns (40)–(45), can be evaluated.
The switching between a limited-N situation and an abundant-N situation occurs when
which is obtained by equating eqns (49) and (50). The curve described by plotting u/g of eqn (51) against p/g of eqn (49) will be referred to as the N border (between the limited-N and abundant-N regions).
Relative growth rate (RGR)
Equation (34), which has just been shown to hold, implies that all constituents grow at the same RGR. Hence, calculating the RGR of just one of them is sufficient. The growth rate of structural carbon is defined by
For the abundant-N case, utilizing eqns (52), (22), (24) and (28), the normalized relative growth rate is
and for the limited-N case, utilizing eqns (52), (18)–(21), (15) and (24), the result is
where the constant approximations of eqns (53) and (54) are valid for young plants, indicating that the growth of young (isolated) plants is exponential (point 1 of the LO). The variable ratio [f/(aMCs)], which equals 1 for young (isolated) plants, decreases as the canopy closes, because the numerator approaches 1 while the denominator continues to grow. In other words, the model predicts a diminishing RGR with time, as is observed in competing plants. Recall, however, that the NICOLET model predicts a constant composition for all crop ages, even after canopy closure (which leads to reduced RGR).
Equation (54) can be simplified if written for a constant value of , by eliminating p/g between eqns (50) and (54). The result is
and it shows that along an iso- line in the N-limiting case the RGR is exactly proportional to the rate of addition of nitrogen, u. Most nutrition experiments are conducted, however, under constant p/g (constant shoot environment) rather than constant (constant composition) conditions; yet when N is strongly limiting growth, the observed results are similar: RCs is roughly proportional to u. This point will be treated numerically later on.
Incidentally, by substituting for u/g in eqn (55) from eqn (51) (on the N border), eqn (55) reduces to eqn (53). Following this example, eqns (40)–(42) and (53) can be shown to be special (N border) cases of eqns (43)–(45) and (54).
Relative nutrient addition rate (RNR)
As already mentioned (point 3 of the LO), experimental evidence indicates that RNR and RGR are essentially the same when growth is nutrient limited. This can be shown to also be the case in the NICOLET model. Starting with the definition of RNR in terms of model variables,
FNu, MNv and MNs are replaced by substituting from eqns (31), (45) and (38), respectively, resulting in an equation that is identical to eqn (54) for RGR, as has just been claimed. This identity is independent of the form of the limitation functions and of the value of .
N : C ratios and water content
In terms of the model, total nitrogen to total carbon ratio is expressed as
structural nitrogen to total carbon ratio [with substitution from eqn (38)] as
and the volumetric water content as
It will be demonstrated numerically below that total N to total C ratio [eqn (57)] is approximately proportional to the nutrient addition rate (point 4 of the LO).
Summary: adherence to the ‘list of observations’ (LO)
All five points of the LO have been accounted for during model development. Points 1–3 and point 5 were shown to be inherently satisfied by the model, independent of coefficient values and specific forms of most functions. Point 4 could not be shown to hold in general and requires quantitative evaluation.
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EQUILIBRIUM GROWTH EXAMPLE |
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TOPABSTRACTINTRODUCTIONMODELEQUILIBRIUM COMPOSITION AND...EQUILIBRIUM GROWTH EXAMPLEDISCUSSIONCONCLUSIONLITERATURE CITED |
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Regions in problem-space
The results of the previous section are now illustrated. The parameter set is the same as used in Seginer (2003) and as presented in Appendix A. Only 12 out of the 19 parameters of the NICOLET model are used, as a result of replacing the five-dimensional environmental space, I, CC, CN, Ta and Tn by the two-dimensional normalized environmental space: p/g and u/g (the inverse transformation is, of course, not unique). The condensed problem-space, where p/g represents the shoot environment and u/g represents the root environment, is shown in Fig. 3. The N-border curve, with co-ordinates derived from eqns (49) and (51), separates the region on the right, where N supply is abundant, from the region on the left, where growth is N-limited. The straight sloping lines, ending at the N border, are iso-
lines, described by eqn (50). Along each of these lines, the ratios MCv/MCs and MNv/MCs remain strictly constant (by definition), MCe/MCs (where MCe 
0) remains nearly constant, and RCs/ag increases linearly [eqn (55)] from zero near the origin to about hg at the N border [eqn (53)]. The positive p/g intercept of the iso- lines is due to some loss of carbon via maintenance respiration (1/
> 0). The E border, between the region where the excess compartment is empty (below the border) and the region where MCe > 0, is close to the iso- = 0.8 line. The horizontal dashed lines in the abundant-N region are iso- as well as iso-composition lines. Every point on these lines has the same properties as the end-point on the N border. Note that if the abscissa, u/g, is interpreted as actual uptake of nitrogen, rather than nitrogen availability, the space to the right of the N border contains no solutions and the horizontal lines have no meaning.
The u/g (availability) versus p/g problem-space is divided by the N border and the E border into four distinct regions. The composition ratios and RGR are calculated for each region by a different set of equations: eqns (40)–(42) and (53) for the region to the right of the N border, eqns (43)–(45) and (54) for the region to the left of the N border; = 0 for the region below the E border and as in the parameter list, for the region above it.
The shape of the N border reflects the characteristics of the limitation functions. Note, however, that the parameter set of Appendix A, used to plot Fig. 3, is not the same as for Fig. 2. In particular, the rise of hg, appropriate for Fig. 3, is between about = 0·1 and = 0·3, and the fall of hp is very steep and around = 0·97. Three distinct ranges are readily discernible in Fig. 3:
1. Approximately 0 < < 0·3: p/g, the carbon source:sink ratio is small; hg < 1 and growth is limited by photosynthesis (‘growth inhibition’). The uptake of N in this range is roughly proportional to photosynthesis, resulting in the almost linear increase of the N border.
2. Approximately 0·3 < < 0·9: both limitation functions are close to 1 (‘balanced growth’) and growth is limited by sink strength, g (e.g. due to low temperature). Increasing the source strength, p, in this range, results in replacement of vacuolar nitrate with carbohydrates. The result is the negative slope of the N border in Fig. 3, reminiscent of the negative correlation between cell sap sugar and nitrate (Seginer, 2003). An approximation to this segment of the N border can be obtained by substituting for the fluxes of eqn (14) from eqns (18)–(22), (25)–(28) and (31), and setting = 0, as well as hp = hg = 1. The result,
has a negative slope of –ßN/ßC.
3. Approximately 0·9 < Cv < 1: the vacuole is nearly saturated with carbohydrates, resulting in hp < 1 (‘photosynthesis inhibition’). As p/g increases in this range, the carbohydrate production is increasingly diverted into the excess-C compartment. Structural growth continues to be limited by sink strength and the uptake of N is limited accordingly. As p/g approaches 
, u/g approaches r, the structural N : C ratio (vertical dashed line).
RGR and N : C ratio
Figure 4 shows the normalized constant RGR of isolated plants [the approximation in eqn (54)], as a function of normalized nutrient availability, u/g, with normalized photosynthesis, p/g, as a parameter. The N border and the E border, as well as the four regions of Fig. 3 are also present here, except that the region where N is abundant and MCe > 0 degenerated into a short segment on the line RCs/ag = 1, left of the E border.
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Where u/g limits growth, RGR is proportional to the nutrient supply and the lines for the four different levels of p/g coincide (line at slope 1/r through origin and asterisk). Under these conditions the vacuole is C-saturated and hp is low. As u/g increases, either p/g or eventually g itself (e.g. via low temperature) becomes the limiting growth factor. The latter happens along the horizontal segment, where RCs/ag = 1.
For the parameter set of Appendix A, normalized photosynthesis, p/g, should be at least about 1·7 to enable maximum RGR when N is abundant. This estimate is obtained by adding to 1 unit which goes to structural growth, also = 0·3 for growth respiration, about 1/ 
0·1 for maintenance respiration, and approx. bg = 0·2 which goes to the vacuole. Maximum normalised growth can be achieved (with sufficient N supply) for any level of p/g above 1·7, but normally there is no incentive to operate at this high level, where the increase in dry matter is not accompanied by an increase of leaf area (structural growth) and fresh mass. In this sense p/g 1 + + 1/ + bg (approx. 1·7 with the current parameter set) represents an optimal normalized shoot environment. Increase of actual (absolute) growth can be achieved if both p and g are increased (e.g. in a glasshouse, by adding supplementary light and heating), while maintaining the desired p/g ratio.
The RGR turns out to be barely affected by the value of . Reducing does, however, reduce total dry matter accumulation (in the excess-C compartment).
Figure 5 shows the total-N : total-C ratio [eqn (57)] as a function of normalized N availability, with normalized photosynthesis as a parameter and two values of , namely = 0·4 and = 1·0. Note that, unlike Figs 3 and 4, here higher values along the ordinate are associated with lower carbon source : sink ratios.
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The curves for constant p/g and
= 1 are almost linear, in agreement with Point 4 of the LO. They do bulge upwards, however, as approaches zero. At = 0·4 the bulge seems still tolerable, but comparison with experimental data is required before a final decision is made regarding the model’s adequacy on this point. |
DISCUSSION |
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TOPABSTRACTINTRODUCTIONMODELEQUILIBRIUM COMPOSITION AND...EQUILIBRIUM GROWTH EXAMPLEDISCUSSIONCONCLUSIONLITERATURE CITED |
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Parameter estimation
The NICOLET model (Seginer, 2003) has 19 parameters, only 12 of which are required for the EG analysis in this paper. The other seven parameters already have adequately established values, based on standard photosynthesis, nutrient uptake and respiration experiments. However, most of the 12 EG parameters, are specific to the NICOLET model and do not yet have established values. The values used in this paper (Appendix A) are based on data from recent experiments (e.g. Broadley et al., 2003), but the number of parameters is too large to be evaluated with confidence from the available experimental data. Some simplification of the parameter estimation task can be achieved by isolating and treating sub-sets of parameters independently of the others. Following are examples of parameter elimination, isolation and grouping.
1. Of the three parameters ßC, ßN and , only two are independent, as is clear from the division of eqn (11) throughout by .
2. Sensitivity analyses, using the method of Ioslovich et al. (2003), have shown that the predictions of the model are insensitive to the slopes, s, of the limitation functions. Hence the accuracy of these parameters is not critical. In the numerical example, rather high values of sg and sp result in sharp (perhaps too sharp) transitions between ranges, as is evident in Figs 3–5.
Certain NICOLET parameters and parameter ratios can be (and have been) obtained reliably from well-established experimental relationships. For instance:
3. From figs 1 and 2 of Seginer (2003), ßC/ßN and /r can be determined independently of any other parameters.
4. The value of can be estimated from the dry matter content of normal lettuce plants. If it is considered that the C content of the vacuole, when saturated with carbon compounds ( = 1), is about 50 % that of the structure and that the C : O ratio of dry matter is the same as that of carbohydrates, then = 0·9 x 10–3 m3 [H2O] mol–1 [C] (Appendix A) is equivalent to the common value of 5 % dry matter content.
5. RGR and the uptake of nitrogen are measurable in many experiments. An experiment with plants growing exponentially at a high p/g ratio (high light, low temperature), may produce a value for RCs at the experimental point indicated by an asterisk in Fig. 4. At this point, (u/g)* = r; hence
‘Mild’ and ‘severe’ N stress
As the availability of N (namely u/g) decreases, plants are exposed to an increasing N stress. In lettuce, there seems to be a clear distinction between ‘mild’ and ‘severe’ stress (Seginer, 2003), the latter setting in when essentially all nitrate has been removed from the vacuole. In terms of Fig. 2, the mild stress is associated with the plateau labelled by ‘balanced growth’, or roughly between = bg and = bp. In this range, small changes in N availability result in large changes of nitrate content, in contrast to the ranges left and right of the plateau, where nitrate content is relatively insensitive to changes in u/g. Left of the plateau, N stress is non-existent, while right of the plateau the stress is ‘severe’. As long as the stress is mild, the plants continue to grow, albeit at a reduced rate, and there is hardly a change in dry matter content, since no excess carbon is produced (MCe 0). When approaches bp (from below), growth decreases rapidly and the dry matter content may double or triple compared with its normal value (Seginer, 2003). In Fig. 3 the mild stress coincides roughly with the range 0.3 
0·9.
Figure 4 shows the effect of N stress on the growth rate. Consider, as an example, the curve for p/g = 1·5, where three distinct segments are evident (parameter values could be chosen to make the transition between ranges more gradual). When N is abundant (u/g 
aprrox. 0·21), the normalized exponential RGR is constant at approximately RCs/ag = 0·95. Between the N border ( 0·25), and the u-limiting line ( 0·96), there is a reduction of growth at a rate of about 2·85 mol [C] mol–1[N]. At a yet lower supply of N, the slope of the curve (loss of growth) is increased to about 6·12 mol[C] mol–1[N]. The slopes of the other iso-p/g curves are similar. For this set of parameters, therefore, the loss of growth may be said to be more than twice as large in the severe-stress range than in the mild-stress range.
An intuitive explanation of this behaviour is as follows. While the excess-C compartment is empty, the available N and C (carbohydrates) are partitioned between the structure (resulting in growth) and the ‘vacuole’ (where C and N accumulate as required by the osmotica balance). When N availability is high, the vacuole is essentially ‘saturated’ with nitrate and most of the carbon is free to participate in the growth (leaf expansion) process. As less N becomes available, the C : N (carbohydrate to nitrate) ratio in the vacuole increases and a larger fraction of the newly produced C compounds is partitioned to the vacuole, leaving less for growth. This explains the moderate changes in growth rate within the ‘mild’ stress range. In the severe-stress range, reduction in the availability of N has only a minor effect on the nitrate content and growth becomes strictly proportional to the supply of N.
Note that when p/g is high (e.g. p/g = 2·0 in Fig. 4) the mild-stress range is limited, because the N border is at a low nitrate level ( 0·7). In such a case, absolute growth rate may be increased by increasing the temperature (that is g), namely by lowering p/g to a level around the optimum of 1·7.
This view of the crop is only partially compatible with the ‘luxury’ consumption concept (e.g. Justes et al., 1994; Grindlay, 1997). The crop described by NICOLET does not take up nitrate just for storage (as it does with carbo hydrates, storing the excess in a special compartment). An increase of nitrogen content is always accompanied by at least some increase of growth rate (as in the mild-stress range).
Optimal operation
It is impossible to quantify the optimal environment for the growing of lettuce when the weather, the engineering and the economics of the system (e.g. glasshouse) are not specified. Nevertheless, it is possible to suggest a few guidelines, as follows.
1. Light and temperature levels should be interdependent such that the ratio p/g is in the neighbourhood of p/g 1 + + 1/ + bg (about 1·7 in this example). With abundant N supply, this insures maximum normalized growth rate, namely RCs/ag = 1 (Fig. 4).
2. The normalized N supply, u/g, should not exceed 0·21 (in this example), to prevent over-supply of N, possibly leading to leakage into the environment.
3. If, as a result, the nitrate concentration in the produce is higher than permissible (by health regulations), p/g should be increased somewhat (e.g. to approx. 2·0), such that the production point moves to the left along the horizontal portion of the N border of Fig. 4. Accordingly, u/g should be somewhat reduced (e.g. to approx. 0·18).
4. The optimal level of g (and hence, via guidelines 1–3, of p and u), depends on the cost involved in changing the environment and on the response (linear or otherwise) of the crop to this change. There may also be some physiological constraints, such as the appearance of leaf-tip burn at high leaf expansion rates.
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CONCLUSION |
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TOPABSTRACTINTRODUCTIONMODELEQUILIBRIUM COMPOSITION AND...EQUILIBRIUM GROWTH EXAMPLEDISCUSSIONCONCLUSIONLITERATURE CITED |
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Many nutrition and growth experiments, as well as some production systems (Both et al., 1999; Albright et al., 2000) subject plants to a constant environment (on a diurnal basis). The resulting equilibrium growth (EG) may, therefore, be studied with the analysis presented in this paper. The special case of balanced growth (BG) is associated with the absence of process inhibition, which is the result of matching strengths of the sources of carbon and nitrogen between them and with the corresponding sink strengths of the growing crop.
The NICOLET model is able to mimic the five central observations of many EG experiments, namely (1) the initial exponential growth, (2) the constant composition, (3) the equality of relative growth rate (RGR) and relative nutrient supply rate (RNR), (4) the proportionality between the N : C ratio and the RNR (and the RGR), and (5) the proportionality between the water content and the reduced N content. Points 1, 2, 3 and 5 are imbedded in the formulation of the model, while point 4 is satisfied approximately.
The model can be used to derive, analytically, the (constant chemical) EG composition and the RGR of a crop under a wide range of environmental conditions, including abundant and limiting N supply. Plotting the results in terms of the normalized shoot environment p/g and the normalized root environment u/g, results in a compact description of the problem-space. A study of these results may suggest the environmental relationships (between light, temperature, carbon dioxide and nitrate concentrations) which maximize productivity. The absolute production level for optimal operation requires additional economic information, which is outside the scope of this paper.
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ACKNOWLEDGEMENT |
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This research has been funded by EU project FAIR6-CT98-4362 (NICOLET).
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, k, 
and others, are omitted from the list. All parameters, except
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TOPABSTRACTINTRODUCTIONMODELEQUILIBRIUM COMPOSITION AND...EQUILIBRIUM GROWTH EXAMPLEDISCUSSIONCONCLUSIONLITERATURE CITED |
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