Simulating Growth and Root-shoot Partitioning in Prairie Grasses Under Elevated Atmospheric CO2and Water Stress

Annals of Botany 81 : 489–501, 1998

Simulating Growth and Root-shoot Partitioning in Prairie Grasses Under Elevated Atmospheric CO2 and Water Stress H. W. H U N T*, J. A. M O R G AN† and J. J. R E A D†‡ * Natural Resource Ecology Laboratory and Department of Rangeland Ecosystem Science, Colorado State UniŠersity, Fort Collins, CO 80523, USA and † USDA-ARS, Crops Research Laboratory, Colorado State UniŠersity, Fort Collins, CO 80526, USA Received : 2 June 1997

Revised for revision : 26 August 1997

Accepted : 4 December 1997

We constructed a model simulating growth, shoot-root partitioning, plant nitrogen (N) concentration and total nonstructural carbohydrates in perennial grasses. Carbon (C) allocation was based on the concept of a functional balance between root and shoot growth, which responded to variable plant C and N supplies. Interactions between the plant and environment were made explicit by way of variables for soil water and soil inorganic N. The model was fitted to data on the growth of two species of perennial grass subjected to elevated atmospheric CO and water stress # treatments. The model exhibited complex feedbacks between plant and environment, and the indirect effects of CO # and water treatments on soil water and soil inorganic N supplies were important in interpreting observed plant responses. Growth was surprisingly insensitive to shoot-root partitioning in the model, apparently because of the limited soil N supply, which weakened the expected positive relationship between root growth and total N uptake. Alternative models for the regulation of allocation between shoots and roots were objectively compared by using optimization to find the least squares fit of each model to the data. Regulation by various combinations of C and N uptake rates, C and N substrate concentrations, and shoot and root biomass gave nearly equivalent fits to the data, apparently because these variables were correlated with each other. A partitioning function that maximized growth predicted too high a root to shoot ratio, suggesting that partitioning did not serve to maximize growth under the conditions of the experiment. # 1998 Annals of Botany Company Key words : Plant growth model, optimization, nitrogen, non-structural carbohydrates, carbon partitioning, elevated CO , water stress, Pascopyrum smithii, Bouteloua gracilis, photosynthetic pathway, maximal growth.

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INTRODUCTION Long term forecasts of the effects of elevated atmospheric CO and climate change on ecosystems require simulation # models. A strictly experimental approach is infeasible because ecological changes involve soil organic matter and nutrient pools with long turnover times, and because future conditions may include new combinations of species, water stress, soil properties and temperature regimes. Predictions of plant responses to these new conditions should be based on sound physiological relationships. Unfortunately, presently available models differ in their predictions of ecological responses to climate change, partly because of our inadequate understanding of how elevated CO affects # water use, nutrient use and allocation among plant organs (Melillo et al., 1995). Plant responses with the greatest potential to affect nutrient cycling include changes in total production, chemical composition and allocation between shoots and roots. These plant responses will affect decomposition rate, immobilization of nutrients into decomposing residues, nutrient mineralization and the future availability of nutrients for plants (Field et al., 1992). Root to shoot ratio (R : S) will affect decomposition because of physicochemical ‡ Current address : USDA-ARS, U.S. Salinity Laboratory, Riverside, CA 92507, USA.

0305-7364}98}040489­13 $25.00}0

differences between root and shoot tissues, and because soil conditions are more favourable for decomposition than surface conditions. R : S is known to be affected by nutrient supply, carbon supply and water stress (Johnson, 1985 ; Sieva$ nen et al., 1988 ; Hilbert, 1990 ; Dewar, 1993 ; Reynolds and Pacala, 1993 ; Cannell and Dewar, 1994). Changes in R : S under elevated CO have varied among studies, ranging # from increases to decreases, or no effects (Hunt et al., 1991 ; Rogers, Runion and Krupa, 1994). The basis for this variation is unclear, but may be related to differences in water or nutrient supply, as well as to species differences. Objectives of this research were to develop a simulation model relating plant growth, root-shoot partitioning and plant chemical composition to CO and water supply, and # to use the model to help evaluate alternative hypotheses for the control of partitioning. We chose a level of resolution appropriate for inclusion in ecosystem level models (e.g. Hunt et al., 1991), rather than more detailed plant physiological models. A variety of root-shoot partitioning models have been proposed. Johnson and Thornley (1987) pointed out that partitioning models have been evaluated qualitatively, but that ‘ little has been done to assess partitioning models experimentally ’. Our model is intended to predict growth of real plants, not merely to serve as a heuristic tool, and accordingly it was objectively evaluated against experimental data from two perennial grasses native to shortgrass steppe (Morgan et al., 1998).

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Hunt et al.—Root-shoot Partitioning in Prairie Grasses MODEL STRUCTURE

State Šariables Our model (Fig. 1) includes separate state variables for carbon (C) and nitrogen (N). ‘ Cell wall ’ includes cellulose, hemicellulose, lignin and any other polymeric C or N not readily remobilized for translocation. ‘ Synthetic machinery ’ is the metabolically active component consisting mostly of proteins and nucleic acids, and is assumed to be mobile and distributed between shoots and roots in proportion to the distribution of cell walls. Cell walls and synthetic machinery each have fixed C}mass and N}mass ratios. Separate state variables are used for substrate C and N ; this allows a variable C}N ratio in substrates as a whole. Substrate C consists of polymeric storage compounds (starch, fructans) and small molecular weight compounds (sugars, amino acids). Substrate N is assumed to consist of 70 % amino acids and 30 % inorganic N, mostly nitrate. A variable corresponding to total non-structural carbohydrates (TNC) is calculated as substrate C minus the small amount of substrate C associated with the amino acid component of substrate N. As with synthetic machinery, substrate C and N also are assumed to be mobile and distributed between shoots and roots in proportion to cell walls. Therefore, plant TNC and N concentrations are calculated on a whole plant basis. The state variable for soil organic N does not include total N, but rather the more labile fraction that contributed to mineralization during the course of the experiment. Depletion of this pool allows the model to represent the decline in mineralization rate typically observed during soil incubations (Bonde and Rosswall, 1987). Soil inorganic N consists of ammonium plus nitrate. There is no state variable for soil water, and observed soil water content was used as a driving variable. Our model was inspired by that of Johnson (1985). Features in common include the definition of C and N substrate pools, the distinction between shoots and roots for structural state variables but not for substrates, and the prediciton of allocation as a function of root and shoot biomass as well as substrate C and N concentrations. The CO2 Ac

Equations for processes Symbols and definitions of variables and parameters used in the model are given in the Appendix. Following Johnson (1985), neither shoot nor root respiration is represented (Fig. 1), and the uptake of CO -C into the substrate C pool # is whole plant net C assimilation, Ac (g C per pot d−") : Wsc ¬ewc, Ac ¯ pac¬Smc¬ Wsc­Wrc

where pac is the maximal specific rate of net C uptake [g C (g C)−" d−"], Smc is whole plant synthetic machinery (g C per pot), Wsc is shoot cell wall (g C per pot), Wrc is root cell wall (g C per pot), and ewc is the effect of soil water content on C uptake. Because net C assimilation is proportional to the shoot’s share [Wsc}(Wsc­Wrc)] of Nrich synthetic machinery, C assimilation depends on plant N status. Johnson (1985) omitted an effect of N on C uptake, since in his model variation in N concentration can only be related to substrate N. It is natural to assume a N effect on net C assimilation since N and water are the main factors limiting production in our system (Hunt et al., 1988). The

B B

ROOT CELL WALL C (Wrc), N

B T

(1)

SHOOT CELL WALL C (Wsc), N

B

SUBSTRATE C (Suc)

primary differences are that the soil environment was made explicit in our model by adding state variables for soil inorganic N and soil organic N, and that Johnson’s ‘ structural ’ state variables are here divided into two components—cell walls and synthetic machinery. Bachelet et al. (1983) also distinguished structural and metabolic components, the former corresponding to our cell walls and the latter to the aggregate of our substrates and synthetic machinery. Thornley (1991) distinguished substrate, machinery and three different categories of cell wall. We have attempted to use the minimal amount of detail in model structure necessary to mimic our experimental data. We initially employed Johnson’s approach of combining cell walls and synthetic machinery into a single state variable, but the model was incapable of simulating the observed reduction in the ratio of synthetic machinery to cell walls, and thus in N concentration, due to aging.

B

SUBSTRATE N (Sun) An

B SYNTHETIC MACHINERY C (Smc), N

T

SOIL INORGANIC N (Sin)

Mn SOIL ORGANIC N (Son) F. 1. State variables (boxes) and processes (arrows) in the model. Symbols for state variables are given in parentheses. The C : N ratios are fixed for cell walls and for synthetic machinery, but not for substrates. Processes include net C assimilation (Ac), biosynthesis (B), turnover (T ), uptake of soil inorganic N (An) and mineralization of soil organic N (Mn).

Hunt et al.—Root-shoot Partitioning in Prairie Grasses 1.0

491

1.0

0.8 0.6 esyn

ewc and ewn

0.8

ewc, P. smithii ewc, B. gracilis ewn, B. gracilis ewn, P. smithii

0.4

0.2

0.0

0.1

0.2 0.3 0.4 Soil water (g g–1 dry soil)

0.5

F. 2. The effects of soil water on reduction factor ewc affecting the rates of net C assimilation and biosynthesis, and reduction factor ewn affecting N uptake. These reduction factors are represented by sine curves and are presented here only over the range of soil water observed in the experiment of Morgan et al. (1998).

specific uptake rate is assigned a higher value in plants grown under elevated CO . The effect of soil water, ewc # (non-dimensional), is a sigmoid function (Fig. 2) constrained by the optimization procedure (see below) to a value of 1±0 in wet soil, equivalent to about 0±35 g water per g dry soil in our experiment. In Johnson’s model, N uptake is a function of root biomass and a ‘ specific ’ (per unit biomass) activity parameter. We added effects of soil water content (swc) and soil inorganic N (Sin, g N per pot). The rate of uptake of soil inorganic N is An (g N per pot d−") : An ¯ anm¬rtc¬ewn¬

Sin ¬ecn, ki­Sin

P. smithii B. gracilis

0.6

(2)

where parameter anm is the maximal specific rate of uptake (g N g root C−" d−"), rtc is root C (cell walls plus the root’s share of synthetic machinery, g C per pot), ewn (Fig. 2) is a non-dimensional reduction factor for the effect of soil water on uptake, Sin is soil inorganic N, and parameter ki is the Michaelis-Menten half saturation constant (g N per pot). The reduction factor ecn (not shown) is a two parameter function that reduces uptake as substrate N becomes large or as substrate C becomes small (Johnson, 1985). The dry mass of the various plant components is calculated by dividing the mass of C (or N) by the mass fraction of C (or N) in the component. The fraction C in cell walls was assumed to be the same as in cellulose (0±45), and the fraction N was estimated from the minimal concentrations (0±0039 N in Bouteloua gracilis, 0±0046 N in Pascopyrum smithii) observed in senescent shoots in autumn (Hunt et al., 1996), when much of the N is retranslocated to perennial organs (Clark, 1977). The fractions C (0±54) and N (0±16) in synthetic machinery are averages for several proteins. The fraction C in substrate C is 0±44 (starch and fructan). The fraction N in substrate N (0±23) is nearly the same in amino acids and nitrate, the principal components of substrate N. The mass of the amino acid fraction of substrate N is included with substrate C.

0.4 0.0

0.1 0.2 Fraction synthetic machinery

0.3

F. 3. The effect of the fraction synthetic machinery [synthetic machinery}(synthetic machinery plus cell walls)] on esyn, the partitioning coefficient for formation of cell walls Šs. synthetic machinery [eqns (6)–(8)]. Coefficient esyn is represented by a sine curve presented in the figure only over the simulated range of fraction synthetic machinery.

Soil organic nitrogen (Son, g N per pot) is mineralized at rate Mn (g N per pot d−") : Mn ¯ pn¬Son¬swc,

(3)

where pn is the maximal specific rate [g N (g N)−" d−"] and swc is soil water content. The trade-off between shoot and root growth is accomplished by assuming a ‘ reciprocal ’ (Johnson, 1985) relationship between partitioning coefficients for shoots (lsh) and roots (lrt) : lsh­lrt ¯ 1.

(4)

The shoot partitioning coefficient is a function of shoot biomass, root biomass, substrate concentrations and a fixed partitioning parameter pf according to : lsh ¯

pf¬nc¬Wrc . cc¬Wsc ­ pf¬nc¬Wrc

(5)

This empirical function serves to increase relative shoot growth when shoot biomass (Wsc) is low relative to root biomass (Wrc), and when substrate N concentration (nc) is high relative to substrate C (cc). Thus, conditions such as high soil inorganic N that tend to increase plant N lead to decreased R : S. Alternative formulations including the effect of water stress on partitioning are presented below. In distinguishing between cell wall and synthetic machinery, our model increases the range of possible plant N concentrations, and creates opportunities for relating rates of C uptake and biosynthesis to plant composition in a physiologically meaningful way. This model structure also requires specification of how plants control the balance between walls and synthetic machinery. We adopted an empirical approach similar to that used for shoot}root partitioning. The partitioning coefficient for synthetic machinery, esyn, is assumed to decrease as the ratio of synthetic machinery to walls increases (Fig. 3). The balance between growth of cell walls and synthetic machinery also is

492

Hunt et al.—Root-shoot Partitioning in Prairie Grasses

affected by the composition of substrates, with wall growth favoured when the C}N ratio of substrates is high. The rate of formation of shoot cell walls, Bswc (g C per pot d−"), is :

for plant composition (Cannell and Dewar, 1994). Specifically, Davidson proposed that :

Bswc ¯ pw¬sw¬Smc¬lsh¬ewc¬(1®esyn),

where sas is the specific activity of shoots [g net C assimilation (g shoot C)−" d−"], sar is the specific activity of roots [g N uptake (g root C)−" d−"], and Bs and Br are the biomasses (g C per pot) of shoots and roots, respectively. Rearranging eqn (9) gives :

(6)

where pw is the rate constant (complex units), ewc (Fig. 2) is the effect of water and sw (fraction dry weight) is the concentration of substrates with C}N ratio matching that of cell walls. If the C}N ratio of substrates (Suc}Sun) is less than that of cell walls, sw equals substrate C concentration. If this ratio is greater than that of cell walls, sw equals substrate N times the C}N ratio of walls. This relationship has the useful property that the rate of formation of shoot cell walls approaches zero if either substrate C or substrate N approaches zero. We used the same soil water reduction factor, ewc, for both photosynthesis [eqn (1)] and biosynthesis [eqn (6)], because using different factors did not improve model performance. The rate of formation of root cell walls, Brwc (g C per pot d−"), is the same as that for shoots, except that lrt [eqn (4)] is substituted for the partitioning coefficient lsh : Brwc ¯ pw¬sw¬Smc¬lrt¬ewc¬(1®esyn),

(7)

where all terms are as defined above. Thus the balance between shoot and root growth is controlled by the values of the partitioning coefficients lsh and lrt, which in turn depend on the partitioning parameter pf [eqn (5)]. The rate of biosynthesis of synthetic machinery is Bsmc (g C per pot d−") : Bsmc ¯ pm¬sm¬Smc¬ewc¬esyn,

(8)

where pm is the rate constant, sm is the concentration of substrates with C}N appropriate to synthetic machinery, defined similarly to sw in eqns (6) and (7), and the other factors are as defined above. Synthetic machinery is thus auto-catalytic, but is kept within bounds by the partitioning coefficient esyn. Transfers of N from substrate N to cell walls and synthetic machinery are calculated by multiplying the transfers of C [eqns (6)–(8)] by the N}C ratio appropriate to either walls or synthetic machinery. Cell walls, once laid down, cannot be catabolized, but synthetic machinery is assumed to turn over at a constant specific rate, pt (d−"). Materials turned over are returned to substrate compartments (flows ‘ T ’ in Fig. 1). Variants on allocation As pointed out by Johnson and Thornley (1987), partitioning models at this level of resolution are fundamentally empirical, as they do not deal with hormonal effects or mechanisms of translocation. To test the following alternative partitioning functions, we incorporated them into the model and used optimization (see below) to derive least squares estimates of parameters. According to the balanced growth hypothesis (Davidson, 1969), R : S is adjusted to achieve a balance between C supply (shoot specific activity¬shoot biomass) and N supply (root specific activity¬root biomass) appropriate

sas¬Bs

£

sar¬Br,

Bs}Br ¯ pd¬sar}sas,

(9)

(10)

where pd is a proportionality constant. Davidson (1969) did not present a dynamic model, but one of several ways in which eqn (10) can be extended to a dynamic model is to assume that the ratio of allocation to shoots Šs. roots equals the ideal ratio (Bs}Br). This assumption leads to an equation for lsh, the shoot partitioning coefficient : lsh ¯

pd¬sar sas ­ pd¬sar

(11)

Equation (11) uses specific activities to control partitioning rather than substrate concentrations and shoot or root biomass as in eqn (5), and the partitioning parameter pd (g C per g N) corresponds to the fixed partitioning parameter pf of eqn (5). This derivation is equivalent to that of Luo, Field and Mooney (1994), if their ‘ plant nitrogen concentration ’ is treated as a fixed partitioning parameter rather than as a dynamic variable. Johnson and Thornley (1987) reviewed arguments that regulation of allocation is more likely based on internal concentrations rather than on rates of uptake as in eqn (11). This idea suggests a third equation for partitioning : lsh ¯

pj¬nc cc ­ pj¬nc

(12)

This equation is similar to eqn (5), except the products of substrate concentration and biomass are replaced with substrate concentrations alone. The partitioning parameter pj has the same units as pf in eqn (5). Minchin, Thorpe and Farrar (1994) hypothesized that partitioning is controlled as a function of substrate C concentration only, since TNC should indirectly reflect N availability under many circumstances. We represented this hypothesis by : ps , (13) lsh ¯ cc­ps where cc is substrate C concentration and ps is a partitioning parameter with the same units as cc. Luo et al. (1994) derived a partitioning model based on the ‘ photosynthesis : growth balance concept ’. Their model was derived for balanced growth. To incorporate their partitioning equation into a dynamic model such as ours, some terms in their equation must be treated as fixed parameters and others as variables. Our interpretation of their model is that allocation to shoots is directly proportional to plant N concentration. To take full advantage of our model structure, we substituted substrate N concentration which should be positively correlated with plant

Hunt et al.—Root-shoot Partitioning in Prairie Grasses N concentration. To insure that the shoot partitioning coefficient be less than one, we used a hyperbolic curve in place of simple proportionality : lsh ¯

1

nc¬pl , ­ nc¬pl

(14)

where nc is substrate N concentration as before and pl is a fixed partitioning parameter. With this formulation, the model of Luo et al. (1994) is the converse of eqn (13), in which partitioning is a function of substrate C concentration only. Growth maximization is among the criteria used to derive partitioning models (Cannell and Dewar, 1994). Johnson and Thornley (1987) applied this criterion during balanced exponential growth to derive the ‘ general partitioning function ’, which is equivalent to eqn (5) with the fixed partitioning coefficient pf replaced by px, a variable function of plant C and N status : px ¯

cc­pcw , nc­pnw

(15)

where cc and nc are substrate C and N concentrations as above, and pcw and pnw are the C and N contents of the cell wall fraction. Gedroc, McConnaughay and Coleman (1996) pointed out that R : S may shift during normal development (ontogenetic drift) independently of any adaptive changes in partitioning arising from resource constraints. Thus, an experimental treatment that affects plant development will also change R : S, even if the plant does not alter partitioning in direct response to the treatment. If plots of R : S Šs. plant size fell along the same line for different treatments, this was taken as evidence that ontogenetic drift accounted for the treatment effect on R : S (Gedroc et al., 1996). Thus we tested a null model of partitioning, in which the partitioning coefficient lsh is linearly related to plant size. However, grass phenological development is usually related to time or ‘ growing degree days ’ rather than size (Gillen and Ewing, 1992). We therefore tested a second null model, another expression of ontogenetic drift which relates lsh linearly to time. In the ultimate null model tested, lsh is a constant independent of time and the size or condition of the plant. Simulation The above expressions for C and N transfers were combined together into separate rate equations for the seven state variables Suc, Wsc, Wrc, Smc, Sun, Sin and Son. State variables for the N contents of shoot and root cell walls and of synthetic machinery are not independent because of fixed C}N ratios, and therefore their dynamics were calculated as proportional to the corresponding C state variable. Together the rate equations conserve mass of both C and N, when a source}sink is included for C. The model was solved as a set of seven difference equations using a variable time step chosen to limit the relative change in the most rapidly changing state variable to less than 3 % per time step. This method of integration gives a good

493

compromise between accuracy and execution time (Hunt et al., 1991).

MATERIALS AND METHODS A detailed description of experimental design and methods is given by Morgan et al. (1998). Briefly, growth conditions were intended to approximate a field environment whilst carefully controlling experimental variables. Plants were grown in field soil in free-draining columns (15 cm diameter¬45 cm tall) under water stress and with no fertilizer added, so that N supply was near that naturally available. The experiment was a full factorial with two perennial grass species, Pascopyrum smithii (Rydb.) A. Love and Bouteloua gracilis (H. B. K. Lag), two watering regimes (moderately and severely water stressed), and two CO levels. Plants # were started in a glasshouse and after 14 d were transferred into two growth chambers with 1000 µmol m−# s−" photosynthetic photon flux density, 28}18 °C day}night temperatures, 14-h photoperiod, and either 375 or 750 µl l−" CO concentrations. Soil water in the columns # was near field capacity at the time CO and water treatments # were started. Periodic irrigations were less frequent in the severely stressed treatment, and were insufficient to reverse a trend of declining soil water throughout the experiment in all treatments. Single columns were sampled weekly for five weeks. The experiment was repeated four times through time for four replications. The weekly destructive samplings determined soil inorganic N, organ dry weight and the concentrations of N and TNC in leaves, stems and roots plus crowns plus rhizomes. For our present purposes, leaves plus stems were combined into shoots, roots plus crowns plus rhizomes referred to as ‘ roots ’, and plant N and TNC concentrations were calculated on a whole plant basis. When soil columns were destructively sampled, representative soil samples were obtained from four depths through the profile, composited, and analysed for available NO − $ and NH + (EPA, 1983). % PARAMETER ESTIMATION We used an optimization procedure to derive estimates of parameter values that minimized the lack-of-fit between data and model predictions. Some researchers object to this procedure on the basis that the modelling activity is degraded to statistical curve-fitting (Romesburg, 1981). For example, Hopkins and Leipold (1996) carried out a simulation study of least-squares parameter estimates in a biochemical model with 19 parameters and 24 data points (not counting replicates). They found that : (1) an error in the estimate of one parameter could be offset by errors in other parameters ; (2) an erroneous model could be made to fit the data well ; and (3) parameter estimates differed from their true values if there was any disparity between the structure of the model and that of the modelled system. The third finding will apply to all models which are abstractions and simplifications of the real world. The first two findings reflect a condition of overdetermination, in which the ratio of data points to fitted parameters (1±3 in their study) is too low. With enough free

494

Hunt et al.—Root-shoot Partitioning in Prairie Grasses

parameters, models of arbitrary structure can be made to fit a data set. In our opinion, the basic distinction between empirical and mechanistic models does not lie in how parameter values are estimated, but rather in the hypothesized parallel between model structure and real world phenomena in mechanistic but not in empirical models. Parameters in mechanistic models have physical meanings, and ideally their values should be based on information independent of the data being simulated (Hopkins and Leopold, 1996). This may be possible in chemical systems in which rate constants can be independently determined for individual reactions, but is less feasible in whole-plant physiology because it is more difficult to break down the system into meaningful components for experimentation. For example, estimates of nutrient uptake parameters of excised roots in solution may not apply to intact mycorrhizal roots growing in soil. In this situation, fitting a mechanistic model to data from a complex experiment may yield better parameter estimates than would a simpler experiment done under artificial conditions. We have addressed the problem of overdetermination by : (1) maximizing the ratio of data points to parameters (5±6 in our experiment, excluding replicates) ; (2) starting with simple models and adding complexity only as necessary to explain the data ; (3) constraining parameter values within reasonable ranges ; and (4) performing sensitivity analyses to determine how parameter values affect model output. With these four measures, leastsquares estimates provide a powerful method for comparing the success of alternative models, since any lack of fit may be ascribed to incorrect model structure rather than to inadequate parameter estimates (Hunt, Cole and Elliott, 1985). Our calculations of model residual error E were based on log transformed data : "! E ¯ 33 ²log(model)®log(data)´# (16) The first summation is over treatments (species, CO and # water), sample dates and five data sets (total weight, R : S, plant N concentration, TNC concentration and soil inorganic N). The second summation is over four replicate pots. The log transform is appropriate when between-replicate variance increases with the mean (Winer, 1971), as in our data. Equation (16) is equivalent to using relative errors between model and data, since log(model)®log(data) ¯ log(model}data). Summing over replicates takes account of the varying precision among dates and data sets. Model parameters were estimated usng Newton-Raphson iteration (Ralston, 1965) to minimize residual error. Minima were confirmed by initializing the optimization routine with different starting parameter values. Because there was no sample date at the initiation of treatments, intiial conditions of several state variables were treated as parameters to be estimated. Fit to data was relatively insensitive to these initial conditions, except for initial total weight. Several parameters were constrained to theoretical values or known relationships. For example, the zero-to-one reduction factors for water effects were required to take a value of 1±0 in wet soil, and the specific rate of C uptake [ pac in eqn (1)] was required to take a value at least 50 % greater in B.

gracilis than in P. smithii, based on photosynthesis data for these species (Morgan et al., 1994 a). Parameters for C and N contents of cell walls, synthetic machinery and substrates were taken as known values as specified above. Values of parameters and initial values of state variables were different in the two species, except for the initial values of soil organic and inorganic N and the specific rate of N mineralization, since the same soil was used for both species. The above procedure for estimating model parameters to achieve a least-squares fit to data was repeated for each different version of the model evaluated [e.g. the standard model with eqn (5) for partitioning replaced with one of eqns (11)–(15)]. Thus, the models could be objectively compared on the basis of residual error. A more intuitive measure of model error than E [eqn (16)] is the average relative error, AE : AE ¯ 10o(E/('!)®1,

(17)

where 760 is the total number of data points. We also employed a more statistically oriented measure of model fit, R #, the ratio of the variance explained by the model to the # non-error variance. Non-error variance was calculated as total variance minus pure error variance, and pure error was calculated as between replicate variance, within treatments and sample dates. RESULTS Experimental results Briefly, we found that both species grew larger under elevated CO and with more water. R : S was greater in P. # T     1. Parameter Šalues and initial Šalues of state Šariables giŠing the best fit of the standard model to the data of Morgan et al. (1998 ) State variable or parameter

Figure or equation

Value P. smithii

B. gracilis

anm ki pac (low CO ) # pac (high CO ) # pesyn1 (esyn)† pesyn2 (esyn)† pewc1 (ewc)† pewc2 (ewc)† pewn1 (ewn)† pewn2 (ewn)† pf pm pn pt pw Son Wsc

Eqn (2) Eqn (2) Eqn (1) Eqn (1) Fig. 3 Fig. 3 Fig. 2 Fig. 2 Fig. 2 Fig. 2 Eqn (5) Eqn (10) Eqn (3) Text Eqns (6), (7) Eqn (3) Eqns (1), (5)

4±68 16±6 1±12 1±31 0±0328 0±737 ®0±0839 0±348 0±0708 0±349 228±0 21±1 0±0255* 0±0377 59±9 0±921* 0±0308

5±40 30±1 1±67 1±72 0±0 0±768 ®0±0246 0±350 0±0164 0±350 451±0 38±2 0±0255* 0±0962 212±0 0±921* 0±0370

* Parameter pn and state variable Son pertain to soil properties, which are the same for both species. † The partitioning factor esyn and the water effects ewc and ewn are presented in Figs 2 and 3. Each curve is defined by two parameters giving the upper and lower argument values returning values of 0±0 and 1±0 in a sine function.

Hunt et al.—Root-shoot Partitioning in Prairie Grasses

495

T     2. Fit of Šarious Šersions of the model to data of Morgan et al. (1998 ) R#

#

Model version† (1) (2) (3) (4)

Standard Constant partitioning Maximize growth Unregulated N uptake

Total weight

Root} shoot ratio

0±98 0±43 0±98 0±98

0±88 0±47 0±19 0±88

Plant N

Plant TNC

Soil inorganic N

0±83 ®0±82 0±87 0±82

0±91 0±64 0±86 0±91

0±75 0±43 0±43 0±73

Average* relative error

0±344 0±856 0±395 0±351

R # is the fraction of maximal possible variance in the data accounted for by the model (see text). A negative value results when model error # exceeds the variance of the data. * Average relative error is given by eqn (17). † Model Version : (1) Our standard model with partitioning according to eqn (5) ; (2) constant partitioning coefficient [eqn (4)] ; (3) partitioning parameter pf [eqn (5)] estimated to maximize growth rather than the fit to data ; (4) effects of plant C and N substrate concentrations on N uptake eliminated [ecn ¯ 1 in eqn (2)].

smithii, and increased with time in all treatments. Plant N concentration was greater in P. smithii, lower under elevated CO , and greater in the severely stressed water treatment. # Plant TNC concentration increased over time, was greater under elevated CO , and was greater in P. smithii. Soil # inorganic N was lower in cores with B. gracilis, lower under elevated CO , and declined over time. Some of these results # are illustrated below. A detailed analysis of these and additional experimental results is given by Morgan et al. (1998).

B. gracilis. The fraction of plant weight in cell walls in the model was fairly constant at about 82 % in B. gracilis and 74 % in P. smithii, dropping slightly between 40 and 60 d due to accumulation of TNC (Fig. 4 D). Estimates of the two parameters in factor ecn [eqn (2)], which regulates plant N uptake as a function of plant C and N substrate concentrations, were such that factor ecn stayed near 1±0 in both species, indicating that demand for N was always high. When this feedback regulation was eliminated from the model, the fit to data was not seriously affected (Table 2, model 4).

Fit of model to data Table 1 presents values of parameters yielding the best fit of the standard model [allocation according to eqn (5)] to the data, and Table 2 gives measures of goodness of fit. The standard model accounted for most of the non-error variation in the data (Table 2, model 1), and correctly predicted many of the significant treatment effects and species differences revealed by ANOVA (Morgan et al., 1998). However, an ANOVA of model error (model®data) revealed significant relationships (not shown) between error and CO treatment for R : S ratio and plant N concentration, # indicating that small systematic departures between model and data remained.

Plant physiology and composition We did not measure substrate N, but model substrate N as a fraction of plant N ranged from 3 to 6 % in P. smithii, and from 5 to 13 % in B. gracilis, after an initial period of equilibration. Substrate N (dry weight of components) as a percentage of plant dry weight ranged from 0±3 to 0±8 % in both species. The function relating the coefficient of partitioning between walls and synthetic machinery to the level of synthetic machinery (Fig. 3) was very similar in both species, but would promote a slightly smaller production of synthetic machinery and greater production of cell walls in

Effects of eleŠated CO # Elevated CO caused greater growth, greater TNC # concentrations, and lower soil and plant N concentrations in both species. Figure 4 illustrates these results for P. smithii and compares model predictions to data. The direct effect of elevated CO was expressed in the model by using # higher values for the rate of carbon uptake (parameter pac, Table 1). An important indirect effect of elevated CO was # expressed through driving the model with observed soil water content, which was significantly greater under elevated CO (Morgan et al., 1998). The model provided estimates of # the system N budget, since it includes N mineralization, plant N uptake and soil inorganic N dynamics. The net outcome of greater growth and lower plant N concentrations under elevated CO in the model was an average (across # water treatments) of 31 % greater N uptake by P. smithii and 7 % greater by B. gracilis. Soil N mineralization was greater under elevated CO because of wetter soil, but this # accounted for only 18 % of the increase in plant N uptake.

Species differences The model represented species differences by assigning different values to 15 species-specific parameters (Table 1). To evaluate which parameters were most responsible for consistently greater growth of B. gracilis compared to P.

496

Hunt et al.—Root-shoot Partitioning in Prairie Grasses 0.2

30

Total dryweight (g per pot)

g N per pot

A

0.1

0.0 B

B. gracilis P. smithii 20

10

5.0

%N

0

C

10

P. smithii, elevated CO2 P. smithii, ambient CO2 B. gracilis, ambient CO2

Root:shoot ratio

0.8

5

0 D

B. gracilis, elevated CO2

0.6

0.4

30

20

0.2

10

0

40 50 60 Days after planting

1.0

15 g per pot

30

F. 5. Species differences in weight of plants grown under elevated CO in the moderately water stressed treatment. Lines are model # output and points are means from the experiment of Morgan et al. (1998).

1.0

% TNC

20

3.0

15

30 45 Days after planting

60

F. 4. Effects of elevated CO on dynamics of soil inorganic N (A), # whole plant N concentration (B), total plant dry weight (C) and wholeplant non-structural carbohydrate (TNC) concentration (D) for P. smithii in the severely water stressed treatment. Lines are model output and points are means from the experiment of Morgan et al. (1998). ——E——, Elevated CO treatment ; – – D – –, ambient treatment.

#

smithii (e.g. Fig. 5), the model was run for each species after substituting values of groups of parameters with values from the other species. Parameters were substituted in groups pertaining to the same physiological function. When parameters controlling C acquisition ( pac, pewc1 and pewc2 ; Table 1) were switched between species, final weight averaged across treatments increased by 28 % in P. smithii and decreased by 27 % in B. gracilis. Thus, the B. gracilis

20

40

60 Days after planting

F. 6. Effect of elevated CO on root to shoot ratio. Lines are # model output and points are data from the experiment of Morgan et al. (1998). Both model and data are averages of the two levels of the water treatment.

parameters confer superior C uptake capacity. When the switch was made for parameters controlling N uptake (anm, ki, pewn1, pewn2, pkc and pkn ; Table 1), final weight decreased by 20 % in P. smithii and increased by 23 % in B. gracilis. Thus, P. smithii has superior N uptake capacity. This advantage of P. smithii does not result from the effect of water on N uptake, for which B. gracilis has a slight advantage (Fig. 2). Because the half-saturation constants for each species (ki, Table 1) are well above the levels of soil inorganic N (e.g. Fig. 4), uptake is approximately proportional to Sin¬anm}ki [cf. eqn (2)], which is greater in P. smithii by 57 %. When initial biomass or the other groups of parameters (biosynthesis}turnover—pm, pw, tsm, pesyn1, pesyn2 ; partitioning—pf ) were switched between species, final weight changed by less than 4 %. Thus, under the

Hunt et al.—Root-shoot Partitioning in Prairie Grasses P. smithii, severe water stress

0.8

Root:shoot ratio

P. smithii, moderate water stress B. gracilis, moderate water stress

0.6

B. gracilis, severe water stress

0.4

0.2

20

30

40

50 60 Days after planting

F. 7. Effect of water treatment (moderate Šs. severe water stress) on root to shoot ratio. Lines are model output and points are data from the experiment of Morgan et al. (1998). Both model and data are averages of the two levels of the CO treatment.

#

conditions of our experiment, species differences in C and N acquisition appear to account for most of the difference in production, with partitioning and biosynthesis having little effect.

497

alternatives yielded fits very similar to the standard model—the average relative error varied only from 0±341 to 0±347, and the R # values differed from the standard model # (Table 2, model 1) by 0±05 or less. However, the null model with constant partitioning coefficient gave an unsatisfactory fit to data (Table 2, model 2). To test whether allocation served to maximize growth, we estimated the value of the partitioning coefficient pf that maximized final plant biomass averaged over all CO and # water treatments, rather than maximizing the goodness-offit to data. All parameters except for pf were kept at their standard values. The resulting value of pf was smaller by 29 % in P. smithii and by 56 % in B. gracilis. In P. smithii, there was a 7 % increase in production, a 10 % increase in root biomass, a slight increase in R : S, and a 4 % increase in N uptake. The average relative error for P. smithii alone increased only from 0±331 to 0±343 ; thus in P. smithii allocation appears not to be far from that which maximizes production. In B. gracilis, maximizing production led to a greater increase in average relative error (from 0±357 to 0±444), due to a significant overestimation of R : S ratio and underestimation of TNC. Root biomass increased by 31 %, but N uptake increased by only 6 % because the lower root biomass in the standard model was already capable of taking up most of the available inorganic N. Production increased by only 1 %, indicating that production in B. gracilis was quite insensitive to partitioning under the conditions of our experiment.

Root : shoot partitioning The observed value of R : S (averaged across water treatments) was greater by 6 % under elevated CO than in # the control in P. smithii and smaller under elevated CO by # 5 % in B. gracilis (P ! 0±10), but these CO effects were not # entirely consistent through time (Fig. 6). In the model, R : S was consistently greater under elevated CO —16 % greater # in P. smithii and 9 % in B. gracilis. As soil water content declined over time (Morgan et al., 1998), R : S increased in both species and in both water treatments (e.g. Fig. 7). However, average R : S in P. smithii in the severely stressed water treatment (0±72) was not significantly greater than in the moderately stressed treatment (0±69), while R : S in B. gracilis was actually smaller in the more severely stressed treatment (0±39 Šs. 0±44, P ! 0±01). Adding a multiplier to eqn (5), which served to increase allocation to roots when the soil was drier, gave only a slight improvement in fit (lowering the standard relative error from 0±344 to 0±343). Interestingly, this version of the model correctly predicted lower R : S of B. gracilis in the severely stressed treatment, indicating that the direct effect of water stress on partitioning was over-ridden by the effects of substrate C and N. In the standard model, partitioning is a joint function of root and shoot biomass and C and N substrate concentrations (Johnson, 1985). Five of the alternative partitioning functions [eqns (11)–(15)] used various combinations of root and shoot biomass, C and N substrate concentrations and root and shoot activities. In the two alternative functions based on ontogenetic drift, the partitioning coefficient was a linear function of total biomass or of time. All seven

DISCUSSION Optimization proved useful for objectively comparing alternative model structures. The power of optimization is in specifying a model that predicts not merely plant growth, but growth plus an array of plant and soil properties across a range of environments. To simultaneously predict these variables requires that all the modelled processes be compatible among themselves ; that is, for the model to be an internally consistent representation of plant growth.

Plant composition Defining the plant in term of cell walls, synthetic machinery and substrate C and N allowed the model to predict much of the observed variation in plant composition. Predicted cell wall contents were higher than reported in B. gracilis shoots (45–59 % ; Wallace, 1969), but the model values include roots which presumably are higher in fibre than are shoots. Production of proportionally more cell wall material and less synthetic machinery in B. gracilis (C ) than % in P. smithii (C ) is consistent with the greater concentration $ of vascular bundles, fewer interveinal mesophyll cells (Morgan and Brown, 1979 ; Wilson, Brown and Windham, 1983), lower mesophyll cell density and substantially lower requirement for carboxylating protein (Conroy, 1992) in leaves of C grasses than C grasses. % $ Johnson (1985) reported a range of 30–50 % for the fraction of plant N in the substrate N pool. However, this

498

Hunt et al.—Root-shoot Partitioning in Prairie Grasses

fraction in well fertilized P. smithii and B. gracilis was between 14 and 41 % in shoots and between 12 and 31 % in roots (H. Skinner, pers. comm.). Our simulated fractions for unfertilized plants (3–13 %) were still lower. It is possible that substrate N could be increased and synthetic machinery N decreased in the model without changing total plant N, but we did not attempt this adjustment because we lacked data for substrate N in our experiment.

N uptake Feedback effects of substrate C and N on N uptake were unnecessary in the model. Although CO enrichment can # affect root N uptake kinetics (Bassirirad et al., 1996 ; Jackson and Reynolds, 1996), this may be unimportant in systems like the shortgrass steppe where soil N levels may limit plant growth (Hunt et al., 1988). Increasing TNC concentrations during the experiment also suggested Nlimitation. Despite lower plant N concentrations, total plant N uptake was greater under CO enrichment, which agrees # with some previous work with these grasses (Hunt et al., 1996). However, Morgan et al. (1994 b) found that CO # enrichment led to reduced plant N uptake in B. gracilis, possibly because the release by the plant of low quality substrates to the soil reduced N mineralization. In our model, increased total plant N did not result from a deregulation of uptake, because the feedback effects of plant N and C uptake were largely inoperative. Some of the additional N uptake under elevated CO resulted from # greater mineralization, but the majority of the increase, at least in P. smithii, was attributable to greater root biomass, and to increased specific uptake rate in wetter soil.

Species comparisons Snaydon (1991) concluded that C species do not typically % have greater production than C species when grown under $ similar conditions, in spite of the apparently higher rates of photosynthesis in C species. Nevertheless, according to our % model analysis, B. gracilis (C ) grew bigger than P. smithii % (C ) because of its superior photosynthetic characteristics, $ and despite its disadvantage in N uptake. The warm daytime temperatures and high growth chamber light fluxes may have favoured growth of B. gracilis over P. smithii (Morgan et al., 1998). Garnier (1991) claimed that fast growing plants were those with a high ratio of root specific activity (N uptake per unit biomass) to shoot specific activity (C uptake per unit biomass), but this relationship is just the opposite for our two species, and may represent a departure due to differing photosynthetic functional groups. C plants tend to have lower tissue N concentrations and % higher nitrogen use efficiency compared to C plants, an $ adaptation due mainly to a more nitrogen-use-efficient photosynthetic metabolism in the former (Brown, 1978). Thus, greater growth and lower specific root N uptake in B. gracilis may reflect those functional group differences, and may explain in part the adaptability of C species to soils of % low N fertility (Epstein et al., 1997).

Partitioning Partitioning models have been derived and evaluated partly on the basis of their performance during balanced exponential growth. Perhaps a more relevant test of partitioning models is their ability to predict transient responses to environmental stresses, as normally encountered by these species in the field (Lauenroth and Milchunas, 1991), and as in our experiment. R : S generally decreases with irrigation (Detling, Parton and Hunt, 1979 ; Gifford, 1979) ; thus, the increase observed in R : S in B. gracilis with additional water was surprising. According to the balanced growth hypothesis, increased allocation to shoots would be expected if added water increases root activity relatively more than shoot activity. Root activity is affected by water directly via the effect of water on uptake of soil inorganic N, and indirectly via the effect on N supply (mineralization). Shoot activity is affected by water directly via the effect of water on C uptake, and indirectly via the effect of water on N supply, since C uptake is proportional to the level of N-rich synthetic machinery. The effects of water treatment on N mineralization (­7 %) were similar for both species. Species differences in the direct effects of water on C and N uptake (Fig. 2) indicate that additional water leads to a greater improvement of root activity relative to shoot activity in P. smithii than in B. gracilis. The net effect of these direct and indirect effects was that the model correctly predicted the small (although slightly positive) relative effect of additional water on R : S ratio in P. smithii, the larger positive relative effect in B. gracilis, and lower plant N concentrations in both species. In the model, the direct effect of CO on shoot activity # exceeded the indirect effect of CO on root activity via soil # water conservation, which resulted in a slightly greater R : S ratio, especially for P. smithii. This was representative of a pattern observed in P. smithii in the middle stages of the study when CO enriched plants displayed higher R : S, # although B. gracilis displayed no such partitioning response (Morgan et al., 1998). The model correctly predicted that elevated CO had a greater effect on R : S ratio in P. smithii # than in B. gracilis. The model was very successful in representing the observed large species differences and temporal trends in R : S, and moderately successful in representing the small treatment (CO and water) effects. Thus, the functional # balance between root and shoot appears to provide a useful perspective for interpreting the dynamic response of R : S to resource constraints, without invoking any arbitrary treatment effects on partitioning. The general equivalence of the various models in which partitioning actively responds to soil water, root and shoot biomass, plant substrate C and N, and shoot and root specific activities indicates that these variables were strongly correlated in our data set, so that any one variable provided virtually the same information as several variables together. The success of the two models in which partitioning was strictly a function of biomass or of time supports the recommendation of Gedroc et al. (1996) that models and experiments need to distinguish ‘ ontogenetic drift ’ from active allocation. Experimental data from a wider range of

Hunt et al.—Root-shoot Partitioning in Prairie Grasses conditions including defoliation and variable N supply might better distinguish partitioning models. A potentially useful model structure not yet evaluated in our research is the separate representation of non-structural pools in shoots and roots (Thornley, 1972 ; Bachelet et al., 1983). It is conceivable that such a structure could explain plant growth and partitioning without resorting to any empirical whole-plant integrating mechanisms such as functional balance between shoots and roots (Cheeseman, 1993). Major departures of partitioning from the optimal must seriously hamper growth (Sieva$ nen et al., 1988). Therefore, the lack of sensitivity of production to large changes in partitioning in B. gracilis was surprising. A likely explanation involves feedbacks between root growth and root specific activity. Final uptake of N in our model was largely determined by the initial N supply plus N mineralization during the experiment. Inorganic N levels declined during the experiment, and additional root growth, while it might temporarily increase the rate of N uptake, would not result in a proportional increase in final N uptake. This explains how a large increase in partitioning to roots can lead to only small increases in N uptake and total plant production. A parallel phenomenon could operate in C uptake, if accumulated photosynthesis is limited by water supply rather than by leaf area. Plants with greater leaf area might have a higher rate of photosynthesis, but would exhaust soil water sooner. Some analyses of optimal partitioning treat shoot and root specific activities as parameters to be varied (Johnson and Thornley, 1987 ; Sieva$ nen et al., 1988 ; Dewar, 1993), rather than as dynamic variables that may interact with the environment, and thus may overestimate the sensitivity of production to partitioning. The large difference between the value of the partitioning coefficient which maximizes production and that which optimizes the fit to data suggests that partitioning in these plants does not serve primarily to maximize production. This is not surprising because production is but one component of fitness (Perrin and Sibly, 1993). Both species appear to allocate more to shoots than needed to maximize growth. Conceivably this is an adaptation to compensate for defoliation which is normally imposed on these two forage species (Lauenroth and Milchunas, 1991). Plant phenological development, reproduction and adaptations for storage and stress survival may further alter the optimal partitioning strategy (Cannell and Dewar, 1994). Finally, plant competition may theoretically favour rapid but inefficient use of resources (Cohen, 1970 ; Eissenstat and Caldwell, 1988). Therefore, optimal allocation could differ in the presence and absence of competitors (Reynolds and Pacala, 1993). The concept of growth maximization is prominent in the development of partitioning models (Hilbert, 1990 ; Cannell and Dewar, 1994), but our results suggest that growth maximization is inadequate as a surrogate for plant fitness. A C K N O W L E D G E M E N TS We thank E. Taylor and G. E. Schuman for soil analyses, D. R. LeCain and B. A. Weaver for plant and soil sampling,

499

M. A. Wheeler for carbohydrate analyses, R. H. Skinner for information on labile plant N compounds and M. C. Fowler for computing support. W. G. Knight and L. M. Dudley contributed to planning the experimental phase of the research. C. A. Jacobs-Carre contributed to data collection and G. V. Richardson advised on statistics. C. V. Cole, R. H. Skinner and J. H. M. Thornley provided insightful reviews. This work was supported by award number 92-37100-7670 of the National Research Initiative Competitive Grants Program of the USDA. LITERATURE CITED Bachelet D, Hunt HW, Detling JK, Hilbert DW. 1983. A simulation model of blue grama biomass dynamics, with special attention to translocation mechanisms. In : Lauenroth WK, Skogerboe GV, Flug M, eds. Analysis of ecological systems : State-of-the-art in ecological modelling. Amsterdam, The Netherlands : Elsevier Scientific Publishing, Inc., 457–466. Bassirirad H, Thomas RB, Reynolds JF, Strain BR. 1996. Differential responses of root kinetics of NH + and NO − to enriched % $ atmospheric CO concentration in field-grown loblolly pine. Plant, # Cell and EnŠironment 19 : 367–371. Bonde TA, Rosswall T. 1987. Seasonal variation of potentially mineralizable nitrogen in four cropping systems. Soil Science Society of America Journal 51 : 1508–1514. Brown RH. 1978. A difference in N use efficiency in C and C plants $ % and its implications in adaptation and evolution. Crop Science 18 : 93–98. Cannell MGR, Dewar RC. 1994. Carbon allocation in trees : a review of concepts for modelling. AdŠances in Ecological Research 25 : 59–104. Cheeseman JM. 1993. Plant growth modelling without integrating mechanisms. Plant, Cell and EnŠironment 16 : 137–147. Clark FE. 1977. Internal cycling of "&nitrogen in shortgrass prairie. Ecology 58 : 1322–1333. Cohen D. 1970. The expected efficiency of water utilization in plants under different competition and selection regimes. Israel Journal of Botany 19 : 50–54. Conroy JP. 1992. Influence of elevated atmospheric CO concentrations # on plant nutrition. Australian Journal of Botany 40 : 445–456. Davidson RL. 1969. Effect of root}leaf temperature differentials on root}shoot ratios in some pasture grasses and clover. Annals of Botany 33 : 561–569. Detling JK, Parton WJ, Hunt HW. 1979. A simulation model of Bouteloua gracilis biomass dynamics on the North American shortgrass prairie. Oecologia 38 : 167–191. Dewar RC. 1993. A root-shoot partitioning model based on carbonnitrogen-water interactions and Mu$ nch phloem flow. Functional Ecology 7 : 356–368. Eissenstat DM, Caldwell MM. 1988. Competitive ability is linked to rates of water extraction. Oecologia 75 : 1–7. Environmental Protection Agency. 1983. Methods for chemical analyses of water and wastes. N-NH method 350.1 and N-NO and N-NO $ $ # method 353.2. In : U.S. EPA Environmental Monitoring and Support Laboratories, Pub. gEPA-600}l. Epstein HE, Lauenroth WK, Burke IC, Coffin DP. 1997. Productivity patterns of C and C functional types in the U.S. Great Plains. $ % Ecology 78 : 722–731. Field CB, Chapin III FS, Matson PA, Mooney HA. 1992. Responses of terrestrial ecosystems to the changing atmosphere : A resourcebased approach. Annual ReŠiew of Ecology and Systematics 23 : 201–235. Garnier E. 1991. Resource capture, biomass allocation and growth in herbaceous plants. Trends in Ecology and EŠolution 6 : 126–131. Gedroc JJ, McConnaughay KDM, Coleman JS. 1996. Plasticity in root}shoot partitioning : optimal, ontogenetic, or both ? Functional Ecology 10 : 44–50. Gifford RM. 1979. Growth and yield of CO -enriched wheat under # water-limited conditions. Australian Journal of Plant Physiology 6 : 367–378.

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Gillen RL, Ewing AL. 1992. Leaf development of native bluestem grasses in relation to degree-day accumulation. Journal of Range Management 45 : 200–204. Hilbert DW. 1990. Optimization of plant root : shoot ratios and internal nitrogen concentration. Annals of Botany 66 : 91–99. Hopkins JC, Leipold RJ. 1996. On the dangers of adjusting the parameter values of mechanism-based mathematical models. Journal of Theoretical Biology 183 : 417–427. Hunt HW, Cole CV, Elliott ET. 1985. Models for growth of bacteria inoculated into sterilized soil. Soil Science 139 : 156–165. Hunt HW, Ingham ER, Coleman DC, Elliott ET, Reid CPP. 1988. Nitrogen limitation of production and decomposition in prairie, mountain meadow and pine forest. Ecology 69 : 1009–1016. Hunt HW, Trlica MJ, Redente EF, Moore JC, Detling JK, Kittel TGF, Walter DE, Fowler MC, Klein DA, Elliott ET. 1991. Simulation model for the effects of climate change on temperate grassland ecosystems. Ecological Modelling 53 : 205–246. Hunt HW, Elliott ET, Detling JK, Morgan JA, Chen D-X. 1996. Responses of a C and a C perennial grass to elevated CO and $ % # temperature under different water regimes. Global Change Biology 2 : 35–47. Jackson RB, Reynolds HL. 1996. Nitrate and ammonium uptake for single- and mixed-species communities grown at elevated CO . # Oecologia 105 : 74–80. Johnson IR, Thornley JHM. 1987. A model of shoot : root partitioning with optimal growth. Annals of Botany 60 : 133–142. Johnson IR. 1985. A model of the partitioning of growth between the shoots and roots of vegetative plants. Annals of Botany 55 : 421–431. Lauenroth WK, Milchunas DG. 1991. Shortgrass steppe. In : Coupland RT, ed. Natural grasslands : Introduction and western hemisphere. New York : Elsevier, 183–226. Luo Y, Field CB, Mooney HA. 1994. Predicting responses of photosynthesis and root fraction to elevated [CO ] a : interactions # among carbon, nitrogen and growth. Plant, Cell and EnŠironment 17 : 1195–1204. Melillo JM, Borchers J, Chaney J, Fisher H, Fox S et al. 1995. Vegetation}ecosystem modeling and analysis project : Comparing biogeography and biogeochemistry models in a continental-scale study of terrestrial ecosystem responses to climate change and CO # doubling. Global Biogeochemical Cycles 9 : 407–437. Minchin PEH, Thorpe MR, Farrar JF. 1994. Short-term control of root : shoot partitioning. Journal of Experimental Botany 45 : 615–622. Morgan JA, Brown RH. 1979. Photosynthesis in grass species differing in carbon dioxide fixation pathways. II. A search for species with intermediate gas exchange and anatomical characteristics. Plant Physiology 64 : 257–262. Morgan JA, Hunt HW, Monz CA, LeCain DR. 1994 a. Consequences of growth at two carbon dioxide concentrations and two temperatures for leaf gas exchange in Pascopyrum smithii (C ) and Bouteloua $ gracilis (C ). Plant, Cell and EnŠironment 17 : 1023–1033. % Morgan JA, Knight WG, Dudley LM, Hunt HW. 1994 b. Enhanced root system C-sink activity, water relations and aspects of nutrient acquisition in mycotrophic Bouteloua gracilis subjected to CO # enrichment. Plant and Soil 165 : 139–146. Morgan JA, LeCain DR, Read JJ, Hunt HW, Knight WG. 1998. Photosynthetic pathway and ontogeny affect water relations and the impact of CO on Bouteloua gracilis (C ) and Pascopyrum # % smithii (C ) Oecologia (in press). $ Perrin N, Sibly RM. 1993. Dynamic models of energy allocation and investment. Annual ReŠiew of Ecology and Systematics 24 : 379–410. Ralston A. 1965. A first course in numerical analysis. New York : McGraw Hill. Reynolds HL, Pacala SW. 1993. An analytical treatment of root-toshoot ratio and plant competition for soil nutrient and light. American Naturalist 141 : 51–70. Rogers HH, Runion GB, Krupa SV. 1994. Plant responses to atmospheric CO enrichment with emphasis on roots and the # rhizosphere. EnŠironmental Pollution 83 : 155–189. Romesburg HC. 1981. Wildlife science gaining reliable knowledge. Journal of Wildlife Management 45 : 293–313.

Sieva$ nen R, Hari P, Orava PJ, Pelkonen P. 1988. A model for the effect of photosynthate allocation and soil nitrogen on plant growth. Ecological Modelling 41 : 55–65. Snaydon RW. 1991. The productivity of C and C plants : a $ % reassessment. Functional Ecology 5 : 321–330. Thornley JHM. 1972. A model to describe the partitioning of photosynthate during vegetative plant growth. Annals of Botany 36 : 419–430. Thornley JHM. 1991. A model of leaf tissue growth, acclimation and senescence. Annals of Botany 67 : 219–228. Wallace JD. 1969. NutritiŠe Šalue of forage selected by cattle on sandhill range. PhD Thesis. Colorado State University. Ft. Collins, CO, USA. Wilson JR, Brown RH, Windham WR. 1983. Influence of leaf anatomy on the dry matter digestibility of C , C , and C }C intermediate $ % $ % types of Panicum species. Crop Science 23 : 141–146. Winer BJ. 1971. Statistical principles in experimental design. McGrawHill, New York.

APPENDIX Symbols Symbol

Equation*

State variables Br 9 Bs 9 Sin 2 Smc 1 Son Suc Sun Wrc Wsc Parameters anm

3 6 6 1 1

ki

2

pac

1

pcw

15

pd

10

pesyn1, pesyn2 pewc1, pewc2 pewn1, pewn2 pf

—

2

Definition and Units

Root C (g C per pot) Shoot C (g C per pot) Soil inorganic N (g N per pot) Whole plant synthetic machinery (g C per pot) Soil organic nitrogen (g N per pot) Substrate carbon (g C per pot) Substrate nitrogen (g N per pot) Root cell wall (g C per pot) Shoot cell wall (g C per pot)

—

Maximal specific rate of N uptake (g N g root C−" d−") Half saturation constant for N uptake (g N per pot) Maximal specific rate of net C uptake [g C (g C)−" d−"] C content of the cell wall fraction (g C g−" d. wt) Fixed partitioning parameter (Davidson, 1969) [g C (g N)−"] Parameters defining esyn (fraction synthetic machinery) Parameters defining ewc (g water g−" dry soil)

—

Parameters defining ewn (g water g−" dry soil)

5

Fixed partitioning parameter (Johnson, 1985) (complex units) Fixed partitioning parameter (Johnson and Thornley, 1987) (g d. wt g−" N) Fixed partitioning parameter (Luo et al., 1994) (g d. wt g−" N) Rate constant for formation of synthetic machinery (complex units)

pj

12

pl

14

pm

8

Hunt et al.—Root-shoot Partitioning in Prairie Grasses Symbol pn

Equation*

Definition and Units

3

Maximal specific rate of N mineralization (g N g−" N d−") N content of the cell wall fraction (g N g−" d. wt) Fixed partitioning parameter (Minchin, 1994) (non-dimensional) Specific turnover rate of synthetic machinery (d−") Rate constant for formation of cell walls (complex units)

pnw

15

ps

13

pt

—

pw

6

Intermediate variables cc 5 Substrate C concentration (g d. wt g−" d. wt) ecn 2 Effect of N and C reserves on N uptake (non-dimensional) esyn 6 Variable partitioning coefficient for synthetic machinery (non-dimensional) ewc 1 Effect of soil water on C uptake and biosynthesis (non-dimensional) ewn 2 Effect of soil water on N uptake (nondimensional) lrt 4 Root partitioning coefficient (nondimensional) lsh 4 Shoot partitioning coefficient (nondimensional) nc 14 Substrate N concentration (g N g−" d. wt)

Symbol

501

Equation*

Definition and Units

px

15

rtc

2

sar sas sm

9 9 8

sw

6

Time-varying partitioning coefficient (Johnson and Thornley, 1986) (g d. wt g−" N) Root C (cell walls plus synthetic machinery) (g C per pot) Specific activity of roots (g N g−" C d−") Specific activity of shoots (g C g−" C d−") Concentration of substrates forming synthetic machinery (fraction d. wt) Concentration of substrates forming cell walls (fraction d. wt)

Driving variable swc — Rates of processes Ac 1 An 2 Brwc 7 Bsmc 8 Bswc

6

Mn

3

Soil water content determining ewc and ewn (g g−" dry soil) Net C assimilation (g C per pot d−") Uptake of soil inorganic N (g N per pot d−") Formation of root cell walls (g C per pot d−") Formation of synthetic machinery (g C per pot d−") Formation of shoot cell walls (g C per pot d−") N mineralization (g N per pot d−")

* Equation where first introduced.