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4 Internal Regulation of Nutrient Uptake

by Relative Growth Rate and Nutrient-Use Efficiency

V.P.Gutschick and J.C.Pushnik

4.1 Introduction

Plants are observed to regulate their uptake of nutrients in elaborate patterns

according to their environment of growth and their stage of development.

Among the common patterns is that plants reduce their nutrient uptake rates

(per mass of root, n, or of the whole plant, nplant) as nutrient concentrations

increase (Clements et al.1979; Godwin and Blair 1991;Youssefi et al.1999; see

Chap. 6, this Vol.). Similarly, changes in the shoot’s environment, such as in

CO2partial pressure,also induce changes in nutrient uptake rates (n).A num-

ber of questions arise – for one,why should a ‘good’such as nutrient acquisi-

tion ever be curtailed,or not expressed at a maximal rate? One must infer that

downregulating the acquisition of a beneficial resource confers a net benefit

in Darwinian fitness,for which most plants or their immediate ancestors have

been heavily selected.Admittedly,long-domesticated plants may diverge from

the fitness functions of wild plants (Gutschick 1987, 1997a,b, 1999; Jackson

and Koch 1997).In some cases,the explanation lies at the immediate physio-

logical level,in that some nutrients in excess are toxic,such as boron (Nable et

al. 1990) and even phosphate (e.g., Romera et al. 1992). Nonetheless, such

downregulation occurs even for nutrients that show no apparent toxicity in

luxury consumption,such as nitrogen.Plant performance does not follow the

guidelines espoused by the actress Mae West,“Too much of a good thing is

wonderful”. Some experimental evidence shows that overexpression of non-

toxic nutrient acquisition is deleterious to plant growth and fitness – witness

the stunting of supernodulating legumes (Carroll et al.1985),which can per-

haps be attributed to excess diversion of photosynthate to N2fixation.

Nonetheless, downregulation occurs even at modest, physiological nutrient

content,most markedly in woody plants (Gessler et al.1998).

Prediction of uptake capacity in changing environments,whether for crops

or wild plants, is highly desirable for studies of global change. We, as a

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H.BassiriRad (Ed.) Nutrient Acquisition by Plants

An Ecological Perspective

© Springer-Verlag Berlin Heidelberg 2005

Page 2

research community, might (and must) achieve description suited to wide

ranges of plants and environments. This is a worthwhile task, but an unend-

ing one,given the infinite continuum of possible environments and of possi-

ble landscapes of Darwinian fitness or agronomic value. Far more useful in

the long term would be a predictive capability based on demonstrably shared

mechanisms (biochemical and genetic),or on a knowledge of the overarching

selection pressures for regulation of uptake mechanisms. A comprehensive

theoretical framework for uptake capacity should be sought simultaneously

on two levels,physiological and ecological/evolutionary.

In the next few pages,we will put forth plausible mass-action forms that fit

observations of how plants respond – in nutrient uptake rate,photosynthetic

rate,tissue nutrient content,and root allocation – particularly to changes in N

availability,or to elevated CO2.The individual rate processes are formulated to

respond properly (e.g.,the fraction of reduced nitrogen (RN) loaded into the

xylem should increase if the root RN increases). The forms we propose are

heuristic (mass actions based on gross pools, not on pools of intermediate

metabolites). These are meant to be guides to the origins of the plant

responses, specifically to promote considerations of how processes must

change relative to each other in order to give observed plant responses. The

resultant model extends simple functional balancemodels,which only resolve

gross root and shoot capture of resources (nutrients, CO2, light) but do not

explain why resource-capture capabilities attain the values observed or how

these capabilities might be regulated (Gutschick and Kay 1995).

Foremost, we attempt here to provide a semi-mechanistic ‘explanation’ of

how the uptake capacity (Vmax) and root:shoot ratio (r) should acclimate to

the growth environment. The functional balance model derived from the

experiments of Gutschick and Kay (1995) sought to identify optimal Vmaxand

r values by fixing either Vmax or r values and varying the other.One intriguing

result was that there should be an optimal root:shoot ratio (r=1),independent

of environment (N only, not considering water). Second, the optimal Vmax

should be infinity: incremental gains in relative growth rate (RGR) continue,

if at smaller rates, for any increase in Vmax. A mechanistic model obviously

would disallow such an extreme; it incorporates responses that evolved

despite not maximizing the relative growth rate.If the predictions are realis-

tic,then one might seek an explanation as to why these mechanisms evolved.

We will start our discussion with the question of ‘what sets nutrient

demand?’. Demand is surprisingly difficult to formulate, at least in terms of

the external environmental variables and the basic growth attributes of the

plant, including physiological capacities for nutrient uptake and photosyn-

thesis,and growth patterns of root and shoot.Most of the literature on plant

function defines demand as a single point value,the current uptake rate that

one might calculate from current growth rate and current tissue nutrient con-

tent,fn,of the plant.This definition is what economists would call the ‘quantity

demanded’. In contrast,‘demand’ is a mathematical function – the quantity

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demanded as a function of price (cost of nutrient acquisition).With this view

of demand,we must inquire why the rate of uptake has its current magnitude,

and what will be the rate if we change environmental conditions,say,CO2lev-

els.

4.2 Phenomenology of Uptake Rate Responding

to Nutrient-Use Efficiency and Growth Rate

4.2.1 The Evolutionary and Ecological Perspective

of Physiological Demand

Economists propose that demand is determined by a rational consumer who

knows the utility of a resource, and seeks to maximize net marginal benefit

(the utility less the cost,as derivative with respect to quantity consumed) of a

resource. For a human consumer, this is monetary value, other values being

converted to this as a medium of exchange. For a plant, demand could be

defined as the continuous function that relates the mass of nutrient acquired

as a function of the cost of acquiring and the benefit of using that nutrient.

Cost is a function of,above all,external nutrient concentration – for example,

it requires more energy in root growth to obtain dilute nutrients.It is neces-

sary that this mass of nutrient optimize the plant’s function,say,its growth or,

more importantly,its Darwinian fitness.

This is where the difficulty arises – in what sense(s) and to what extent do

plants optimize their performance? Fitness can never be fully maximized

(Stenseth and Maynard Smith 1984). Even if fitness is nearly optimized, the

relation to readily measured physiology or even simple growth in mass is

complicated.Greatest vegetative biomass is not equivalent to greatest fitness.

For example,the dispersal dynamics of seeds does matter,and for perennials,

reproduction in any one year involves tradeoffs with future reproductive out-

put (Boutin and Harper 1991; Edwards and Crawley 1999). Consider further

the relation of total biomass growth to nutrient acquisition.Fast early growth

may deplete the soil of nutrients for later reproductive growth.For nutrients

that are poorly mobile in plants, this may more than negate benefits of early

growth, and may explain why early RGR is manifestly held below maximal

RGR in some plants.Moreover,any prediction of optimal uptake capacity for

a specified environment requires that we possess information on aspects such

as the availability of nutrients at all future times,especially at times of repro-

duction.Yet there are uncertain,risky,or stochastic elements in the availabil-

ity of a nutrient,which may depend on,e.g.,weather (precipitation,tempera-

ture). The same risk and uncertainty are associated with the utility of a

nutrient. Changes in the concurrent availability of water modify the photo-

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synthetic utility of nitrogen,as discussed in the following sections.On an eco-

logical level, high N content brings stochastic risks of herbivory; these risks

may be the most potent limiters of high nutrient content,particularly for N.

Even for deterministic costs and benefits,the plant cannot accurately sense

current availability in a distributed soil volume.Future availability depends as

well on other organisms for nutrient mobilization,such as by microbial min-

eralization.Of course,there are broad,predictable patterns of future availabil-

ity, such as seasonal flushes, and plants have been naturally selected to

respond effectively to these patterns,but the risk lies in the details,especially

in competitive growth.Many of the limitations and promises of optimization

theory applied to plants have been discussed by,e.g.,Bloom et al.(1985) and

Gutschick (1987).We shall provisionally accept a number of limitations, and

explore some details of what is involved in optimizing RGR alone with a

known external nutrient concentration,known cost of acquiring and metab-

olizing the nutrient, and known benefit (current photosynthetic utility) as a

function of tissue nutrient content (fn).

4.2.2 A View from Physiology:Response of Growth Rate

to Nutrient Availability,Cost,and Utility

An understanding of the regulation of nutrient acquisition and use requires a

physiological perspective.Considerable detail at the levels of physiology and

gene expression has been achieved in model systems, such as Arabidopsis.

However,these model systems are somewhat limiting,in that they do not ade-

quately represent the range of selection pressures that have molded diverse

physiological patterns among different species and environments.Ecologists

and whole-plant physiologists are positioned to broaden this view,generating

clues as to the nature of regulatory signals and regulatory actions.One exam-

ple is in elucidating the roles of nutrient costs (for acquisition and metabo-

lism), and of saturation of benefits in making it beneficial to limit nutrient

uptake.Nutrients are costly,in several ways – for example,in the construction

of roots, and in metabolism for uptake, reduction, and maintenance

(Gutschick 1987; Zerihun et al. 1998). Moreover, as nutrients become more

abundant,there is a decline in their marginal benefit,as measured by photo-

synthetic utility, p* (mass of photosynthate made per gram of nutrient per

day; Gutschick and Kay 1995).The photosynthetic utility of nitrogen,in par-

ticular,is much increased when the shoot is exposed to high CO2.

As a first accounting of such costs and benefits, we present here a model

applicable primarily to growth in a steady environment, and omitting the

complication of transitions from vegetative to reproductive growth. In this

model (elaborated from Gutschick and Kay 1995,Appendix III therein),nutri-

ent uptake and nutrient use for photosynthesis achieve a functional balance –

that is, root and shoot functions attain a balance. Such models do appear to

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%fn,

correctly predict the trends in fnand in RGR at high CO2(Zerihun et al.2000;

BassiriRad et al.2000).Can such models also predict trends in Vmax? They can

do so only on the basis that a given Vmaxmay maximize growth rate, rather

than giving a mechanistic explanation.However,this view must be accommo-

dated, if only to explain why the physiological mechanisms exist and act as

they do.We show that yet more physiological and developmental traits must

be considered.

Consider the rate of dry-matter gain of a plant limited by its nutrient

uptake.Uptake occurs at velocity n per mass of root.Total root mass is mr,so

that the uptake rate of the whole plant is mrn.New plant tissue has a fractional

nutrient content,where the tilde indicates increment in new tissue. The

uptake-limited growth rate is then

(4.1)

and the relative growth rate of the whole-plant is then divided by whole-plant

mass:

(4.2)

Here,r is the root:shoot ratio,as usual.

The photosynthesis-limited growth rate is simply the whole-plant photo-

synthetic rate,Aplant,multiplied by the conversion efficiency from raw photo-

synthate to dry matter,b.Now,Aplantequals the average photosynthetic rate of

leaves per mass, AL,m, multiplied by leaf mass, mL. Much photosynthesis is

done at light saturation, where the rate per leaf area is proportional to the

nitrogen mass per leaf area. Equivalently, then, the rate per leaf mass is pro-

portional to the mass fraction of nitrogen (and perhaps also of phosphorus;

see Gutschick 1993), that is, AL,m=p*fn. The leaf mass may be expressed as a

fraction of shoot mass,aL=mL/ms.Thus,the growth rate is

(4.3)

and the photosynthesis-limited relative growth rate is the above divided by

total plant mass,

m

mm

sr

+

(4.4)

If root and shoot are in functional balance,the two RGR expressions above

are equal.Dividing out a common factor of 1/(1+r),we obtain

(4.5)

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&

%

fmm

ul

r

ν /

n

=

RGR

m

+

mmf

%

r

r f

)

ul

r

ν

rsnn

=

()

=

+

(

ν

%

1

& mpfm

pl

nLs

= βα

*

RGRpfpfr

pl

L

s

nLn

==+

βαβα

**

/()1

rv fpf

nLn

/

*

= βα

Page 6

rvp

L

/.

*

βα

()

In functional balance, the incremental and average nutrient contents are

equal,so that we can solve for fnas

by functional balance between the root and shoot,and is not freely adjustable

as a plant response.Substituting this expression for fninto either RGR expres-

sion gives an expression for RGR in terms of physiology,allocation,and envi-

ronment:

Note that it is determined

(4.6)

This growth rate is realistic,except that it increases indefinitely as n or p*

increases.In order for n,hence,Vmax,to have an optimum (that is,for RGR to

decline beyond that),it is necessary that there be both a cost to nutrient acqui-

sition and a saturation of the benefit of the nutrient in photosynthesis. The

costs have been discussed at length for N by,e.g.,Gutschick (1981) and Zeri-

hun et al.(1998).We incorporate them as changing the whole-plant photosyn-

thetic rate in Eq. (4.4) above, introducing a factor (1–cNfN), with cNbeing the

energetic cost of acquiring and metabolizing N.The saturation of photosyn-

thetic carbon assimilation with fNis also observed empirically (Sinclair and

Horie 1989).It is also interpretable in theory.Consider leaves of a given mass

per area – roughly,their thickness;specifying fNthen determines the carboxy-

lation capacity VC,max. For a specified leaf irradiance IL, one can derive esti-

mates of leaf (and also whole-plant) assimilation per unit area and per unit

mass of N. Both of these functions begin as linear in fNat low fN, and then

approach an asymptote as light,rather than N,begins to be limiting the pho-

tosynthetic rate per mass of leaf. We thus introduce another factor that

decreases the whole-plant photosynthetic rate in Eq.(4.4);one of the simplest

mathematical forms with a single saturation parameter q is exp(–qfN).Here,q

is the inverse of a ‘saturating’N content in the shoot,fN,s,sat.(Other forms,such

as Michaelis-Menten,give very similar results.)

4.2.3 Does a Simple Model Predict an Optimal Uptake Capacity and Root

Allocation?

We now use this model, with the modified costs and benefits of nutrients, to

predict the optimal value of Vmax, for nitrogen in particular.We estimate the

other physiological and environmental variables, and then vary Vmaxto find

the value that maximizes RGR.We consider various values of the N-reduction

cost,cN,as well.The solution is obtained numerically (we use a binary search

in fN;note that our Fortran programs are freely available).The other variables

we must set are: the nutrient concentration at the root surface, ce(a nominal

100 µM here); the Michaelis constant, Km(taken as 30 µM), in order to com-

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RGR

r

+

r

p

L

=

ν

βα

1

*

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pute nas Vmaxce/(ce+Km);the root:shoot ratio,r(0.4 is taken as a typical value);

the biosynthetic conversion efficiency, b (almost invariant among plants, so

that we fix it at 0.77, excluding the N-reduction cost; BassiriRad et al. 2000);

the fraction of shoot mass as leaves,aL(fixed at 0.55,appropriate for a young

plant); and p*,the photosynthetic utility of the nutrient (fixed at 15.26 g car-

bohydrate g–1dry matter day–1,to represent a modest competitor).

Figure 4.1 presents the results. Note first that RGR responds more weakly

than linearly to changes in uptake capacity.Following the elementary model

of Eq. (4.6), it rises initially as the square root of n, and hence as the square

root of Vmax. It then flattens out as N uptake increases, mostly because the

photosynthetic capacity saturates in fN. Increasing costs of N reduction also

exact a penalty in RGR, as expected. For all cases with relatively high costs

(>6 g glucose per g nutrient, there is a clear optimum value of Vmax, above

which RGR declines.At uptake capacities significantly higher than the opti-

mum and at high nutrient cost, however, no functional balance can be

attained. For such cases, but also in general, one needs a model with realis-

tic repression of uptake by the accumulation of N in tissue, as we develop

below.

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Fig.4.1.Predictions of the functional balance model for responses of relative growth rate

(RGR) and tissue fractional N content (fN) to left combinations of N-uptake capacity and

nitrate-reduction cost (cn), and right combinations of uptake capacity and root:shoot

ratio. The model equations are presented in Section 4.2.1.At high Vmaxand high cn, no

functional balance is possible; results are correspondingly absent. Thick lines indicate

the optimal Vmax (left) and optimal root:shoot ratio (right)

Page 8

Predicted tissue N content is only realistic in some regions of the simula-

tions presented here.When Vmax is low or moderate, the tissue content, fn, is

realistic (below 5% N), but not for high Vmax. Surprisingly, unrealistically

high fNoccurs when there is a high cost of N that might be assumed to sup-

press N accumulation.However,this cost penalizes carbon accumulation,not

uptake. Feedback to uptake is clearly not represented, nor is it warranted if

increased N were really to confer a growth benefit, even a small one. Also

unrealistic is that at high Vmax, the limited diffusibility of N in soil (even as

nitrate) draws down the concentration of N at the root. In essence, Vmaxis

partly ineffective. This phenomenon can be modeled readily, for a specified

root geometry and diffusivity (Gutschick and Kay 1995). Figure 4.2 shows

such a calculation.

On the right-hand side of Fig.4.1 are results in which the root:shoot ratio

is varied as well as is Vmax.This simulation is aimed at determining if there is

an optimal value of r (as real plant behavior implies),when realistic costs and

realistic saturation of benefits occur.Indeed,for any Vmaxthere is an optimal

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Fig.4.2. Diminishing returns in actual uptake rate at high N-uptake capacity,when dif-

fusibility of N in soil is limited at the stated realistic value.The steady-state uptake rate

was computed for cylindrical roots of 20-µm radius with a depletion zone 0.6 mm wide,

using standard equations (see Gutschick and Kay 1995). The achieved uptake rate, n,

equals Vmaxca/(ca+Km), where cais the concentration at the root surface. This rate is

equated to an effective Vmax(Vmaxeff) operating at the N concentration in bulk soil,ce,as

Vmaxeffce/(ce+Km)

Page 9

value of r,and it declines at high uptake rates,following one’s intuition that N

demand is more readily satisfied in such cases.

Another interesting perspective from physiology is how the benefit of a

nutrient is affected by other nutrients,as well as by water and other environ-

mental factors.Consider how a decrease in water availability induces stomatal

closure. The ensuing decrease in leaf-interior partial pressure of CO2(Ci)

reduces the carboxylation rate per Rubisco enzyme by well-known formulae

(Farquhar et al. 1980), thus reducing the photosynthetic gain per mass of N

per unit time.In the long term (weeks),plants can approach an optimal mixed

usage of water and N.We have not entered this consideration in our model.

For greater realism and more insight, we must propose that the uptake

capacity Vmaxand root allocation are not free parameters,but are both tied to

internal indicators of plant status,to achieve optimal values (maximal RGR),

or to meet other physiological,developmental,and environmental constraints

(including risks) that we have not yet considered.Uptake then should respond

to N status,although not to that alone,since the optimal RGRis attained at dif-

ferent magnitudes of fNwhen the photosynthetic utility,p*,changes:the mag-

nitude of fNis predicted to decrease with increasing p*,as is seen in elevated-

CO2experiments almost uniformly.Evaluating if this is optimal requires more

detail.We have presented some detail,but have not answered the question of

how optimal uptake shifts as p* changes,in other publications (BassiriRad et

al.2000).With the realization that internal N and carbohydrate (CH) statuses

both represent signals of sub- or supra-optimality, we must proceed to a

model that resolves internal pools of N and CH. Such a model is presented

below.Before proceeding to this larger model,we analyze another,little appre-

ciated limit on the utility of nutrients, namely, internal developmental limits

on growth rate. This surely must be a strong modifier of growth rate, and of

any near-optimizing responses in Vmaxand in root allocation.

4.3 Toward a Model of Uptake Regulation in Response

to Nutrient Utility

4.3.1 Predicted Response to an Intrinsic,Physiological or Developmental

Limit on Relative Growth Rate

A phenomenon that is insufficiently appreciated as determining RGR and

nutrient content or uptake rates is the maximal relative growth rate,RGRmax.

This has been measured by, e.g., Poorter and Remkes (1990), and has been

incorporated into a model of RGR by BassiriRad et al. (2000). Presumably, it

arises from the existence of limits on the number of growing points (meris-

tems) in a plant, and of limits on organ expansion rates (e.g., Tardieu and

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Granier 2000).This may be taken as a quantitative expression of the qualita-

tive term ‘vigor’. As the environment becomes more favorable, RGR is

approaches RGRmax.As the utilization of growth substrates nears saturation,

presumably the use of growth substrates stagnates and their internal pools

rise markedly. These pools should then function as feedback signals. In our

simple model of functional balance with finite RGRmax(BassiriRad et al.2000;

see Fig.4 therein),the bulk nutrient content,fn,does rise sharply,although we

do not explain this rise mechanistically.

To explore how the existence of RGRmaxis predicted to affect optimization

(and thus regulation) of Vmax,we use an expanded functional balance model

(BassiriRad et al.2000).To express the origin of RGR limitation,we resolve a

new variable,fC,the nominal (molar) concentration of nonstructural carbohy-

drates internal to the plant. RGR initially increases with fCbut reaches an

asymptote because growing points saturate in their ability to use carbohy-

drates. We express RGR, then, as RGRmaxfC/(fC+Ke), with kCas a parameter.

Photosynthesis, too, is affected (repressed) by high fC, being decreased by a

factor 1/(1+kCfC).As in BassiriRad et al.(2000),we choose values of kC=1,and

Ke=0.25; both are unitless.The value of fCitself is set by the balance between

photosynthetic production and consumption in growth – that is, functional

balance occurs in carbohydrate generation and use,analogous to the balance

for N. As in the model without this developmental limitation, we then

explored predicted performance with varied uptake capacity and cost of N

reduction.Figure 4.3 shows that RGR is curtailed even when it is significantly

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Fig.4.3. Predictions of growth rate and

tissue N content for the functional bal-

ance model with a developmental limit

(RGRmax=0.20 day–1) imposed.Physio-

logical parameters are as in Fig.4.1,

with the addition of growth-limiting

parameters described in the text.Thick

lines indicate the optimal Vmaxvalues.

Shaded lines represent the simulations

of Fig.4.1 for comparison

Page 11

below RGRmax,while tissue N content is higher because use of N for growth is

increasingly restricted.Optimal values of Vmaxremain apparent,and at closely

the same values as for unlimited growth in Fig.4.1.More signatures of devel-

opmental limitation may be discerned if internal pools of metabolites are

resolved,as we now proceed to do.

4.3.1.1 Do Internal N and Carbohydrate Pools Explain Responses of

Vmax to Photosynthetic N Utility and to Developmental Limitations on

Relative Growth Rate?

A model with an account of developmental limitations must resolve an inter-

nal pool of nonstructural carbohydrates. Further insight might be gained by

resolving additional internal pools. The predicted plant performance might

thus be more realistic.Conversely,such a model might be useful in predicting

the ‘signatures’ (patterns of changes in internal pools) that indicate that a

plant is responding either to developmental limitations (RGRmax) or to

changes in photosynthetic N utility (p*),or to additional internal factors.

There is substantial qualitative evidence, and significant quantitative evi-

dence,that uptake (as Vmax,and Km,etc.) responds to internal pools,which are

reflected as well in tissue nutrient concentration,fn(see Chap.6,this Vol.).At

the root, one observes downregulation of n by high concentrations of the

nutrient itself,or of some key pool involving the nutrient.High [K+] appears

to downregulate Vmax of K+(Glass and Dunlop 1979; Siddiqi and Glass 1987);

soluble K+is,of course,the only significant component of total tissue K+.For

nitrogen, the situation is more complex; n is regulated by a small pool of

reduced-N compounds (King et al.1992),which broadly track total tissue con-

tent. However, rapid transients occur in the regulatory pool in response to

sudden (and normal) changes in shoot photosynthesis with time of day, for

example (Morcuende et al. 1998), and these are important manifestations of

regulation that must occur faster than changes in bulk tissue content.Similar

small pools appear to regulate iron uptake,for example (Schmidke et al.1999).

In steady growth, bulk tissue content is related to the regulatory metabolite

pools,and thus is correlated with Vmax(see Chap.6,this Vol.).

Also relevant to the role of internal pools in regulation and any associated

optimization are the changes in fninduced by changes in the shoot environ-

ment, such as light (IL=irradiance in the photosynthetically active radiation

region or PAR) or CO2(expressed as Ca=partial pressure in ambient air) or

temperature.Increases in Cacause increases in carboxylation rate per Rubisco

molecule (Farquhar et al. 1980), and thus in carbohydrate pools. These com-

monly cause significant decreases in nitrogen and sulfur contents (Penuelas

and Matamala 1990;Peterson et al.1999).In previous papers (Gutschick 1993;

Gutschick and Kay 1995),we expressed this effect solely with the physiologi-

cal parameter p*, the nutrient-use efficiency (or efficacy) in photosynthesis.

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In functional balance, RGR is predicted (and observed; ibid., and Zerihun et

al. 2000) to respond directly to

Here,to explain optimal performance,we must resolve more detailed internal

pools of metabolites.

The most parsimonious set of pools is composed of the following: nitrate

in the root (Nro,‘o’representing the oxidized form),reduced N in the root and

shoot (Nrrand Nsr, respectively), and nonstructural carbohydrate in the root

and shoot (Cr, and Cs, respectively). Correspondingly, we must resolve the

processes of: N uptake; N reduction (taken to be all in the root; this may be

generalized readily); growth of root tissue – with attendant incorporation of

nitrate, reduced N, and carbohydrate; xylem loading of reduced N; shoot

growth with incorporation of reduced N and CH; phloem loading of CH; and

phloem loading (recycling) of reduced N back to the root.We derive a set of

kinetic equations for all these processes in terms of the metabolite pools and

rate constants for the net metabolic reactions. The rate constants are esti-

mated numerically from ‘gross’observables:growth rates under baseline con-

ditions; uptake rate per mass of root and its response to reduced-N accumu-

lation in the root; tissue N contents in the root and shoot (Nro,Nrr,Nsr) under

baseline conditions; the fraction of reduced N recycled to the root; the

response of xylem and phloem loading to reduced-N pools; and photosyn-

thetic CO2assimilation per mass of N,and its response to tissue N (fN) and to

CH accumulation.These data are not trivial to obtain,but they are much sim-

pler than a more detailed biochemical dataset. Once the rate constants are

computed,it is assumed that all acclimations of the plant (to altered N avail-

ability,shoot CO2or temperature,etc.) are represented by the kinetic scheme

with these rate constants fixed – that is, the kinetic scheme captures all the

acclimation of the plant.

Many kinetic schemes have been proposed for the various facets of root N

uptake,its feedback regulation,root and shoot growth,photosynthesis,and its

regulation.(Le Bot et al.1998).We propose here a similarly parsimonious syn-

thesis of these ideas, and some new ideas. First, we take the evidence that

internal or cytoplasmic pools of reduced N, as they increase, can downregu-

late N uptake from soil (King et al.1992; Kronzucker et al.1995; Sivasankar et

al.1997),and propose the simplest functional form for whole-plant N uptake,

U,to be:

while fnresponds inversely as.

(4.7)

Here, n0is the Michaelis-Menten form,Vmax0ce/(ce+Km), where Vmax0is the

fully unrepressed value.The denominator is a downregulating factor,with aU

being a physiological constant (parameter). Next, we assume that our plant

reduces all its nitrate in the root. (The model can readily be generalized to

include nitrate reduction partially in the shoot,and also uptake of some N as

reduced N.Roughly,the partial uptake of N as reduced N can be simulated by

V.P.Gutschick and J.C.Pushnik74

Ecological Studies Vol 181, BassiriRad (Ed.) – Page Proofs as of 02/10/2005

Ecological Studies Vol 181, BassiriRad (Ed.) – Page Proofs as of 02/10/2005

p*,

p*

Uma

r

ν0/ 1

Ur

=+

()

N

Page 13

decreasing the cost of N reduction, ignoring the difference in nitrate and

ammonium uptake kinetics.) The reduction rate,RNR,is taken as simple mass

action,responsive to both nitrate and carbohydrate in the root:

(4.8)

Root growth,Gr=dmr/dt,similarly follows mass action in both substrates,

(4.9)

The individual rates of incorporation of reduced N and of carbohydrates

must be specified,in a form that generates consistent fractions of N in tissue.

For reduced-N incorporation into root tissue,IN,r,many empirical forms may

be proposed,in advance of detailed experimental evidence.Having tested the

behavior of many polynomial forms, we propose a Michaelis-Menten form

that realistically generates increased incorporation of N into tissue (increased

fN) as the availability of reduced N increases (that is,as Nrincreases):

(4.10a)

(4.10b)

Here, the rate constant for incorporating reduced N into growing tissue,

kNRG, is proportional to the rate constant kRGfor total growth, with a propor-

tionality that produces the proper fNat low N availability.A selectivity coeffi-

cient, Ksel, prevents excessive incorporation of N relative to C. Similarly, the

incorporation of carbohydrate into new tissue,IC,r,must be consistent with the

overall growth rate and with the fraction of biomass deriving from reduced N.

We derived the form

(

+

(

⎣

(4.11)

The third term enforces mass balance of reduced and nitrate N in new tis-

sue.Here,fN,RNand fN,NOare the mass fractions of N in the form of reduced N

(about 0.18 for amino acids) and of nitrate (18/62=0.29), respectively; FDMis

that fraction of fresh mass represented by dry matter.

Proceeding toward the shoot,we describe xylem loading of reduced N,XN,

as an attenuating quadratic in Nr,

(

(4.12)

Internal Regulation of Nutrient Uptake75

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Ecological Studies Vol 181, BassiriRad (Ed.) – Page Proofs as of 02/10/2005

Rm k

r

C

NRNRr

o

r

=

N

G m k C

rr RGrr

r

=

N

IR K C

sel,rNRr

r

rr

N

NN

=+

()

/

=

()

+

()

kC K C

selNRGrrrr

NN

2/

I k C

RG

k

C

N

K C

sel

ff

C rrr

r

NRG

rr

r

rrRN NO

,

,,

=−

)

)

+−

⎡

⎢

⎢

⎤

⎥

⎦

N

N

NN

2

11

⎥ ⎥

k C

RG

F

rr

r

DMr

o

NN

Lk a

Xr

r

Xr

r

N

NN

=

)

+

()

2

1/

Page 14

This is a modification of simple linear kinetics by a Michaelis-Menten fac-

tor, Nrr/(1+aXNrr), which increases the export fraction at high Nrr. Shoot

growth in mass,Gs=dms/dt,is described as by simple mass action,

(4.13)

Note that the ratio of shoot growth to root growth remains constant

(root:shoot fraction is constant) if the C and N pools (which are concentra-

tions) are stable. The incorporation of reduced N into shoot biomass has a

form analogous to that in roots, excluding the nitrate term. The shoot as

source of carbohydrates is described as in our basic model,with total carbo-

hydrate production rate A given as

(4.14)

To account for repression by accumulated carbohydrate, we add a simple

factor,

(4.15)

The factor in the numerator simply normalizes A to its value at a chosen

reference environment in which A attains a known rate A0and Cshas its base

value, Cs0. The parameter aPSdescribes the sensitivity of photosynthesis to

repression by free carbohydrate.To describe phloem loading of carbohydrate

to circulate to roots, PC, we use a model based on the pressure-flow model

(admittedly of modest accuracy: Farrar and Jones 2000),

(4.16)

Here,Dy is the water-potential difference from shoot to root (<0),R is the

gas constant,and T the absolute temperature.Partly as a signal for regulating

N uptake,reduced N is also loaded into phloem to return to roots.We take the

rate,PN,as

(

(4.17)

This form makes the fractional return of reduced N increase as reduced N

increases – that is,when N supply exceeds growth demand.

These equations were developed by some trial and error in order to get

realistic responses of growth, fN, and uptake to changes in external N avail-

ability,in p*(hence,in CO2or irradiance),and in growth potential.It is impor-

tant to note that a maximal RGR is implicit in the rate constants kRG,kSG.Most

of the rate constants can be fitted from observed values of RGR, fN, and the

pool sizes for a given environment. Other parameters may be estimated to

V.P.Gutschick and J.C.Pushnik 76

Ecological Studies Vol 181, BassiriRad (Ed.) – Page Proofs as of 02/10/2005

Ecological Studies Vol 181, BassiriRad (Ed.) – Page Proofs as of 02/10/2005

Gm k C

ss SGss

r

=

N

A p fmp fm

,sL ,sLs

==

∗∗

NNα

AA a C

PS

+

a C

PSss

→

()

+

()

00

11/

Lm k

s

RT CC

CPCsr

=+−

()

⎡⎣⎤⎦

∆ψ

LLK C

PNCs

r

s

r

s

N

NN

=+

)

/

Page 15

explore the sensitivity of growth and fNto partitioning and feedback parame-

ters such as Kseland aX.The model is coded in Fortran 77 and will be provided

to anyone interested (please contact the authors). The code is voluminously

commented; the output is pseudo-graphic,providing a schematic printout of

pools and flows to a plain ASCII terminal.

4.3.2 Applying the Model:Differences from Simple Functional Balance

This explicit carbon–nitrogen pool model (CN model, or CN) has been

applied to a few illustrative cases (Fig.4.4).These are (1) a base case with the

same physiology as that used in the FB model above; (2) the same,but with a

developmental limit, RGRmax=0.15 per day; (3) doubled p*, as would be the

case if atmospheric CO2approximately doubled. The other parameters are

similar to the other simulations here: shoot leaf fraction aL=0.55; root:shoot

ratio r=0.4; external solution concentration of nitrate ce=100 mmol m–3

(100 µM); uptake Michaelis constant Km=30 mmol m–3; biosynthetic effi-

ciency is 0.77; cost of nitrate reduction cN=6 g carbohydrate g–1N; saturation

of photosynthetic N utility given with fN,sat=0.03 g N g–1dry matter;p*=15.3 g

carbohydrateg–1Nday–1;fresh mass/dry mass ratio=4;free nitrate concentra-

tion in roots in base case=12.5mM;repression parameter for uptake aU=100g

reduced N g–1fresh mass; repression parameter for xylem loading aXL=500 g

N g–1fresh mass; recycling ratio in base case=0.33; Dy=0.1 MPa; aPS=5.19 in

the CN model, to give a fractional expression of photosynthesis equal to

1/1.3=0.77 in the base case. For N incorporation into new root growth, the

parameter Kselis set to 0.01,making the fraction N in new tissue fairly sensi-

tive to the reduced-N:CH balance in the root.

Several features immediately distinguish the CN model from the simple

functional balance model. Most notably, there is no optimal Vmaxwhen

expressed as unrepressed value, Vmax0– RGR increases smoothly toward an

asymptote, and never decreases (Fig. 4.4A, top row of graphs). Even as Vmax0

grows very large, the buildup of reduced N (RN) in the root represses Vmax

(Eq. 4.7) and forces its decline. The effect is shown in Fig. 4.4C for all three

physiological constitutions modeled.The modest repression of root growth at

high N availability (Fig.4.4C,bottom row) acts to further limit N uptake.This

consequently limits the associated reduction costs and tissue N buildup. In

the shoot,the benefit of increased N is progressively but not inordinately cut

by the inefficient partitioning of N (the saturation of photosynthetic rate, as

fN,sincreases;Fig.4.4B,middle row of graphs).Thus,increased N availability –

achieved either by high external concentration or by high uptake capacity as

Vmax0– again gives no optimal value of uptake capacity as Vmax0.The observed

control of uptake capacity,particularly to low values,must originate in selec-

tion on some traits coupled to nutrition,such as the cation–anion balance that

might be adversely affected at high N uptake. Even more likely is the rising

Internal Regulation of Nutrient Uptake77

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Ecological Studies Vol 181, BassiriRad (Ed.) – Page Proofs as of 02/10/2005