Planta (2001) 214: 220±234
DOI 10.1007/s004250100602
O R I GI N A L A R T IC L E
J.P. Comstock
Steady-state isotopic fractionation in branched pathways
using plant uptake of NO3± as an example
Received: 26 October 2000 / Accepted: 13 February 2001 / Published online: 11 August 2001
Ó Springer-Verlag 2001
Abstract A steady-state isotopic discrimination model is
developed for material transfer between a single source
and two distinct sinks arising from an internal branching
of the uptake pathway. Previous analyses of isotopic
discrimination in multistep processes are extended to
include the e€ects of the interacting sinks. The theory is
®rst developed as a set of generic expressions allowing
for ¯exibility in the de®nition of intermediate pools in all
parts of the branched transport pathways, and then
applied to a case study of nitrate uptake by plants. The
isotopic composition of assimilated nitrate may be
evaluated with the model for either contrasting root
versus shoot assimilate pools, each with a unique
isotopic signature, or as a single mean value for wholeplant nitrate reduction. The theory is further developed
to indicate how isotopic measurements may be used to
infer (i) e‚ux:in¯ux ratios at the root plasma membrane,
(ii) partitioning of assimilate capture between root and
shoot reduction sites, and (iii) mixing of root and shoot
assimilate pools in sink tissues due to whole-plant circulation of organic nitrogen in both xylem and phloem.
Keywords E‚ux:in¯ux ratio á Isotopic
discrimination á Isotopic heterogeneity á Nitrate
assimilation á Nitrogen partitioning á Steady-state
fractionation
Introduction
A concise theory of isotopic fractionation during carbon
capture by plants has been of tremendous value in
understanding carbon dynamics in physiological,
J.P. Comstock
Cornell and Boyce Thompson Institute Stable Isotope
Laboratory (CoBSIL), Boyce Thompson Institute
for Plant Research, Tower Road, Ithaca, NY 14853, USA
E-mail: [email protected]
Fax: +1-607-2541242
ecological, and ecosystem studies, but a similar theoretical grounding has not yet been developed with regards to nitrogen assimilation (Handley et al. 1998).
This has been the case in part because plants can utilize
multiple forms of nitrogen in the environment, and no
single uptake process may fully account for the isotopic
compositions observed. Also important, however, is the
fact that the physiological processes of nitrogen assimilation within the plant have a more complex pattern of
internal compartmentation than is the case with carbon.
In the case of nitrate, the single most important form of
mineral nitrogen for most plants, compartmentation
includes uptake by the root system and an internal
branching of the nitrate ¯ux between separate assimilation sites in the root and shoot tissues (Fig. 1, sinks 1
and 2). The branching of the nitrate assimilation pathway at the root cytoplasm with partial reduction of nitrate to organic products and partial transfer of
unreacted nitrate to the shoot creates a single uptake
process with intermediate pools distributed among
multiple organs within the plant. The need to account
for the integrated dynamics of such a branched uptake
pathway has prevented direct transfer of algorithms
developed to describe carbon isotope discrimination to
the understanding of nitrogen isotopic behavior in
plants. Models have been presented for uptake to a
single sink (Mariotti et al. 1982; Shearer et al. 1991), but
these have not been expanded to include the multiple
sinks found in most plants. Robinson et al. (1998) presented a compartment-based process model for numerical solution of the ¯uxes controlling nitrogen isotopic
composition in plants, but simple expressions relating
isotopic behavior to uptake characteristics are still
lacking.
A theoretical analysis of isotopic fractionation during
transfer from a single source to multiple sinks by a
branched uptake pathway is presented below. The ®rst
half of this paper provides a general treatment to
describe the isotopic fractionation in a bifurcated pathway. The theory is then speci®cally applied to the model
of nitrate uptake for plants growing under pure nitrate
221
dˆ
Rsample Rstd
Rstd
…1†
where Rstd refers to a known reference material.
If an isotope e€ect is present, then the product of a
reaction takes on a di€erent isotopic abundance ratio
than the reactant. The magnitude of the isotope e€ect is
described by the fractionation factor (a):
aˆ
Rreactant
Rproduct
…2†
It is often useful to de®ne discrimination as the deviation
of a from 1. In the treatments below, the value a±1 associated with a unidirectional ¯ux (i.e. single transport
step or discreet reaction) is a constant designated by a
lower-case letter (a±z), while discrimination by an entire
process is designated by the Greek letter D. Despite
similar de®nitions, these two variable types have an
important distinction in that D is not a constant, but
varies with ¯ux dynamics. Note that Farquhar et al.
(1989) is followed in omitting the factor of 1,000 associated with units of &(per million) from the mathematical de®nitions of D and discrimination constants.
Fig. 1 Modeled ¯ow-chart of nitrogen exchanges during the
uptake of nitrate from a source pool in the soil (ellipse) and
reductive assimilation into organic sink pools (trapezoids) in both
root and leaf tissues. Intermediate nitrate pools in various tissues
(boxes), are all considered to be in a steady state with respect to
both total mass and isotopic composition. Polygons represent the
bulk organic nitrogen of root and shoot tissues. In all cases, the
pool number is the same as the subscripted index of R, the isotopic
abundance ratio of each respective pool. Arrows refer to total
unidirectional ¯uxes of both isotopes, with magnitude Fi. Lowercase letters refer to discrimination constants
nutrition, and allows prediction of the isotopic ratios of
the assimilatory sinks as well as intermediate pools
(Fig. 1, R1±R5). The predicted values of the isotopic
composition of the immediate assimilatory products also
permits an estimate of the mixing of root and shoot
assimilation in the bulk organic nitrogen of various tissues. The speci®c application of the model to nitrate
assimilation makes the most parsimonious interpretation possible of the number of subpools within the plant
required to explain the isotopic behavior. The theoretical background is presented with sucient generality,
however, to facilitate adjustment of the discrimination
model for di€erent hypotheses of internal compartmentation, or variation in compartmentation among
di€erent plants.
General discrimination theory
Terminology and basic concepts
Variation in isotopic composition is characterized by the
isotopic abundance ratio, R, de®ned as [heavy]/[light]
isotopes in a sample. R is often reported using d notation
(Farquhar et al. 1989; see Table 1 for detailed de®nitions):
Assumptions and applicability
The basic question is: ``How does variation in discrimination constants among di€erent steps of multi-component processes interact with ¯ux dynamics to generate
variable discrimination (d…source:sink† ˆ Rproduct =Rsource 1)
in the overall movement from source to ®nal sink(s)
(Figs. 2, 3, ellipses and trapezoids, respectively)?'' The
processes are modeled as series of pools, with gross,
unidirectional ¯uxes passing in both directions between
consecutive pools (Fig. 2). Each unidirectional ¯ux
(Fig. 2, arrows) is associated with a discrimination
constant (e.g. a=aa±1). The assumption of a steady state
applies to both total mass and isotopic composition of
any intermediate pools (Fig. 2, boxes), which are considered to be internally well mixed with respect to the
isotopic e€ects of all ¯uxes. A net ¯ux occurs only from
the initial or source pool (Fig. 2, ellipses), and into the
®nal sink (Fig. 2, trapezoid). It is also assumed that, at
steady state, {Ri} of intermediate pools may have any
needed values due to previous, transient enrichment or
depletion, but is then fully stable at the predicted steadystate solution. The mass balance of transient conditions
leading to this steady state is not treated in the current
analyses.
R0 is assumed to have a stable value, but the mass
balance of the source pool is not incorporated into the
following models. It is thus assumed either that the
source pool size is very large relative to the ¯uxes
modeled, such that it behaves as an `in®nite source' over
a limited time interval, or that other, unlabeled processes
actually balance the ¯uxes described such that R0 is
really in a steady state like other modeled pools. The
predicted {Ri} for all intermediate pools, as well as the
222
Table 1 List of variables used in the text
Variables
De®nitions
Ri
Ratio of heavy to light isotopes in a given pool. Subscripts 1, 2, 3... used with R are the same as the pool numbers,
and pools are shown as closed ®gures: circles, squares, and trapezoids, in the ®gures. Pool numbers may
sometimes be replaced with speci®c names such as source or sink. In a few cases the subscript net is used to
indicate R of material transferred in a net ¯ux rather than resident in a speci®c pool
Isotopic composition of a given sample expressed relative to an arbitrary standard (see Eq. 1 for de®nition).
Subscript i, when used, refers to same pools as Ri
Concentration of molecule of interest in di€erent pools. Subscripts match Ri to designate speci®c pools
The gross, unidirectional ¯ux indicated by an arrow between two pools in any of the ®gures. Subscripts 1, 2,
3...refer to the respective ¯ux arrows in the ®gure under discussion. The subscript net refers to the di€erence
between opposing gross ¯uxes between the same two pools
Fractionation factor describing the isotope e€ect inherent in a single-step unidirectional process. It is de®ned as
Rreactant/Rproduct. Subscripts a, b, c..., when used, refer to fractionation in distinct steps. Most rigorously, this
implies a unique a for each gross ¯ux arrow in a ®gure (e.g. Fig. 1, pathway D), but often the forward and
backward transfers between adjacent pools are taken to have the same a (e.g. Fig. 1a±c).
Discrimination constants of a single unidirectional transfer or reaction de®ned as the deviation of a from unity (i.e.
a±1). The symbols match the subscripts of the a notation when referring to the same steps
Discrimination, as de®ned above for a, b, c¼, only referring to a process with at least one reversible step in which
component discrimination constants may have variable expression dependent on ¯ux dynamics. De®ned as
Rsource/Rsink±1, where subscripts i and j refer to source and sink pools, respectively (e.g. ®gures). When numeric
subscripts are used, they match the subscripts of Ri in designating the same pools. Several speci®c formulae are
developed in the text to de®ne D in terms of component ¯uxes and discrimination constants for di€erent
transport pathways
A shorthand notation referring to a speci®c formula previously de®ned for D in unbranched chains of pools, but
used as part of a more complex formula for branched pathways in which H(i:j) does not de®ne the full isotopic
relationship between pools i and j
Branchpoint Interaction Factor. Subscripts i and j refer to the pool numbers of the source and branchpoint,
respectively. B(i...j) de®nes the probablitity (0±1) that a heavy isotope excluded from the branch to sink 1 by
discrimination processes, will end up in sink 2 as opposed to e‚uxing back to the source. B(i...j) depends on the
number of distinct pools and their ¯ux relationships, but not on the values of any discrimination constants
Basal Cross-Product Factor de®ned by discrimination constants and ¯uxes in shared portion of a branched
pathway, and used in the most precise formulation of D (Eq. B6). Subscripts i and j refer to source and
branchpoint, respectively. See Eq. B7 and Table 2 for numeric calculation of F(i...j). The cross-product e€ects of
the discrimination constants are minor compared to the direct additive e€ects, and F(i...j) is deleted from the
generally more useful approximation in Eq. 11 in the main text
`E‚ux product'/`in¯ux product'. The e‚ux product calculated for each pool in a pathway is the product of the
chain of unidirectional ¯uxes leading back from that pool to the original source pool of the material transfer.
The in¯ux product is the product of all unidirectional ¯uxes leading inward from the source to the designated
pool. The value of EP:IP is always between 0 and 1, and its contrasting values for each pool in a pathway
determines the relative expression of all component discrimination constants in D (e.g. Eq. 7).
di
Ci
Fi
ai
a, b, c¼
D(i:j)
H(i:j)
B(i¼j)
F(i¼j)
EP:IP
sink, is inherent in the model solutions based on massbalance analyses.
Discrimination in an unbranched pathway
The simplest ¯ow pathway has just source and sink with
no intermediates (Fig. 2, pathways A.1 and A.2). The
derivation of D(0:1) for this system begins with the massbalance relationship between net transfer and component ¯uxes:
F net ˆ F 1
F2
…3†
where `Fi' are total ¯uxes including both heavy and light
isotopes and subscripts refer to Fig. 2, pathway A.1.
This system must exhibit not only conservation of total
mass, but also conservation of overall isotopic abundance
(Hayes 1982) expressed by multiplying each of the ¯uxes
from Eq. 3 by its expected isotopic ratio (Fig. 2, pathway A.2):
R0
R1
Fnet R1 ˆ F1
F2
…4†
1‡a
1‡a
R0 and R1 are the isotopic ratios of source and sink,
respectively, and `a' is the discrimination associated with
the transfer in either direction. The left-hand term is
simply net ¯ux times Rsink. The isotopic compositions of
the individual ¯uxes F1 and F2, however, are subtler.
They are calculated using Eq. 2 with aa equal to 1+a,
and Rsource being that of the originating pools (i.e. R0
and R1 for F1 and F2, respectively) (Fig. 2, pathway A.2).
To derive the expression for D0:1, Eqs. 3 and 4 are
solved simultaneously by ®rst substituting F2±F1 for Fnet
in Eq. 4, and then collecting all terms for R0 and R1:
R1 F1
F2
F2 ‡
1‡a
ˆ R0
F1
1‡a
…5†
Algebraic rearrangement and the de®nition of D(0:1) as
Rsource/Rproduct±1 leads directly to:
D…0:1†
R0
ˆ
R1
1ˆa 1
F2
F1
…6†
Discrimination by the whole pathway has a maximum
value of a, the discrimination inherent in unidirectional
223
Fig. 2 Linear unbranched pathways. Shapes representing pools
and all symbols have the same meanings as in Fig. 1. A.1 Simplest
possible pathway with only source (ellipse) and sink (trapezoid)
pools. A.2 The same as A.1, but indicating the isotopic composition
of the instantaneous ¯uxes F1 and F2. The solid-edged ellipse and
trapezoid labeled R0 and R1 represent distinct pools, while the boxes
with dashed outlines are just mathematical concepts of the isotopic
composition of the instantaneous ¯uxes without any necessary
assumption that they exist as separate, discreet pools. B Similar to
A.1, but with an inde®nite number of potential intermediate pools
(square rectangles). C The same as B, but with ¯uxes all determined
by pool sizes and ®rst-order rate constants. D Same as B only with
di€erent fractionation constants for forward and backward
reactions at each transfer step
transfer. This is fairly intuitive since the inward ¯ux
contributing to R1 is isotopically lighter than R0 to a
degree determined by `a', but the back-reaction discriminates in the same manner, leaving the heavy isotopes behind to enrich R1. If F2=F1, the e€ects cancel,
and D=0. Note that Eq. 6 (as well as all subsequent
algorithms for D) could be rewritten using d notation as:
D…0:1†
d0 d1
ˆ
d0
1 ‡ d1
d1 a 1
F2
F1
…6b†
In all subsequent derivations, we begin as illustrated
above with two equations to describe (i) conservation of
total mass and (ii) isotopic conservation during steady
state, and solve them jointly. Expressions analogous to
Eq. 6, but for linear pathways with one or two intermediate pools, are derived in Appendix AI, and it is
concluded by induction that D for a pathway with n
pools beyond the source (Fig. 2, pathway B) is:
Fig. 3 A pathway with one source, two potential sinks, and a
branchpoint. Symbols labeling individual ¯uxes and pools have the
same de®nitions as in Fig. 1
D…0:n†
ˆa 1
F2
F2 F2 F4
F2 F4 F2 F4 F6
‡b
‡c
F1
F1 F1 F3
F1 F3 F1 F3 F5
F2 F4 F6 . . . F2…n 1†
F2 F4 F6 . . . F2n
‡...z
F1 F3 F5 . . . F2…n 1† 1 F1 F3 F5 . . . F2n 1
…7†
Expressions related to Eq. 7 can be found in several
places in the literature (Farquhar et al. 1982 , 1989;
Handley and Raven 1992). The form given here is highly
generic.
Some important points can be made about Eq. 7.
The equation for D(source:sink) across a linear string of
pools is the set of associated discrimination constants
weighted by a predictable series of progressive `e‚ux
product'/'in¯ux product' (EP:IP). For any transfer
step in the chain, full expression of its particular discrimination constant in D(source:sink) requires, ®rst of
all, that the EP:IP of the immediate source pool in the
transfer must be near 1. Further, if the EP:IP calculated for the recipient pool of the same transfer is also
near 1, then the potential discrimination of that step
cancels out (see explanation of Eq. 6). A general way
to think about this is that, whenever the EP:IP of a
pool approaches 1, bidirectional mixing, including any
224
pools between it and the source, is the dominant
process, and the pool approaches equilibrium with R0.
This behavior is also consistent with the observation
that full expression of the di€erent discrimination
constants for di€erent steps in a pathway is mutually
exclusive, and irreversible steps suppress all subsequent
potential for discrimination in the ®nal product
(O'Leary 1981).
Also implicit in the derivation of D is the {Ri} for
all intermediate pools. For illustration, consider a
linear pathway with a source (R0), two intermediate
pools (R1 and R2) and a ®nal sink (R3) (i.e. Fig. 2,
pathway B, with R3 taken as the terminal sink). From
known {'ai±1'} and {Fi}, the value of R3 is calculated
as R0/(1+D0:3) using Eq. 7. R1 and R2 are then
calculated relative to R3, treating each respective
intermediate as the `source pool' while employing
Eq. 7. This is consistent with earlier comments that R0
might itself represent a steady-state intermediate in a
larger process. We can thus use appropriate forms of
Eq. 7 to work backwards, giving R3á(1+D1:3) and
R3á(1+D2:3), for R1 and R2, respectively (i.e. R3á(1+
b(1±F4/F3)+c(F4/F3±F4F6/F3F5)) and R3á (1+c(1±F6/
F5), respectively).
Equation 7 is the most general form, developed in
regard to the gross ¯uxes. However, if (i) ¯uxes are dependent on the concentrations of the respective pools,
(ii) the forward and backward rate constants at a given
transfer step are equal, and, ®nally, (iii) all the rate
constants are ®rst order (see Fig. 2, pathway C), then
Eq. 7 reduces to an important special case:
C0 C 1
C1 C2
D…0:n† ˆ a
‡b
C0
C0
C…n 1† Cn
C2 C3
‡c
...z
…8†
C0
C0
In familiar contexts such as a di€usion gradient, the
discrimination constants most expressed by the overall
pathway will be those associated with the largest concentration drops (i.e. most limiting resistances to ¯ux)
(Farquhar et al. 1982).
Finally, in many processes the discrimination
constants may not be equal for the forward and
backward ¯uxes between two contiguous pools (Fig. 2,
pathway D). The analogue to Eq. 7 developed with
unique discrimination constants for each unidirectional
¯ux is:
aa F2
0 aa F 2
0 aa ab F2 F4
D…0:n† ˆ a a
‡ b
b
aa0 F1
aa0 F1
aa0 ab0 F1 F3
aa ab F2 F4
a
a
a
F
F
0 a b c 2 4 F6
‡ c
c
aa0 ab0 F1 F3
aa0 ab0 ac0 F1 F3 F5
aa ab F2 F4 ::::F2…n 1†
‡ ::::: z
aa0 ab0 F1 F3 :::F…2…n 1† 1†
aa ab :::az F2 F4 :::F2n
…9†
z0
aa0 ab0 :::az0 F1 F3 :::F2…n 1†
where aa and aa' etc. are 1+a and 1+a', respectively.
Because they are close to 1, these extra factors have only
a small in¯uence on the ®nal value of D. Omitting them
is often acceptable (see `model validation' below) and
leads to:
F2
F2
F2 F4
D…0:n† ˆ a a0
b0
‡ b
F1
F1
F1 F3
F2 F4
F2 F4 F6
‡ c
c0
F1 F3
F1 F3 F5
F2 F4 :::F2…n 1†
F2 F4 :::F2n
…10†
‡ ::::: z
z0
F1 F3 :::F…2…n 1† 1†
F1 F3 :::F2…n 1†
Despite the similarity to Eq. 7, the forward and backwards discriminations at a given transfer step need not
cancel out when EP:IP equals 1. This could represent
equilibrium rather than kinetic isotope e€ects, or any
real system where forward and backward ¯ux arrows
represent inherently di€erent processes (e.g. C4 photosynthesis with a 4-carbon pump into the bundle sheath
and a leakage of CO2 back-di€using).
Equations 1±10 summarize a very general treatment of
fractionation in unbranched pathways, but do not represent radically new expressions for fractionation. For
example, the well-known models of discrimination during
C3 photosynthesis (Farquhar et al. 1982, 1989) are
equivalent to Eqs. 7 and 8, while the commonly used
expressions for C4 photosynthesis (Farquhar 1983) and
during the `leaky pump' assimilation of HCO3± by algae
(Sharkey and Berry 1985) are closely related to Eqs. 9 and
10. The same approaches can be extended to include novel
expressions for discrimination in branched pathways.
Branched pathways with multiple sinks
Consider a generic pathway as before, only with a
branch point at some intermediate pool and then separate branches leading to two distinct sinks (Fig. 3). The
derivation for either sink alone is developed in Appendix B culminating in Eq. A.II.6. An often more useful,
very close approximation to the full equation is:
D…source:sink1† ˆ H…source:sink1† ‡ B…source:::bp†
…D…bp:sink1†
D…bp:sink2† †
…11†
where bp stands for branchpoint (e.g., pool 4 in Fig. 3),
B() refers to a branchpoint interaction factor with a
range of possible values from zero to one (Table 2), and
H and D both represent a similar algebraic shorthand to
be expanded for calculation by application of Eqs. 7 or 9
(D for an unbranched pathway) with source:sink de®ned
by the respective subscripts. The distinction of the two
symbols, H and D, is conceptual in that, where D is used,
the expression is not only part of the needed total
equation, but also expresses a necessarily true isotopic
relationship between the subscripted pools, while H is
used to represent a similar algebraic expression from
225
Table 2 Branch interaction factors as a€ected by the length of the
common path below the branchpoint. The numbering of ¯uxes and
isotope ratios corresponds to that of Fig. 3. The ®rst column gives
the notation used in Eqs. 11 and B6 to refer to branch interaction
factors for basal stem pathways of di€erent length. The middle two
columns provide orientation between the table and Fig. 3
Notation used in Eq. 11
Source pool R
Number of intermediate pools
in basal path
Branch interaction factor, B()
B(3..4)
R3
0
…F15
B(2..4)
R2
1
B(1..4)
R1c
2
B(0..4)
R0
3
B(source..branchpoint)
Rsource
n
F16 †
…1†
…F15 F16 †
F6
1‡
F5
F7
…F15 F16 †
F4 F4 F6
1‡ ‡
F5 F5 F7
F3
…F15 F16 †
F2 F2 F4 F2 F4 F6
1‡ ‡
‡
F1
F3 F3 F5 F3 F5 F7
Fnet:other:sink
…1 ‡ n flux quotients†
primaryinflux
F1
Eqs. 7 or 9 when that expression does not necessarily
de®ne the full isotopic relationship.
The branchpoint interaction factor, B(), describes the
relative likelihood (potential values range 0±1) for heavy
isotopes excluded from one branch to end up in the alternative sink rather than e‚uxing back to the original
source. The precise value of B() depends on two
component factors (Table 2). The ®rst is the ratio of the
net ¯ux into the alternative branch (making the value of
B() sink-speci®c) relative to the total, unidirectional in¯ux entering the bottom of the pathway from the ultimate source, and the second is a progressive sum of ¯ux
quotients related to the shared pathway and the same for
both sinks. Reduced total e‚ux, longer shared pathways, and/or ¯ux dynamics in the shared pathway that
inhibit transference through the multiple mixing steps of
the intervening pools make it more likely for heavy
isotopes excluded from one branch to end up in the
other sink, and this results in a value of B() closer to 1.
The exact expression for B() is not fully intuitive, but
follows a regular pattern that can be expanded for a
common basal pathway of any length (Table 2).
Equation 11 indicates that, for two sinks originating
from a common source, the expected isotope ratio of
just one of the two sinks considered alone is related,
®rst of all, to the discrimination between the source and
that sink as would be predicted by Eq. 7 while disregarding the presence of the other branch [i.e.
H(source:sink1)]. To this is added a second term consisting of the branch interaction factor [B(source..bp); Table 2]
multiplied by the di€erence in discrimination expected
in the two diverged paths beyond the branchpoint. The
positive sign associated with D(bp:sink1) in Eq. 11 indicates that the expected fractionation past the branchpoint leading to the sink under evaluation (i.e. sink 1)
can be enhanced by the presence of branch 2. This can
be understood as an e€ective increase in the EP:IP for
sink 1 since material moving from the branchpoint to
sink 2 can carry with it some of the excess heavy isotopes failing to enter sink 1, removing them from the
system and allowing the pathway to sink 1 to more
fully express its potential discrimination. In contrast,
D(bp:sink2), has a negative sign in Eq. 11, because any
heavy isotopes left preferentially at the branchpoint by
the ¯ux leading to sink 2 will necessarily tend to isotopically enrich the ¯ux to sink 1. If D(bp:sink1) and
D(bp:sink2) are equal in value, all interactive e€ects will
cancel regardless of the partitioning of ¯uxes between
sinks or other ¯ux dynamics contributing to B().
As discussed previously in regard to Eq. 7, the values
of R for all intermediate pools are also implicit in Eq. 11
for a branched pathway. The required formulas and
order of calculation to predict R for both sinks 1 and 2
and all intermediate pools in Fig. 3 are summarized in
Table 3.
Interestingly, the subtler interactions between the
branches cancel out in the mean D of the combined sinks
de®ned as:
D…source:combined sinks 1&2†
.R
sink1 Fnet:sink1 ‡ Rsink2 Fnet:sink2
ˆ R0
Fnet:total
1
…12†
The derivation is again left to Appendix B, but the ®nal
formula, based on Eq. 11 to de®ne Rsink, is:
D…source:combined sinks 1&2†
Fnet:sink1 H…source:sink1† ‡Fnet:sink2 H…source:sink2†
ˆ
Fnet:total
…13†
where H has the same de®nition as in Eq. 11. The reciprocal interactions of the branches, which were important
in Eq. 11 for de®ning D of individual sinks, cancel out in
this weighted mean of the respective H values.
Model validation and application
Validation of isotopic theory
The process of isotopic discrimination is inherently
expressed, from ®rst principles, as a change in the
226
Table 3 Calculation of {Ri} for Fig. 3 with the order of calculations
and appropriate formulae. R0, all discrimination constants (a±i), and
{Fi} are given as inputs. The order of calculation is: (i) R for sink 1 is
calculated using the appropriate form of Eq. 11, (ii) R of intermediates between sink 1 and branchpoint (including the branchpoint
itself) using Eq. 7, (iii) calculation of all basal stem pools relative to
sink 1 using Eq. 11 with the appropriate form of the branch interaction factor (Table 2), (iv) calculation of sink 2 relative to the
branchpoint with Eq. 7, and (v) R of second branch intermediates
using Eq. 7. Both Eq. 7 and Eq. 11 are used to ®rst calculate R of
terminal sinks relative to a source, and also to back-calculate R for
intermediate pools. In each instance where an intermediate pool is
treated as Rsource, D…source:sink† ˆ Rsource =Rsink 1, and therefore
Rsource ˆ Rsink …1 ‡ D…source:sink† † where D(source:sink) is de®ned by the
appropriate equation identi®ed in the right-hand column
Pool label
Source
Branchpoint
Sink 1
Pool R
No.
0
1
2
3
4
5
6
7
8
9
Sink 2
10
Calculation D calculated
order
using:
De®ned input
R7 1 ‡ D…1:7†
7
R7 1 ‡ D…2:7†
6
R7 1 ‡ D…3:7†
5
4
R7 1 ‡ D…4:7†
R7 1 ‡ D…5:7†
3
2
R7 1 ‡ D…6:7†
R0
1
1 ‡ D…0:7†
R10 1 ‡ D…8:10† 10
R10 1 ‡ D…9:10†
9
R4
8
1 ‡ D…4:10†
Eq.
Eq.
Eq.
Eq.
Eq.
Eq.
11
11
11
7
7
7
Eq. 11
Eq. 7
Eq. 7
Eq. 7
abundance ratio, [heavy]/[light], as one isotopic form
reacts faster than another, but de®nition of precise massbalance relationships requires the use of atom fractions,
[heavy]/([heavy]+[light]). Discrimination algorithms
combining both features are needlessly complex, however, and, when working with natural abundance levels,
the error generated by the approximation of treating an
expression based on the abundance ratio as if it
expressed a true mass balance is usually well below the
resolution of measurement (Hayes 1982). The initial
equations above constraining `conservation of isotopic
abundance' (e.g. Eq. 4 or A.I.2) incorporate this common, minor imprecision.
All ®nal algorithms discussed above were tested for
mass-balance consistency. The de®ned inputs in all
simulations were: (i) Rsource, (ii) {Fi}, and (3) {a, b, c, ...}.
The value of {Ri} for all pools was then predicted from
the algebraic solutions (Table 3 for Fig. 3, Table 4 for
Fig. 1). All intermediate pools were then tested for mass
Table 4 List of equations used
to de®ne the isotopic composition of all pools for nitrate
assimilation (Fig. 1)
balance using two approaches, illustrated below for the
®rst intermediate pool of any straight-chain (e.g. Fig. 2,
pathway B, or Fig. 3):
R0
R1
R1
R2
F1
F2
F3
F4
ˆ0
…1 ‡ a†
…1 ‡ a†
…1 ‡ b†
…1 ‡ b†
…14†
and
R0
R1
F2 A
F1 A
…1 ‡ a†
…1 ‡ a†
R1
R2
F4 A
ˆ0
F3 A
…1 ‡ b†
…1 ‡ b†
…15†
where the quotients
inside the angled brackets are R¯ux.i,
and each A Rflux:i is to be calculated as R¯ux.i/(R¯ux.i+1)
(i.e. the conversion from isotope ratio to the corresponding atom fraction). Equation 14 is a `mass balance'
based on isotopic ratios. It is not perfectly accurate, but
matches the assumptions of the derivations. Equations 7, 9, and A.II.6 all had zero residuals for all pools
when tested by Eq 14, indicating algebraic accuracy in
the derivations. Equation 15 always indicated small but
non-zero residuals. To evaluate the true accuracy of the
predicted {Ri}, a set of equations analogous to Eq. 15 for
each intermediate pool in the pathway was solved
simultaneously by numerical techniques, along with a
related statement that the isotopic composition of the
product pool must equal the net contribution from the
source. This numerical solution identi®ed the unique
{Ri} for which all mass-balance residuals were <1á10±16.
These were compared to the predicted {Ri} for (i) sensitivity analyses of accuracy across a range of input
parameters, and (ii) testing the accuracy of algebraically
`simpli®ed' expressions from which `trivial' terms and
factors had been deleted (i.e. Eqs. 10 and 11).
For most parameterizations relevant to natural
abundance contexts, the maximum error in d units for
the predicted R of any pool in a pathway modeled by
Eq. 7, 9, or A.II.6, was in the order of 0.001±0.01&.
Dependence of accuracy on {Fi} was negligible. Dependency of accuracy on the value of R0 from 0.001 to
0.5 was also almost negligible when {ai} values were in
the range observed in biological transformations. A
small dependency was evident on the magnitude of the
{ai}, but with Rsource de®ned at a value realistic for
natural abundance nitrogen (0.0036), errors as large as
0.1& were likely only when discrimination constants
were over 300&. If {ai} and Rsource both had high values,
Pool label
Pool No.
R
Soil NO3± (source)
Root NO3± (branchpoint)
Root assimilated N (sink 1)
Xylem NO3±
Leaf NO3±
Leaf assimilated N (sink 2)
0
1
2
3
4
5
De®ned input
=R5=R3
=R0/(1+D0:2)
=R5=R1
=R5(1+D5:4)
=R0/(1+D0:5)
Calculation
order
D calculated
using:
3
1
3
4
2
Eq.
Eq.
Eq.
Eq.
Eq.
7
19
7
7
20
227
the resulting variability in R among di€erent pools seriously compromised its suitability as a proxy for massbalance calculations and led to inaccurate predictions,
but such parameterization is not relevant to actual natural abundance studies in nitrogen or carbon.
The simpli®ed equations with `trivial' terms deleted
also performed well, but rigorous tests revealed some
caveats. Equation 10 is a simpli®ed version of Eq. 9, and
lacks the ratios aa/aa', ab/ab' etc. The magnitude of error
associated with this deletion is sensitive to the EP:IPs. If,
for example, a in Fig. 2, pathway D, di€ered from a' by
as much as 20&, and the ®rst step was at full equilibrium
(i.e. F 2 =F 1  1), then the expected error (a worst-case
scenario with respect to F2/F1) could be as large as 0.4&.
Moreover, if several such steps with unequal fractionation constants for forward and backwards reactions
occur in sequence (and with the same polarity in terms of
the unequal isotope e€ects), the errors are multiplicative
and even larger. However, although the errors might
seem signi®cant in such a case, approaching a few per
million, several additive equilibrium fractionations of
20& would give an overall all-isotope e€ect almost two
orders of magnitude larger than this error. In most
practical cases, the use of Eq. 10 gives values correct to
within 0.1&, but when EP:IP ratios approach 1 and very
precise values are needed, it is better to use Eq. 9.
Equation 11 is also a simpli®ed approximation, in this
case of Eq. A.II.6. Similar caveats hold as discussed for
Eqs. 9 and 10. A worst-case scenario includes (i) a large
value for Dbp:sink2 which forms the factor outside the
square brackets in A.II.6, (ii) little or no e‚ux back to the
source, (iii) ¯uxes into the branches largely irreversible,
and (iv) most net ¯ux into the alternate pathway.
Quantitative examples are given later in regard to the
nitrate-assimilation model described below (Table 5).
Table 5 Predicted isotopic behavior of plant pools during nitrate
assimilation. Subscripted variables refer to Fig. 1, and it is assumed that F5=F7=F9 at all times, that only the assimilation step
shows discrimination (i.e. a=c=d=0), and that b=e=16&. The
upper ®ve rows show variable inputs de®ning contrasting ¯ux
dynamics. d calculations follow Table 4, only Eq. B6 was used
instead of Eq. 11. The `maximum d error' gives the maximum
In¯ux (F1)
E‚ux (F2)
Root assimilation (F3)
Leaf assimilation (F9)
d Soil NO3± (R0), &
d Root NO3± (R1), &
d Root assimilate (R2), &
d Xylem NO3± (R3), &
d Leaf NO3± (R4), &
d Leaf assimilate (R5), &
d Plant ((F3R2+F9R5)/(F3+F9)), &
d maximum error for any pool, &
E‚ux/in¯ux (F2/F1)
E‚ux/in¯ux % Error using Eq. 22
Assimilation in root (F3/(F3+F9))
Root assimilation % Error using Eq. 23
Expected isotopic behavior during uptake of
NO3±
The processes of nitrate uptake and reduction are
likely to be near to the isotopic steady state most of
the time and thus amenable to the kind of discrimination model developed above. Cytoplasmic pools are
small, with turnover half-life estimates most commonly
between 2 and 5 min (Lee and Clarkson 1986; Siddiqi
et al. 1991). Labeled nitrate applied externally to the
root can be traced through the root cytoplasmic pool
into the xylem sap and leaf cytoplasmic pools all on a
timescale of minutes in herbaceous species (Lazof et al.
1992; Hayashi et al. 1997).
Discrimination against 15N during nitrate assimilation in plants can be described by applying Eqs. 11
and 13 to the conceptual ¯ow chart describing uptake
(Fig. 1). Several key steps are assumed to be irreversible, including the actual reduction of NO3± to
organic nitrogen in either root or shoot, and the
transport of NO3± in the xylem. Since NO3± is not
phloem mobile, it is assumed that there is no backtransport of NO3± from the shoot back to the root
(Pate 1973; Smirno€ and Stewart 1985; Jeschke and
Pate 1991). This means EP:IP from the leaf nitrate
nitrate pool is zero, and there can be no expression of
discrimination by leaf NO3± reductase activity against
the soil NO3± pool signature (i.e. all NO3± entering the
leaf is eventually ®xed or retained in the vacuolar
NO3± pool). The NO3± pool acting as substrate for the
root nitrate reductase, however, can exhibit e‚ux back
to the soil, and thus allow for the possibility of net
discrimination during plant uptake of NO3±. From
Eq. 11 and the diagram of Fig. 1 we therefore state
that discrimination by the individual root and leaf
sinks, respectively, must be:
discrepancy when the same table is compiled using Eq. 11. Based
on presumed measurement of the isotopic values in an experiment,
the theoretical accuracy of Eqs. 22 and 23 is assessed for evaluating the underlying ¯ux dynamics. Five cases are considered: 1±3
have high e‚ux:in¯ux but vary in fraction of assimilation in root,
while 4 and 5 show the same contrasts at low e‚ux:in¯ux
Case 1
Case 2
Case 3
Case 4
Case 5
1
0.9
0.09
0.01
0
1.42
±14.35
1.42
17.44
1.42
±12.77
0.02
0.9
0.0
0.9
1.3
1
0.9
0.05
0.05
0
0.79
±14.97
0.79
16.80
0.79
±7.09
0.06
0.9
0.0
0.5
0.7
1
0.9
0.01
0.09
0
0.16
±15.59
0.16
16.16
0.16
±1.42
0.02
0.9
0.0
0.1
0.2
1
0.1
0.81
0.09
0
12.92
±3.03
12.92
29.13
12.92
±1.44
0.21
0.1
0.0
0.9
0.3
1
0.1
0.09
0.81
0
1.42
±14.35
1.42
17.44
1.42
±0.16
0.02
0.1
0.0
0.1
0.1
228
D…0:2† ˆ a 1
F2
F1
and
D…0:5† ˆ a 1
F2
F1
‡
F2
F5
b ‡ ‰b
F1
F1
cŠ
…16†
‡
F2
F3
c ‡ ‰c
F1
F1
bŠ
…17†
and, by Eq. 13, the net fractionation by the whole
plant to:
F3 a 1 F2 ‡ F2 b ‡ F5 a 1 F2 ‡ F2 c
F1
F1
F1
F1
D…0:2&5† ˆ
F1 F2
…18†
These expressions cover the full range of possible
fractionation in a system with this branched structure,
but most of these steps are not thought to show
substantial isotope e€ects during nitrate assimilation.
Di€usion of a charged ionic species in water is little
in¯uenced by a single isotopic substitution because of
the large sphere of hydration. No current evidence yet
supports discrimination by either initial uptake from
the soil NO3± pool across the plasma membrane in
root tissues, or during xylem loading (Mariotti et al.
1982; Handley et al. 1998). The only established point
of expressed discrimination during uptake and assimilation of NO3± is thought to be the ®rst step of reduction from NO3± to nitrite (Handley et al. 1998;
Yoneyama et al. 1998). Both the further reduction of
nitrite and incorporation into an organic molecule
may be discriminating reactions, but are tightly linked
to the ®rst irreversible step such that all nitrite formed
is converted to organic product (Ledgard et al. 1985).
If we set a=c=d=0, then Eqs. 16, 17 and 18 further
reduce to:
F2 ‡ F5
D…0:2† ˆ
b
…19†
F1
D…0:5† ˆ
F3
b
F1
…20†
and
D…0:2&5† ˆ
F3
…F1
F2
b
F2 † F1
…21†
Equations 19±21 permit rather simple interpretations.
Expression of the potential discrimination associated
with NO3± assimilation in the root (Eq. 19) is
dependent directly on the total e¯ux:in¯ux for the root
NO3± pool, with the added insight that transport up to
the shoot in the xylem (F5 in Fig. 1) counts similarly
to actual e‚ux back to the soil NO3± pool (F2). This
makes sense, since either ¯ux can remove accumulating heavy isotopes.
Discrimination between leaf and soil pools (Eq. 20) is
actually a balancing re¯ection of root discrimination in
this model. If there is no discrimination in the irreversible xylem transport process, then the ¯ux into leaf assimilate must, at steady state, equal the signature of
what is loaded into the xylem, which is R1. R1 is enriched
rather than depleted in the heavy isotope by a degree
determined by the fraction of total in¯ux from the soil
that is assimilated in the root (F3/F1). A tendency for the
shoot to have a more positive d15N than the root is a
common feature in plants growing under pure nitrate
nutrition (Yoneyama et al. 1998).
Finally, the mean isotopic discrimination by the
whole plant (Eq. 21) is a weighted average of Eqs. 19
and 20. Since Eqs. 19 and 20 are both re¯ections of the
root assimilation process, but have unlike signs, mean
plant discrimination will always be less than that by the
root sink alone. The ®nal result (Eq. 21) indicates that
expression of b is related to two factors: (i) the EP:IP of
the root NO3± pool with respect to the soil only and
excluding xylem loading as an e‚ux (in contrast
to Eq. 19), and (ii) the fraction of all assimilation that
actually occurs in the root.
Quantitative examples of how such contrasting ¯ux
dynamics a€ect the total isotopic behavior of the plantsoil system (Fig. 1) are given in Table 5. The {Ri} for
®ve di€erent scenarios are given as d values. Calculations follow Table 4, but displayed values are the result
of using Eq. A.II.6 directly, and not the simpli®ed
Eq. 11 (i.e. Eqs. 19 and 20 for this model pathway and
speci®cally for root and leaf sinks, respectively). The
maximum error for any pool in the model resulting
from using Eq. 11 is given in the last line of the d values
of Table 5, and is only about 0.2& in the worst-case
scenario. This error arises exclusively from Eq. 20, D(soil
NO3:leaf assimilate). This is because a=c=0 in Fig. 1
causes all of the cross-products inside the square
brackets of Eq. A.II.6 to be zero, and only when calculating D(soil NO3:leaf assimilate) does D(bp:alternate sink) have
a non-zero value (i.e. `b') causing the outer factor in
Eq. A.II.6 to be other than 1 and giving a di€erent
result than Eq. 11. The fully precise version of Eq. 20
would have another factor of 1/(1+b). While this might
seem simple enough to include, it would needlessly
complicate the expression developed in Eq. 21 for the
whole plant, and other expressions to be developed
below, and the added accuracy is barely within most
measurement precisions.
Discussion
This model allows for clari®cation of several points regarding 15N discrimination during plant assimilation of
NO3±. First, as has been emphasized by previous authors
(Mariotti et al. 1982; Shearer et al. 1991; Handley and
Raven 1992), there can only be net discrimination with
uptake at the whole-plant level if there is a substantial
e‚ux term between the root tissue and the soil NO3± pool
229
(Table 5, Cases 1 vs. 4). This is because the discriminating step is the ®rst reduction step, which utilizes an
intermediate pool, cytoplasmic NO3±, as its substrate. If
there is no e‚ux then the intermediate root NO3± pool
undergoes transient enrichment until the elevated R
cancels out the discrimination of reduction during steady-state uptake. For the same reason, net discrimination
during assimilation is only possible when a substantial
fraction of the NO3± reduction occurs in the root, because
only the root NO3± pool can have e‚ux back to the soil
(Table 5, Cases 1 vs. 3). This is a consequence of the nonmobility of NO3± in phloem. Actual discrimination at the
whole-plant level depends, therefore, on both e‚ux:in¯ux at the root plasma membrane, and the partitioning
of total reduction between root and shoot tissues
(Eq. 21). Many di€erent combinations of these two factors could give the same overall net discrimination, but
with di€erent implications for isotopic heterogeneity
between root and shoot (Table 5, Cases 3 vs. 4).
This isotopic behavior can potentially be used to
advantage in future studies of nitrogen metabolism in
plants. If it is assumed that (i) the plant is dependent
solely on NO3± uptake for its nutrition, (ii) the bulk root
NO3± pool accurately re¯ects the mean isotopic composition of the root nitrate-reductase substrate pool, and
(iii) the plant has grown for an extended period of time
under constant nutritional conditions so that the isotopic composition of accumulated biomass is consistent
with current physiological activity, then we can use
isotopic measures of standing biomass to evaluate mean
values of e‚ux/in¯ux (F2/F1 in Fig. 1), partitioning
of overall NO3± reduction between root and shoot
(F3/(F3+F5) in Fig. 1), and an estimate of the net mixing
of the organic nitrogen pools originally reduced in root
versus shoot due to the circulation of organic nitrogen in
both phloem and xylem (Fig. 1, transfer from leaf and
root assimilatory NO3± sinks to bulk organic root and
shoot pools)(see Appendix C for derivations).
efflux dsoil:NO3
ˆ
influx droot:NO3
dplant
dplant
…22†
The formula is given in terms of d to emphasize the
nature and number of empirical measurements needed.
dplant refers to the bulked organic nitrogen of the whole
plant. Only if all reduction is con®ned to the foliage is
there no isotopic record of the e‚ux:in¯ux ratio at the
root.
The partitioning of NO3± reduction between root and
shoot is highly variable among species and environmental conditions (Smirno€ and Stewart 1985; Andrews
1986). By this model it can be estimated from isotopic
data as:
root assimilated N droot:NO3
ˆ
total assimilated N
b
dplant
…23†
where b is the discrimination constant for NO3± reduction (Fig. 1). Estimates of this parameter in the literature
are somewhat variable (Handley and Raven 1992), but
average between 15 and 17&. Equations 22 and 23 are
developed based on Eq. 11, which is an approximation
as discussed above. The theoretical limits of accuracy
due to this are illustrated for various ¯ux dynamics in
Table 5.
Predicting the isotopic composition of the root and
leaf assimilate pools requires measurement of only
droot.NO3. The irreversible NO3± reduction reaction in the
root should produce an assimilate pool which di€ers
from droot.NO3 by b. Since the root cytoplasmic pool is
both the substrate for reduction and the branchpoint for
xylem loading (which does not discriminate), the NO3±
transported into the leaf should have the same signature
as the root cytoplasmic pool. In the leaf, d of residual
NO3± should become further enriched, again by b, while
the nitrogen ¯ux assimilated into organic compounds
has the same signature as xylem NO3± input at steady
state. This leads to:
droot assimilate ˆ droot:NO3
b
…24†
and
dleaf: assimilate ˆ droot:NO3
…25†
Assuming that transport of organic nitrogen occurs
without discrimination, the degree to which leaf and
root assimilate pools are mixed in the growth of new
tissues throughout the plant can then be estimated as a
two-ended mixing model after measuring d for the sink
of interest:
dsink ˆ droot:assimilate ‡ …1
r†dleaf: assimilate
…26†
which can be solved for the only unknown, r, the fraction of nitrogen in the sink originating from root reduction.
The data of Yoneyama and Kaneko (1989) and Evans et al. (1996) for Brassica campestris (L.) var. rapa
and Lycopersicon esculentum Mill. cv. T-5), respectively,
allow us to evaluate the above expressions for some
empirical measurements. Based on Eq. 23, the fraction
of NO3± assimilated in the root in these studies was 13
and 58%, respectively, and this root assimilate pool
contributed 36 and 53% (Eq. 26), respectively, to the
total nitrogen in the organic fraction of the root. These
numbers indicate extensive xylem and phloem circulation and mixing of leaf and root assimilate pools.
It is of interest that the data of Yoneyama and
Kaneko (1989) have an isotopic pattern throughout the
whole plant that is largely consistent with the model
presented here. This includes a leaf NO3± pool enriched
by the leaf reductase discrimination constant over the
root NO3± pool (26.1 vs. 14.2&, compare with Table 5).
In contrast, the data of Evans et al. (1996) indicate an
enrichment of only 3& for leaf NO3± over the root NO3±
pool, much less than predicted. The latter observation
could indicate either a spatial or temporal compartmentation of the leaf nitrate pool in L. esculentum that is
not accounted for in the present model, such as a storage
230
pool in isotopic equilibrium with incoming xylem nitrate
more than the leaf reductase substrate pool.
Equations 22±25 may give only rough estimates of
the parameters in question, and are the subject of current experiments. The key pool to all of these expressions is the root cytoplasmic NO3±, the branchpoint pool
in the model (Fig. 1) and the substrate for root nitrate
reductase (Granstedt and Hu€aker 1982). This pool,
however, is small, and very dicult to measure directly.
Measures of bulk root tissue NO3± will generally be
dominated by the much larger vacuolar pool (Ferrari
et al. 1973; Martinoia et al. 1973; Granstedt and
Hu€aker 1982; Miller and Smith 1996).
Estimates for turnover rates of vacuolar nitrate have
been in the order of 10±14 h (MacKown et al. 1981;
Yoneyama et al. 1987; Volk and Jackson 1993) though
these values could vary dramatically depending on nutritional status and the variable vacuolar pool size. Since
the cytoplasmic pool is the branchpoint for vacuolar
exchange as well, the vacuolar pool should re¯ect the
cytoplasmic pool isotopically over time. The slower exchange rate and turnover times of the vacuolar pool
should result in a time-averaged record of the isotopic
value of the cytoplasmic pool. This may be advantageous since the cytoplasmic pool might vary substantially in response to diurnal patterns in nitrate uptake
and assimilation (Scaife and Schloemer 1994; Delhon
et al. 1995). While there are numerous questions to be
answered regarding the accuracy of this average, it can
potentially give a value more appropriate for comparison with organic nitrogen in bulk plant tissues than
would direct measurement of the more labile cytoplasmic pool. These relationships are currently being evaluated in comparative time courses of d15N for bulk root
NO3± and xylem NO3±. The latter should re¯ect the
full variation of any diurnal pattern in d15N for the
cytoplasmic pool itself.
The most comparable model in the literature predicting d15N patterns in plants with multiple nitrate
reductase sinks is a compartment-based process model
solved by numerical techniques (Robinson et al. 1998).
Their model goes beyond just the NO3±assimilation
process, and includes explicit terms for storage, both
phloem and xylem transport of reduced nitrogen compounds, and the hypothesized excretion of organic N
from the root to the soil, but it is consistent with
assumptions made above in that only the reduction of
nitrate was treated as exhibiting isotopic discrimination.
Numerical solution in Robinson et al. (1998) was based
on a simultaneous best-®t of the entire {Ri} as opposed
to the small number of selected Ri evaluated above in
Eqs. 22±24 and yielding speci®c insights into the nitrogen metabolism. The method used by Robinson et al.,
when applied to the data of Evans et al. (1996) and
Yoneyama and Kaneko (1989), resulted in multiple
solutions, at least in part because they included in the
analysis unmeasured ¯uxes and Ri, such as the excretion
of organic N. Of current interest is that the model
solutions they concluded were most `biologically rele-
vant' were in close agreement the estimates developed
here and reported above for these same datasets.
The numerically solved process-model approach
permits virtually unlimited amounts of detail to be included, and has essential applications in rigorously
testing integrated models of nitrogen metabolism and
quantifying speci®c hypotheses. An example is the
evaluation of the putative excretion of organic N discussed in Robinson et al. (1998). The algebraic solutions
presented here have di€erent inherent utility, and are
likely to be most insightful when keeping the number of
recognized pools to a necessary minimum. The simple
expressions in Eqs. 22±26, for example, indicate important insights that may be gained from a limited number
of measurements, and might be more useful in evaluating larger numbers of plants for the selected characteristics.
In the context of the model presented above, the plant
nitrate pools are treated as intermediates of small size
that can be ignored in the overall de®nition of massbalance between soil and plant, and only organic
nitrogen is allowed to be a net sink. The isotopic values
of these intermediate pools are predicted, but not accounted for. For most plants, nitrate storage accounts
for less than 1% of total plant nitrogen content
(Smirno€ and Stewart 1985; Pate et al. 1993; Grindlay
1997). In such plants, this simplifying assumption is
probably justi®ed, but in some species grown at high
external nitrate, vacuolar nitrate pools can be quite
variable in size and can account for up to 10% of total
plant nitrogen. In such plants, the model would need to
be expanded to recognize the vacuolar nitrate pool as a
separate sink for mass balance.
Acknowledgements I thank Dave Evans for ®rst interesting me in
this problem, Lou Derry, Dennis Swaney, Roman Pausch,
and David Robinson for helpful comments on the manuscript,
and David Robinson for also sharing the Excel spreadsheet on
isotopic fractionation during nitrate assimilation related to
Robinson et al. (1998). The author was partially supported by EPA
grant #826531-01-0 during preparation of this manuscript.
Appendix A. Unbranched pathways
One intermediate pool
Consider pathway B in Fig. 2, but let pool 2 be the ®nal
sink (i.e. F5=F6=0). Setting the net ¯ux moving from
pool 0 to pool 1 equal to that from pool 1 to pool 2
de®nes a steady state for total mass in the transport
pathway:
F1
F2 ˆ F3
F4
…A1†
Isotopic steady state for the same process is de®ned by
multiplying each ¯ux term in Eq. A2 by its isotopic
abundance ratio (see previous explanation for Eq. 4):
F1 R0
1‡a
F2 R1
F 3 R1
ˆ
1‡a 1‡b
F4 R2
1‡b
…A2†
231
To derive an expression for R0/R2 from Eq. A2, R1 must
®rst be eliminated. Based on Eq. 6, the following
expression holds between pools 1 and 2:
F4
R1 ˆ R2 1 ‡ b 1
…A3†
F3
Equation A3 treats the intermediate pool as the source
during application of Eq. 6. Substituting A3 into A2
yields:
h i3
2 F4
F4
R
F
1‡b
1
F
1‡b
1
3
2
F 1 R0 4 2
F3
F3
5R2
ˆ
‡
…1‡a†
…1‡a†
…1‡b†
…A4†
The numerator over 1+b further simpli®es to
(F3±F4)(1+b), and (1+b) cancels. From Eq. A1, replace
(F3±F4) with (F1±F2):
2 3
F4
F
1‡b
1
2
F 1 R0 4
F3
ˆ
‡…F1 F2 †5R2
…A5†
…1‡a†
…1‡a†
Multiply both sides by (1+a) and divide by F1R2:
h
i
F4
F
F
‡a…F
F
†‡F
1‡b
1
1
2
1
2
2
R0
F3
ˆ
…A6†
R2
F1
The ®rst term on the right equals 1, which we subtract
from both sides, and the F2 terms not being multiplied
by either a or b cancel out:
R0
F2
F2 F2 F4
D…0:2† ˆ
1ˆa 1
‡b
…A7†
R2
F1
F1 F1 F3
and from Eq. A6:
F4
F4
R1 ˆ R3 1 ‡ b 1
‡c
F3
F3
F4 F6
F3 F5
…A11†
The substitutions and collection of terms proceeds
exactly as above, only the analogue of Eq. A4 has an
additional term on the right-hand side over the
denominator (1+c). Simpli®cation and reduction leads
to:
" F4 ‡ c F4 F4 F6
2F
1
‡
b
1
2
2F1 R0
F3
F3 F3 F5
ˆ
…1 ‡ a†
…1 ‡ a†
#
‡ …F3
F4 ‡ F5
F 6 † R3
…A12†
From Eq. A8, substitute 2(F1±F2) for (F3±F4+F5±F6)
and then proceed as before to the ®nal expression:
R0
F2
F2 F2 F4
1ˆa 1
D…0:3† ˆ
‡b
R3
F1
F1 F1 F3
F2 F4 F2 F4 F6
…A13†
‡c
F1 F3 F1 F3 F5
Appendix B. Derivations of branched pathways
First consider the simplest possible branched pathway
(Fig. 4: note that pools and ¯uxes are labeled to be
consistent with the larger pathway of Fig. 3 for later
comparison). The goal is to derive an expression for
D3:5, the discrimination between the source and sink 1
alone.
Derivation for two intermediate pools
As in Eq. A1, the starting point is a de®nition of massbalance including all intermediate pools:
2…F 1
F 2 † ˆ …F 3
F 4 † ‡ …F 5
F 6†
…A8†
and an isotopic mass balance de®ned by multiplying A8
by the isotopic ratios of each unidirectional ¯ux:
F1 R0
F2 R1
F 3 R1
F4 R2
F5 R2
F6 R3
ˆ
‡
2
1‡a 1‡a
1‡b 1‡b
1‡c 1‡c
…A9†
This time, substitutions are needed for both R1 and R2 in
terms of R3.
As before, from Eq. 6:
F6
R2 ˆ R3 1 ‡ c 1
…A10†
F5
Fig. 4 Reproduction of central 4 pools surrounding the branchpoint in Fig. 3. Source and sink relationships (ellipse and
trapezoids), however, are rede®ned as required in the derivation
leading to Eq. B5. Additional pools were then added to branches
and basal shared pathway, and source, intermediate, and sink pools
rede®ned as appropriate to each stage, in a sequential manner to
derive the general expressions in Eqs. B6 and B11, which apply to
the entirety of Fig. 3
232
Begin as before with statements of total mass balance:
F7
F8 ˆ F9
F 10 ‡ F 15
F 16
…B1†
and isotopic conservation:
F7 R3
…1 ‡ d†
F 8 R4
F9 R4
ˆ
…1 ‡ d† …1 ‡ e†
F10 R5
F15 R4
‡
…1 ‡ e† …1 ‡ h†
F16 R8
…1 ‡ h†
…B2†
Similar derivations must be repeated many times
while sequentially adding more intermediate pools in the
branches towards each individual sink, and within
the shared basal portion of the pathway prior to the
branchpoint. Each solution provides additional identities needed to eliminate all R but the source and sink of
interest from the equation. The ®nal result is a general
expression applicable to Fig. 3 or even larger systems:
2
D…source:sink1†
H…source:sink1† ‡ B…source...bp† D…bp:sink1†
6 ‡ Fnet:sink2 U…source...bp† D…bp:sink1†
6 primary influx
ˆ6
Fnet:sink1
4 ‡ primary
influx U…source...bp† D…bp:sink2†
efflux product
‡BP influx
product D…bp:sink1† D…bp:sink2†
To derive an expression for R3/R5, R4 and R8 must be
eliminated. These substitutions are done sequentially
using Eq. 7, ®rst eliminating R8 in terms of R4:
F15
…B3†
R8 ˆ R4
…F15 ‡ …F15 F16 †h†
and then R4 in terms of R5:
R4 ˆ R5
…F9 ‡ …F9 F10 †e†
F9
…B4†
Since more-detailed algebra has been presented in previous examples, the steps are merely outlined here. After
making the indicated substitutions for R4 and R8 into
Eq. B2, it is helpful to ®rst group the terms with common factors of 1/(e+1) and 1/(h+1). This permits some
reduction of the denominators. Then combine all terms
over a common denominator, isolate the quotient R3/R5
on one side of the equality, and subtract 1 from each
side. Expand all products, collect all terms multiplied by
d, e, or h, and reduce each of the three resulting expressions using repeated ¯ux substitutions from Eq. B1
to give:
d 1
F8
F8 ‡ F15 F16
‡
e 1
F7
F7
…F15 F16 †
‡
de…1 F10 =F9 †
F7
D…3:5† ˆ
‡
‡
…F9
F10 †
F7
F8
e …1
F7
dh…1
F16 =F15 †
F10 =F9 †h…1
F16 =F15 †
F10
F9
D…bp:sink2†
3
7
1
7
7
5 1 ‡ D…bp:sink2†
…B6†
where H and D both refer to equivalent algebraic
expansions using Eq. 7 and the indicated subscripted
pools for `source':sink. The essential di€erence between
H and D is that while D indicates a true isotopic relationship between two pools in a subsection of the
overall pathway, H alone does not. The term `primary
in¯ux' refers to the unidirectional ¯ux from the ultimate source in the model to the ®rst intermediate pool
of the uptake pathway, `BP(e‚ux product)/(in¯ux
product)' is the now familiar e‚ux/in¯ux product calculated between the ultimate source pool and the
branchpoint, and B() and F() refer to a `branchpoint
interaction factor' and a new function analogous to
Eq. 7 for D, respectively, both to be discussed in detail
below.
In the case of the pathway in Fig. 4, B(source..bp) is
equal to Fnet_sink2/primary in¯ux (i.e. [F15±F16]/F7) and
F(source..bp) is d, the discrimination constant of the
transfer step from source to branchpoint. Using these
identities, Eqs. B5 and B6 are identical algorithms for
D(3:5) in Fig. 4. However, both B() and F() add additional terms with each additional intermediate pool
added to the shared basal path in multi-pool pathways.
The form of B() for all possible starting points before
F15
F16
F7
h 1
F16
F15
1
1 ‡ h …1
F16 =F15 †
…B5†
233
the branch point in Fig. 3 are given in Table 3. B()
actually has two component factors, one of which is
always Fnet_sink2/primary in¯ux while the other is a
progressive sum of ¯ux quotients. This sum of quotients is not intuitive in form, but has a predictable
pattern for any number of pools (Table 2). F() is calculated by taking the same terms listed in Table 2, and
weighting each one by the sequence of discrimination
constants in the basal shared path. For Fig. 4 this gave
simply d, and for the full basal path shown in Fig. 3
this gives:
U…0:4† ˆ a ‡
F2
F2 F4
F2 F4 F6
b‡
c‡
d
F3
F3 F5
F3 F5 F7
…B7†
Happily, much of the complexity of Eq. B6 has an
almost negligible e€ect on the ®nal calculated value.
The factor 1/(1+D(bp:sink2)) outside the large square
brackets is always very close to 1, while the last three of
the four terms inside the square brackets are all discrimination cross-products with magnitudes similar to
D2, which is close to zero. Moreover, these subtle e€ects
of the outer factor and inner cross-products tend to
cancel each other. Eliminating these extraneous terms
and factors is often justi®ed and leads to Eq. 11 in the
main text.
The mean D for a branched pathway
This de®nition in Eq. 12 is essentially equivalent to:
D…0:combined:sinks† ˆ
Fnet:sink1 D…0:sink1† ‡ Fnet:sink2 D…0:sink2†
Fnet:total
…B8†
Substituting for D(source:sink) using Eq. 11 for each of the
two sinks into Eq. B8 gives:
Fnet:sink1 ‰H…0:sink1† ‡B…0::bp† …D…0:sink1† D…0:sink2† †Š
Fnet:total
net:sink2 ‰H…0:sink2† ‡B…0::bp† …D…0:sink2† D…0:sink1† †Š
Fnet:total
D…0:sink1&2† ˆ ‡ F
…B9†
All terms associated with the branch interaction factor
cancel out, leaving us with a simple ¯ux-weighted average of H for each branch as shown in Eq. 13 of the
main text.
D…soil:nitrate:mean:plant†
F2
ˆ
D…source:nitrate:leaf:assimlate†
F5 ‡ F3
…C1†
As discussed in the main text, D(source.nitrate:leaf.assimilate) is
equivalent to D(source.nitrate:root nitrate). F3+F5, the total
assimilation by both sinks, is equivalent to F1±F2, the net
uptake of nitrate at the root surface. These two substitutions into C1 give:
!
F2
F1
D…soil:nitrate:mean:plant† ˆ
D…soil:nitrate:root:nitrate†
F2
1 ‡ F1
…C2†
Solving for F2/F1:
D…soil:nitrate:mean:plant†
F2
ˆ
F1 D…soil:nitrate:mean:plant† D…soil:nitrate:root:nitrate†
Using d  dsource
…C3†
dproduct (see Eq. 6b) gives Eq. 22.
Partitioning NO3± reduction between root and shoot
Again substituting into Eq. 21 de®ning net discrimination at the whole-plant level, but this time replacing F2/
F1 based on Eq. C3 gives:
D…soil:nitrate:mean:plant†
D…soil:nitrate:mean:plant†
ˆ
D…soil:nitrate:mean:plant† D…soil:nitrate:root:nitrate†
F3
b
…C4†
F3 ‡ F5
F3/(F3+F5) is the fraction of total nitrate assimilation
taking place in the root (Fig. 1). Solving for this ¯ux
quotient yields:
root assimilation
total assimilation
D…soil:nitrate:mean:plant†
ˆ
D…soil:nitrate:root:nitrate†
b
…C5†
Equation 23 is rewritten from Eq. C5 in terms of d
based on D  dsource dproduct .
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