The distribution of nitrogen isotopes in the biosphere has the potential to offer insights into the past, present and future of the nitrogen cycle, but it is challenging to unravel the processes controlling patterns of mixing and fractionation. We present a mathematical model describing a previously overlooked process: nitrogen isotope fractionation during leaf-atmosphere NH3(g) exchange. The model predicts that when leaf-atmosphere exchange of NH3(g) occurs in a closed system, the atmospheric reservoir of NH3(g) equilibrates at a concentration equal to the ammonia compensation point and an isotopic composition 8.1‰ lighter than nitrogen in protein. In an open system, when atmospheric concentrations of NH3(g) fall below or rise above the compensation point, protein can be isotopically enriched by net efflux of NH3(g) or depleted by net uptake. Comparison of model output with existing measurements in the literature suggests that this process contributes to variation in the isotopic composition of nitrogen in plants as well as NH3(g) in the atmosphere, and should be considered in future analyses of nitrogen isotope circulation. The matrix-based modelling approach that is introduced may be useful for quantifying isotope dynamics in other complex systems that can be described by first-order kinetics.
The isotopic composition of nitrogen in the terrestrial biosphere is striking in two ways: first, the biosphere as a whole is enriched in 15N relative to N2(g) in the atmosphere (Hoering & Ford 1960); and second, the individual reactive inorganic and organic nitrogen pools within the biosphere are extremely variable. The former pattern is well-understood. The overall average isotopic composition of nitrogen in the biosphere is generated by the balance of the isotope effects associated with the primary input and output fluxes that interconvert reactive nitrogen with inert N2(g) [i.e. the combination of small discrimination in nitrogen fixation and large discrimination against 15N in denitrification, per Hoering & Ford (1960)]. In contrast, the spatial and temporal variability of the individual nitrogen pools within the biosphere is generated by a myriad of physical, chemical and biological processes that are much more complex and imperfectly understood. The goal of this article is to explore whether leaf-atmosphere NH3(g) exchange, a process that currently is overlooked in most treatments of nitrogen cycling, may in fact have an important influence on the distribution of nitrogen isotopes along the soil-plant-atmosphere continuum.
N2(g) in the atmosphere is comprised of 99.6% of the lighter nitrogen isotope (14N), 0.4% of the heavier nitrogen isotope (15N), and serves as the standard reference for the nitrogen isotope system, that is δ15N = 0 (Mariotti 1983). Measurements of the natural abundance of δ15N in bulk plant tissue typically range between −20 and +20‰ (Peterson & Fry 1987; Handley & Raven 1992; Hogberg 1997). While some broad trends in plant δ15N have been explained by systematic variation in the openness of nitrogen cycling with climatic gradients (Austin & Vitousek 1998; Handley et al. 1999; Martinelli et al. 1999; Schuur & Matson 2001; Amundson et al. 2003; Craine et al. 2009; Houlton & Bai 2009), a large fraction of the global range of δ15N values is typically represented in samples collected from a single location at a single point in time (Garten 1993; Schulze, Chapin & Gebauer 1994; Handley et al. 1999; Ometto et al. 2006; Aranibar et al. 2008).
Efforts to explain between-plant and within-plant variability in nitrogen isotope composition have focused on the isotopic variability of nitrogen sources available to plants, fractionations associated with nitrogen acquisition and fractionations associated with subsequent metabolism. Variability between plants has been attributed to three processes that influence the isotopic composition of acquired nitrogen: (1) preferential uptake from different inorganic and organic source pools (Herman & Rundel 1989; Schulze et al. 1994; Vallano & Sparks 2012); (2) differences in uptake mechanisms, for example between roots and mycorrhizae (Hobbie, Macko & Shugart 1999a); and (3) the degree to which uptake is limiting (Mariotti et al. 1982). Variability within an individual plant has been attributed to fractionations associated with assimilatory, dissimilatory and transamination reactions within nitrogen metabolism, and the subsequent distribution of relatively enriched or depleted metabolites to certain organelles, organs or tissue types (Evans et al. 1996; Werner & Schmidt 2002; Cernusak, Winter & Turner 2009; Gauthier et al. 2012).
These and other studies investigating the nitrogen isotope composition of plants have relied on the assumption that, once a plant acquires and assimilates nitrogen, the nitrogen isotopic composition of the whole plant is stable and equal to that of the assimilated nitrogen. In this view, when fractionations occur within metabolism, the nitrogen isotopes are simply redistributed between internal pools in a zero-sum game (Mariotti et al. 1982; Handley & Raven 1992; Robinson, Handley & Scrimgeour 1998; Hobbie, Macko & Shugart 1999b; Evans 2001; Robinson 2001; Werner & Schmidt 2002; Cernusak et al. 2009; Tcherkez 2011; Gauthier et al. 2012). This paradigm is at odds with the observations that NH3(aq) in leaves continually exchanges with NH3(g) in the atmosphere, and that the exchange is associated with a number of isotope effects (Fig. 1).
Nitrogen metabolism maintains a pool of NH3(aq) within the mesophyll cells of leaves. The NH3(aq) pool is controlled by metabolic processes that produce, consume and transport NH3(aq) (Martin et al. 1983; Joy 1988) (Fig. 2). Major pathways producing ammonia/ammonium include import from xylem cells, nitrate/nitrite reduction within chloroplasts and photorespiratory glycine deamination within mitochondria (Schjoerring et al. 2002). On the consumption side, primary assimilation and photorespiratory reassimilation of ammonia/ammonium primarily occur through chloroplastic glutamine synthetase (EC 126.96.36.199), with additional assimilation occurring through the cytosolic isoenzyme (Keys 2006; Bernard & Habash 2009). Under ordinary conditions, the flux of NH3(aq) through photorespiratory metabolism is likely to be an order of magnitude higher than the flux through primary assimilation (Keys 2006). Accordingly, the majority of nitrogen assimilated by a plant is likely to pass through the foliar NH3(aq) pool under two scenarios: first, when primary nitrogen assimilation occurs predominantly in leaves, and amino acids are then exported to sinks within the shoot and roots; second, when primary nitrogen assimilation occurs predominantly in roots, and the amino acids imported to leaves mix with the pools of amino acids that are photorespiratory intermediates before being incorporated into protein, nucleic acids and other nitrogen-containing compounds (e.g. chlorophyll, coenzymes, secondary metabolites, etc.).
The foliar NH3(aq) pool is critical because NH3(g) volatilizes at air-water interfaces, which generates a NH3(g) pool in the intercellular airspaces. Whenever the stomata are open, the NH3(g) in the intercellular airspaces can exchange with NH3(g) in the atmosphere (Farquhar et al. 1980a). The larger the rate of leaf-atmosphere NH3(g) exchange relative to the rate of primary nitrogen assimilation, the greater the potential for leaf-atmosphere NH3(g) exchange to influence the nitrogen balance and nitrogen isotopic composition of the whole plant. Typical rates of NH3(g) uptake or loss are several nanomoles of NH3(g) m−2 s−1 (Farquhar et al. 1980a). While these instantaneous fluxes appear small, they can be equivalent to several percent of the rate of primary nitrogen assimilation. For example, consider a plant with an average net photosynthetic rate of 10 umol CO2(g) m−2 s−1, and average loss of 2.5 nmol NH3(g) m−2 s−1; if the plant spent half of the assimilated carbon for cellular maintenance, and invested the remainder in structural material with a molar C:N ratio of 100, then the rate of primary nitrogen assimilation would average 50 nmol N m−2 s−1. Thus, about 5% of the total nitrogen originally assimilated would have volatilized as NH3(g).
Fluxes of this magnitude are important isotopically because each of the physical, chemical and biological processes involved in the exchange of NH3(g) between a leaf and the atmosphere is characterized by an isotope effect. Diffusion drives the transport of NH3(g) through the leaf boundary layer, stomatal pore and intercellular air spaces. Theoretical calculations predict a kinetic isotope effect of 17.6‰ for NH3(g) diffusion through still air (Farquhar, Wetselaar & Weir 1983), and 11.7‰ for diffusion through the boundary layer (Supporting Information Appendix S1). At gas-liquid interfaces on the interior (i.e. at the apoplast) and exterior (i.e. surface films) of the leaf, NH3(g) equilibrates with NH3(aq). Norlin, Irgum & Ohlsson (2002) have measured an isotope effect of 5.4‰ associated with the gas phase-aqueous phase equilibrium. In solution, NH3(aq) is rapidly protonated and equilibrates with . Hermes, Weiss & Cleland (1985) measured an isotope effect of 19.2‰ associated with the equilibrium between protonation and deprotonation. Finally, enzymatic transformations of ammonia are also associated with isotope effects. Yoneyama et al. (1993) measured a kinetic isotope effect of 16.5‰ for fixation of ammonium through glutamine synthetase. Other reactions within nitrogen metabolism are also expected to be associated with kinetic isotope effects, although many have not yet been measured [e.g. see Tcherkez (2011) for isotope effect estimates for glycine decarboxylase and glutamate:glyoxalate aminotransferase].
These observations led a few researchers to hypothesize that leaf-atmosphere NH3(g) exchange might contribute to the distribution of nitrogen isotopes that is observed in plants and the environment (Farquhar et al. 1980a; Handley & Raven 1992; Harper & Sharpe 1998). In this article, we make that hypothesis quantitative by building a model that represents foliar nitrogen metabolism as a system with open boundaries, such that fractionation and mixing resulting from the exchange between NH3(aq) in leaves and NH3(g) in the atmosphere have the potential to influence the nitrogen isotopic composition of protein in leaves as well as NH3(g) in the atmosphere. We apply the model to predict the equilibrium isotope distribution resulting from three different nitrogen nutrition regimes. Finally, we assess the contribution the model might make to explain a range of δ15N measurements previously reported in the plant physiology, biogeosciences and atmospheric sciences literature.
Diverse approaches have been used to model isotope mixing and fractionation (O'Leary 1981; Hayes 1983; Rooney 1988; Berry 1989; Comstock 2001; Fry 2003; Ogle, Wolpert & Reynolds 2004; Currie 2008). Here, we formulate a system of linear, first-order, ordinary differential equations that represents nitrogen metabolism and NH3(g) exchange at the scale of a single C3 leaf (Table 1, Fig. 3). Our notation is based on chemical kinetics, following Bigeleisen & Wolfsberg (1958), and is described in Appendix I.
Table 1. Mathematical terms used in this paper
aat T = 298.15K
bEquivalent to Vo/2
cCalculation based on measured value of 1.0165; see Supporting Information Appendix S1 for details.
dFarquhar et al. (1983) derived 1.018; we recalculate with an additional significant digit.
eCalculated using Mason & Marrero (1970) and Kays & Crawford (1980).
GGAT: glutamate:glyoxylate aminotransferase; SHMT: serine hydroxymethyltransferase; SGAT: serine:glyoxylate aminotransferase; NR: nitrate reductase; NiR: nitrite reductase; GS: glutamine synthetase; calculation based on measured value of 1.0165; see Supporting Information Appendix S1 for details.
The model includes 10 pools (Pi): five inorganic pools [P1: , aqueous nitrate within leaf cells; P2: NH3(aq), aqueous ammonia within leaf cells; P3: , ammonium within leaf cells; P9: NH3(g)(i), gaseous ammonia in intercellular airspaces; P10: NH3(g)(a), gaseous ammonia in the atmosphere] and five organic pools [P4: Gln, glutamine; P5: Glu, glutamate; P6: protein-N; P7: Gly, glycine; P8: Ser, serine]. The pools are connected by 15 fluxes (Fj), each expressed as having a first-order (or pseudo first-order) dependence on the size of the pool that it originates from, such that Fj = kjPi, where kj is a rate constant.
There are four transboundary fluxes connecting the internal pools to sources and/or sinks. Nitrogen enters the leaf as nitrate () imported from the xylem, reduced to nitrite () by nitrate reductase [EC 188.8.131.52], and converted from to NH3(aq) by nitrite reductase [EC 184.108.40.206] with rate constant k1. In addition, nitrogen enters as gaseous ammonia (NH3(g)) diffusing in from the atmosphere with rate constant k15. Nitrogen leaves as organic N in the form of protein biosynthesis with rate constant k7, or as gaseous ammonia diffusing out into the atmosphere with rate constant k14. Since Glu is the basic building block for the other amino acids that comprise proteins, we follow Werner & Schmidt (2002) in specifying Glu as the source for protein biosynthesis.
There are four internal inorganic fluxes. Aqueous ammonia exchanges with protonated ammonium, with rate constants k2 for protonation and k3 for deprotonation. These fluxes are temperature- and pH-dependent. Aqueous ammonia exchanges with gaseous ammonia in the intercellular air spaces, with rate constants k12 for volatilization and k13 for dissolution. These fluxes are also temperature-dependent.
There are seven internal organic fluxes representing primary nitrogen assimilation and the photorespiratory nitrogen cycle. In the chloroplast, ammonia is fixed with rate constant k4 into glutamine via glutamine synthetase (GS; EC 220.127.116.11) and then converted with rate constant k5 to glutamate via glutamine:2-oxoglutarate aminotransferase (GOGAT; EC 18.104.22.168), also known as glutamate synthase. Fixation of ammonia via GS occurs in proportion to the availability of Glu provided by GOGAT; thus, F4 = k4P2 is equal to F6 = k6P5. Note that, biochemically, GS binds to ammonium and representing the substrate as ammonia is a mathematical abstraction; more detail is provided in Supporting Information Appendix S1. Using 2-oxoglutarate supplied either by photorespiration or the citric acid cycle, GOGAT converts each molecule of Gln to two molecules of Glu. One of these cycles back to GS (k6), and the other is recycled for photorespiration (k8) or exported for protein biosynthesis (k7). Since the activities of glutamate:glyoxylate aminotransferase (EC 22.214.171.124; k8) and serine:glyoxylate aminotransferase (EC 126.96.36.199; k10) are balanced during photorespiration, the net flux of Gly they produce is equal to the rate of oxygenation at Rubisco (Vo; also see Fig. 1). Similarly, the activities of serine hydroxymethyltransferase (EC 188.8.131.52; k9) and glycine decarboxylase (EC 184.108.40.206; k11) are balanced such that each accounts for one-half of the overall photorespiratory nitrogen flux.
Aspects of nitrogen metabolism beyond primary assimilation and photorespiratory recycling have not been represented explicitly (e.g. metabolism of other amino acids, purines and pyrimidines, ureides, etc.).
Model solution and parameterization
We use two approaches to solve the system. First, we use substitution to eliminate intermediate pools in a sequential chain of reactions, as originally applied by O'Leary (1981) and Berry (1989) and many others thereafter. Second, we use matrix algebra. Matrix algebra methods are widely utilized in other branches of the natural sciences and engineering, and adopting this approach to solve isotope systems offers several advantages over substitution. Both methods are described in detail in Appendix II.
Briefly, the change in pool sizes over time is written as a linear combination of the fluxes contributing material to, and removing material from each pool:
Here, is a column vector with i rows representing the changes in pool size for the entire system, P is a column vector with i rows representing pool sizes at any time t, and B is a column vector with i rows representing constant inputs at every time t. A is a square matrix populated by rate constants k(i,j) such that each element of is equal to the product of the corresponding row of A and the vector P, plus the corresponding element of B. When , that is at equilibrium, the equilibrium pool sizes () are given by:
where A−1 represents the inverse of A. By repeating the above procedure for 14N and 15N and dividing the corresponding pool sizes, we solve for the equilibrium isotopic composition of all of the pools:
In addition, we use the eigenvalue decompositions of the turnover matrices (Al, Ah) to calculate the isotopic response time (τ, also called the e-folding or relaxation time), which describes the timescale required to approach equilibrium:
To parameterize the model, we specify rate constants for all fluxes, for both the heavy and light isotopes. Some values are derived from existing measurements in the literature, some from new measurements in our laboratory and others from assumptions. Due to the separation of timescales between the relevant physiochemical, physiological and environmental processes, we focus on the physiological timescale, and specify that the physiochemical fluxes are so rapid as to be at equilibrium and the environmental processes so slow as to be abstracted as parameters. The full parameterization is described in Supporting Information Appendix S1. The model is implemented in MATLAB (2012a, The MathWorks, Natick, MA, USA). A complete list of the equations required to run the model is given in Supporting Information Appendix S2.
We examine the effects of leaf-atmosphere NH3(g) exchange on nitrogen isotope distribution in three scenarios with different configurations of nitrogen supply:
Two sources of nitrogen, derived from the soil and NH3(g) in the atmosphere. The concentration of the atmospheric NH3(g) source is exactly equal to the NH3(g) compensation point, such that foliar efflux and uptake of NH3(g) are balanced.
One source of nitrogen, derived from the soil. The NH3(g) compensation point is higher than the atmospheric concentration of NH3(g), such that the leaf only loses NH3(g).
One source of nitrogen, NH3(g) in the atmosphere. The NH3(g) compensation point is lower than the atmospheric concentration of NH3(g), such that the leaf only takes up NH3(g).
For each scenario, we assume that the concentration and isotopic composition of NH3(g) supplied by the atmosphere is unaffected by the efflux of NH3(g) from the leaf. We specify that the source pool of NH3(g) in the atmosphere has an isotopic composition of −8.1‰, and the source pool of from the soil has an isotopic composition of 0‰. The model is run under constant light (1500 umol PAR m−2 s−1) and CO2(g) (394 ppm), while varying combinations of three parameters: stomatal conductance (0–1 mol m−2 s−1), leaf temperature (10–40 °C) and nitrogen import rate (50, 100, 200 nmol N m−2 s−1). These ranges of conductance and temperature were chosen to encompass a wide range of the conditions on the land surface. The nitrogen import rates were chosen to correspond to the theoretical rates of nitrogen assimilation that must be maintained for plant functional types photosynthesizing at 20 umol CO2 m−2 s−1, annually spending half of the acquired photosynthate on respiration, and maintaining whole-plant molar C:N stoichiometry of 200:1 (i.e. tree), 100:1 (i.e. shrub) or 50:1 (i.e. herbaceous plant), assuming a residence time of 1 year for nitrogen in tissue.
Across the three scenarios, the response time of the entire system, τ, varies from tens of seconds to several minutes. Under most conditions, however, the response time is less than 2 min. This indicates that, following a perturbation, the most slowly responding pool returns to within 1/e4, or 1.8%, of its equilibrium isotopic composition after four response times, that is 8 min. Accordingly, isotopic equilibrium is achieved on a physiologically relevant timescale in this system. We focus our analysis on the equilibrium results.
To evaluate the sensitivity of the equilibrium results to key empirical uncertainties, we also simulate several alternative model structures and parameter sets. With respect to the structure of the model, we consider the effects of (1) the identity of the amino acids exported for protein synthesis and (2) the accumulation of photorespiratory intermediates. With respect to the parameterization, we consider the effects of estimates from Tcherkez (2011) of the isotope effects for key reactions where no measurements are available (i.e. glutamate:glyoxalate aminotransferase and glycine decarboxylase).
Relationship between NH3(g) compensation point and fluxes
Across the leaf temperatures considered, the NH3(g) compensation point averages 14 ppb and ranges from less than one ppb NH3(g) to 38 ppb NH3(g) (Fig. 4). In Scenario 1, when nitrogen is supplied from the soil and the [NH3(g)] in the atmosphere is at the leaf's NH3(g) compensation point, uptake and efflux of NH3(g) are balanced, that is net NH3(g) exchange is zero. In Scenario 2, when all nitrogen is supplied from the soil, and the atmosphere is a sink for NH3(g), the leaf only loses NH3(g). In Scenario 3, when all nitrogen is supplied from the atmosphere, the atmosphere is both a source and a sink for NH3(g). Thus, the leaf always has net uptake of NH3(g) when the [NH3(g)] in the atmosphere is above the leaf's NH3(g) compensation point, and net loss when the atmospheric concentration is below the compensation point.
Scenario 1: atmospheric [NH3(g)] equal to [NH3(g)(i)]
In this scenario, a leaf has two potential sources of nitrogen (i.e. imported from root uptake (at 0‰) and NH3(g) in the atmosphere (at −8.1‰) (Fig. 5). When the concentration of NH3(g) in the atmosphere is exactly at the leaf's NH3(g) compensation point, the rate of gaseous influx is equal to the efflux and the rate of soil nitrogen import is equal to the rate of protein synthesis. The isotopic composition of the protein always matches the soil source (i.e. 0‰) and the gaseous efflux always matches the gaseous influx (i.e. −25.7‰). In this unique situation, the nitrogen assimilation system appears to be unaffected by the gaseous exchange because the isotopic composition of NH3(g) in the atmosphere matches the isotopic compensation point of the leaf, −8.1‰. When the stomata are closed, the isotopic composition of NH3(g) in the intercellular air spaces is lighter than by 8.1‰ due to the fractionations associated with gaseous/aqueous equilibration and fixation through glutamine synthetase (i.e. 5.4 + 2.7 = 8.1‰). If the stomata remain closed, this tiny, captive pool of NH3(g) has no effect on the isotopic composition of the nitrogen assimilated into protein. If the stomata open, the NH3(g) inside the leaf can exchange with NH3(g) in the free atmosphere, making it possible to have isotopic ‘leakage’ to or from the assimilation stream. However, when the stomata are open and the atmosphere is at the isotopic compensation point, the isotopic compositions of the gaseous source and sink fluxes are identical, so as the isotopic compositions of the fluxes from the soil source and to the protein sink. The absolute difference between the isotopic compositions of the two sinks reflects the intervening fractionations (i.e. for diffusion, gaseous/aqueous equilibration and fixation through glutamine synthetase: |Δsinks| = 17.6 + 5.4 + 2.7 = 25.7‰).
While Scenario 1 technically describes an open system (i.e. the atmospheric NH3(g) source is not modified by NH3(g) efflux from the leaf), it is notable that this is precisely the dynamic that would develop if the atmospheric reservoir was a closed system (i.e. such that atmospheric NH3(g) could mix with NH3(g) efflux from the leaf). In a closed system, the concentration of NH3(g) in the atmosphere would evolve to match the leaf's compensation point, and its isotopic composition would be depleted by 8.1‰ relative to the isotopic composition of nitrogen cycling in the plant-soil system. Does this occur in the atmosphere? Most of the modern troposphere has several part-per-billion concentrations of NH3(g) (Beer et al. 2008; Clarisse et al. 2009). While these NH3(g) levels are similar to the compensation points of leaves, the residence time of NH3(g) in the atmosphere is orders of magnitude shorter than the timescale on which global mixing could homogenize NH3(g) concentrations. Thus, while the conditions considered in Scenario 1 are realistic, they are not general; they represent a special case. Atmospheric concentrations of NH3(g) are sometimes above, sometimes at, and sometimes below the compensation point of plants, and the isotope dynamics associated with net uptake as well as net efflux are both relevant. In Scenario 1, net uptake marginally depletes the nitrogen in protein, whereas net efflux marginally enriches this pool (Fig. 5a,c,e). The relative magnitude of the discrimination is proportional to the net uptake or efflux; the greater the imbalance between the two gaseous fluxes, the greater the isotopic enrichment or depletion of the sinks due to the kinetic fractionation associated with diffusion. The following scenarios explore the limits where foliar efflux vastly exceeds uptake, and uptake vastly exceeds efflux.
When the atmosphere is ammonia-free, the soil supplies all of the nitrogen available to a plant, and whenever the stomata are open, efflux dominates leaf-atmosphere NH3(g) exchange. Relative to the soil source (here, δ15N = 0‰), the nitrogen assimilated into protein is always isotopically enriched (Fig. 6a,c,e) and the efflux is always isotopically depleted (Fig. 6b,d,f). The isotopic composition of the two sinks is offset by 25.7‰. The relative magnitude of enrichment and depletion vary with factors that control the partitioning of source nitrogen to the two sinks: stomatal conductance, leaf temperature and the nitrogen import rate. At low stomatal conductance, low temperature or high rates of nitrogen import, the leaf is essentially an unbranched system. While a pool of NH3(g) exists in the intercellular air spaces at δ15N = −8.1‰, the majority of nitrogen imported from the soil is assimilated into organic nitrogen; little escapes from the leaf as NH3(g). Close to this limit, the protein formed in the leaf approaches the isotopic composition of the source nitrogen, and the NH3(g) that escapes approaches δ15N = −25.7‰. As conductance and/or temperature increases, or the nitrogen import rate decreases, the system begins to branch, with proportionally more NH3(g) diffusing into the atmosphere. As this occurs, depleted NH3(g) escapes from the leaf, and all of the leaf internal pools become relatively enriched. When nearly all of the imported nitrogen escapes from the leaf as NH3(g), the efflux approaches the isotopic composition of the source nitrogen and the organic nitrogen remaining the leaf approaches δ15N = 25.7‰. While these limits are informative for understanding how the model works, most real leaves would be expected to have intermediate levels of partitioning between the two sinks, and therefore isotopic compositions that range between the above minima and maxima (i.e. for protein, 0 − 25.7‰; for NH3(g) efflux, −25.7 − 0‰). Under the range of conditions specified in Scenario 2, the isotopic composition of nitrogen in protein ranges between 0 and 12‰, and the escaping NH3(g) ranges between −25.7 and −14‰ (Fig. 6).
When the atmosphere is ammonia-rich and the soil supplies no nitrogen to the plant, uptake dominates leaf-atmosphere NH3(g) exchange (Fig. 7). Under these conditions, the NH3(g) diffusing into the leaf has an isotopic composition of −25.7‰ due to the isotopic composition of the source nitrogen in the atmosphere (here, δ15N = −8.1‰) and the kinetic fractionation associated with diffusion. Relative to this influx, the organic nitrogen assimilated into protein is always isotopically enriched (Fig. 7a,c,e), the efflux is isotopically depleted (Fig. 7b,d,f), the two sinks are offset by 25.7‰ and the actual isotopic composition of each sink is controlled by the factors that control partitioning of the influx, just as in Scenario 2.
In this case, leaf temperature and nitrogen import rate continue to control partitioning, but stomatal conductance does not, because it has an equal effect on uptake and loss of NH3(g) (Fig. 6 versus Fig. 7). When leaf temperature is low and the net rate of nitrogen import is high (i.e. uptake is occurring from an atmospheric reservoir much higher in concentration than the leaf's compensation point), the majority of nitrogen imported from the atmosphere is assimilated into protein and little escapes from the leaf as NH3(g). Close to this limit, the protein formed in the leaf approaches the isotopic composition of the nitrogen supplied by the atmosphere, and the NH3(g) that escapes approaches δ15N = −51.4‰. As temperature increases or the net rate of nitrogen import decreases (i.e. the leaf-atmosphere concentration gradient is weakened), proportionally more NH3(g) diffuses back into the atmosphere. Again parallel to Scenario 2, this enriches all of the leaf internal pools. As the rate of gaseous efflux approaches the rate of influx, that is the leaf's compensation point, the isotopic composition of the efflux approaches that of the influx, and the protein approaches δ15N = 0‰. Thus, the isotopic composition of protein ranges from a minimum of −25.7‰ to a maximum of 0‰ and NH3(g) efflux from −51.4 to −25.7‰, depending on partitioning between sinks. Under the conditions specified in Scenario 3, the isotopic composition of nitrogen in protein ranges between −25.7 and −8.5‰ (Fig. 7). Note that these dynamics are identical to those of carbon isotope fractionation in the simplified C3 photosynthesis model of Farquhar, O'Leary & Berry (1982).
Sensitivity to model structure and parameterization
Changes to the model structure and/or parameterization can introduce offsets to the original difference in nitrogen isotope composition between organic (15N-Glu and/or 15N-Gly) and inorganic (15N-NH3(g)) sinks, |Δsinks| (Table 2). Within the original model structure, including an estimate of the isotope effect for glutamate:glyoxalate aminotransferase (α8 = 1.005) proportionally increases |Δsinks|. When the model structure is revised so that Gly is the primary export, including an estimate of the isotope effect for glycine decarboxylase (α11 = 0.995) proportionally suppresses |Δsinks|. Individually, the marginal effect of each of these perturbations is to shift |Δsinks| by ± 20%. However, when the model structure is revised to accommodate simultaneous accumulation of Gly and export of Glu, including the same estimate of either α8 or α11 alone shifts |Δsinks| by only ± 10%. When estimates for both α8 and α11 are included together, the perturbations fully cancel one another out, and |Δsinks| converges on the original, unperturbed value.
Table 2. Sensitivity of results to model structure and parameterization
Original parametersα8 = 1.000 α11 = 1.000
Alternative Iα8 = 1.005 α11 = 1.000
Alternative IIα8 = 1.000 α11 = 0.995
Alternative IIIα8 = 1.005 α11 = 0.995
For a leaf under a single set of initial conditions (i.e. 25 °C, stomatal conductance 0.5 mol m s−1, imported at 50 nmol m−2 s−1, atmospheric [NH3] = 5 ppb), the isotopic composition of the sinks is sensitive to model structure and parameterization. In the original structure, Glu is exported for protein synthesis; in alternative 1, Gly is exported for protein synthesis; in alternative 2, export is split between 50% accumulation of Gly and 50% protein synthesis from Glu. In the original parameterization, the kinetic isotope effects for GGAT (α8) and GDC (α11) were set to 1.000; in alternative I, an estimate was included for GGAT; in alternative II, an estimate was included for GDC; and in alternative III, both estimates were included.
Alternative structure 1
Alternative structure 2
Thus, the isotope effects of reactions that are not situated at metabolic branch points are expressed exclusively in the isotopic composition of intermediate pools; they have no effect on the isotopic composition of the final sinks, and therefore the model is insensitive to assumptions about the magnitude of these parameters. However, the isotope effects of reactions at metabolic branch points are expressed in the isotopic composition of the final sinks, and therefore the model is sensitive to assumptions about the location of branch points and the associated isotope effects. With respect to the specific alternatives considered here, the original finding of relative isotopic depletion of NH3(g) and relative isotopic enrichment of protein-N appears to be robust, likely because the estimates of the unknown enzymatic isotope effects are relatively small compared to the isotope effects associated with volatilization, dissolution and diffusion.
In most empirical studies, definitive identification of the processes controlling δ15N is a highly underdetermined problem due to incomplete information about nitrogen sources and fractionations. By quantifying the expected influence of leaf-atmosphere NH3(g) exchange on the isotopic composition of nitrogen in plants and the atmosphere, this model may either add to or winnow down candidate interpretations for processes controlling δ15N of organic nitrogen in plants, as well as the δ15N of atmospheric NH3(g). In the following examples, we test the model's ability to explain a range of δ15N measurements previously reported in the plant physiology, biogeosciences and atmospheric sciences literature. The examples span the range of nitrogen source scenarios considered above (i.e. including situations where exchange is dominated by volatilization, uptake, or mixing).
Example 1: Comparisons of the δ15N of leaves from a wide variety of species experiencing ambient and doubled CO2(g) in free-air carbon enrichment (FACE) experiments have indicated that leaves are relatively more depleted under doubled relative to ambient CO2(g) (BassiriRad et al. 2003). The depletion has a mean of around 1‰, and to date, the mechanism has not been clear. In the model developed here, simulating the suppression of photorespiration by the FACE CO2(g) increase from 350 to 550 ppm leads to a depletion of organic nitrogen that varies from 0–1‰, depending on the leaf temperature and stomatal conductance. For a constant leaf temperature of 25 °C and stomatal conductance of 0.5 mol m−2 s−1, the predicted depletion is 0.3‰. Species-specific responses to elevated CO2(g) (e.g. in stomatal conductance, leaf temperature, nitrogen import, degree of nitrate storage in vacuoles and/or recycling of protein) could further deplete or enrich foliar nitrogen relative to this value. Therefore, it seems likely that this mechanism contributes to, although is not wholly responsible for, the shift in δ15N seen in FACE experiments.
Example 2: A recent analysis of δ15N in C3 and C4 grasses growing in arid environments across Australia indicates that δ15N is enriched by 0−4.4‰ in C3 relative to C4 species, with the magnitude of the enrichment increasing with aridity (Murphy & Bowman 2009). Co-occurring C3 and C4 grasses would be expected to differ in total photorespiratory flux, stomatal conductance, leaf temperature, nitrogen import rate and nitrogen partitioning, each of which would influence the net fractionation. In the model developed here, we simulated the suppression of photorespiration in a C4 leaf by increasing CO2(g) from 350 to 2000 ppm. This leads to a depletion of organic nitrogen that varies from 0 to 4.8‰, depending on the leaf temperature and stomatal conductance. For a constant leaf temperature of 25 °C and stomatal conductance of 0.5 mol m−2 s−1, the predicted depletion is 1.0‰. These results indicate that leaf-atmosphere NH3(g) exchange could strongly contribute to, or be primarily responsible for, the shift in δ15N between co-occurring C3 and C4 grasses.
Example 3: Tozer et al. (2005) measured δ15N in atmospheric NH3(g), rainfall and the thallus of several lichens as well as the terrestrial algae Trentepohlia growing close to natural and anthropogenic sources of NH3(g). While atmospheric and rainfall δ15N fell in the range of −1 to −8‰, lichen and Trentepohlia δ15N averaged −20‰, that is a net fractionation on the order of −15.5‰. In the model developed here, we simulated diffusive uptake of NH3(g) from atmospheric reservoirs ranging from 20 to 80 ppb, the range of concentrations reported by Tozer et al. (2005). The model predicts that the observed net fractionation would occur at a thallus temperature between 13 °C (at 20 ppb) and 20 °C (at 80 ppb). Since lichens and green algae lack stomata, diffusive uptake across the thallus could take place during the night or day, possibly accounting for the relatively cool temperatures inferred for uptake. In this case, the model clearly confirms that the observations are consistent with a source and sink connected by diffusive uptake of NH3(g).
Example 4: Fogel et al. (2008) measured highly depleted δ15N in the leaves of Rhizopora mangle (range: −21.6 to 4.0‰) on an island where atmospheric NH3(g), rainfall and porewater δ15N averaged −18.5, −9.6 and 5.3‰, respectively. If we simplify the mixing problem by omitting potential contributions from rainfall , we can use the other constraints to estimate the relative contributions of porewater and atmospheric NH3(g) to the trees' foliar nitrogen budget. For example, if all of the nitrogen in the most depleted R. mangle leaves, at δ15N = −21.6‰, had been derived from atmospheric NH3(g) at −18.5‰, then the net fractionation on uptake would have been −3.1‰. The model predicts that this magnitude fractionation would be consistent with a leaf temperature of 25 °C, stomatal conductance of 0.5 mol m−2 s−1, and source reservoir concentration of 17 ppb NH3(g). Alternatively, the same foliar nitrogen isotope signature could be consistent with a smaller proportion of NH3(g)-derived nitrogen that was acquired with a stronger fractionation on uptake, for example, due to a higher-concentration atmospheric reservoir.
Example 5: Ehleringer et al. (2010) measured carbon and nitrogen concentrations and stable isotope ratios in leaves from tropical forest canopy profiles in Brazil. In this dataset, foliar δ15N and δ13C become increasingly enriched with height in the canopy. Based on δ15N alone, this pattern could be generated by increases in either leaf temperature or stomatal conductance from the understory to the top of the canopy (Supporting Information Fig. S1). However, the paired measurements of foliar δ13C provide an additional constraint that can help differentiate between these possibilities. The δ13C pattern is most likely driven by two mechanisms: one, declines in the Ci/Ca ratio with height which shift control of foliar δ13C from the large isotope effect associated with Rubisco towards the smaller isotope effect associated with CO2 diffusion; and two, declines in the refixation of respired CO2 with height (Supporting Information Fig. S1). Based on the Buchmann et al. (1997) estimate that about 80% of the δ13C gradient within a forest canopy in French Guiana was due to changes in Ci/Ca and 20% due to source air effects, we calculated the total discrimination against 13C in the Ehleringer et al. (2010) data set using the expression:
where Δ13C is the leaf-level discrimination dependent on Ci/Ca; δ13Ctroposphere is the isotopic composition of CO2 in the troposphere, estimated to have been −8‰ when these samples were collected; and δ13Cleaf are the measured foliar values. The parameter c is 1.6‰, which is equal to 20% of the average measured 0 to 40 m δ13Cleaf gradient in the data set (8.0‰). The parameters h and hmax represent the height of a given leaf and the maximum height of the canopy, respectively. Assuming that foliar nitrogen sources did not vary systematically with canopy position, we estimated the total discrimination against 15N and regressed Δ13C against Δ15N using geometric mean regression. The Δ13C was significantly positively correlated with Δ15N (Δ15N = β * Δ13C + α: β = 0.71, α = −22.7, t = 666.8, d.f. = 685, P < 0.001). Thus, the calculated δ13C gradient from the understory to the top of the 40 m tall canopy could explain 4.8‰ enrichment in δ15N, approximately 40% of the total range of δ15N measurements (Supporting Information Fig. S2). This relationship is consistent with a single environmental driver that varies with height and influences both of the isotopes, such as a leaf temperature gradient. Increasing leaf temperatures with height could cause volatilization and decreases in Δ15N, while simultaneously reducing the relative humidity, inducing stomatal closure, lowering Ci/Ca, and decreasing Δ13C. In general, however, the independence of controls on Δ15N and Δ13C implies that these two isotope signatures should not always be expected to co-vary positively in leaf samples. In existing data sets, variation in the magnitude and direction of correlation between foliar carbon and nitrogen isotope signatures is the rule rather than the exception. Indeed, in the data set of Fogel et al. (2008) discussed above, Δ15N and Δ13C were negatively, rather than positively, correlated, in all likelihood reflecting different mechanisms contributing to the observed patterns. An approach that utilizes the model presented here in combination with the model of Farquhar et al. (1982) may have increased power for constraining candidate interpretations of mechanisms responsible for variation in foliar Δ15N as well as foliar Δ13C.
Example 6: While the isotopic composition of plant organic nitrogen is of interest to the plant physiologists, the isotopic composition of ammonia in the atmosphere is also of interest to the atmospheric sciences and biogeosciences communities. While ammonia is aloft, it is a key player in tropospheric chemistry (Dentener & Crutzen 1994); when deposited to the land or ocean surface, ammonia modulates the dynamics of the terrestrial and marine nitrogen cycles (Galloway et al. 2008). Measurements of the isotopic composition of gaseous ammonia are sparse, and a great deal of variability occurs over time and between sites. In unpolluted areas, there is a clear trend for depletion of δ15N in atmospheric ammonia as well as ammonium and nitrate deposited in precipitation (Wada, Shibata & Torii 1981; Freyer 1991; Garten 1992; Heaton et al. 1997). In contrast, ammonia, ammonium and nitrate resulting from urban and agricultural emissions tend to be somewhat more enriched in δ15N (Elliott et al. 2007). The depletion and high variability in ammonia δ15N in unpolluted areas have typically been ascribed to volatilization from soils that generates isotopically depleted ammonia (Vitousek, Shearer & Kohl 1989; Austin & Vitousek 1998) as well as precipitation events that scavenge isotopically enriched ammonium and deposit it to the land surface, leaving residual, isotopically depleted ammonia aloft (Moore 1974, 1977). Based on the analysis above, exchange with plant canopies can also generate gaseous ammonia with highly variable but generally depleted δ15N (Figs 5, 6, 7b,d,f).
Overall, the model predictions are consistent with a range of measurements in the literature of the plant sciences, biogeosciences and atmospheric sciences. In some cases, quantifying the expected influence of leaf-atmosphere NH3(g) exchange provides a satisfying interpretation for an enigmatic δ15N pattern, or convincingly winnows down multiple candidate interpretations for processes controlling δ15N. However, in other cases, it simply adds yet another plausible mechanism to an already underdetermined problem. Therefore, for the model to be applied in a precise, quantitative manner to studies of δ15N, two conditions must be met. First, further experimental tests are needed to resolve questions about the physiological processes controlling leaf-atmosphere NH3 exchange and the magnitude of isotope effects within nitrogen metabolism. Second, a quantitative understanding of the pools and fluxes involved in nitrogen metabolism should be developed for the system in question, and the model structure and/or parameterization adapted accordingly. If both of these conditions are met, then the model may yield further insights into a range of natural abundance and tracer δ15N patterns.
With respect to plants, the model may be useful for studying nitrogen nutrition and metabolism, accounting for surplus and/or deficit in nitrogen budgets, and interpreting naturally occurring patterns of variation in foliar nitrogen isotope composition. With respect to the atmosphere, the model may be useful for teasing apart the sources and sinks for tropospheric ammonia, studying the atmospheric reactive nitrogen cycle, and interpreting environmental records of the isotopic composition of nitrogen-containing atmospheric gases (e.g. N2O) and particles (e.g. and aerosols). In addition, the matrix-based modelling approach that is introduced may be useful for quantifying isotope dynamics in other complex systems, especially when the classical approach for determining equilibrium solutions becomes unwieldy or dynamics are of particular interest.
C. Field, D. Menge, M. O'Leary, P. Vitousek and W. Watt provided constructive guidance during model development. We also thank G. Farquhar, A. Kornfeld and two anonymous reviewers for valuable suggestions on earlier versions of the manuscript. NH3 measurements were supported by grants from the National Science Foundation (Award #1040106 to C.B.F and J.A.B.) and the Air Liquide Foundation (to J.A.B.). Funding for J.E.J. was provided by a Graduate Research Fellowship from the National Science Foundation and the Bing-Mooney Fellowship in Environmental Science from Stanford University.
Appendix: Appendix I: Isotope notation
In general, the isotope literature is characterized by variable terminology. While recent efforts have been made to standardize notation, the new standards are not consistent with any single one of the conventions that have been used historically in plant physiology or ecology. Thus, some discussion of the past conventions and the terminology used in this analysis is necessary.
A variety of analytical techniques are now used to measure isotope composition, from the standard isotope ratio mass spectrometers to the more recently introduced optical spectrometers. All retrieve a measurement of the isotope ratio, R:
In plant physiology and ecology, this is commonly expressed as a δ15N value, relative to a recognized standard:
Rstandard for nitrogen isotopes is the isotopic composition of atmospheric N2(g), with isotope ratio 15N/14N = 0.003677 (Mariotti 1983). Organic nitrogen in plants can be either enriched or depleted in 15N relative to the atmospheric N2(g) standard. Positive δ15N values mean that a sample is richer in 15N (‘heavier’); negative δ15N values mean that it is poorer in 15N (‘lighter’). While the isotope delta is a dimensionless quantity, δ15N values are small and therefore are typically reported in parts per thousand (i.e. per mille or ‰).
Differences in the δ15N value between a source and product indicate that the transformation is accompanied by an isotope effect that partially separates, or fractionates, the isotopes. An isotope effect is defined as the alteration of the rate constant of a reaction when an atom in a reactant molecule is replaced by one of its isotopes. Isotope effects are often classified as being either kinetic or thermodynamic, which represents a distinction between non-equilibrium and equilibrium situations. Kinetic isotope effects occur because different isotopic species are transformed at different rates, due to their mass differences; thermodynamic isotope effects represent the balance of two kinetic isotope effects for processes at equilibrium.
A number of different conventions have been used to quantify isotope effects and the isotopic fractionation that results from them. One convention is that an ‘isotope effect’ is synonymous with an ‘isotopic fractionation factor’ (α) and is defined by the ratio of rate constants for reactions involving 14N and 15N, respectively:
While much of the geochemical literature follows an alternative convention in which α = 15k/14k, the notation in Eqn. (8) has been widely used in plant physiology and we adopt it here. Using Eqn. (10), consider a given transformation, A B, where k1 is the rate constant for A → B, and k2 is the rate constant for B → A. Here, the kinetic isotope effect associated with A → B is given by the ratio of rate constants for reactions involving 14N and 15N, respectively: α1 = 14k1/15k1. The kinetic isotope effect associated with B → A is α2 = 14k2/15k2. When A B is at equilibrium, the thermodynamic isotope effect is:
Two concepts follow from this derivation: first, if the thermodynamic isotope effect and the kinetic isotope effect in one direction are known, it is possible to calculate the kinetic isotope effect in the other direction (O'Leary 1981). Second, if the concentration of the source is large enough that the isotopic composition of the source reservoir is insignificantly altered by the reaction (O'Leary 1981) or if the isotopic composition of the product is measured within an infinitely short time period (Mariotti et al. 1981), then a kinetic isotope effect describes the offset between the isotopic composition of the source and product:
However, if the isotopic composition of the source reservoir is significantly altered by the reaction, then the offset between the isotopic composition of the residual source and the cumulative product varies based on the fraction, f, of the original source that has reacted, as well as 14k/15k. Expressions describing this quantitatively were derived by Bigeleisen & Wolfsberg (1958).
As with the isotope effect, multiple conventions have been used to describe the ‘isotopic fractionation,’ or the change in the distribution of isotopes between substances that is caused by an isotope effect. In the plant physiology literature, the tendency has been to use the term ‘discrimination’ as synonymous with ‘isotopic fractionation.’ O'Leary (1981) defined discrimination (Δ) as:
For numerical convenience, Farquhar & Richards (1984) proposed an alternative expression for discrimination shortly thereafter:
For values of α < 1.050 encountered with the nitrogen isotopes, the two expressions are approximately equal, yielding Δ within about 2‰. The latter expression is dominant in the plant physiological literature today, and can be interpreted as follows. When the light isotope reacts more rapidly than the heavy one, the isotope effect calculated as α = 14k/15k is greater than unity, and the corresponding value for discrimination (Δ) is positive, indicating ‘more discrimination against the source’ and ‘more depletion of the product.’ Accordingly, the equation for discrimination is linked to the isotopic composition of sources and products by the expression given by Farquhar & Richards (1984):
Since the denominator is typically quite close to 1, the expression is often simplified to:
In contrast, in the geochemistry literature, the alternative α = 15k/14k has lent itself to a different way of describing isotopic fractionation. Here, the term ‘enrichment factor,’ ε, is used to describe the quantity α − 1 = (15k/14k) − 1. In turn, the enrichment factor is linked to the isotopic composition of sources and products by:
In this approach, more positive values of ε indicate ‘more enrichment of the product.’ This alternative convention has been adopted by a number of authors working with nitrogen isotopes in the context of plant and microbial physiology (Mariotti et al. 1981; Fogel & Cifuentes 1993; Hayes 2001).
The geochemical convention for describing isotopic fractionation is intuitively satisfying insofar as positive values of ε indicate enrichment of the product, whereas negative values indicate depletion. In contrast, the physiological convention assigns positive values of Δ to indicate depletion of the product, and negative values to indicate enrichment. While this quirk facilitates dealing with carbon isotope fractionation during photosynthesis (i.e. where the product is always depleted), it somewhat complicates analyses of other fractionating processes, in other isotope systems.
Nevertheless, we follow the conventions that plant physiologists have used for the isotope effect (α = 14k/15k) and for isotopic fractionation (Δ = α−1). We hope that this maximizes the clarity of the present analysis for readers who are already familiar with the physiological treatments of carbon isotope discrimination during photosynthesis (Farquhar et al. 1982; Farquhar 1983; Farquhar, Ehleringer & Hubick 1989).
Appendix: Appendix II: Solution methods
Solution via substitution
At steady state, the flux of protein generated through net biosynthesis must be equal to the sum of the inorganic nitrogen imported and the net atmospheric exchange:
Conservation of mass requires that 14N and 15N each must be conserved:
The quantities that are actually measured experimentally (15k = 14k/α and 15P = 14P · R) can be substituted into the previous equation:
Through substitution, we can write an expression for the isotopic composition of NH3(g)(i), R9, in terms of the isotopic composition of the sources (R1, R10), and the intervening rate constants and fractionations:
By substituting this equation into the previous one, rearranging, and assuming that α7 = 1, we arrive at an expression for the isotopic composition of glutamate (R5), which is equal to the isotopic composition of protein (R6):
Thus, once fractionations are accounted for, the isotopic composition of organic nitrogen assimilated by the plant is equal to the flux-weighted average of inputs (from the soil and the atmosphere), less losses (to the atmosphere).
Solution via matrix algebra
Alternatively, the isotope system can be solved using matrix algebra. The two approaches give analytically identical equilibrium solutions. However, the matrix-based approach also makes it possible to solve for the complete dynamics of the system. In this approach, d(Pi)/dt is not set to zero (as above), but is written as a linear combination of the fluxes contributing material to, and removing material from, Pi:
Here, is a column vector with i rows representing the changes in pool size for the entire system, P is a column vector with i rows representing pool sizes at any time t, and B is a column vector with i rows representing constant inputs at every time t, i.e.
A is a square matrix with dimensions i × j. The matrix is populated by rate constants k(i,j) such that each element of is equal to the product of the corresponding row of A and the vector P, plus the corresponding element of B. For example, if the expressions that describe the change in pool sizes are:
then the corresponding matrix of rate constants must be:
When , that is, at equilibrium, the equilibrium pool sizes () are given by:
where A−1 represents the inverse of A (Brannan & Boyce 2010; Kreyszig 2011). By repeating the above procedure for each isotope of interest, it is possible to solve for the isotopic composition of the pools at equilibrium. For 14N and 15N, we use the isotopic composition of the source pools, Rsource = 15N/14N, to obtain the 15N pool sizes in B, and introduce the fractionation factors α(i,j) = 14k(i,j)/15k(i,j) to the 15N matrix Ah. The subscripts ‘l’ and ‘h’ differentiate between arrays corresponding to the light and heavy isotopes, respectively. By applying element-wise division to the pool size vectors, we solve for the equilibrium isotopic composition of all of the pools:
If the dynamics of pool size and isotopic composition are of interest, this notation can be extended to derive an expression for the complete solution to the system. The key to the complete solution is the eigenvalue decomposition of A. For any normal (i.e. non-defective) square matrix A, eigenvalues and eigenvectors can be computed efficiently using any number of programming environments (e.g. MATLAB, R, etc.). Given the eigenvalues and eigenvectors of A, a general solution can be expressed for P (t) based on the principle that when V is an eigenvector of A, then AV = λV, where λ is an eigenvalue of A (Trefethen & Bau 1997; Banerjee 2005; Brannan & Boyce 2010; Kreyszig 2011).
The eigenvalues provide insight into the timescale on which equilibrium is reached. Following a perturbation, mass equilibrium will be approached much more quickly than isotopic equilibrium. The time that is required for a system to return to a steady-state is governed by the most slowly decaying eigenvalue (Banerjee 2005). In a system comprised of normal isotope effects, the most slowly decaying eigenvalue will be controlled by the heavier isotope, that is, here by 15N. However, in a system comprised of inverse isotope effects, the most slowly decaying eigenvalue will be controlled by the lighter isotope, that is, here by 14N. In any situation, the approach to isotopic equilibrium can be quantified using the response time (τ, also called the e-folding or relaxation time), which is calculated by taking the inverse of the absolute value of the smallest non-zero eigenvalue:
Comparison of the two approaches
In some respects, the two solution processes are complementary alternatives. For relatively small systems of equations, solution of the equilibrium via substitution can lead to mechanistic insights that may be somewhat obscured by the matrix algebra-based solutions. For example, several insights into factors that influence the net fractionation can be gleaned from Eqn. (22). The rate constants for photorespiration (k8) and nitrogen export (k7) affect the isotopic composition of NH3(g) efflux from the leaf. Temperature will also have direct effects on the net fractionation, because it controls the solubility of NH3(g) in solution, and therefore k13/k12. In addition, the pH of the cell controls k2/k3; its effects on the size of the NH3(aq) pool influence the magnitude of NH3(g) uptake (F15) and loss (F14). These insights are not as intuitive in the mathematically equivalent matrix-based expressions, Eqn. (29) and Eqn. (30).
Nonetheless, the matrix-based approach does have several distinct advantages. For larger and more complex systems, the matrix-based approach significantly simplifies the process of obtaining an equilibrium solution. For any size system, the matrix-based approach provides insight into the characteristic response time and therefore, the timescale on which equilibrium is achieved. Such understanding can be applied to identify whether transient or steady-state dynamics are most biologically (or ecologically) relevant. In addition, the matrix-based computations tend to be fast, efficient and readily reproducible. Finally, while the method described here applies only to systems that can be adequately described by first-order or pseudo-first-order rate equations, it is often possible to reduce systems described by more complex equations to a linear form which is tractable with matrix algebra.