the impact of metabolism on stable isotope dynamics: a...

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Phil. Trans. R. Soc. B (2010) 365, 3455–3468

doi:10.1098/rstb.2010.0097

The impact of metabolism on stable isotopedynamics: a theoretical framework

Laure Pecquerie1,2,*, Roger M. Nisbet1, Ronan Fablet3,4,

Anne Lorrain5 and Sebastiaan A. L. M. Kooijman6

1Ecology, Evolution and Marine Biology Department, University of California Santa Barbara,Santa Barbara, CA 93106-9620, USA

2IRD, UMR 212 EME, Centre de Recherche Halieutique Mediterraneenne et Tropicale, Av. Jean Monnet,BP 171, 34203 Sete cedex, France

3Telecom Bretagne/LabSTICC, Technopole Brest-Iroise – CS 83818, 29238 Brest Cedex 3, France4Universite Europeenne de Bretagne, 5 Bd Laennec, 35000 Rennes, France

5IRD/LEMAR, Centre IRD de Brest, BP 70, 29280 Plouzane, France6Vrije Universiteit, Department of Theoretical Biology, de Boelelaan 1087, 1081,

Amsterdam, The Netherlands

Stable isotope analysis is a powerful tool used for reconstructing individual life histories, identifyingfood-web structures and tracking flow of elemental matter through ecosystems. The mechanismsdetermining isotopic incorporation rates and discrimination factors are, however, poorly understoodwhich hinders a reliable interpretation of field data when no experimental data are available. Here,we extend dynamic energy budget (DEB) theory with a limited set of new assumptions and rules inorder to study the impact of metabolism on stable isotope dynamics in a mechanistic way. We cal-culate fluxes of stable isotopes within an organism by following fluxes of molecules involved in alimited number of macrochemical reactions: assimilation, growth but also structure turnover thatis here explicitly treated. Two mechanisms are involved in the discrimination of isotopes: (i) selec-tion of molecules occurs at the partitioning of assimilation, growth and turnover into anabolic andcatabolic sub-fluxes and (ii) reshuffling of atoms occurs during transformations. Such a frameworkallows for isotopic routing which is known as a key, but poorly studied, mechanism. As DEB theoryspecifies the impact of environmental conditions and individual state on molecule fluxes, we discusshow scenario analysis within this framework could help reveal common mechanisms across taxa.

Keywords: stable isotope ratios; discrimination mechanisms; dynamic energy budget theory;metabolism; reshuffling; molecule selection

1. INTRODUCTIONIn ecological studies, stable isotope analysis (SIA) hasbeen successfully applied to reconstruct diet andmigration patterns of organisms, identify food-webstructures and track flows of elemental matter withinan ecosystem (Fry 2006). It relies on the existence ofchemical elements with two or more stable isotopesthat are unevenly distributed among compounds orcompartments. Heavy isotopes of the commonestelements in the biosphere (13C, 2H, 18O, 15N) arerare, representing less than 1 per cent of the carbon,hydrogen, oxygen and nitrogen on earth. Yet there arewell-documented, though small, variations in their pro-portions: e.g. leaves are typically depleted in 13Ccompared to atmospheric CO2 (Farquhar et al. 1989).

The observation that animal tissues reflect the isoto-pic composition of their diet but are typically enriched

r for correspondence ([email protected]).

ic supplementary material is available at http://dx.doi.org//rstb.2010.0097 or via http://rstb.royalsocietypublishing.org.

tribution of 14 to a Theme Issue ‘Developments in dynamicudget theory and its applications’.

3455

in 13C and 15N (DeNiro & Epstein 1978, 1981) iskey to most SIA applications in ecology. However, theunderlying mechanisms are still poorly understood(Martınez del Rio et al. 2009). A large number offactors, all relating to metabolism, could explain theoverall trophic increase in rare isotopes in animal tis-sues: nutritional status, body size, diet quality,assimilation efficiency, excretion form, protein turnoverrates (Minagawa & Wada 1984; Adams & Sterner 2000;Vanderklift & Ponsard 2003; Barnes et al. 2007). Sev-eral models have been developed to investigate someof these factors (e.g. Ponsard & Averbuch 1999;Harvey et al. 2002; Olive et al. 2003; Marin-Leal et al.2008). But the number of processes involved and thevariability of observed patterns led Wolf et al. (2009)to renew their call both for more experiments and thedevelopment of theoretical models.

In the present study, we develop mechanistic theorythat relates metabolism to isotope fluxes within anorganism. Isotopes of atoms are embedded into mol-ecules. To mechanistically understand isotopedynamics, we need to follow the fate of moleculeswithin an organism and how atoms are redistributedamong molecules during chemical transformations.

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macrochemical reaction: S1 + S2 → P1 + P2

1. mobilization

S1

P1

P1

P2

P1

S2

2. selection 3. reshuffling

energy source

energy

energy source

proportion of heavy isotopes- +

catabolicroute

catabolicroute

building blocks

building blocks

anabolicroutes

Figure 1. Proposed mechanisms for stable isotope discrimination in a transformation: the transformation of one generalized

compound into other compounds occurs in several steps. Mobilization does not modify the isotopic composition of the remain-ing compound. Discrimination occurs during (i) selection (circled cross symbol) of molecules according to their allocation tothe catabolic (energy) and the anabolic routes (building blocks) and (ii) the reshuffling of atoms during the chemical reaction,if there is more than one substrate. If only one substrate is involved in the macrochemical reaction, then the products have the

same isotopic composition as the substrate after selection: product 1 (P1) has the same isotopic composition as substrate 1 (S1)in the catabolic route. If two or more substrates and two or more products are involved in the reaction, then the isotopic ratio ofe.g. P2 depends on the fraction of atoms that originate from S1 and S2. Here, most of S2 atoms went to P2. Colours describethe proportion of heavy isotopes.

3456 L. Pecquerie et al. Stable isotope dynamics in an organism


Previous approaches made the implicit assumption thatmolecules during transformations were disassembledinto their elemental components (Martınez del Rioet al. 2009) and that fluxes of elements could be studiedindependently. We aim at developing a theoreticalframework that relaxes this assumption.

We work within the framework of dynamic energybudget (DEB) theory (Kooijman 2010), which pro-vides us with a framework that already specifies theimpact of metabolism on molecule fluxes. DEBtheory defines metabolism as a set of three chemicaltransformations—assimilation, growth and dissipa-tion—where substrate molecules are transformed intoproduct molecules. These transformations fully specifyall mass fluxes within an organism, and between anorganism and its environment. To follow the isotopiccomposition of these fluxes, new developments arenonetheless required. The turnover process, criticalto understand isotopic incorporation rates (Martınezdel Rio et al. 2009), is implicit in the standard DEBframework and needs careful attention. Mechanismsfor discrimination, i.e. change in isotopic composition,also need to be defined.

This paper first presents the extensions required tointroduce stable isotopes within the DEB frameworkand the proposed mechanisms for isotopic discrimi-nation in this framework. Second, we describe theresulting dynamic isotope budget (DIB) model. Wethen illustrate with numerical simulations the impactof body length and nutritional status on the isotopiccomposition of organisms experiencing constant andfluctuating isotopic composition of their food. Finally,we discuss how the framework can be used to explain

Phil. Trans. R. Soc. B (2010)

the observed variability in isotopic incorporation ratesand discrimination factors among individuals andamong species.

2. INTRODUCING STABLE ISOTOPE DYNAMICSWITHIN DEB THEORYDEB theory provides a conceptual and quantitativeframework for studying isotopic incorporation rateswithin an organism and discrimination mechanisms.To this end, the standard DEB model (Sousa et al.2010) must be extended with a set of consistentassumptions at a more detailed level of organization.These assumptions should specify how isotopes ofatoms in molecules travel through the organism‘jumping’ from one molecule to another. In thissection, we summarize the existing DEB framework(see table 1) and introduce the two DEB principlesdeveloped to study stable isotope dynamics: turnoverof structure and recognition of anabolic and catabolicroutes in transformations. We then detail themechanisms for isotopic discrimination (figure 1).

(a) Standard DEB framework

DEB theory describes the rates at which an organismassimilates and uses energy for maintenance, growthand reproduction as a function of its state and itsenvironment (i.e. food density and temperature;Kooijman 2010; Sousa et al. 2010). Each metabolicfunction defines a chemical transformation, which iskey for a mechanistic understanding of stable isotopedynamics. These transformations involve four organicgeneralized compounds with constant elemental


proportion of heavy isotopes

structure(t + dt)

structure(t) dVG

dVLr

dVLr

dVL

dVL – +

+

Figure 2. Representation of the processes that impact the isotopic composition of structure in a time step (dt), (i) growth

(standard DEB model), (ii) turnover and (iii) recycling, with dVG, dVL and dVLr the amounts of structure producedduring these processes, respectively.

Table 1. DEB assumptions and definitions on compounds,

transformations and fluxes within an organism, as relevantfor stable isotope dynamics.

DEB theory delineates four organic generalizedcompounds: food X, reserve E, structure V and faeces P.The biomass of an individual is composed of reserve Eand structure V.

The elemental composition of the organic compounds isspecies-specific, but does not change throughout the lifeof the organism (strong homeostasis assumption).

The number of elements per compound is the same for all

organic compounds. The standard DEB model deals withthe four most abundant elements in living organisms:carbon C, hydrogen H, oxygen O and nitrogen N.

Mass of compounds is given in C-moles, i.e. the amount ofeach element is expressed relative to the amount of

carbon in this compound. Compound formula is thusgiven by CHnHiOnOiNnNi, with nxi the proportion ofatoms of element x(¼ H, O, N) relative to carbon incompound i(¼ X, E, V, P).

The number of elements per compound fixes (and is equalto) the number of mineral compounds in eachtransformation. Hence, the mineral compounds that areassociated with each transformation are: carbon dioxideC (CO2), water H (H2O), dioxygen O (O2) and N-waste

N (NH3, urea, etc.)A transformation describes an irreversible conversion of

substrates into products and is represented by amacrochemical reaction equation for which stoichiometryprinciples apply.

Fluxes of compounds can participate in three types ofmetabolic transformations only: assimilation, growth anddissipation.

Assimilation is the transformation of two substrates, food Xand dioxygen O, into five products, reserve E, carbon

dioxide C, water H, N-waste N and faeces P.Growth is the transformation of two substrates, reserve E

and dioxygen O, into four products, structure V, carbondioxide C, water H and N-waste N.

Dissipation encompasses the transformations of twosubstrates, reserve E and dioxygen O, into three products,carbon dioxide C, water H and N-waste N.

L. Pecquerie et al. Stable isotope dynamics in an organism 3457


composition: food, reserve, structure and faeces, andfour mineral compounds: carbon dioxide, water,dioxygen and N-waste (NH3, urea, etc.; Kooijman2010, pp. 83–89). Three transformations are definedin the standard DEB framework:

— assimilation: the conversion of food to reserve andproducts (including faeces),

— growth: the conversion of reserve to structure andproducts,

— dissipation: the conversion of reserve to products.

The products leave the organism, and all three trans-formations require (environmental) dioxygen assubstrate, which is here supposed to be non-limiting.The three transformations can be presented in theform of macrochemical reaction equations (Kooijman2010, pp. 94–95), where we follow four elements (C,H, O and N). These macrochemical reactionequations specify the appearance and disappearanceof all compounds. However, they do not provide thedestination of each atom and new developments arerequired to follow atoms and isotopes of these atomsin transformations.

(b) New requirements within the DEB

framework

Although DEB theory provides us with a detailed fra-mework to study mass fluxes, new developments arerequired to follow isotope fluxes. In particular, weneed to carefully specify one transformation that iskey to understand isotopic incorporation rates: theturnover of structure. In order to study isotopic dis-crimination, we also need to specify the anabolic andthe catabolic routes of each transformation for whichmolecules can be selected.

(i) Turnover of structureIn order to study the changes in the isotopic compo-sition of structure, we need to define the turnover ofstructure. Unlike reserve, structure requires mainten-ance. This process is a significant part of thevolume-specific somatic maintenance processes. It is,however, not explicitly described in the standard


DEB model as there is no net production of structure:the incoming flux of renewed structure is compensatedby the outgoing flux of degraded structure. However,




these two fluxes may have different isotopic compo-sition (figure 2). Thus, we need to quantify thesefluxes and their isotopic composition. We furtherassume that part of the degraded structure can berecycled to form renewed structure (figure 2). Forexample, non-functional proteins could be brokendown into amino acids that are reused to form newfunctional proteins.

(ii) Catabolic/anabolic routes of a transformationWe detail the catabolic and anabolic routes of eachchemical transformation to allow selection of substratemolecules depending on their isotopic composition(figure 1). The catabolic route of a transformationuses substrates to produce energy for the anabolic routeof the transformation. The anabolic route of a trans-formation uses this energy and substrates as a source ofbuilding blocks to produce a given compound. In thegrowth transformation for instance, reserve is used asa source of energy as well as building blocks toproduce structure.

(c) Discrimination mechanisms during

transformations

The standard DEB framework and the new develop-ments presented in the previous section define aframework in which mass fluxes within an organismare carefully specified. This section presents ourassumptions for the change in isotopic compositionof these fluxes, i.e. discrimination. We define threesteps in a transformation: mobilization, selection ofmolecules for anabolic and catabolic routes andreshuffling of atoms (figure 1). We assume that onlythe two last steps of a transformation could result ina change in isotopic composition.

H1. Mobilization. Mobilization of a compound from apooldoesnotmodify the isotopiccompositionof thispool.

This assumption follows from the strong homeo-stasis assumption. For example, if we think of twolarge starch molecules, one that contains one 13Cand one with only 12C, we can reasonably assumethat their mass difference is not sufficiently differentto allow selection.

H2. Molecule selection. Mobilized molecules havedifferent probabilities to be selected for the anabolicor the catabolic route of a transformation dependingon their isotopic composition (figure 1). Thenumber of molecules with one rare isotope in theanabolic route of a transformation is given by themean of a Fisher’s noncentral hypergeometricdistribution.

At this step, we assume that molecules can beselected according to their isotopic composition andthat the probability of selecting one type of moleculefor a given metabolic route depends on its fate. Forexample, once mobilized and monomerized, starchbecomes available as small molecules of glucose; amolecule of glucose that contains a 13C has a signifi-cant different weight compared with a molecule ofglucose with only 12C. A possible mechanism forselecting molecules with only light isotopes for thecatabolic route for instance could be that light isotopes


have weaker binding and could be more easilybroken down.

Formally, we cannot select molecules in a flux toseparate the flux into two sub-fluxes (the anabolicand catabolic). We can only select from a pool, andconvert the flux to an infinitesimally small pool bymultiplication with an infinitesimally small timeincrement dt. On the assumption that selection is atrandom, and that molecules with a heavy isotopehave a deviating probability to be selected, thenumber of molecules with heavy isotopes followsFisher’s noncentral hypergeometric distribution(McCullagh & Nelder 1989). We choose to refrainfrom a full stochastic formulation of the selectionprocess and use the mean value of this distributionto follow the fate of isotopes. In particular, we sup-pose that selection occurs at the separation of thecatabolic and the anabolic fluxes. Further detailscan be found in Kooijman (2010, pp. 97–100).

H3. Atom reshuffling. Allocation of atoms of asubstrate molecule to products is not random.

The concept of atom reshuffling recognizes thatmolecules are not completely disassembled intoelements during chemical reactions. To illustrate whyatom reshuffling might impact stable isotope ratios,we use the photosynthesis example: CO2 þH2O!CH2O þ O2. The atoms of O2 produced duringphotosynthesis for instance all originate from water,implying that the isotopic composition of O2 dependsonly on the isotopic composition of the water, with theoxygen isotopic composition of CO2 being irrelevant.To further illustrate this effect of atom reshuffling ofsubstrates on the isotopic composition of a product,we detail the fate of the atoms from substrates to pro-ducts in a transformation involving two substrates andtwo products: S1 þ S2! P1 þ P2 (figure 1). We pre-sent the theory for the general case of a transformationwith multiple substrates and products in appendix Cand further details can be found in Kooijman (2010,pp. 96–97). The reshuffling concept is particularlyrelevant to discrimination in assimilation in caseswhere the reserve and structure of food aredistinguished or where two types of food areconsidered.

3. DIB MODELThe DIB model describes the changes in reserve andstructure of an organism, the amount of organic andmineral compounds exchanged with the environment,and the isotopic compositions of these different com-pounds. These changes depend on the state of theorganism and the environmental conditions it experi-enced: food density, temperature and isotopiccomposition of its food. The standard DEB modeldetermines the different mass fluxes involved in thesechanges. Our contribution specifies the proportionsof isotopes in these fluxes.

(a) Standard DEB model

The full description of the standard DEB model, itsnotation, and the scheme of the different fluxes canbe found in Sousa et al. (2010). An individual isdescribed by two state variables: the reserve mass ME


Table 2. State and forcing variables, parameter values and standard DEB equations for the dynamics of reserve and

structure. Parameters are taken from Kooijman (2010) for a generalized animal with organic compounds CH1.8O0.5N0.2.Rates are given at the reference temperature T1 ¼ 293 K (¼208C).

variables interpretation

ME reserve mass (mol)MV structural mass (mol)e ¼MEv/(L3fJEAmg) scaled reserve densityL ¼ (MV/[MV])1/3 volumetric length (cm)Lf ¼ L/dM length (cm)

X food density (mmol l23)T temperature (K)f(X ) ¼ X/(X þ K) scaled functional responsec(T ) ¼ exp(TA/T1 2 TA/T ) temperature correction

parameters value interpretation

TA 8000 Arrhenius temperature (K)K 0.0159 saturation coefficient (mmol l23)fJEAmg 0.0826 maximum surface-area-specific assimilation rate (mmol cm22 d21)[JEM] 0.033 volume-specific somatic maintenance rate (mmol cm23 d21)

[MV] 4 volume-specific structural mass (mmol cm23)v 0.02 energy conductance (cm d21)k 0.8 fraction of used reserve to growth þmaintenanceyVE 0.8 yield of structure from reserve in growthLm ¼ kfJEAmg/[ JEM] 2 maximum volumetric length (cm)

dM 0.2 shape coefficientg ¼ v[MV]/(kfJEAmgyVE) energy investment ratio

equations

d

dtME ¼ _J EAþ _J EC; ð3:1Þ

d

dtMV ¼ _J VG ¼ �ðk _J EC � _J EMÞyVE ð3:2Þ

with _J EA ¼ cðTÞf ðXÞf _J EAmgL2 ð3:3Þ

_J EC ¼ �cðTÞf _J EAmgL2 ge

g þ e1þ L

gLm

� �ð3:4Þ

and _J EM ¼ �cðTÞ½ _J EM �L3: ð3:5Þ



(mol) and the structural mass MV (mol). Forsimplicity, we do not discuss development and repro-duction but the reasoning for stable isotopedynamics also applies. The equations for the dynamicsof the mass of reserve and structure are givenin table 2.

Assimilation (figure 3a) is defined by the followingmacrochemical reaction equation:

YAXXX þ YA

OXO! YAEXE þ YA

PX P þ YAHXH þ YA

NX N

þ YACX C ð3:6Þ

The stoichiometry of each transformation is speci-fied by the different yield coefficients Yij

k. Theydefine the number of C-moles of compound i pro-duced per C-mole of compound j in transformationk and are ratios of two fluxes: Yij

k ¼ Jik/Jjk, with Jik theflux of compound i and Jjk the flux of compound jtaken as the reference rate of the transformation.Using the flux of food JXA as the reference rate inassimilation, YEX

A defines the number of moles of Eproduced per mole of X. As some compounds (e.g.H2O) can serve both as products and substrates, weneed a sign convention for the calculation of the yield


coefficients. Substrate and product fluxes should haveopposite signs and a substrate flux is taken to be negative(see appendix C). For example, the flux of reserve (pro-duct) in assimilation JEA is positive while JXA, the fluxof food (substrate), is negative. The yield coefficientYXX

A equals 1 by definition and we have YEXA ¼ JEA/

JXA ¼ 2yEX and YPXA ¼ 2yPX, with yEX and yPX two

model parameters. The elemental compositions ofcompounds are required for the calculation of the min-eral yield coefficients and are model parameters(tables 1 and 2).

Using the flux of reserve JEG as the reference rate,growth (figure 3b) is defined by

YGEEE þ YG

OEO! YGVEV þ YG

CEC þ YGHEH

þ YGNEN ð3:7Þ

with YEEG ¼ 1 and YVE

G ¼ yVE the yield of structure overreserve in the growth transformation, a model par-ameter. Calculation of the yield coefficients of thegrowth transformation is detailed in the electronicsupplementary material as an example.


energy

O

energy

XAa = kAa XAEA = –yEX

E

VG = –yVE EG

V

O

(1–kAa ) XA

XA

EG

X

E

(a) assimilation

reshuffling

selection

(c) volume-specific somatic maintenance

structureturn-over

(b) growth

C, H, N, P

J JJ

JJ

J

J J JJ J

J

J J JJ J

J

J

J

J J

J

J

C, H, N

H, N

O

(1–kGa ) EG

O

catabolic

catabolic

catabolic

anabolic

EGa = kGa EG

H, N

anabolic

energy

VL1

VrL

V

E

L2

O

O

C, H, N

energy

O

C, H, N

C, H, N

(1–kL1a) EL

(1–kL) EM

OL1

ELa = kL1a EL

VrL = kL2a VL2

VL2

EL = kL EMEM

H, N

anabolic

XA

Figure 3. Scheme of the metabolic transformations in a dynamic isotope budget (DIB) model: (a) assimilation, (b) growth and(c) volume-specific somatic maintenance: turnover of structure and other volume-specific somatic maintenance processes. Iso-topic composition of a flux can change during: (circled cross symbol) selection of molecules for catabolic and anabolic routes

and (filled star) reshuffling of atoms during transformation of substrates into products. X food, E reserve, V structure, P faeces,O dioxygen, C carbon dioxide, H water and N N-waste (e.g. NH3, urea).



Dissipation encompasses the transformation ofreserve into mineral products in the following pro-cesses: somatic and maturity maintenance,development and the conversion of the reproductionbuffer into offspring (overheads of reproduction).Using the flux of reserve JED as the reference rate, weobtain:

YDEEE þ YD

OEO! YDCEC þ YD

HEH þ YDNEN ð3:8Þ

with YEED ¼ YCE

D ¼ 1 as reserve E is the only substrateand carbon dioxide C the only product containingcarbon atoms. Among the dissipation processes, struc-ture turnover does not modify structural mass butimpacts its isotopic composition and should bedetailed (figure 3c).

(b) Structure turnover

The turnover of structure can be described as twocoupled macrochemical reactions: the production of


renewed structure L1 and the degradation of structureL2 (figure 3c). Using the flux of renewed and degradedstructure JVL1 and JVL2 (¼ 2JVL1) as reference rates,we obtain:

L1 :YL1EV E þ YL1

OV Oþ YL1VrV

V ! V

þ YL1CV Cþ YL1

HV Hþ YL1NV N

ð3:9Þ

and

L2 :V þ YL2OV O! YL2

VrVV

þ YL2CV Cþ YL2

HV Hþ YL2NV N

ð3:10Þ

with

_J EL ¼ kL_J EM; ð3:11Þ

YL1EV ¼

_J EL

_J VL1

¼ � 1

yLVE

ð3:12Þ




and

YL1VrV¼

_J VrL

_J VL1

¼_J VrL

_J VL2

¼ YL2VrV¼ �kLr ð3:13Þ

where JEL is the flux of reserve allocated to the turn-over of structure, which is assumed to be a largefraction of JEM the volume-specific somatic mainten-ance costs, JVL1 and JVL2 the renewed structure fluxand the degraded structure flux, respectively, andJVr

L the (negative) recycled structure flux. Thefraction of reserve allocated to the turnover of struc-ture kL ([ [0,1]), the yield of structure from reservein the turnover transformation yVE

L and the fractionof degraded structure that is recycled kLr are newmodel parameters.

(c) Isotope dynamics in reserve and structure

For clarity, we write the equations for 14N and 15N iso-topes but the equations are general for any element.We define the 15N isotope fraction in the reservegNE

15 , in the structure gNV15 and in the whole body

(reserve þ structure) gNW15 as follows:

g15NE ¼

n15NE

nNE

ð3:14Þ

g15NV ¼

n15NV

nNV

ð3:15Þ

and

g15NW ¼

n15NW

nNW

¼ n15NEME þ n15

NV MV

nNEME þ nNV MV

ð3:16Þ

with nNE15 and nNV

15 the proportion of 15N atoms inreserve and structure, respectively. Thus, nNE

15 ME

and nNV15 MV are the total number of 15N atoms in

reserve and structure, respectively.In the literature, the variable d quantifies the differ-

ence between the isotope ratio of a sample and areference isotope ratio relative to that reference:

d15N ¼ 1000Rs � Rr

Rr

� �ð3:17Þ

with Rs ¼15N/14N the ratio of the frequencies of the

heavy and light isotopes in the sample and Rr the inter-national reference standard atmospheric N2 fornitrogen, Rr ¼ 0.0036765 (Fry 2006). As a ratio ofratios, the variable d is appropriate to compare smalldifferences but not to study dynamics. The ratio Rs

relates nonetheless to the variable gNj15 ( j ¼ E, V, W )

as follows:

Rs ¼g15

Nj

1� g15Nj

: ð3:18Þ

We use the d notation in our results and we definethe discrimination factor between compartments Aand B as the following: DAB ¼ d15NA 2 d 15NB.

The changes in isotope fraction in the reserve gNE15 is

obtained by assuming that there is no isotope selectionduring mobilization of reserve (H1. Mobilization):

d

dtg15

NE ¼n15A

NE

nNE

� g15NE

� � _J EA

ME

ð3:19Þ


with nNE15A the proportion of 15N in the reserve flux in

assimilation JEA (equation (3.3)). The expression fornNE

15A is given in equation (A 1) and the full derivationfor equation (3.19) is given in the electronicsupplementary material.

Only the flux of renewed structure JVL1 impacts theisotopic composition of structure (H2. moleculeselection), not the degraded flux of structure JVL2

(H1. mobilization):

d

dtg15

NV ¼n15G

NV

nNV

� g15NV

� � _J VG

MV

þ n15L1NV

nNV

� g15NV

� � _J VL1

MV

ð3:20Þ

with nNV15G the proportion of 15N in the flux of structure

in growth JVG (equation (3.2)) and nNV15L1 the pro-

portion of 15N in the flux of renewed structure JVL1

(equation (3.11)). The expression for nNE15G and nNV

15L1

are given in equations (A 2) and (A 3) and the full deri-vation for equation (3.20) is given in the electronicsupplementary material.

4. SIMULATIONSIn this section, we illustrate the implementation of thetheory by studying three scenarios, each involving twoidentical organisms experiencing changes in the isoto-pic value of their food. We assume in all simulationsthat assimilation discriminates against heavy isotopeswhile growth and structure turnover discriminateagainst light isotopes. We follow individual growth,changes in isotopic values in reserve and structure,and the resulting changes at the whole body level.Temperature is kept constant in all simulations, butfood levels can be different. At ‘high’ food level,the scaled functional response f is set to 0.9; at ‘low’food level, f ¼ 0.2. Parameters are given in tables 2and 3.

(a) Incorporation rate depends

on individual length

Simulation 1 illustrates that large individuals haveslower isotopic incorporation rates (Martınez del Rioet al. 2009 and references therein). We study theeffect of a switch in the d15N of the food for individualsof different lengths. Both individuals experience highfood level but individual 2 is born earlier than individ-ual 1. Hence, at the time of diet switch, individual 2 islarger than individual 1 (figure 4a). When the switchoccurs, d15N in reserve is the same for both individuals(figure 4b). As recycling of structure during turnoverdiscriminates against light isotopes and individual 2 isolder than individual 1 at the switch, d15N values instructure are different between the two individuals(figure 4c). Although individual 1 has a lower isotopicvalue in structure than individual 2, individual 1 reachessubsequently a new ‘equilibrium’ value faster(figure 4d).

(b) Discrimination factor depends

on the reserve density

Observations made at the individual level showed anincrease of the diet–tissue discrimination factor in


Table 3. Specific state variables and parameter values for the DIB model. State variables and parameters are dimensionless.

state variables interpretation

nNE15 proportion of 15N atoms in reserve

nNV15 proportion of 15N atoms in structure

gNE15 ¼ nNE

15 /nNE fraction of 15N in reserve

gNV15 ¼ nNV

15 /nNV fraction of 15N in structure

parameters value interpretation

kL 0.8 fraction of volume-specific somatic maintenance allocated to structure turnoverkLr 0.4762 fraction of structure turnover that is recycledyEX 0.8 yield of reserve from food in assimilation

yPX 0.1 yield of faeces from food in assimilation

yVEL 0.63 yield of structure from reserve in turnover of structure

bNX15Aa 0.999 odds ratio in assimilation

bNE15Ga 1.005 odds ratio in growth

bNE15L1a 1.007 odds ratio in the production of renewed structure

bNV15L2a 1.007 odds ratio in the degradation of structure

aEXNA 0.8 reshuffling coefficient for nitrogen from X to E in assimilation

aVENG 0.8 reshuffling coefficient for nitrogen from E to V in growth

aVENL1 0.4737 reshuffling coefficient for nitrogen from E to V in turnover

aVVNL1 0.4737 reshuffling coefficient for nitrogen from V to V in turnover



starvation conditions (Adams & Sterner 2000;Oelbermann & Scheu 2002; Gaye-Siessegger et al.2007). In simulation 2, we illustrate this effect in lowfood conditions rather than starvation conditions asthe model is not specified for strong starvation con-ditions, i.e. when the reserve is not sufficient to covermaintenance costs. We study the effect of nutritionalstatus after a switch in the d15N of the food: afterthe switch, individual 2 experiences low food levelssuch that no growth occurs subsequently (figure 5a).

In low food conditions, isotopic incorporation ratesare slower both in reserve and structure (figure 5b,c).As no growth occurs after the switch for individual 2,the recycling of structure also results in a larger dis-crimination factor in structure after day 475.Interestingly, the d15N value for the whole body inindividual 2 becomes larger than individual 1 earlierthan in structure (day 325, figure 5d). This patternis explained by the differences in isotopic values inreserve and structure and the relative contributionsof reserve and structure to the total biomass. In lowfood conditions, the reserve to structure ratio, i.e.the reserve density, is lower. As structure is heavierthan reserve, it results in a higher isotopic value inthe total biomass in individual 2.

(c) Fluctuations in the isotopic composition

of the food

Seasonal isotopic variation at the base of a food web iscommonly propagated to higher levels of a food web,but with decreasing amplitude (e.g. Kiriluk et al.1995; Harvey et al. 2002). In simulation 3, we studythis pattern by simulating a seasonal variation in thed15N of the food. Individuals are grown in high andlow food levels, respectively. Individual 2 is thus


smaller than individual 1 throughout the experiment(figure 6a) and its reserve density is lower. The signa-ture of their food is fluctuating (sinusoid) but is thesame for the two individuals (figure 6b).

We observe the same effects of body length as insimulation 1: the larger individual has a slower isotopicincorporation rate which results in a lag between thereserve and the food isotopic value and this lagincreases as the individual is growing (figure 6b). How-ever, the isotopic composition of reserve inindividual 2 (low food conditions) follows closely theisotopic composition of the food. The individual issmall enough, which compensates the effect of aslower incorporation rate when reserve density issmall (simulation 2). The isotopic composition ofreserve in turn impacts the isotopic composition ofstructure (figure 6c). The amplitude of the variationsof the isotopic composition of both reserve and struc-ture is decreasing as the individuals grow (figure 6b,c)which is consistent with observed patterns (Harveyet al. 2002, and references therein). The differencesin d15N at the whole body level are also explained bythe differences in reserve density between the twoindividuals as in simulation 2 (figure 6d).

This simulation shows that we can simulate smallindividuals with d15N levels equal or greater thanlarger conspecifics (Kiriluk et al. 1995). In reserve,this pattern is owing to faster incorporation rates insmall individuals (figure 6b). At the whole bodylevel, this pattern is explained by lower reserve densityin individual 2 (figure 6d). This simulation emphasizesthe value of a bioenergetic approach that integrates thedynamics of the environment with no a priori fixeddiscrimination factor values to understand observedpatterns. We show that the model can be used todisentangle different factors.


0

1

2

3

4

5

6

7

leng

th (

cm)

(a) (b)

(c) (d)

d15N

str

uctu

re (

‰)

0 200 400days since diet switch

0 200 400days since diet switch

0

2

4

6

8

10

d15N

who

le b

ody

(‰)

0

2

4

6

8

10

d15N

res

erve

(‰

)

0

2

4

6

8

10 indiv 1indiv 2d15N food

Figure 4. Simulation 1. (a) Growth in length of individuals born on different dates but experiencing the same conditions

(temperature, food density). Changes in d15N in (b) reserve, (c) structure and (d) the whole body (reserve and structure)after a switch in the food d15N.



5. DISCUSSIONIn this research, we introduced stable isotopes withinDEB theory, in order to study in a mechanistic waythe impact of metabolism on isotopic incorporationand discrimination within an organism. The newsteps involve careful bookkeeping of atoms and iso-topes embedded into molecules duringtransformations. We also required some new com-ponents beyond ‘standard’ DEB, notably recognitionof anabolic and catabolic routes in transformations,so as to allow molecule selection and an explicit treat-ment of structure turnover. Our simulations illustratethat part of the variability in isotopic incorporationrates and discrimination factors among individualscan be explained by differences in body length andnutritional status. We also show the potential of themodel to explain patterns in dynamic environments.

We make two important steps towards a mechanis-tic understanding of the impact of metabolism onstable isotope dynamics. First, we formulate discrimi-nation mechanisms for each of three metabolicfunctions: assimilation, growth and turnover. The pro-duction of excreted products such as CO2 or N-wasteinvolves a linear combination of these three functions,and can therefore be derived. Here, our approach dif-fers from Ponsard & Averbuch (1999) who treateddiscrimination during excretion as a single mechan-ism. In our representation, changes in the isotopiccomposition of the total N-waste flux could be


explained by changes in the relative contribution ofeach metabolic function, which may offer newinterpretations of observed patterns. Our approach isalso consistent with an observation by Carleton &Martınez del Rio (2005) who observed that variationsin respiration rate, taken as a proxy for metabolic rate,and incorporation rate of 15N and 13C weredecoupled. These authors suggested that turnover pro-cesses should not be estimated by the total respirationflux. Our approach recognizes that respiration has con-tributions not only from turnover processes but alsofrom other dissipation processes (development, main-tenance of gradients, movement, etc), and fromgrowth and assimilation in growing and feeding indi-viduals. Thus, respiration is not an explanatoryvariable in our approach.

Second, the new approach allows us to recognize thatmolecules are not disassembled into their elementalcomponents during transformations. This step wasidentified by Martınez del Rio et al. (2009) as importantfor improving our understanding of stable isotopedynamics in an organism. Other approaches implicitlyassume complete mixing of atoms during transform-ations, thereby preventing the study of isotopicrouting. To our knowledge, only one previous studyincorporated routing into a mixing model for d13C byassuming that carbon in food proteins was routedpreferentially into tissue proteins (Martınez del Rio &Wolf 2005). In our approach, the reshuffling principle


0

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5

6

leng

th (

cm)

(a) (b)

(c) (d)

0 200 400 600

0

2

4

6

8

10

days since diet switch0 200 400 600

days since diet switch

d15N

str

uctu

re (

‰)

0

2

4

6

8

10

d15N

who

le b

ody

(‰)

0

2

4

6

8

10 indiv 1indiv 2d15N food

d15N

res

erve

(‰

)

Figure 5. Simulation 2. (a) Growth in length of individuals grown in the same conditions until diet switch. Individual 2 experi-

ences a decrease in food density after the switch. Changes in d15N in (b) reserve, (c) structure and (d) the whole body after aswitch in the food d15N are represented.



allows specification of a non-random allocation ofatoms of a particular substrate to a particular pro-duct. This mechanism impacts isotopic compositionof a given product when two substrates are involvedin the transformation. This mechanism is particularlyrelevant when several food items are considered orwhen reserve and structure of the food items areconsidered (e.g. Kuijper et al. 2004; Kooijmanet al. 2007).

In this paper, we adhere rigorously to the assump-tions of standard DEB theory, and require that anyextensions are consistent with standard DEB prin-ciples. This philosophy determines ourrepresentation of recycling of structure. A readercould reasonably ask why we do not return therecycled part to reserve first; the answer is that itwould require an additional parameter to describethe transformation of structure into reserve (over-head), but much more important it would changethe reserve dynamics. For similar reasons, we assumethat mobilization does not impact the isotopic compo-sition of the pool from which molecules are mobilizedbecause of the strong homeostasis assumption. Theequation describing reserve dynamics (equation (3.1))is derived from the weak homeostasis assumptionKooijman (2010) and departing from this assumptionhas far-reaching consequences, e.g. for determiningthe chemical composition of reserve and structurefrom data on body composition or to explain the


transition from multiple reserves to single reservesystems in evolutionary contexts (Kooijman 2010).

Although mobilization does not change the isotopiccomposition of substrate, our simulations lead to thesame conclusions to those in Ponsard & Averbuch(1999). These authors found that a dynamic equilibriumbetween assimilation and excretion could explain trophicenrichment in nitrogen in a fully grown animal: ifexcretion favours the dissipation of light isotopes in theenvironment, assimilation should be less biased towardslight isotopes. In our simulations, assimilation also dis-criminates against heavy isotopes and growth andstructure turnover discriminate against light isotopeswith a higher probability, i.e. odds ratio in assimilation iscloser to 1 with the value 1 corresponding tono-selection situations (see table 3 and appendix B).The result is that total biomass, composed of reserveand structure, is heavier than food. Odds ratio values arechosen such that the biomass–food discriminationfactors are in the range of observed values (Vanderklift &Ponsard 2003).

Our simulations show that the model can capturethe known effects of body length and nutritionalstatus on incorporation rates and discrimination fac-tors among conspecifics in constant and variablefood isotopic ratios. As our framework is fully consist-ent with standard DEB theory, we can fully exploit itsproperties to further study a variety of scenarios whereconfounding factors might be involved. At present, we


0

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leng

th (

cm)

(a) (b)

(c) (d)

−4

−2

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−4

−2

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−4

−2

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500 1000 1500

time (d)500 1000 1500

time (d)

d15N

str

uctu

re (

‰)

d15N

who

le b

ody

(‰)

d15N

res

erve

(‰

)

indiv 1indiv 2d15N food

Figure 6. Simulation 3. (a) Growth in length of individuals grown in the same temperature conditions but different food levels.Both individuals experience fluctuations in the isotopic composition of their food. Changes in d15N in (b) reserve, (c) structureand (d) the whole body are represented.



can study the impact of variable temperature and vari-able food densities. Short-term starvation situationswhere maintenance costs can be paid from reservecan also be studied, an issue that was not taken intoaccount in the bioenergetic approach developed byHarvey et al. (2002) for fish muscle for instance.Strong starvation situations where maintenance costscannot be covered by reserve are, however, more com-plex. Responses to these situations can be typicallyspecies-specific. Our treatment of structure turnoveropens, however, possibilities to treat structure shrink-ing as a mechanism to cope with strong starvation.The degraded structure might only be partiallyrenewed during structure turnover which results inshrinking. This approach is consistent with theapproach developed by Tolla et al. (2007) wheresomatic maintenance is covered by both reserve andstructure.

In the present framework, we can relate model out-puts to measurements at the whole body level, whichintegrate tissues with fast and slow turnover rates. Tofurther link model variables to sub-individual data,new developments are nonetheless required. Tissuescould be modelled as proportion of both reserve andstructure with variable maximum reserve densitiesand structure turnover rates. Moreover, studiesreport non-destructive methods that follow stable iso-tope ratios in time for a given individual in particulartissues such as blood cells or protein plasma, and


also in hairs and feathers (Martınez del Rio et al.2009). Measurements in time are of particular impor-tance for the application of DEB theory (Kooijmanet al. 2008). Application of the theory to such datasetscould be a fruitful strategy for better understandingthe impact of dynamic environments. Measurementsof the exchanges with the environment together withgrowth are, however, required, e.g. food, faeces, butalso N-waste production and reproductive outputsfor instance. Our framework could guide the designof such experiments.

Searching for common mechanisms across taxa isperhaps the most promising contribution of this newtheory. DEB theory is built on the premise that themechanisms for the organization of metabolism arenot species-specific (Sousa et al. 2010). Body size scal-ing relationships in DEB theory provide a mechanismfor the decrease in the specific respiration rate withspecies maximum size (Kooijman 2010; Sousa et al.2010). These relationships could help reveal whichphysiological traits are key to explain common patternsacross taxa and which traits could be responsible forvariations across taxa. As emphasized by Ponsard &Averbuch (1999), the inter-species variability of thetissue–diet discrimination factor may not be random.Understanding the mechanisms responsible for thisvariability would increase the potential applicationsof SIA in field studies to species for which no exper-imental data are available. Following DEB principles




and body size scaling relationships in particular, wealso suggest that the variability of incorporation ratesof stable isotopes in comparable tissues may also notbe random among species as they are determined bymetabolic rates. Furthermore, our framework mayalso account for differences in C : N ratios in reserveand structure and differences in biochemical form ofexcretion as the chemical composition of moleculesis specified. Scenarios could be designed to studywhy ammonotelic organisms show lower d15N enrich-ment than ureotelic or uricotelic organisms(Vanderklift & Ponsard 2003) or why herbivorousfish tend to have a more variable food-to-biomassdiscrimination factor (Mill et al. 2007).

This work presents new theory for stable isotopedynamics within an organism in the context of DEBtheory. Important steps are made towards a mechanis-tic understanding of isotopic incorporations rates anddiscrimination factors among individuals and amongspecies. The theory provides mechanisms for discrimi-nation and deals with dynamic environments, therebyproviding a sound basis for further scenario analysis tounderstand patterns in data and learn more aboutmetabolic organization in general.

This research was supported by the NSF grants EF-0742521and DEB-0717259. We are grateful to J. Allen,S. Bonhommeau, T. Klanjscek, S. Lefebvre, E. Muller,H. M. Page, T. Sousa, K. Taulbee and three anonymousreferees for valuable suggestions and comments.

APPENDIX A. ISOTOPIC COMPOSITION OFFLUXES

This section describes how we obtain nNE15A, nNV

15G and nNV15L,

the proportions of 15N that enter the pools of reserve andstructure during assimilation, growth and structure turn-over (equations (3.19) and (3.20)). Reserve and structureare here products of these transformations.

n15ANE ¼ aNA

EX n15AaNX

1

yEX

; ðA 1Þ

n15GNV ¼ aNG

VE n15GaNE

1

yVE

ðA 2Þ

and

n15L1NV ¼ aNL1

VE n15L1aNE

1

yLVE

þ aNL1VV n15L2a

NV

1

kLr

: ðA 3Þ

The dimensionless reshuffling coefficients apsNk (0�

apsNk � 1) specify what fraction of N atoms in substrate s

ends up in product p in transformation k (k ¼ A, G,L1, L2). For instance, the number of N atoms thatgoes in E from X is fixed by the reshuffling coefficientaEX

NA. This coefficient does not depend on the isotopefraction of X. Reshuffling coefficients can be defined asmodel parameters or given by stoichiometryrequirements if we assume complete reshuffling (seeappendix C and the electronic supplementary material).Note that nNV

15L1 the proportion of 15N that enter struc-ture during turnover depends on two substrates,reserve and structure, and nNV

15L2a the proportion of 15Nin the recycled flux of structure JVrL.


The coefficients nNs15ka specify the proportion of 15N

in substrate s (s ¼ X, E, V ) in the anabolic componentof transformation k, i.e after selection (A2. moleculeselection). To study the properties of the model in asimple and deterministic way, we evaluate nNs

15ka withan approximation of the mean of a Fisher’s noncentralhypergeometric distribution using the odds ratios bNs

15ka

(see equation (B 1)) and kka the different fractions ofthe substrate fluxes that go into the anabolic routes:

kAa ¼_J XAa

_J XA

¼ yEX

YaEX

ðA 4Þ

kGa ¼_J EGa

_J EG

¼ yVE

YaVE

ðA 5Þ

kL1a ¼_J ELa

_J EL

¼ yLVE

YL1aVE

ð1� kLrÞ ðA 6Þ

and

kL2a ¼ �kLr ðA 7Þ

with for instance JXAa ¼ 2JEA/YEXa and YEX

a ¼ 1 bydefinition of a building block, i.e. all carbon atoms ofthe food compound in the anabolic component ofassimilation are allocated to reserve. Similarly YVE

a ¼

YVEL1a ¼ 1 by definition of the anabolic component.

Values for parameters yEX, yVE and yVEL are provided

in tables 2 and 3.

APPENDIX B. SELECTION FROM FLUXESTransformations ka and kc correspond to the anabolicand catabolic routes of transformation k(k ¼ A, G, L),respectively.

We must have nNs15k Jsk¼ nNs

15kaJska þ nNs15kcJskc or

nNs15k ¼ nNs

15kakkaþ nNs15kc (1 2 kka). We have nNs

15k ¼ gNs15

nNs (as mobilisation does not change the isotope ratioof a given compound) and introduce an odds ratiobNs

15ka for the 15N in compound s (E or V for instance)in transformation ka. The number of isotopes in theanabolic flux times a time increment follows Fisher’snoncentral hypergeometric distribution (H2. moleculeselection) with approximate mean:

n15kaNs ≃ 2n15k

Ns b15kaNsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

B2 þ 4ð1� b15kaNs Þb15ka

Ns n15kNs kka

q� B

;

n15kcNs ¼

n15kNs � n15ka

Ns kka

1� kka

ðB 1Þ

for B ¼ nNs15k 2 (1 2 kka) 2 (nNs

15k þ kka)bNs15ka, with par-

ameters bNs15ka and kka, the two new parameters for

transformation k. We must have

n15kNs � n15ka

Ns kk and

B2 þ 4ð1� b15kaNs Þb15ka

Ns n15kNs kka � 0:

ðB 2Þ

If bNs15ka ¼ 1, we have nNs

15ka ¼ nNs15k and the process is

unselective.For further details, the DEBtool routine fnchd.m

provides the expected value of Fisher’s non-centralhypergeometric distribution, its approximation, andthe corresponding mean of the binomial distribution




for small samples sizes for comparison purposes(http://www.bio.vu.nl/thb/deb/deblab/debtool/).

APPENDIX C. RESHUFFLINGThe dimensionless reshuffling coefficient aps

Nk (0 �aps

Nk � 1) specifies what fraction of N atoms in sub-strate s ends up in product p in transformation k(H3. Atom reshuffling).

Given nNs15k the relative frequency of 15N in all sub-

strates s [ S in transformation k, the proportionsnNp

15k are given for p [ P by

n15kNp ¼ �

Xs[S

aNkps n15k

Ns

_J sk

_J pk

¼ �Xs[S

aNkps n15k

Ns

1

Ykps

ðC 1Þ

with S and P the substrates and products intransformation k, respectively, and

Pp[P aps

Nk ¼ 1. Ifns substrates and np products exist, the number ofreshuffling parameters is (np21)ns.

The isotope fraction in the flux of structure duringgrowth for instance is given by

n15GNV ¼ �aNG

VE n15GNE

_J EG

_J VG

¼ aNGVE n15G

NE

1

yVE

: ðC 2Þ

If we suppose complete reshuffling, i.e mixing, of allsubstrate atoms of nitrogen in a given transformation,the isotope ratios of that element are equal in allproducts, so for product p [ P we have

n15kNp ¼ nNp

Xs[S

_J sk n15kNs

_J sk nNs

¼ nNp

Xs[S

n15kNs

nNs

ðC 3Þ

and aNkps ¼

nNp_J pkP

p[P nNp_J pk

¼nNpY

kpsP

p[P nNpYkps

ðC 4Þ

with Jsk taken as the reference rate for the yieldcoefficients Yps

k .

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