Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Modeling the dynamics of stable isotope tissue-diet enrichment Christopher H. Remien n University of Idaho, Department of Mathematics, Moscow, ID 83844, Russia
H I G H L I G H T S
Stable isotope ratios of animal tissue are used to estimate diet. Estimates rely on a predictable offset between isotope ratios of diet and tissue. The isotope ratio offset from diet to tissue depends on diet, metabolism. We model the incorporation of carbon and nitrogen isotopes in diet to tissues. Model predicts how diet and energy expenditure shape isotope ratios of tissue.
art ic l e i nf o
a b s t r a c t
Article history: Received 31 October 2013 Received in revised form 15 November 2014 Accepted 19 November 2014
Reconstructions of dietary composition and trophic level from stable isotope measurements of animal tissue rely on a predictable offset of stable isotope ratios from diet to tissue. Physiological processes associated with metabolism shape tissue stable isotope ratios, and as such the spacing between stable isotope ratios of diet and tissue may be influenced by processes such as growth, nutritional stress, and disease. Here, we develop a model of incorporation stable isotopes in diet to tissues by coupling stable isotope dynamics to a model of macronutrient energy metabolism. We use the model to explore the effect of changes in dietary intake, both composition and amount, and in energy expenditure, on body mass and carbon and nitrogen stable isotope ratios of tissue. & 2014 Published by Elsevier Ltd.
Keywords: Fractionation Metabolism Stable isotope ratio
1. Introduction Natural abundance stable isotope ratios of animal tissues are powerful indirect markers of diet and migration and are of particular importance when direct measurements are impossible or impractical (Martinez del Rio et al., 2009). Through metabolism, ingested nutrients are incorporated into tissues, recording and integrating the carbon, nitrogen, oxygen, hydrogen, and sulfur isotopic signal of diet. Animal diets can be highly variable, and if dietary sources vary in stable isotope ratios, tissue stable isotope ratios can provide a means to quantify changes in dietary sources over space and time. The stable isotope ratios of dietary sources are the major determinant of the stable isotope ratios of consumer tissues, but isotopes do not act as pure tracers, even with constant source isotopic composition. Oxygen and hydrogen stable isotope ratios in proteinaceous tissues such as hair are primarily determined by the stable isotope ratio of drinking water (Ehleringer et al., 2008), but the quantity of ingested water affects the stable isotope shift from
n
Tel.: þ 208 885 5901. E-mail address: [email protected]
drinking water to tissue (O'Grady et al., 2010). Carbon and nitrogen stable isotope ratios in tissue are primarily determined by the stable isotope ratio of dietary sources, but physiological and metabolic processes can lead to a difference between the stable isotope ratio of consumer tissue and diet, termed consumer tissue-diet discrimination (Vanderklift and Ponsard, 2003). Nitrogen consumer tissue-diet discrimination, Δδ15 N, is typically about 3–4‰ and has been used to estimate trophic position and food chain length (Post et al., 2000; Post, 2002). Carbon consumer tissue-diet discrimination is variable, typically about 0–1‰ (Caut et al., 2009), with variations in lipid content (Post et al., 2007) and amino acid composition (O'Brien et al., 2005) affecting the carbon stable isotope ratio of the tissue. Unfortunately, variability in stable isotope discrimination leads to uncertainties when estimating diet and trophic position using stable isotopes (Caut et al., 2009, 2008; Bond and Diamond, 2011). Linear mixing models that are used to reconstruct the relative contribution of dietary sources from stable isotope measurements rely on predictable consumer tissue-diet discrimination to create meaningful reconstructions (Phillips, 2012). Isotopic discrimination depends on the details of macronutrient metabolism. Δδ15 N likely results from the preferential excretion of 14N due to fractionation during transamination and deamination of amino acids (Macko et al., 1986; Chikaraishi
http://dx.doi.org/10.1016/j.jtbi.2014.11.018 0022-5193/& 2014 Published by Elsevier Ltd.
Please cite this article as: Remien, C.H., Modeling the dynamics of stable isotope tissue-diet enrichment. J. Theor. Biol. (2014), http://dx. doi.org/10.1016/j.jtbi.2014.11.018i
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et al., 2007). Interesting feedbacks exist between Δδ15 N and dietary protein quality and quantity that depend on physiology, metabolism, and physiological state (Martinez del Rio et al., 2009; Robbins et al., 2010). Carbon consumer tissue-diet discrimination, Δδ13 C, varies with amino acid, with the δ13C of essential amino acids tracking that of dietary protein and the δ13C of non-essential amino acids reflecting a mix of the δ13C of dietary protein and the δ13C of nonessential amino acids that were synthesized using carbon that originated from other sources such as carbohydrates or fat (O'Brien et al., 2005). A quantitative understanding of the mechanisms of stable isotope discrimination will aid diet and trophic level reconstructions. Furthermore, tissue stable isotope ratio measurements may become a useful proxy for metabolic and nutritional state if the mechanisms controlling discrimination are better understood. Here, we present a model of macronutrient energy metabolism that has been adapted to study how stable isotope ratios of tissues vary with changes in body weight, body composition, and diet. The model describes the basic macronutrient fluxes of an animal. Isotopes in macronutrients are tracked as they are ingested, incorporated into tissues, and excreted. The model predicts trajectories for body composition and tissue stable isotope ratios following changes in intake and expenditure.
2. Energy partition stable isotope model Changes in stored energy are determined by energy balance. When total energy intake is greater than energy expenditure, energy is stored, while stored energy is lowered when energy intake is less than expenditure. Stored energy can be converted to body mass by accounting for energy density. Mathematical models have a long history in the study of energy metabolism and body weight change (Antonetti, 1973; Alpert, 1979; Guo and Hall, 2009). We will study the influence of energy metabolism on stable isotope dynamics by tracking stable isotope dynamics in the context of the macronutrient energy partition model developed by Chow and Hall (2008).
compartments, such as gluconeogenesis and lipogenesis, to be accounted for in the functional forms of fF and fC. Nitrogen is found in protein, but not in fat or glycogen. Fractionation of nitrogen is thought to occur with deamination during catabolism so that 15N is disproportionally retained relative to 14N (Macko et al., 1986; Chikaraishi et al., 2007). 14N follows the dynamics of body protein so that 14 N ¼ κ P, where κ ¼ 0:15 is the fraction of nitrogen in protein. 15N also closely follows protein trajectories, but with fractionation in deamination:
ρP =κ
The parameter Rin is the molar ratio of 15N to 14N in dietary protein, and α accounts for the fractionation associated with deamination. The losses in the 15N equation reflect fractionation from deamination in protein energy metabolism, but no fractionation with losses of intact amino acids and proteins. Following simplifications in Chow and Hall (2008), we define lean mass as L ¼ M F where M is total mass, so that dL dP dG ¼ ð1 þ hP Þ þ ð1 þ hG Þ ; dt dt dt with hydration coefficients for intracellular water associated with protein and glycogen hP ¼1.6 and hG ¼ 2.7, respectively. The small storage capacity of glycogen allows the model to be reduced by taking dG=dt ¼ 0, so that f C ¼ I C =E, P ¼ ðL cÞ=ð1 þ hp Þ, and 14 N ¼ κ ðL cÞ=ð1 þ hp Þ, where the mass c is the lean mass that is not associated with protein. Again following Chow and Hall (2008), we assume L F trajectories for further reduction of the macronutrient model. Let ðρF =ρL ÞdF=dL ¼ γ ðF; LÞ. The function γ varies for different species and can be approximated by cross-sectional measurements. Dividing dF=dt by dL=dt, imposing L F trajectories, and solving for the fraction of energy expenditure derived from metabolism of fat, f F ¼ I F =E ðγ ð1 þ γ ÞÞðI E hÞ=E, where I ¼ I F þ I C þ I P . With trajectories imposed, the nitrogen stable isotope model reduces to the two dimensional system
ρL
2.1. Nitrogen stable isotope discrimination Tracking the dynamics of the three primary macronutrients fat, F, glycogen, G, and protein, P, leads to flux balance equations:
ρF
dF ¼ IF f F E dt dG ρG ¼ IC f C E dt dP ρP ¼ IP ð1 f F f C ÞE h: dt
The equations for fat, glycogen, and protein are almost identical to those used by Chow and Hall (2008), but, following Martinez del Rio and Wolf (2005), have an additional protein loss term, h, that accounts for protein and amino acids that are lost without deamination, such as metabolic fecal nitrogen, hair, and shed skin. The rate of intact protein and amino acid loss is typically small compared to the excretion rate of nitrogen from amino acid catabolism, but it is potentially important in influencing Δδ15 N. The parameters ρF ¼ 9400 kcal=kg, ρG ¼ 4200 kcal=kg, and ρP ¼ 4700 kcal=kg (Chow and Hall, 2008) represent the energy densities of fat, glycogen, and protein, respectively, and IF, IC, and IP are the energy intake rates of fat, carbohydrates, and protein, respectively. The fractions of energy expenditure derived from metabolism of fat, carbohydrates, and protein, fF, fC, and 1 f F f C , respectively, are functions of body composition and intake rates. For now, consider conversions between body macronutrient
15 15 d15 N N N ¼ Rin I P α14 1 f F f C E 14 h: dt N N
dL 1 ¼ ðI E hÞ dt 1 þ γ
ρP =κ
15 15 d15 N N γ N ¼ Rin I P α14 E ðI I P Þ þ ðI E hÞ 14 h dt 1þγ N N
where ρL ¼ ρP =ð1 þ hP Þ ¼ 1800 kcal=kg. 2.2. Carbon stable isotope discrimination We track the dynamics of the three primary macronutrients fat, F, glycogen, G, and protein, P, as we did above with nitrogen, but because carbon is found in all three primary macronutrients, we must explicitly consider conversions between macronutrient compartments:
ρF
dF ¼ I F þ sGF sFG f F E dt dG ρG ¼ I C sGF þ sFG sGP þ sPG f C E dt dP ρP ¼ IP þ sGP sPG ð1 f F f C ÞE h: dt
The parameters sGF, sFG, sGP, and sPG represent the rate of lipogenesis, gluconeogenesis from fat, amino acid synthesis, and gluconeogenesis from protein, respectively. Just as in the 15N model, 13C follows the macronutrient dynamics so that 12 CF ¼ κ F F, 12 CG ¼ κ G G, and 12 CP ¼ κ P P where κ F , κ G , and κ P are the fraction of carbon in fat, glycogen, and protein, respectively. The 13C in fat, glycogen, and protein closely track the
Please cite this article as: Remien, C.H., Modeling the dynamics of stable isotope tissue-diet enrichment. J. Theor. Biol. (2014), http://dx. doi.org/10.1016/j.jtbi.2014.11.018i
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C.H. Remien / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 12 C trajectories, but with potential fractionation in conversions between macronutrient compartments and excretion:
13
13 13 13 C C C C ρF =κ F d F ¼ Rin;F IF þ αGF 12 G sGF αFG 12 F sFG αout;F 12 F f F E dt CG CF CF
ρG =κ G d
13 13 13 CG C C C ¼ Rin;G I C αGF 12 G sGF þ αFG 12 F sFG αGP 12 G sGP dt CG CF CG 13
þ αPG 12
ρP =κ P d
CP CP
13
sPG αout;G 12
CG CG
f CE
13
13 13 CP C C ¼ Rin;P I P þ αGP 12 G sGP αPG 12 P sPG dt CG CP 13
αout;P 12
CP CP
13
ð1 f F f C ÞE 12
CP CP
instead of enrichment, defined as
ΔδX tissue diet ¼ δX tissue δX diet : It can be shown that ΔδX tissue diet ¼ ϵtissue diet only when Rdiet ¼ Rstandard so that though ΔδX tissue diet is convenient to calculate, it is an approximation that varies with the isotopic composition of the diet, whereas ϵntissue diet is independent of the isotopic composition of the diet (Auerswald et al., 2010). Throughout the modeling, we use differential equations to track the amounts of heavy and light isotope in tissue as they vary dynamically. The ratio of heavy to light isotope in tissue, the stable isotope ratio of tissue in delta notation, the apparent fractionation factor, the apparent enrichment, and the apparent discrimination factor can be easily calculated from the amounts of heavy and light isotope in tissue.
13
h:
The parameters Rin;F , Rin;G , and Rin;P are the molar ratio of 13C to C in dietary fat, glycogen, and protein, respectively. The parameters αGF , αFG , αGP , and αPG account for fractionation in lipogenesis, gluconeogenesis from fat, amino acid synthesis, and gluconeogenesis from protein, respectively, and αout;F , αout;G , and αout;P account for fractionation from catabolism of fat, glycogen, and protein, respectively. 12
2.3. Notation 2.3.1. Delta notation Stable isotope ratios are commonly reported as parts per million relative to a standard using δ notation: Rsample δX ¼ 1 n1000; Rstandard where Rsample and Rstandard are the molar ratios of heavy to light isotope in the sample and standard, respectively. For nitrogen, Rsample ¼ 15 N=14 N, and the established standard is air with Rstandard ¼ 0:0036765. For carbon, Rsample ¼ 13 C=12 C, and the established standard is Pee Dee Belemnite with Rstandard ¼ 0:0112372. 2.3.2. Isotope enrichment, discrimination Following established terminology (Passey et al., 2005; Cerling and Harris, 1999; Craig, 1954), an apparent fractionation factor is defined as
αntissue diet ¼ Rtissue =Rdiet ¼ ð1000 þ δX tissue Þ=ð1000 þ δX diet Þ: The superscript asterisk indicates that isotopic equilibrium is not assumed. The apparent isotope enrichment is given by
ϵntissue diet ¼ ðαntissue diet 1Þn1000: For convenience, many studies, including common linear mixing models (Phillips et al., 2014), use a discrimination factor 8
3.1. Nitrogen stable isotopes At steady state with dL=dt ¼ 0 and d15 N=dt ¼ 0, the model has a unique steady state where E ¼ I h and 15 N=14 N ¼ Rin I P =ðαI P þ ð1 αÞhÞ, so that αntissue diet ¼ I p =ðαI P þ ð1 αÞhÞ and ϵntissue diet ¼ ð1 αÞðI P hÞ=ðαI P þð1 αÞhÞ. With the fraction of nitrogen excreted as urea defined at steady state as ðI P hÞ=I P , the nitrogen stable isotope enrichment increases approximately linearly with ðI P hÞ=I P . Because the model structure assumes a single pool of protein with one fractionated and one unfractionated outflow, the nitrogen stable isotope ratio steady state is the same as in Martinez del Rio and Wolf (2005) (Fig. 1A). With h and α fixed, 15 N=14 N increases with protein intake, but quickly saturates. For parameters estimated for a 65 kg female human so that protein loss without deamination is h ¼ 35 kcal=day (estimated from Chacko and Cummings, 1988) and α ¼ 0:995 (estimated from Macko et al., 1986; Chikaraishi et al., 2007) a diet consisting of 50 g/day of protein results in δ15 N ¼ 4:1‰ while a high protein diet of 150 g/day results in δ15 N ¼ 4:7‰ (Fig. 1B). In addition to steady state values, the model also predicts time courses of body mass and tissue nitrogen stable isotope ratio following a change in diet or energy expenditure. To predict nitrogen stable isotope time courses, we need estimates for γ and energy expenditure. Following Chow and Hall (2008), for humans we can utilize cross-sectional data, imposing L F trajectories specified by cross-sectional data of body weights of women, γ ¼ ðρF =ρL ÞF=10:4, so that F ¼ DexpðL=10:4Þ (Forbes, 1987). Energy expenditure consists of basal metabolic rate and energy expended in physical activity. In principle, energy expenditure is a function of body composition, macronutrient intake rates, and physical
5
N
4
4
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N
3. Results
=0.992 =0.995 =0.998
6 15
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3
3 2
2
h=4.4 g/day h=8.8 g/day h=17.5 g/day
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0 0.0
0.2
0.4
0.6
(Ip h) Ip
0.8
1.0
0
50
100 150 200 250 300
protein intake (g/day)
Fig. 1. Steady state tissue δ15N is determined by the fraction of nitrogen excreted as urea, which at steady state is ðI P hÞ=I P . For fixed h, the rate of protein loss without deamination (e.g., metabolic fecal nitrogen, shed skin and hair), δ15N increases with protein intake and quickly saturates. Dietary intake is set to δ15 Ndiet ¼ 0‰ so that tissue δ15N¼ Δδ15 N.
Please cite this article as: Remien, C.H., Modeling the dynamics of stable isotope tissue-diet enrichment. J. Theor. Biol. (2014), http://dx. doi.org/10.1016/j.jtbi.2014.11.018i
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activity. We take energy expenditure from the Katch–McArdle formula with E ¼ β ð370 þ21:6LÞ where the parameter β has a value between 1.2 and 1.9 and accounts for a person's physical activity level (Katch et al., 1996). A phase plane for body mass and δ15N is shown in Fig. 2 with dietary intake of I ¼ 2050 kcal=day, mass of lean mass not associated with protein c¼2 kg, rate of protein loss without deamination h¼35 kcal/day (estimated from Chacko and Cummings, 1988), physical activity constant β ¼ 1:5, fractionation factor from deamination α ¼ 0:995 (estimated from Macko et al., 1986; Chikaraishi et al., 2007), and constant D¼0.26. 15N does not affect body composition dynamics, so the L-nullcline is a horizontal line. Body composition dynamics are entirely determined by the energy partition model, with Lean-Fat trajectories specified by Forbes' curve. With Lean-Fat trajectories specified by Forbes' curve and energy expenditure by the Katch–McArdle formula, body composition
dynamics affect δ15N dynamics, and hence observed enrichment (Figs. 3 and 4). During weight loss, energy intake is less than expenditure and stored lean mass is mobilized for energy use. Nitrogen in lean mass that is metabolized for energy is fractionated from deamination and is excreted as urea. The fractionation from deamination initially leads to higher δ15N during weight loss, but as the mass equilibrates to its new steady state, δ15N decreases to its steady state value. Similarly, during weight gain, intake is greater than expenditure so that more lean mass is stored than used for energy compared to steady state conditions. Nitrogen follows protein dynamics, and, because fractionation occurs with deamination, growth initially leads to lower δ15N values than at steady state, before increasing to the steady state value. The change in δ15N due to weight gain or loss is due to N imbalance and thus depends on diet composition (Fig. 4). If protein intake is high, δ15N closely tracks the steady state value while larger deviations occur if protein intake is low. In the model, an increase of 150 kcal/day in energy intake leads to a nearly 0:5‰ deviation in Δδ15 N with low protein intake of 26 g/day, five percent of the total caloric intake.
3.2. Carbon stable isotopes Because carbon is found in all three of the macronutrients fat, glycogen, and protein, there is potential for fractionation not just in ingestion and excretion, but also in conversions between macronutrient compartments. Consider the case with a diet that contains sufficient energy intake, so that lipogenesis and gluconeogenesis are low, sGF ¼ sFG ¼ sPG ¼ 0, but low in some nonessential amino acids, so that sGP 4 0. Then at steady state, 13 CF =12 CF ¼ Rin;F =αout;F , 13
CG
12
CG
78
60
5.0 4.8
74 72
4.6
70
4.4
68 50
N
70
Rin;G I C ; ðαGP αout;G ÞsGP þ αout;G I C
Mass (kg) Tissue 15N
76 Mass (kg)
Mass (kg)
80
¼
15
Fig. 2. Nullclines and possible trajectories in the body mass-tissue δ15N phase plane. Dietary intake is set to δ15 Ndiet ¼ 0‰ so that tissue δ15 N ¼ Δδ15 N.
4.2
66 4.0 2.5
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Mass (kg)
Mass (kg)
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3.0
3.5
4.0 15
N
4.5
5.0
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0
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600
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1000
Time (days)
Fig. 3. Weight gain (weight loss) for a typical 65 kg woman caused by a 150 kcal/day increase (decrease) in dietary intake leads to a decrease followed by increase (increase followed by decrease) of δ15N. Protein intake is assumed to be 15% of calories consumed. Dietary intake is set to δ15 Ndiet ¼ 0‰ so that tissue δ15N¼ Δδ15 N.
Please cite this article as: Remien, C.H., Modeling the dynamics of stable isotope tissue-diet enrichment. J. Theor. Biol. (2014), http://dx. doi.org/10.1016/j.jtbi.2014.11.018i
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C.H. Remien / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
80
N(0)
0.0
15
70
fraction protein in diet = 0.05 fraction protein in diet = 0.15 fraction protein in diet = 0.25
N(t)-
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0.0 fraction protein in diet = 0.05 fraction protein in diet = 0.15 fraction protein in diet = 0.25
55 50 2
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4 15
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Time (days)
N
Fig. 4. The deviation in δ15N caused by weight gain and loss is dependent upon the fraction of protein in the diet, with higher protein content in diet leading to lower deviation in δ15N. Dietary intake is set to δ15 Ndiet ¼ 0%.
Cases with only lipogenesis or gluconeogenesis follow similarly. If lipogenesis is occurring so that sGF 40 and there is no gluconeogenesis or amino acid synthesis, sGP ¼ sFG ¼ sPG ¼ 0, at steady state
-10 13
C
GP =1.005
-15
13
C
Dietary Carbohydrate
Tissue Protein
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
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GP=1
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GP =0.995
12
CF
13
CG
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CG
-20
-25
Dietary Protein
13
0.2
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1.0
sGP sGP + Ip Fig. 5. The steady state tissue protein δ13C is intermediate between the δ13C of ingested protein and carbohydrate and depends on the fraction of amino acids that are synthesized by the organism sGP =ðsGP þ I P Þ.
13
CP
12
CP
13
Rin;P I P þ αGP 12
¼
13
CF
12
CF
¼
sGF ;
Rin;G I C ; ðαGF αout;G ÞsGF þ αout;G I C
If there is no fractionation from amino acid synthesis or excretion so that αGP ¼ αout;G ¼ αout;P ¼ 1, then 13 CF =12 CF ¼ Rin;F , 13 CG =12 CG ¼ Rin;G and 13 CP =12 CP ¼ ðRin;P I p þ Rin;G sGP Þ=ðI P þ sGP Þ (Fig. 5). Similar to Martinez del Rio and Wolf (2005), the steady state values are intermediate between ingested protein and ingested glycogen, depending on the rate of amino acid synthesis, because of isotopic routing.
Rin;P I P
:
αout;P IP þ ð1 αout;P Þh
Rin;F I F
αout;F I F þ ðαFG αout;F ÞsFG 13
CG
sGP CG : ¼ 12 CP αout;P ðI P þ sGP Þ þ ð1 αout;P Þh CP
¼
CG αout;F ðI F þ sGF Þ
If there is no fractionation so that αout;P ¼ αout;G ¼ αout;F ¼ αGF ¼ 1, then at steady state 13 CF =12 CF ¼ ðRin;F I F þRin;G sGF Þ=ðI F þ sGF Þ, 13 CG = 12 CG ¼ Rin;G , and 13 CP =12 CP ¼ Rin;P . With gluconeogenesis from both fats and proteins but no amino acid synthesis or lipogenesis so that sPG 4 0, sFG 40 and sGF ¼ sGP ¼ 0, at steady state,
and 13
¼
CG
and
C
-30 0.0
CF
Rin;F I F þ αGF 12
13 12
CG CG
Rin;G I C þ αFG 12 ¼
CF
; 13
sFG þ αPG 12
CP
CF CP αout;G ðI C þ sFG þ sPG Þ
sPG ;
and 13
CP
12
CP
¼
Rin;P I P
:
αout;P IP þ ðαPG αout;P ÞsPG þ ð1 αout;P Þh
If there is no fractionation so that αout;P ¼ αout;G ¼ αout;F ¼ αFG ¼ αPG ¼ 1, then 13 CF =12 CF ¼ Rin;F , 13 CG =12 CG ¼ ðRin;G IC þ Rin;F sFG þ Rin;P sPG Þ=ðI C þ sFG þ sPG Þ, and 13 CP =12 CP ¼ Rin;P .
Please cite this article as: Remien, C.H., Modeling the dynamics of stable isotope tissue-diet enrichment. J. Theor. Biol. (2014), http://dx. doi.org/10.1016/j.jtbi.2014.11.018i
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4. Discussion Time-varying estimates of diet can be constructed by sequentially measuring the stable isotope ratios of directionally growing tissues such as hair, baleen, nails, and teeth as well as through repeated measurement of tissues that can be sampled nonlethally, such as blood plasma (Martinez del Rio et al., 2009; Wolf et al., 2009; Crawford et al., 2008; Cerling et al., 2009). Dynamic models are necessary to understand how changes in diet and metabolism affect the incorporation of dietary stable isotopes into animal tissues over time (Martinez del Rio and Carleton, 2012). We built a whole organism dynamic model of isotope incorporation by coupling stable isotope dynamics to an existing model of macronutrient dynamics. The model predicts how the time course of body mass and the stable isotope ratio of tissues vary with diet intake and composition and provides a conceptual framework to begin to elucidate the effects of diet and physiology on stable isotope ratios of animal tissues. The model tracks the fate of stable isotopes as they are ingested, incorporated into tissues, and excreted. Nitrogen isotope dynamics are determined by dietary intake and energy expenditure. Changes in protein and energy intake are predicted to affect stable isotope ratios of animal tissue, even with constant stable isotope composition of diet. The model is clearly a simplification of reality, assuming a single pool of fat, protein, and glycogen. Stable isotope turnover rates of individual tissues will vary from the tissue protein turnover rate predicted by the model. The effects of changes in protein and energy intake may be accentuated in some tissues, such as those with rapid turnover rates. Different tissues vary in Δδ15 N, and amino acid synthesis, degradation, and transport may further mix stable isotopes within an animal and complicate the temporal dynamics of isotopes in tissues. Because different tissues vary in turnover rate, the isotopic composition of tissues with different turnover rates may provide information regarding the timing of diet history (Martinez del Rio et al., 2009). There is often high total protein turnover (c.a. 300 g/day for humans) but only modest protein intake and excretion (c.a. 75 g/day for humans) which implies a high level of recycling of amino acids within and between tissues to supply amino acids for protein synthesis. The model suggests that isotopic turnover rates depend not just on the intrinsic turnover rate of the tissue, but also on the diet, and this is corroborated by a recent study in rats (Braun et al., 2013). Carbon isotope dynamics are more complicated than nitrogen because carbon is found in all three macronutrients. The carbon from all dietary sources is not likely to be perfectly mixed, and this is termed isotopic routing (Martinez del Rio and Carleton, 2012). For example, only carbon atoms from ingested essential amino acids are incorporated into essential amino acids in tissue protein, since these amino acids cannot be synthesized by the organism using carbon from other sources. Carbon in non-essential amino acids may have been derived from ingested protein or may have been synthesized from carbon that was ingested as carbohydrates or fat, depending on the diet and nutritional requirements of the animal. There are many opportunities for potential fractionation in conversions between macronutrients. If the fractionation factors are small, then at steady state the stable isotope signature of one macronutrient compartment is predicted to be a mix of the carbon stable isotope ratios of ingested macronutrients. For example, the carbon stable isotope ratio of tissue protein may be a mix between the carbon isotope ratio of ingested protein and that of fats and carbohydrates that are incorporated into amino acids through amino acid synthesis. There may be additional information in sampling the stable carbon isotope ratios of animal fat, glycogen, and protein because if they vary, conversions between macronutrient compartments may have a large effect on tissue protein stable isotope ratios.
Amino acids vary in Δδ15 N (Macko et al., 1986; McClelland and adn Montoya, 2002), and the variation is likely caused by differences in the biochemistry of amino acid metabolism (Chikaraishi et al., 2007). A quantitative, mechanistic understanding of how Δδ15 N varies with amino acid intake will require a model based on ingestion and metabolism of individual amino acids, including transfers of nitrogen between amino acids. More detailed models could also be developed to account for fluxes between various tissue compartments. Ideally, models should be developed to the level of measurements, and amino acid level models will be needed as amino acid level stable isotope measurements become more common. Protein quality, defined as the relative abundance of the most limiting amino acid, is also an important determinant of Δδ15 N (Florin et al., 2011). It is thought that as protein quality decreases, Δδ15 N increases. This could possibly be explained by a mismatch in the amino acid profile of diet and tissue needs. If the amino acid profile of diet has an excess of some amino acids, amino acid catabolism of these amino acids may increase. Caut et al. suggest that stable isotope discrimination factors vary with diet stable isotope value, though admit a lack of theoretical foundation (Caut et al., 2009). While this may be empirically true, stable isotope ratios of diet are not likely mechanistically driving changes in stable isotope discrimination. Instead, the mechanisms driving stable isotope discrimination are likely related to metabolism, tissue turnover, isotopic routing, variation in food macronutrient composition, heterogeneity in source isotope ratios, or some other source (Auerswald et al., 2010; Codron et al., 2012). Because of their intimate relationship with metabolism, tissue stable isotope ratios are related to human health, disease, and nutrition (Petzke and Fuller, 2012; Reitsema, 2013). Changes in nitrogen balance during malnutrition, growth, and pregnancy are recorded in tissue δ15N (Fuller et al., 2004, 2005). Tissue δ15N is related to animal protein intake and has been suggested as a possible biomarker for long term exposure to specific nutrients and nutritional habits related to disease (Petzke et al., 2010; Nash et al., 2009). Amino acid level measurements may again provide more detailed information but measurements are still relatively uncommon in the context of health and disease (Petzke and Fuller, 2012). Models describing stable isotope dynamics of animal tissues can be classified as forward and inverse models. Forward models determine the stable isotope ratio of animal tissue for given diet and energy dynamics. Previously developed forward models have been used to predict the influence of protein intake (Martinez del Rio and Wolf, 2005; Schoeller, 1999; Balter et al., 2006) and energy metabolism (Pecquerie et al., 2010) on the stable isotope ratios of animal tissues. The model developed here is also a forward model, and builds on previous models by connecting stable isotopes to a model of macronutrient energy dynamics, explicitly including changes in body composition associated with body weight change and variability in diet composition. The model is quite general, as all animals ingest, use for energy, and are composed of the three macronutrients fat, carbohydrate, and protein. To parameterize the model for a specific animal, one must estimate γ, the lean-fat body composition curve that is specific to the animal, and E, the energy expenditure, which scales with body mass and activity level. Fractionation factors in excretion and conversion between macronutrient compartments may also vary between organisms, but this variation may be small because of conserved biochemical reactions. Forward tissue stable isotope models are typically mechanistic and attempt to account for relevant features of physiology and ecology. They are useful to characterize how dietary and physiological variation affect incorporation of stable isotope ratios to tissue. Inverse models and statistics are used to better interpret stable isotope measurements of animal tissue. Linear mixing models are used to estimate the relative contribution of multiple sources to diet assuming steady state conditions (Moore and Semmens, 2008; Parnell
Please cite this article as: Remien, C.H., Modeling the dynamics of stable isotope tissue-diet enrichment. J. Theor. Biol. (2014), http://dx. doi.org/10.1016/j.jtbi.2014.11.018i
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et al., 2010; Phillips et al., 2014). Carbon and nitrogen tissue-diet discrimination are key parameters in these models, as well as models to estimate trophic level and food chain length (Post et al., 2000; Post, 2002). Stable isotope models that are used to estimate diet composition assume simple functional forms, while mechanistic forward models of stable isotope dynamics often have more parameters but account for additional factors that affect stable isotope discrimination dynamics. If there is sufficient knowledge of the study system, it may be possible to use a more detailed mechanistic forward model such as the one presented here rather than a linear mixing model to estimate diet. In cases where there is not sufficient knowledge of the study system to parameterize a detailed model, mechanistic forward models may be useful to determine the uncertainty in parameters in simpler models, such as stable isotope discrimination of linear mixing models, or to assess how deviation from assumptions such as the steady state assumption of linear mixing models affects stable isotope discrimination. Beyond their utility in data analysis, mechanistic models provide a conceptual framework, incorporating key elements such as turnover, isotopic routing, and fractionation, to elucidate the effects of diet and physiology on the stable isotope ratios of animal tissues. Funding C.H.R. was funded as a University of Utah Research Training Group Fellow through NSF award EMSW21-RTG and as a Postdoctoral Fellow at the National Institute for Mathematical and Biological Synthesis, an Institute sponsored by the National Science Foundation, the U.S. Department of Homeland Security and the U.S. Department of Agriculture through NSF Award EF-0832858, with additional support from the University of Tennessee, Knoxville. References Alpert, S.S., 1979. A two-reservoir energy model of the human body. Am. J. Clin. Nutr. 32, 1710–1718. Antonetti, V.W., 1973. The equations governing weight change in human beings. Am. J. Clin. Nutr. 26, 64–71. Auerswald, K., Wittmer, M.H.O.M., Zazzo, A., Schäufele, R., Schnyder, H., 2010. Biases in the analysis of stable isotope discrimination in food webs. J. Appl. Ecol. 47, 936–941. Balter, V., Simon, L., Fouillet, H., Lecuyer, C., 2006. Box-modeling of δ15N/δ14N in mammals. Oecologia 147, 212–222. Bond, A.L., Diamond, A.W., 2011. Recent Bayesian stable-isotope mixing models are highly sensitive to variation in discrimination factors. Ecol. Appl. 21 (4), 1017–1023. Braun, A., Auerswald, K., Vikari, A., Schnyder, H., 2013. Dietary protein content affects isotopic carbon and nitrogen turnover. Rapid Commun. Mass Spectrom. 27, 2676–2684. Caut, S., Angulo, E., Courchamp, F., 2008. Caution on isotopic model use for analyses of consumer diet. Can. J. Zool. 86, 438–445. Caut, S., Angulo, E., Courchamp, F., 2009. Variation in discrimination factors (δ15N and δ13C): the effect of diet isotopic values and applications for diet reconstruction. J. Appl. Ecol. 46, 443–453. Cerling, T.E., Harris, J.M., 1999. Carbon isotope fractionation between diet and bioapatite in ungulate mammals and implications for ecological and paleoecological studies. Oecologia 120, 347–363. Cerling, T.E., Wittemyer, G., Ehleringer, J.R., Remien, C.H., Douglas-Hamilton, I., 2009. History of animals using isotope records (HAIR): a 6-year dietary history of one family of African elephants. Proc. Natl. Acad. Sci. U. S. A. 106, 8093–8100. Chacko, A., Cummings, J.H., 1988. Nitrogen losses from the human small bowel: obligatory losses and the effect of physical form of food. Gut 29, 809–815. Chikaraishi, Y., Kashiyama, Y., Ogawa, N.O., Kitazato, H., Ohkouchi, N., 2007. Metabolic control of nitrogen isotope composition of amino acids in macroalgae and gastropods: implications for aquatic food web studies. Mar. Ecol. Prog. Ser. 342, 85–90. Chow, C.C., Hall, K.D., 2008. The dynamics of human body weight change. PLoS Comput. Biol. 4 (3), e1000045. Codron, D., Sponheimer, M., Codron, J., Newton, I., Lanham, J.L., Clauss, M., 2012. The confounding effects of source isotopic heterogeneity on consumer-diet and tissue-tissue stable isotope relationships. Oecologia 169 (4), 939–953. Craig, H., 1954. Carbon 13 in plants and the relationship between carbon 13 and carbon 14 variations in nature. J. Geol. 2, 115–149.
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Please cite this article as: Remien, C.H., Modeling the dynamics of stable isotope tissue-diet enrichment. J. Theor. Biol. (2014), http://dx. doi.org/10.1016/j.jtbi.2014.11.018i
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Contents lists available at ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Modeling the dynamics of stable isotope tissue-diet enrichment Christopher H. Remien n University of Idaho, Department of Mathematics, Moscow, ID 83844, Russia
H I G H L I G H T S
Stable isotope ratios of animal tissue are used to estimate diet. Estimates rely on a predictable offset between isotope ratios of diet and tissue. The isotope ratio offset from diet to tissue depends on diet, metabolism. We model the incorporation of carbon and nitrogen isotopes in diet to tissues. Model predicts how diet and energy expenditure shape isotope ratios of tissue.
art ic l e i nf o
a b s t r a c t
Article history: Received 31 October 2013 Received in revised form 15 November 2014 Accepted 19 November 2014
Reconstructions of dietary composition and trophic level from stable isotope measurements of animal tissue rely on a predictable offset of stable isotope ratios from diet to tissue. Physiological processes associated with metabolism shape tissue stable isotope ratios, and as such the spacing between stable isotope ratios of diet and tissue may be influenced by processes such as growth, nutritional stress, and disease. Here, we develop a model of incorporation stable isotopes in diet to tissues by coupling stable isotope dynamics to a model of macronutrient energy metabolism. We use the model to explore the effect of changes in dietary intake, both composition and amount, and in energy expenditure, on body mass and carbon and nitrogen stable isotope ratios of tissue. & 2014 Published by Elsevier Ltd.
Keywords: Fractionation Metabolism Stable isotope ratio
1. Introduction Natural abundance stable isotope ratios of animal tissues are powerful indirect markers of diet and migration and are of particular importance when direct measurements are impossible or impractical (Martinez del Rio et al., 2009). Through metabolism, ingested nutrients are incorporated into tissues, recording and integrating the carbon, nitrogen, oxygen, hydrogen, and sulfur isotopic signal of diet. Animal diets can be highly variable, and if dietary sources vary in stable isotope ratios, tissue stable isotope ratios can provide a means to quantify changes in dietary sources over space and time. The stable isotope ratios of dietary sources are the major determinant of the stable isotope ratios of consumer tissues, but isotopes do not act as pure tracers, even with constant source isotopic composition. Oxygen and hydrogen stable isotope ratios in proteinaceous tissues such as hair are primarily determined by the stable isotope ratio of drinking water (Ehleringer et al., 2008), but the quantity of ingested water affects the stable isotope shift from
n
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drinking water to tissue (O'Grady et al., 2010). Carbon and nitrogen stable isotope ratios in tissue are primarily determined by the stable isotope ratio of dietary sources, but physiological and metabolic processes can lead to a difference between the stable isotope ratio of consumer tissue and diet, termed consumer tissue-diet discrimination (Vanderklift and Ponsard, 2003). Nitrogen consumer tissue-diet discrimination, Δδ15 N, is typically about 3–4‰ and has been used to estimate trophic position and food chain length (Post et al., 2000; Post, 2002). Carbon consumer tissue-diet discrimination is variable, typically about 0–1‰ (Caut et al., 2009), with variations in lipid content (Post et al., 2007) and amino acid composition (O'Brien et al., 2005) affecting the carbon stable isotope ratio of the tissue. Unfortunately, variability in stable isotope discrimination leads to uncertainties when estimating diet and trophic position using stable isotopes (Caut et al., 2009, 2008; Bond and Diamond, 2011). Linear mixing models that are used to reconstruct the relative contribution of dietary sources from stable isotope measurements rely on predictable consumer tissue-diet discrimination to create meaningful reconstructions (Phillips, 2012). Isotopic discrimination depends on the details of macronutrient metabolism. Δδ15 N likely results from the preferential excretion of 14N due to fractionation during transamination and deamination of amino acids (Macko et al., 1986; Chikaraishi
http://dx.doi.org/10.1016/j.jtbi.2014.11.018 0022-5193/& 2014 Published by Elsevier Ltd.
Please cite this article as: Remien, C.H., Modeling the dynamics of stable isotope tissue-diet enrichment. J. Theor. Biol. (2014), http://dx. doi.org/10.1016/j.jtbi.2014.11.018i
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et al., 2007). Interesting feedbacks exist between Δδ15 N and dietary protein quality and quantity that depend on physiology, metabolism, and physiological state (Martinez del Rio et al., 2009; Robbins et al., 2010). Carbon consumer tissue-diet discrimination, Δδ13 C, varies with amino acid, with the δ13C of essential amino acids tracking that of dietary protein and the δ13C of non-essential amino acids reflecting a mix of the δ13C of dietary protein and the δ13C of nonessential amino acids that were synthesized using carbon that originated from other sources such as carbohydrates or fat (O'Brien et al., 2005). A quantitative understanding of the mechanisms of stable isotope discrimination will aid diet and trophic level reconstructions. Furthermore, tissue stable isotope ratio measurements may become a useful proxy for metabolic and nutritional state if the mechanisms controlling discrimination are better understood. Here, we present a model of macronutrient energy metabolism that has been adapted to study how stable isotope ratios of tissues vary with changes in body weight, body composition, and diet. The model describes the basic macronutrient fluxes of an animal. Isotopes in macronutrients are tracked as they are ingested, incorporated into tissues, and excreted. The model predicts trajectories for body composition and tissue stable isotope ratios following changes in intake and expenditure.
2. Energy partition stable isotope model Changes in stored energy are determined by energy balance. When total energy intake is greater than energy expenditure, energy is stored, while stored energy is lowered when energy intake is less than expenditure. Stored energy can be converted to body mass by accounting for energy density. Mathematical models have a long history in the study of energy metabolism and body weight change (Antonetti, 1973; Alpert, 1979; Guo and Hall, 2009). We will study the influence of energy metabolism on stable isotope dynamics by tracking stable isotope dynamics in the context of the macronutrient energy partition model developed by Chow and Hall (2008).
compartments, such as gluconeogenesis and lipogenesis, to be accounted for in the functional forms of fF and fC. Nitrogen is found in protein, but not in fat or glycogen. Fractionation of nitrogen is thought to occur with deamination during catabolism so that 15N is disproportionally retained relative to 14N (Macko et al., 1986; Chikaraishi et al., 2007). 14N follows the dynamics of body protein so that 14 N ¼ κ P, where κ ¼ 0:15 is the fraction of nitrogen in protein. 15N also closely follows protein trajectories, but with fractionation in deamination:
ρP =κ
The parameter Rin is the molar ratio of 15N to 14N in dietary protein, and α accounts for the fractionation associated with deamination. The losses in the 15N equation reflect fractionation from deamination in protein energy metabolism, but no fractionation with losses of intact amino acids and proteins. Following simplifications in Chow and Hall (2008), we define lean mass as L ¼ M F where M is total mass, so that dL dP dG ¼ ð1 þ hP Þ þ ð1 þ hG Þ ; dt dt dt with hydration coefficients for intracellular water associated with protein and glycogen hP ¼1.6 and hG ¼ 2.7, respectively. The small storage capacity of glycogen allows the model to be reduced by taking dG=dt ¼ 0, so that f C ¼ I C =E, P ¼ ðL cÞ=ð1 þ hp Þ, and 14 N ¼ κ ðL cÞ=ð1 þ hp Þ, where the mass c is the lean mass that is not associated with protein. Again following Chow and Hall (2008), we assume L F trajectories for further reduction of the macronutrient model. Let ðρF =ρL ÞdF=dL ¼ γ ðF; LÞ. The function γ varies for different species and can be approximated by cross-sectional measurements. Dividing dF=dt by dL=dt, imposing L F trajectories, and solving for the fraction of energy expenditure derived from metabolism of fat, f F ¼ I F =E ðγ ð1 þ γ ÞÞðI E hÞ=E, where I ¼ I F þ I C þ I P . With trajectories imposed, the nitrogen stable isotope model reduces to the two dimensional system
ρL
2.1. Nitrogen stable isotope discrimination Tracking the dynamics of the three primary macronutrients fat, F, glycogen, G, and protein, P, leads to flux balance equations:
ρF
dF ¼ IF f F E dt dG ρG ¼ IC f C E dt dP ρP ¼ IP ð1 f F f C ÞE h: dt
The equations for fat, glycogen, and protein are almost identical to those used by Chow and Hall (2008), but, following Martinez del Rio and Wolf (2005), have an additional protein loss term, h, that accounts for protein and amino acids that are lost without deamination, such as metabolic fecal nitrogen, hair, and shed skin. The rate of intact protein and amino acid loss is typically small compared to the excretion rate of nitrogen from amino acid catabolism, but it is potentially important in influencing Δδ15 N. The parameters ρF ¼ 9400 kcal=kg, ρG ¼ 4200 kcal=kg, and ρP ¼ 4700 kcal=kg (Chow and Hall, 2008) represent the energy densities of fat, glycogen, and protein, respectively, and IF, IC, and IP are the energy intake rates of fat, carbohydrates, and protein, respectively. The fractions of energy expenditure derived from metabolism of fat, carbohydrates, and protein, fF, fC, and 1 f F f C , respectively, are functions of body composition and intake rates. For now, consider conversions between body macronutrient
15 15 d15 N N N ¼ Rin I P α14 1 f F f C E 14 h: dt N N
dL 1 ¼ ðI E hÞ dt 1 þ γ
ρP =κ
15 15 d15 N N γ N ¼ Rin I P α14 E ðI I P Þ þ ðI E hÞ 14 h dt 1þγ N N
where ρL ¼ ρP =ð1 þ hP Þ ¼ 1800 kcal=kg. 2.2. Carbon stable isotope discrimination We track the dynamics of the three primary macronutrients fat, F, glycogen, G, and protein, P, as we did above with nitrogen, but because carbon is found in all three primary macronutrients, we must explicitly consider conversions between macronutrient compartments:
ρF
dF ¼ I F þ sGF sFG f F E dt dG ρG ¼ I C sGF þ sFG sGP þ sPG f C E dt dP ρP ¼ IP þ sGP sPG ð1 f F f C ÞE h: dt
The parameters sGF, sFG, sGP, and sPG represent the rate of lipogenesis, gluconeogenesis from fat, amino acid synthesis, and gluconeogenesis from protein, respectively. Just as in the 15N model, 13C follows the macronutrient dynamics so that 12 CF ¼ κ F F, 12 CG ¼ κ G G, and 12 CP ¼ κ P P where κ F , κ G , and κ P are the fraction of carbon in fat, glycogen, and protein, respectively. The 13C in fat, glycogen, and protein closely track the
Please cite this article as: Remien, C.H., Modeling the dynamics of stable isotope tissue-diet enrichment. J. Theor. Biol. (2014), http://dx. doi.org/10.1016/j.jtbi.2014.11.018i
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C.H. Remien / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 12 C trajectories, but with potential fractionation in conversions between macronutrient compartments and excretion:
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13 13 13 C C C C ρF =κ F d F ¼ Rin;F IF þ αGF 12 G sGF αFG 12 F sFG αout;F 12 F f F E dt CG CF CF
ρG =κ G d
13 13 13 CG C C C ¼ Rin;G I C αGF 12 G sGF þ αFG 12 F sFG αGP 12 G sGP dt CG CF CG 13
þ αPG 12
ρP =κ P d
CP CP
13
sPG αout;G 12
CG CG
f CE
13
13 13 CP C C ¼ Rin;P I P þ αGP 12 G sGP αPG 12 P sPG dt CG CP 13
αout;P 12
CP CP
13
ð1 f F f C ÞE 12
CP CP
instead of enrichment, defined as
ΔδX tissue diet ¼ δX tissue δX diet : It can be shown that ΔδX tissue diet ¼ ϵtissue diet only when Rdiet ¼ Rstandard so that though ΔδX tissue diet is convenient to calculate, it is an approximation that varies with the isotopic composition of the diet, whereas ϵntissue diet is independent of the isotopic composition of the diet (Auerswald et al., 2010). Throughout the modeling, we use differential equations to track the amounts of heavy and light isotope in tissue as they vary dynamically. The ratio of heavy to light isotope in tissue, the stable isotope ratio of tissue in delta notation, the apparent fractionation factor, the apparent enrichment, and the apparent discrimination factor can be easily calculated from the amounts of heavy and light isotope in tissue.
13
h:
The parameters Rin;F , Rin;G , and Rin;P are the molar ratio of 13C to C in dietary fat, glycogen, and protein, respectively. The parameters αGF , αFG , αGP , and αPG account for fractionation in lipogenesis, gluconeogenesis from fat, amino acid synthesis, and gluconeogenesis from protein, respectively, and αout;F , αout;G , and αout;P account for fractionation from catabolism of fat, glycogen, and protein, respectively. 12
2.3. Notation 2.3.1. Delta notation Stable isotope ratios are commonly reported as parts per million relative to a standard using δ notation: Rsample δX ¼ 1 n1000; Rstandard where Rsample and Rstandard are the molar ratios of heavy to light isotope in the sample and standard, respectively. For nitrogen, Rsample ¼ 15 N=14 N, and the established standard is air with Rstandard ¼ 0:0036765. For carbon, Rsample ¼ 13 C=12 C, and the established standard is Pee Dee Belemnite with Rstandard ¼ 0:0112372. 2.3.2. Isotope enrichment, discrimination Following established terminology (Passey et al., 2005; Cerling and Harris, 1999; Craig, 1954), an apparent fractionation factor is defined as
αntissue diet ¼ Rtissue =Rdiet ¼ ð1000 þ δX tissue Þ=ð1000 þ δX diet Þ: The superscript asterisk indicates that isotopic equilibrium is not assumed. The apparent isotope enrichment is given by
ϵntissue diet ¼ ðαntissue diet 1Þn1000: For convenience, many studies, including common linear mixing models (Phillips et al., 2014), use a discrimination factor 8
3.1. Nitrogen stable isotopes At steady state with dL=dt ¼ 0 and d15 N=dt ¼ 0, the model has a unique steady state where E ¼ I h and 15 N=14 N ¼ Rin I P =ðαI P þ ð1 αÞhÞ, so that αntissue diet ¼ I p =ðαI P þ ð1 αÞhÞ and ϵntissue diet ¼ ð1 αÞðI P hÞ=ðαI P þð1 αÞhÞ. With the fraction of nitrogen excreted as urea defined at steady state as ðI P hÞ=I P , the nitrogen stable isotope enrichment increases approximately linearly with ðI P hÞ=I P . Because the model structure assumes a single pool of protein with one fractionated and one unfractionated outflow, the nitrogen stable isotope ratio steady state is the same as in Martinez del Rio and Wolf (2005) (Fig. 1A). With h and α fixed, 15 N=14 N increases with protein intake, but quickly saturates. For parameters estimated for a 65 kg female human so that protein loss without deamination is h ¼ 35 kcal=day (estimated from Chacko and Cummings, 1988) and α ¼ 0:995 (estimated from Macko et al., 1986; Chikaraishi et al., 2007) a diet consisting of 50 g/day of protein results in δ15 N ¼ 4:1‰ while a high protein diet of 150 g/day results in δ15 N ¼ 4:7‰ (Fig. 1B). In addition to steady state values, the model also predicts time courses of body mass and tissue nitrogen stable isotope ratio following a change in diet or energy expenditure. To predict nitrogen stable isotope time courses, we need estimates for γ and energy expenditure. Following Chow and Hall (2008), for humans we can utilize cross-sectional data, imposing L F trajectories specified by cross-sectional data of body weights of women, γ ¼ ðρF =ρL ÞF=10:4, so that F ¼ DexpðL=10:4Þ (Forbes, 1987). Energy expenditure consists of basal metabolic rate and energy expended in physical activity. In principle, energy expenditure is a function of body composition, macronutrient intake rates, and physical
5
N
4
4
15
N
3. Results
=0.992 =0.995 =0.998
6 15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
3
3 2
2
h=4.4 g/day h=8.8 g/day h=17.5 g/day
1
0
0 0.0
0.2
0.4
0.6
(Ip h) Ip
0.8
1.0
0
50
100 150 200 250 300
protein intake (g/day)
Fig. 1. Steady state tissue δ15N is determined by the fraction of nitrogen excreted as urea, which at steady state is ðI P hÞ=I P . For fixed h, the rate of protein loss without deamination (e.g., metabolic fecal nitrogen, shed skin and hair), δ15N increases with protein intake and quickly saturates. Dietary intake is set to δ15 Ndiet ¼ 0‰ so that tissue δ15N¼ Δδ15 N.
Please cite this article as: Remien, C.H., Modeling the dynamics of stable isotope tissue-diet enrichment. J. Theor. Biol. (2014), http://dx. doi.org/10.1016/j.jtbi.2014.11.018i
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activity. We take energy expenditure from the Katch–McArdle formula with E ¼ β ð370 þ21:6LÞ where the parameter β has a value between 1.2 and 1.9 and accounts for a person's physical activity level (Katch et al., 1996). A phase plane for body mass and δ15N is shown in Fig. 2 with dietary intake of I ¼ 2050 kcal=day, mass of lean mass not associated with protein c¼2 kg, rate of protein loss without deamination h¼35 kcal/day (estimated from Chacko and Cummings, 1988), physical activity constant β ¼ 1:5, fractionation factor from deamination α ¼ 0:995 (estimated from Macko et al., 1986; Chikaraishi et al., 2007), and constant D¼0.26. 15N does not affect body composition dynamics, so the L-nullcline is a horizontal line. Body composition dynamics are entirely determined by the energy partition model, with Lean-Fat trajectories specified by Forbes' curve. With Lean-Fat trajectories specified by Forbes' curve and energy expenditure by the Katch–McArdle formula, body composition
dynamics affect δ15N dynamics, and hence observed enrichment (Figs. 3 and 4). During weight loss, energy intake is less than expenditure and stored lean mass is mobilized for energy use. Nitrogen in lean mass that is metabolized for energy is fractionated from deamination and is excreted as urea. The fractionation from deamination initially leads to higher δ15N during weight loss, but as the mass equilibrates to its new steady state, δ15N decreases to its steady state value. Similarly, during weight gain, intake is greater than expenditure so that more lean mass is stored than used for energy compared to steady state conditions. Nitrogen follows protein dynamics, and, because fractionation occurs with deamination, growth initially leads to lower δ15N values than at steady state, before increasing to the steady state value. The change in δ15N due to weight gain or loss is due to N imbalance and thus depends on diet composition (Fig. 4). If protein intake is high, δ15N closely tracks the steady state value while larger deviations occur if protein intake is low. In the model, an increase of 150 kcal/day in energy intake leads to a nearly 0:5‰ deviation in Δδ15 N with low protein intake of 26 g/day, five percent of the total caloric intake.
3.2. Carbon stable isotopes Because carbon is found in all three of the macronutrients fat, glycogen, and protein, there is potential for fractionation not just in ingestion and excretion, but also in conversions between macronutrient compartments. Consider the case with a diet that contains sufficient energy intake, so that lipogenesis and gluconeogenesis are low, sGF ¼ sFG ¼ sPG ¼ 0, but low in some nonessential amino acids, so that sGP 4 0. Then at steady state, 13 CF =12 CF ¼ Rin;F =αout;F , 13
CG
12
CG
78
60
5.0 4.8
74 72
4.6
70
4.4
68 50
N
70
Rin;G I C ; ðαGP αout;G ÞsGP þ αout;G I C
Mass (kg) Tissue 15N
76 Mass (kg)
Mass (kg)
80
¼
15
Fig. 2. Nullclines and possible trajectories in the body mass-tissue δ15N phase plane. Dietary intake is set to δ15 Ndiet ¼ 0‰ so that tissue δ15 N ¼ Δδ15 N.
4.2
66 4.0 2.5
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65
Mass (kg)
Mass (kg)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
4.0
45 2.5
3.0
3.5
4.0 15
N
4.5
5.0
5.5
0
200
400
600
800
1000
Time (days)
Fig. 3. Weight gain (weight loss) for a typical 65 kg woman caused by a 150 kcal/day increase (decrease) in dietary intake leads to a decrease followed by increase (increase followed by decrease) of δ15N. Protein intake is assumed to be 15% of calories consumed. Dietary intake is set to δ15 Ndiet ¼ 0‰ so that tissue δ15N¼ Δδ15 N.
Please cite this article as: Remien, C.H., Modeling the dynamics of stable isotope tissue-diet enrichment. J. Theor. Biol. (2014), http://dx. doi.org/10.1016/j.jtbi.2014.11.018i
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80
N(0)
0.0
15
70
fraction protein in diet = 0.05 fraction protein in diet = 0.15 fraction protein in diet = 0.25
N(t)-
65 60
-0.2
15
Mass (kg)
75
55
-0.4
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3
Tissue
4 15
0
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200
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600
800
1000
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1000
Time (days)
N
80 0.4
N(0) 15
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N(t)-
65 60
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Mass (kg)
75
0.0 fraction protein in diet = 0.05 fraction protein in diet = 0.15 fraction protein in diet = 0.25
55 50 2
3
Tissue
4 15
5
0
200
400
600
Time (days)
N
Fig. 4. The deviation in δ15N caused by weight gain and loss is dependent upon the fraction of protein in the diet, with higher protein content in diet leading to lower deviation in δ15N. Dietary intake is set to δ15 Ndiet ¼ 0%.
Cases with only lipogenesis or gluconeogenesis follow similarly. If lipogenesis is occurring so that sGF 40 and there is no gluconeogenesis or amino acid synthesis, sGP ¼ sFG ¼ sPG ¼ 0, at steady state
-10 13
C
GP =1.005
-15
13
C
Dietary Carbohydrate
Tissue Protein
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
5
13
GP=1
13
GP =0.995
12
CF
13
CG
12
CG
-20
-25
Dietary Protein
13
0.2
0.4
0.6
0.8
1.0
sGP sGP + Ip Fig. 5. The steady state tissue protein δ13C is intermediate between the δ13C of ingested protein and carbohydrate and depends on the fraction of amino acids that are synthesized by the organism sGP =ðsGP þ I P Þ.
13
CP
12
CP
13
Rin;P I P þ αGP 12
¼
13
CF
12
CF
¼
sGF ;
Rin;G I C ; ðαGF αout;G ÞsGF þ αout;G I C
If there is no fractionation from amino acid synthesis or excretion so that αGP ¼ αout;G ¼ αout;P ¼ 1, then 13 CF =12 CF ¼ Rin;F , 13 CG =12 CG ¼ Rin;G and 13 CP =12 CP ¼ ðRin;P I p þ Rin;G sGP Þ=ðI P þ sGP Þ (Fig. 5). Similar to Martinez del Rio and Wolf (2005), the steady state values are intermediate between ingested protein and ingested glycogen, depending on the rate of amino acid synthesis, because of isotopic routing.
Rin;P I P
:
αout;P IP þ ð1 αout;P Þh
Rin;F I F
αout;F I F þ ðαFG αout;F ÞsFG 13
CG
sGP CG : ¼ 12 CP αout;P ðI P þ sGP Þ þ ð1 αout;P Þh CP
¼
CG αout;F ðI F þ sGF Þ
If there is no fractionation so that αout;P ¼ αout;G ¼ αout;F ¼ αGF ¼ 1, then at steady state 13 CF =12 CF ¼ ðRin;F I F þRin;G sGF Þ=ðI F þ sGF Þ, 13 CG = 12 CG ¼ Rin;G , and 13 CP =12 CP ¼ Rin;P . With gluconeogenesis from both fats and proteins but no amino acid synthesis or lipogenesis so that sPG 4 0, sFG 40 and sGF ¼ sGP ¼ 0, at steady state,
and 13
¼
CG
and
C
-30 0.0
CF
Rin;F I F þ αGF 12
13 12
CG CG
Rin;G I C þ αFG 12 ¼
CF
; 13
sFG þ αPG 12
CP
CF CP αout;G ðI C þ sFG þ sPG Þ
sPG ;
and 13
CP
12
CP
¼
Rin;P I P
:
αout;P IP þ ðαPG αout;P ÞsPG þ ð1 αout;P Þh
If there is no fractionation so that αout;P ¼ αout;G ¼ αout;F ¼ αFG ¼ αPG ¼ 1, then 13 CF =12 CF ¼ Rin;F , 13 CG =12 CG ¼ ðRin;G IC þ Rin;F sFG þ Rin;P sPG Þ=ðI C þ sFG þ sPG Þ, and 13 CP =12 CP ¼ Rin;P .
Please cite this article as: Remien, C.H., Modeling the dynamics of stable isotope tissue-diet enrichment. J. Theor. Biol. (2014), http://dx. doi.org/10.1016/j.jtbi.2014.11.018i
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4. Discussion Time-varying estimates of diet can be constructed by sequentially measuring the stable isotope ratios of directionally growing tissues such as hair, baleen, nails, and teeth as well as through repeated measurement of tissues that can be sampled nonlethally, such as blood plasma (Martinez del Rio et al., 2009; Wolf et al., 2009; Crawford et al., 2008; Cerling et al., 2009). Dynamic models are necessary to understand how changes in diet and metabolism affect the incorporation of dietary stable isotopes into animal tissues over time (Martinez del Rio and Carleton, 2012). We built a whole organism dynamic model of isotope incorporation by coupling stable isotope dynamics to an existing model of macronutrient dynamics. The model predicts how the time course of body mass and the stable isotope ratio of tissues vary with diet intake and composition and provides a conceptual framework to begin to elucidate the effects of diet and physiology on stable isotope ratios of animal tissues. The model tracks the fate of stable isotopes as they are ingested, incorporated into tissues, and excreted. Nitrogen isotope dynamics are determined by dietary intake and energy expenditure. Changes in protein and energy intake are predicted to affect stable isotope ratios of animal tissue, even with constant stable isotope composition of diet. The model is clearly a simplification of reality, assuming a single pool of fat, protein, and glycogen. Stable isotope turnover rates of individual tissues will vary from the tissue protein turnover rate predicted by the model. The effects of changes in protein and energy intake may be accentuated in some tissues, such as those with rapid turnover rates. Different tissues vary in Δδ15 N, and amino acid synthesis, degradation, and transport may further mix stable isotopes within an animal and complicate the temporal dynamics of isotopes in tissues. Because different tissues vary in turnover rate, the isotopic composition of tissues with different turnover rates may provide information regarding the timing of diet history (Martinez del Rio et al., 2009). There is often high total protein turnover (c.a. 300 g/day for humans) but only modest protein intake and excretion (c.a. 75 g/day for humans) which implies a high level of recycling of amino acids within and between tissues to supply amino acids for protein synthesis. The model suggests that isotopic turnover rates depend not just on the intrinsic turnover rate of the tissue, but also on the diet, and this is corroborated by a recent study in rats (Braun et al., 2013). Carbon isotope dynamics are more complicated than nitrogen because carbon is found in all three macronutrients. The carbon from all dietary sources is not likely to be perfectly mixed, and this is termed isotopic routing (Martinez del Rio and Carleton, 2012). For example, only carbon atoms from ingested essential amino acids are incorporated into essential amino acids in tissue protein, since these amino acids cannot be synthesized by the organism using carbon from other sources. Carbon in non-essential amino acids may have been derived from ingested protein or may have been synthesized from carbon that was ingested as carbohydrates or fat, depending on the diet and nutritional requirements of the animal. There are many opportunities for potential fractionation in conversions between macronutrients. If the fractionation factors are small, then at steady state the stable isotope signature of one macronutrient compartment is predicted to be a mix of the carbon stable isotope ratios of ingested macronutrients. For example, the carbon stable isotope ratio of tissue protein may be a mix between the carbon isotope ratio of ingested protein and that of fats and carbohydrates that are incorporated into amino acids through amino acid synthesis. There may be additional information in sampling the stable carbon isotope ratios of animal fat, glycogen, and protein because if they vary, conversions between macronutrient compartments may have a large effect on tissue protein stable isotope ratios.
Amino acids vary in Δδ15 N (Macko et al., 1986; McClelland and adn Montoya, 2002), and the variation is likely caused by differences in the biochemistry of amino acid metabolism (Chikaraishi et al., 2007). A quantitative, mechanistic understanding of how Δδ15 N varies with amino acid intake will require a model based on ingestion and metabolism of individual amino acids, including transfers of nitrogen between amino acids. More detailed models could also be developed to account for fluxes between various tissue compartments. Ideally, models should be developed to the level of measurements, and amino acid level models will be needed as amino acid level stable isotope measurements become more common. Protein quality, defined as the relative abundance of the most limiting amino acid, is also an important determinant of Δδ15 N (Florin et al., 2011). It is thought that as protein quality decreases, Δδ15 N increases. This could possibly be explained by a mismatch in the amino acid profile of diet and tissue needs. If the amino acid profile of diet has an excess of some amino acids, amino acid catabolism of these amino acids may increase. Caut et al. suggest that stable isotope discrimination factors vary with diet stable isotope value, though admit a lack of theoretical foundation (Caut et al., 2009). While this may be empirically true, stable isotope ratios of diet are not likely mechanistically driving changes in stable isotope discrimination. Instead, the mechanisms driving stable isotope discrimination are likely related to metabolism, tissue turnover, isotopic routing, variation in food macronutrient composition, heterogeneity in source isotope ratios, or some other source (Auerswald et al., 2010; Codron et al., 2012). Because of their intimate relationship with metabolism, tissue stable isotope ratios are related to human health, disease, and nutrition (Petzke and Fuller, 2012; Reitsema, 2013). Changes in nitrogen balance during malnutrition, growth, and pregnancy are recorded in tissue δ15N (Fuller et al., 2004, 2005). Tissue δ15N is related to animal protein intake and has been suggested as a possible biomarker for long term exposure to specific nutrients and nutritional habits related to disease (Petzke et al., 2010; Nash et al., 2009). Amino acid level measurements may again provide more detailed information but measurements are still relatively uncommon in the context of health and disease (Petzke and Fuller, 2012). Models describing stable isotope dynamics of animal tissues can be classified as forward and inverse models. Forward models determine the stable isotope ratio of animal tissue for given diet and energy dynamics. Previously developed forward models have been used to predict the influence of protein intake (Martinez del Rio and Wolf, 2005; Schoeller, 1999; Balter et al., 2006) and energy metabolism (Pecquerie et al., 2010) on the stable isotope ratios of animal tissues. The model developed here is also a forward model, and builds on previous models by connecting stable isotopes to a model of macronutrient energy dynamics, explicitly including changes in body composition associated with body weight change and variability in diet composition. The model is quite general, as all animals ingest, use for energy, and are composed of the three macronutrients fat, carbohydrate, and protein. To parameterize the model for a specific animal, one must estimate γ, the lean-fat body composition curve that is specific to the animal, and E, the energy expenditure, which scales with body mass and activity level. Fractionation factors in excretion and conversion between macronutrient compartments may also vary between organisms, but this variation may be small because of conserved biochemical reactions. Forward tissue stable isotope models are typically mechanistic and attempt to account for relevant features of physiology and ecology. They are useful to characterize how dietary and physiological variation affect incorporation of stable isotope ratios to tissue. Inverse models and statistics are used to better interpret stable isotope measurements of animal tissue. Linear mixing models are used to estimate the relative contribution of multiple sources to diet assuming steady state conditions (Moore and Semmens, 2008; Parnell
Please cite this article as: Remien, C.H., Modeling the dynamics of stable isotope tissue-diet enrichment. J. Theor. Biol. (2014), http://dx. doi.org/10.1016/j.jtbi.2014.11.018i
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et al., 2010; Phillips et al., 2014). Carbon and nitrogen tissue-diet discrimination are key parameters in these models, as well as models to estimate trophic level and food chain length (Post et al., 2000; Post, 2002). Stable isotope models that are used to estimate diet composition assume simple functional forms, while mechanistic forward models of stable isotope dynamics often have more parameters but account for additional factors that affect stable isotope discrimination dynamics. If there is sufficient knowledge of the study system, it may be possible to use a more detailed mechanistic forward model such as the one presented here rather than a linear mixing model to estimate diet. In cases where there is not sufficient knowledge of the study system to parameterize a detailed model, mechanistic forward models may be useful to determine the uncertainty in parameters in simpler models, such as stable isotope discrimination of linear mixing models, or to assess how deviation from assumptions such as the steady state assumption of linear mixing models affects stable isotope discrimination. Beyond their utility in data analysis, mechanistic models provide a conceptual framework, incorporating key elements such as turnover, isotopic routing, and fractionation, to elucidate the effects of diet and physiology on the stable isotope ratios of animal tissues. Funding C.H.R. was funded as a University of Utah Research Training Group Fellow through NSF award EMSW21-RTG and as a Postdoctoral Fellow at the National Institute for Mathematical and Biological Synthesis, an Institute sponsored by the National Science Foundation, the U.S. Department of Homeland Security and the U.S. Department of Agriculture through NSF Award EF-0832858, with additional support from the University of Tennessee, Knoxville. References Alpert, S.S., 1979. A two-reservoir energy model of the human body. Am. J. Clin. Nutr. 32, 1710–1718. Antonetti, V.W., 1973. The equations governing weight change in human beings. Am. J. Clin. Nutr. 26, 64–71. Auerswald, K., Wittmer, M.H.O.M., Zazzo, A., Schäufele, R., Schnyder, H., 2010. Biases in the analysis of stable isotope discrimination in food webs. J. Appl. Ecol. 47, 936–941. Balter, V., Simon, L., Fouillet, H., Lecuyer, C., 2006. Box-modeling of δ15N/δ14N in mammals. Oecologia 147, 212–222. Bond, A.L., Diamond, A.W., 2011. Recent Bayesian stable-isotope mixing models are highly sensitive to variation in discrimination factors. Ecol. Appl. 21 (4), 1017–1023. Braun, A., Auerswald, K., Vikari, A., Schnyder, H., 2013. Dietary protein content affects isotopic carbon and nitrogen turnover. Rapid Commun. Mass Spectrom. 27, 2676–2684. Caut, S., Angulo, E., Courchamp, F., 2008. Caution on isotopic model use for analyses of consumer diet. Can. J. Zool. 86, 438–445. Caut, S., Angulo, E., Courchamp, F., 2009. Variation in discrimination factors (δ15N and δ13C): the effect of diet isotopic values and applications for diet reconstruction. J. Appl. Ecol. 46, 443–453. Cerling, T.E., Harris, J.M., 1999. Carbon isotope fractionation between diet and bioapatite in ungulate mammals and implications for ecological and paleoecological studies. Oecologia 120, 347–363. Cerling, T.E., Wittemyer, G., Ehleringer, J.R., Remien, C.H., Douglas-Hamilton, I., 2009. History of animals using isotope records (HAIR): a 6-year dietary history of one family of African elephants. Proc. Natl. Acad. Sci. U. S. A. 106, 8093–8100. Chacko, A., Cummings, J.H., 1988. Nitrogen losses from the human small bowel: obligatory losses and the effect of physical form of food. Gut 29, 809–815. Chikaraishi, Y., Kashiyama, Y., Ogawa, N.O., Kitazato, H., Ohkouchi, N., 2007. Metabolic control of nitrogen isotope composition of amino acids in macroalgae and gastropods: implications for aquatic food web studies. Mar. Ecol. Prog. Ser. 342, 85–90. Chow, C.C., Hall, K.D., 2008. The dynamics of human body weight change. PLoS Comput. Biol. 4 (3), e1000045. Codron, D., Sponheimer, M., Codron, J., Newton, I., Lanham, J.L., Clauss, M., 2012. The confounding effects of source isotopic heterogeneity on consumer-diet and tissue-tissue stable isotope relationships. Oecologia 169 (4), 939–953. Craig, H., 1954. Carbon 13 in plants and the relationship between carbon 13 and carbon 14 variations in nature. J. Geol. 2, 115–149.
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Please cite this article as: Remien, C.H., Modeling the dynamics of stable isotope tissue-diet enrichment. J. Theor. Biol. (2014), http://dx. doi.org/10.1016/j.jtbi.2014.11.018i
