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Louisiana State UniversityLSU Digital Commons
LSU Doctoral Dissertations Graduate School
8-28-2018
High-Dimensional Isotope RelationshipsYuyang HeLouisiana State University and Agricultural and Mechanical College, [email protected]
Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations
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This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion inLSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please [email protected].
Recommended CitationHe, Yuyang, "High-Dimensional Isotope Relationships" (2018). LSU Doctoral Dissertations. 4695.https://digitalcommons.lsu.edu/gradschool_dissertations/4695
HIGH-DIMENSIONAL ISOTOPE RELATIONSHIPS
A Dissertation
Submitted to the Graduate Faculty of the
Louisiana State University and
Agricultural and Mechanical College
in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
in
The Department of Geology & Geophysics
by
Yuyang He
B.S., Ocean University of China, 2011
M.S., Ocean University of China, 2014
December 2018
ii
CONTENTS
ABSTRACT ................................................................................................................................... iii
CHAPTER 1. OVERALL INTRODUCTION ............................................................................... 1
CHAPTER 2. PREDICTING HIGH-DIMENSIONAL ISOTOPE RELATIONSHIPS FROM
DIAGNOSTIC FRACTIONATION FACTORS IN SYSTEMS WITH MASS TRANSFER ....... 5
2.1. Introduction .................................................................................................................... 5
2.2. NO3- Cycling Models ..................................................................................................... 9
2.3. Results and Discussions ............................................................................................... 12
2.4. Implications .................................................................................................................. 18
2.5. Conclusions .................................................................................................................. 21
2.6. Supporting Information ................................................................................................ 22
CHAPTER 3. IDENTIFYING APPARENT LOCAL STABLE ISOTOPE EQUILIBRIUM IN A
COMPLEX NON-EQUILIBRIUM SYSTEM ............................................................................. 39
3.1. Introduction .................................................................................................................. 39
3.2. Methods For Evaluating a System at Disequilibrium States ........................................ 41
3.3. Applications .................................................................................................................. 43
3.4. Implications and Future Works .................................................................................... 46
3.5. Conclusions .................................................................................................................. 47
CHAPTER 4. EQUILIBRIUM INTRAMOLECULAR ISOTOPE DISTRIBUTION IN LARGE
ORGANIC MOLECULES ........................................................................................................... 48
4.1. Introduction .................................................................................................................. 48
4.2. Method .......................................................................................................................... 53
4.3. Results .......................................................................................................................... 55
4.4. Discussions and Applications ....................................................................................... 58
4.5. Conclusions .................................................................................................................. 61
4.6. Support Information ..................................................................................................... 62
CHAPTER 5. OVERALL CONCLUSION .................................................................................. 67
REFERENCES ............................................................................................................................. 68
APPENDIX; COPYRIGHT INFORMATION ............................................................................. 81
VITA ............................................................................................................................................. 88
iii
ABSTRACT
High-dimensional isotope relationships describes the relationships of two or more element or
position-specific (PS) elements in the same molecule or ion. It provides us more powerful tools
to study reaction mechanisms and dynamics. Chapter 1 is about dual or multiple stable isotope
relationship on δ-δ (or δ'-δ') space. While temporal data sampled from a closed-system can be
treated by a Rayleigh Distillation Model (RDM), spatial data should be treated by a Reaction-
Transport Model (RTM). Here we compare the results of a closed-system RDM to a RTM for
systems with diffusional mass transfer by simulating the trajectories on nitrate's δ'18O-δ'15N space.
Our results highlight the importance of linking the underlying physical model to the plotted data
points before interpreting their high-dimensional isotope relationships. Chapter 2 proposed a
rigorous approach that can describe isotope distribution among biomolecules and their apparent
deviation from equilibrium state. Applying the concept of distance matrix in graph theory, we
propose that apparent local isotope equilibrium among a subset of biomolecules can be assessed
using an apparent fractionation difference (|Δα|) matrix. The application of |Δα| matrix can help
us to locate potential reversible reactions or reaction networks in a complex system like a
metabolic system. Chapter 3 calculated the equilibrium PS isotope composition for large organic
molecules. A prevailing idea is that each of the positions can reach equilibrium with each other,
if a reaction is fully reversible. However, such an equilibrium intramolecular isotope distribution
(Intra-ID) can only be achieved when every carbon atom of different positions exchange with
each other within a molecule. Equilibrium Intra-IDs (reduced partition function ratios, β) can
serve as a fixed reference for measured Intra-ID. The analysis of calculated PS 13β factors of
acetate and C16 fatty acid showed that equilibrium isotope effect can produce fatty acid with
alternating Intra-ID from disequilibrium precursors.
1
CHAPTER 1. OVERALL INTRODUCTION
Isotope composition can be reported using the classical δ-notation (𝛿 ≡ (𝑅𝑠𝑎𝑚𝑝𝑙𝑒
𝑅𝑟𝑒𝑓− 1)) or δ'-
notation (𝛿′ ≡ ln (𝑅𝑠𝑎𝑚𝑝𝑙𝑒
𝑅𝑟𝑒𝑓)), where R denotes the ratio between the number of moles of a rare
isotope and a common isotope of an element (e.g. 18R = 18O/16O for oxygen). The processes that
affect isotope compositions is described by isotope fractionation (isotope fractionation factor is defined as
𝛼𝐴−𝐵 ≡𝑅𝐴
𝑅𝐵, where A and B denote different phases in a reaction). Isotope composition and
fractionation describe the relationship of two isotopes of the same element and has been used to solve bio-
and geochemistry problems. Stable isotope composition and fractionation is a powerful tool that has
been applied to reconstruct paleo-environment, to trace global element cycling, to study reaction
kinetics, and to be used as geothermometers. Isotope composition provides unique information
from a higher-dimension that is not available from the study of element concentrations only.
In addition, high-dimensional isotope relationships provide us more powerful tools to study
reaction mechanisms and dynamics, which cannot be deciphered from a single isotope
composition. The isotope relationships of three isotope of an element are used to study mass
independent processes. For instance, Δδ33S (= δ33S - 0.515 * δ34S) anomaly of sulfur revealed the
rise of atmosphere oxygen at 2.3 Ga (1). The isotope relationships of two or more elements in the
same molecule ion can be used to characterize reaction mechanisms. For instance, the deviation
of Δδ18O and ΔδD relationship in water from the global meteoric water line (ΔδD = 8 * Δδ18O +
10) is used to study local evaporation (2).
The study of high-dimensional isotope relationship, including triple-isotope system (e.g.
mass independent isotope effect Δ17O for oxygen), clumped isotope effect (e.g. Δ47 for
carbonate), dual-element isotope relationship (e.g. global meteoric water line), and position-
2
specific isotope compositions (site preference of N2O), is starting. In my dissertation, I addressed
my attempt to understand high-dimensional isotope effect, focusing on dual- or multiple-element
isotope relationship and position-specific isotope compositions.
Dual- or multiple-element isotope relationship describes the isotope fractionation
relationship of two or more elements during a process. For ions or molecules with two elements,
like nitrate, sulfate, and perchlorate, dual-element isotope composition has been used to decode
systems, in which multiple reactions are involved. Position-specific isotope compositions study
the isotope relationship of the same element at different positions in the same molecule. Nitrogen
isotope site preference (SP) of nitrous oxide (NNO) uses the difference between δ15Nα (center
nitrogen) and δ15Nβ (terminal nitrogen) to study its formation pathways. If the Nα and Nβ have
independent sources and reaction paths, and there is no equilibration among them, we cannot
treat them as the same element in the molecule. In fact, the two nitrogens at different sites in the
mineral behave like two different elements, just like as if they were sulfur and oxygen in a
sulfate ion. In my understanding, these two sets of problems are both study of multiple unique
but coupled isotope system in the same ion or molecule. Therefore, in this dissertation, the two
problems are grouped together as a same kind of problem.
In section 1, I started from the isotope relationship between two elements in the same ion or
molecules. The stable isotope composition relationships of two or more elements in a molecule
or ion are often characteristic of specific biogeochemical pathways, e.g. reduction of oxyanions
or degradation of organic contaminants. Because of the complexity of system dynamics and
reaction kinetics, the isotope relationships between the two elements (Δδ18O/Δδ15N in case of
NO3-) can hardly be calibrated in laboratory, when multiple reactions are involved. However,
using calibrated α's for the involving reactions, we can use numerical model to predict the dual-
3
element isotope relationship. Current field practices often neglect the influence of mass transfer
and directly compare the observed dual-element isotope relationships with closed-system batch
experiments. The validity of such practice has not been carefully examined. It could result in
incorrect estimation of the degree of isotope fractionation and of the underlying mechanisms.
Therefore, in section 1, using a set of diagnostic α’s, rate constants, initial concentrations and
isotope compositions, we built a closed-system time-dependent Rayleigh Distillation Model and
a one-dimensional diffusion, steady-state, single-source Reaction Transport Model to predict the
changes of concentrations for 14N, 15N, 16O, 17O, and 18O of NO3-, NO2
- and NH4+ through time
(RDM) or depth (RTM). Using a revised nitrogen cycling model as an example, we show that the
degree of diagnostic isotope fractionation would be significantly underestimated but the dual
isotope slope of δ'18O-δ'15N slightly overestimated. Our analysis demonstrates that a trajectory in
dual or multiple isotope space is informative for the underlying fractionation mechanism only if
the physical processes are understood a priori.
The development of position-specific isotope analysis technique is rocketing, but the
understanding is not advancing in the same speed. Large organic molecules have multiple sites
for carbon, hydrogen, oxygen and other elements. The dual- or tri-isotope relationship can be
presented as a trajectory on 2D or 3D space. For higher dimension, such trajectory cannot be
visually interpreted. In section 2, |Δα| matrix has been introduced to describe the inter- or intra-
molecular isotope composition distribution of a system, and evaluating the deviation of the
system from its equilibrium state. Because the equilibrium isotope fractionation data is lacking
for position-specific isotope compositions, i.e. intramolecualr isotope distribution (Intra-ID), we
first apply |Δα| matrix on compound-specific isotope compositions, i.e. intermolecular isotope
4
distribution. We re-analyzed published data of different amino acids (AA) in potato and in green
alga.
The applications of |Δα| matrix and measured Intra-ID are based on the understanding the
equilibrium Intra-ID. Predicted reduced partition function ratio (RPFR or β factor) can give the
equilibrium isotope fractionation factor (αeq) between a PS carbon in a molecule and its atomic
form. In section 3, position-specific 13β factors for a set of organic molecules at 25℃ are
provided by using density functional theory (DFT) and Urey-Bigeleisen-Mayer model. Current
computational resources limit the ability of calculating RPFR for large organic molecules. Since
the vibrational frequencies of an atom can only be influenced by its close surroundings, we also
apply cluster model to the calculation of Coenzyme A. The result shows that cluster model
calculation provides accurate RPFRs for large organic molecules and drastically simplifies the
calculation for RPFR of large organic molecules. In addition, C16 fatty acid (FA) is used as an
example to discuss the Intra-ID of C16 FA produced in equilibrium from acetate. The case
illustrate that intra-ID of a molecule is only informative when the underlying fractionation
processes are independently determined.
5
CHAPTER 2. PREDICTING HIGH-DIMENSIONAL ISOTOPE
RELATIONSHIPS FROM DIAGNOSTIC FRACTIONATION FACTORS IN
SYSTEMS WITH MASS TRANSFER
2.1. Introduction
Two or more elements in an ion or molecule can each have their own stable isotope behavior that
is diagnostic of specific chemical or physical processes. These dual or multiple elemental isotope
relationships have been used to identify biodegradation pathways, characterize source and sink,
and uncover enzymatic reaction mechanisms of oxyanions and organic molecules. For instance,
an approximate linear relationship with a regression slope of ~1 exists on δ18O-δ15N space for
residual nitrate samples collected at various stages of laboratory denitrification experiments (3).
Batch experiments show that residual ClO4- collected at different stages of biodegradation
display a δ18O-δ37Cl regression slope of 2.5 (4). Different biodegradation pathways of benzene,
toluene, ethylbenzene, xylenes, 1,2-dichloroethane, and methyl tert-butyl ether have shown
distinctive δD-δ13C and/or δD-δ13C-δ37Cl trajectories from batch experiments as well (5-9).
These linear trajectories exist because the isotope fractionation factors (α’s) of two or more
elements in an ion or molecule behave in a way that is related. If the isotope ratios are sampled
from the residues in a closed-system at various times, a Rayleigh Distillation Model (RDM) can
accurately quantify the process, and a diagnostic, constant α value can be obtained via:
𝑅𝑡 𝑅0⁄ = 𝑓𝛼−1 (2.1)
in which Rt and R0 are isotope ratios of the residue at two separate times t and 0, f is the
ratio of molar abundance of the major isotopologue at time t over time 0, which is very close to
the fraction of whole sample left at time t over time 0. Because we know well when a RDM
should apply, we know what we need to measure in the experiment and what such determined α
means.
6
It is important to note that this diagnostic α may not be an intrinsic α. We reserve “intrinsic”
only for kinetic isotope effect (KIE) or equilibrium α (αeq) of a defined process. Apparent α value
(αap) can be obtained from observed data. It could describe a set of processes. Some enzymatic
reactions can be well-coordinated, resulting in a relative constant αap value under different
reaction conditions. Such a constant αap is not intrinsic, but it can be “diagnostic” for a set of
tightly-bonded processes.
Obviously, if we link two samples and measure their concentration and isotope composition
differences, we cannot assume a priori that these two data points are linked by a closed-system
Rayleigh Distillation process, and therefore a diagnostic αap obtained from the two samples using
Eq. 2.1 is not informative on the underlying processes that connect them, if any.
It has been reported that, if a system does not fit strictly a RDM, such as the case of a
concentration gradient in sediment pore water or ground water systems, plotting spatially
distributed data points using a RDM usually underestimates the degree of the diagnostic
fractionation (10-11). The reason for the underestimation is because a direct application of a
RDM to spatially distributed data neglects mass transfer via diffusion and/or advection that
occurs in, for example, laboratory sediment columns, natural water bodies, or seafloor sediment
porewater profiles. These open-system spatially distributed samples should be treated with a
Reaction-Transport Model (RTM).
As we know, a trajectory on δ-δ space by plotting residues collected at different stages from
a closed-system RDM can ultimately be linked to the relationships between diagnostic α’s
(Δδ18O/Δδ15N ≈ (18α-1)/(15α-1), in the case of NO3-) (11-12). Some researchers have noticed that
plotting spatially distributed samples on δ-δ space would yield the same trajectory as if the
spatial samples were snapshots of different temporal states in a closed-system RDM. For
7
instance, a relationship of Δδ18O/Δδ15N ≈1 has also been seen among NO3- samples collected at
various depths in seawater columns (13-14). Similarly, a relationship of Δδ18O/Δδ37Cl ≈2.6 has
been observed among samples collected from different sites in a ground water system (11).
Because of the apparently identical trajectories obtained from a RDM and a RTM, most in the
community often plotted mixed spatial and/or temporal data on δ-δ space and to deduce the
underlying diagnostic α relationships, which can reveal mechanisms of the processes of interest.
However, conceptually, spatial data should not be treated with a RDM, and the RDM deduced α
relationships should not be taken as the correct one a priori. This issue has been raised in
numerous publications and has been addressed to certain degrees in theoretical studies (10, 12,
15) as well as in parts of specific case studies (11). However, the impact on the slope value of a
trajectory on δ-δ space has not been examined. Such slope values are widely used in discussing
mechanisms in the literature (13-14, 16-36).
Furthermore, laboratory experiments and natural processes are rarely a single step process.
Taking oxyanion reduction as an example, a re-oxidation process often introduces a new flux of
the same oxyanion and results in an increase or a slower decrease of its concentration. For
instance, in freshwater column, the observed nitrate δ18O-δ15N trajectories for samples collected
at different depths commonly have a slope smaller than 1. Such deviation from 1 has been
interpreted as the result of different extents of nitrification and has been used to estimate relative
N fixation fluxes (14, 17, 37-39). Considering the widespread practice of mixed plotting of
spatial and/or temporal data on δ18O-δ15N space, this deviation from slope of 1 could be the
combination result of factors other than the nitrification flux alone. Without a careful
examination of the factors for deviation, especially for a system that has mass transfer, any
interpretation on the deviation is unfounded.
8
One way to approach this problem is to simulate processes by given a set of parameters,
such as reaction rates, initial NO3- concentrations and isotope compositions, and α’s to model
trajectories we obtained on δ-δ space for a system. Such “bottom-up” approach can help to reveal
the factors that affect the final trajectories on δ-δ space. For the nitrogen cycle, most of these
parameters are now available, thanks to research done by many groups over the years (3, 40-50).
Simulation work has been performed on nitrate δ18O-δ15N space, notably the Granger-Wankel’s
(GW) model (39) and Casciotti-Buchwald’s (CB) model (51), both being a RDM. However,
trajectories on nitrate δ18O-δ15N space for spatially different data points in systems influenced by
mass transfer have not been explored.
RTM has been applied to studies of SO42- reduction rate in sediment profile. REMAP
program is built to simulate isotope change in sediments (52), which has been applied to explain
the SO42- isotope profile in sediment porewater (53-54). To apply REMAP, the net reduction
rates at different depth are needed. Another RTM for SO42- reduction assumed SO4
2- reduction
rate as constant and simulated the δ18O and δ34S changes in sediments (55). The modeled results
are almost identical for two systems with different temperature, sedimentation rate, and SO42-
reduction rate. Thus, the two SO42- reduction RTM cases were only used as a supporting
evidence for the validity to apply RDM to data collected from RTM system (55).
In this study, giving a set of diagnostic α’s, k’s, initial concentrations and isotope
compositions, we built a closed-system time-dependent RDM and a one-dimensional diffusion,
steady-state, single-source RTM to predict the changes of concentrations for 14N, 15N, 16O, 17O,
and 18O of NO3-, NO2
- and NH4+ through time (RDM) or depth (RTM). We first highlight the
principal differences between the trajectories on δ'-δ' space generated by a RDM and a RTM,
9
mainly through a set of analytical solutions. Next, we explore and compare NO3- trajectories on
δ'-δ' space predicted by a RDM versus by a RTM using a revised nitrogen cycling model.
2.2. NO3- Cycling Models
Our N cycling model is adopted and revised from GW model (39). In brief, NO3- is reduced to
NO2- by denitrification (NAR). NO2
- is consumed by nitrification (NXR), during which NO2- is
oxidized to NO3-. NO2
- can be further reduced (NIR) to other nitrogen pool. NH4+ is consumed to
produce NO2- by aerobic ammonia oxidation (AMO) while being oxidized by anammox (AMX)
to other nitrogen pool (Fig. 2.1.).
Figure 2.1. Schematic of reaction pathways for Nitrogen cycling (Adapted from (15)).
Our model differs from GW or CB model in three aspects. 1) We considered both reaction
and diffusional mass transfer in a RTM, which should in principle be applied to samples
collected from different depths of water/ sediment pore water column, or surface/ ground streams
with single point source. 2) GW model assumed constant reaction rates (39). CB model
considered reaction rate as first-order kinetics with unchanged (38, 51) or decayed (14) rate
constants (k). Our model used CB model’s assumption and our k’s do not change over time. For
NAR, this assumption is valid under low NO3- concentration (56). In addition, k’s for forward
and backward nitrite-water exchange reactions (kexch.f and kexch.b) and oxygens of NO3- produced
10
from NO2- and H2O during NXR (kNXR and kNXR.H2O) are assumed to be the same in CB model.
Such assumptions will cause concentration imbalance. Therefore, in our model, we only gave
kexch.f and kNXR, from which kexch.b and kNXR.H2O are calculated respectively. Similarly, k’s for
ammonia oxidized by O2 and H2O during AMO (kAMO.H2O and kAMO.O2) are calculated from kAMO
in our model (2.6.3. Equations). And 3) our models include 17O, which can be used to predict
triple oxygen isotope effect. Due to a lack of further calibration, all processes are assumed to be
mass-dependent with 17θ = 0.528 (17θ≡ ln17α/ln18α in the case of triple oxygen, see 2.6.2.
Notations). Thus, we will only discuss the influence of diffusional mass transfer on the observed
slope on δ'17O-δ'18O space for NAR process. Since all 17θ’s are assigned to be 0.528, the
trajectories on δ'17O-δ'18O space are identical between a RDM and a RTM and under different
dynamics. Thus the δ'17O-δ'18O trajectory will not be discussed further in this paper. Finally, our
model script in R language is posted in 2.6.4. Model details and script in R language.
δ' notation is used here to simplify mathematical treatment (see 2.6.2. Notations), which are
calculated from predicted concentrations. In this paper, δ' notation refers to our results, and δ
notation refers to previous studies. δ'-δ' trajectories in this paper refer only to the trajectory of
NO3- data points. Because δ' ≈ δ, our RDM results are comparable with those of GW and CB
models.
Table 2.1. Parameters used in models
Parameters Descriptions Values Refs
[NO3-]ini Initial nitrate concentration 10 μmol/L
[NO2-]ini Initial nitrite concentration 0 μmol/L
[NH4+]ini Initial ammonium concentration 10 μmol/L
𝛿′15𝑁𝑁𝑂3−
𝑖𝑛𝑖 Initial nitrate 15N isotope composition 0 ‰
𝛿′18𝑂𝑁𝑂3−
𝑖𝑛𝑖 Initial nitrate 18O isotope composition 0 ‰
𝛿′15𝑁𝑁𝐻4+
𝑖𝑛𝑖 Initial ammonium N isotope composition 0 ‰
𝛿′18𝑂𝐻2𝑂 Water 18O isotope composition 0 to -20‰
(table cont’d.)
11
Parameters Descriptions Values Refs
𝛿′18𝑂𝑂2 O2 18O isotope composition 23 ‰
17θ Triple oxygen isotope component 0.528
kNAR Rate constant for denitrification (NAR) (day-
1)
10-3 (51)
kNXR Rate constant for nitrification (NXR) (day-1) = 0 to 2× kNAR
kNIR Rate constant for nitrite reduction (NIR)
(day-1)
= 1 to 100× kNAR
kAMO Rate constant for aerobic ammonia oxidation
(AMO) (day-1)
= 0 to 100× kNAR
kAMX Rate constant for anammox (AMX) (day-1) = 1 to 100× kNAR
kexch.f Rate constant for nitrite exchange with water,
forward reaction (day-1)
= 0 to 1000× kNAR See 2.6.3.
Equations 15αNAR NAR 15N isotope fractionation factor 0.985 (3) 18αNAR NAR 18O isotope fractionation factor = 15αNAR (3) 18αNAR.BR NAR branching 18O isotope fractionation
factor
0.975 (49)
15αNXR NXR 15N isotope fractionation factor 1.015 (40) 18αNXR.NO2 NXR 18O isotope fractionation factor for
nitrite oxidation
1.004 (41)
18αNXR.H2O NXR 18O isotope fractionation factor for
water oxidation
0.986 (41)
15αNIR NIR 15N isotope fractionation factor 0.995 (49) 18αNIR NIR 18O isotope fractionation factor = 15αNIR (49) 15αAMO AMO 15N isotope fractionation factor for
ammonia oxidation
0.936 (44)
18αAMO.H2O AMO 18O isotope fractionation factor for
water oxidation
0.986 (42)
18αAMO.O2 AMO 18O isotope fractionation factor for O2
oxidation
0.986 (42)
15αAMX AMX 15N isotope fractionation factor 1.035 (45) 18αeq Nitrite-water equilibrium 18O isotope
fractionation factor
0.9865 (43)
18KIEexch.f Nitrite-water exchange 18O forward kinetic
isotope effect
1
𝐷𝑁𝑂3− diffusion coefficient for nitrate (m2 day-1) 0.1728 (57)
𝐷𝑁𝑂2− diffusion coefficient for nitrite (m2 day-1) = 1 to 5× 𝐷𝑁𝑂3
−
𝐷𝑁𝐻4+ diffusion coefficient for ammonium (m2 day-
1) = 𝐷𝑁𝑂3
−
Notes: 1. All processes are assumed to be mass dependent with 17θ = 0.528 (δ'17Oini = δ'18Oini×17θ
and 𝛼 = 𝛼 𝜃171817 )
2. Isotope compositions of NO2- are not applicable, since [NO2
-]ini are set as 0 μmol/L. However,
the initial isotope compositions and concentrations of NO2- are adjustable in our giving script
(S2.4. Model details and Script in R language).
12
3. Diagnostic fractionation factors for 15N and 18O isotopes are adopted from the "standard" GW
model (39), kNAR is adopted from CB model (51).
4. We set 𝛿′18𝑂𝑁𝑂3−
𝑖𝑛𝑖 = 0‰. Therefore, 𝛿′18𝑂𝐻2𝑂 actually represents the initial δ'18O difference
between water and NO3- (∆𝛿′18𝑂𝑁𝑂3
−−𝐻2𝑂).
2.3. Results and Discussions
2.3.1 Denitrification
Figure 2.2. Predicted denitrification trajectories on δ'17O-δ'18O (A, C) and δ'18O-δ'15N (B, D)
space for nitrate by a RTM (top) and a RDM (bottom) (kNXR/kNAR = 0), with the given diagnostic 18α = 15α 0.985, 17α = 0.992, and 17θ = 0.528. We set 𝐷𝑁𝑂2
−/𝐷𝑁𝑂3− = 1 for the RTM.
The trajectory on δ'-δ' space describes the relationship between changes of two isotope
compositions (e.g. δ'18O and δ'15N, or δ'17O and δ'18O) with time (RDM) or space at isotope
steady-state (RTM). Fig. 2.2. illustrates NO3- trajectories on δ'-δ' space predicted by a RDM and
a RTM for the scenario of NAR only (kNXR/kNAR = 0). The RTM and the RDM resulted in almost
the same slope (S) in δ'17O-δ'18O space (Fig 2.2. A, C) or numerically identical ones in δ'18O-
δ'15N space (Fig 2.2. B, D).
13
Here, we examine how the resulting trajectories on δ'-δ' space produced by a RTM and a
RDM differ in a single-source system with a given pair of diagnostic α’s.
Assuming a reaction rate (R) is the first-order kinetics of concentration (C, i.e. R = -k×C),
the diagnostic α is defined as:
𝛼𝑖𝑛 ≡𝑘𝑟
𝑘𝑐 (2.2)
where r denotes rare isotope, c denotes common isotope.
For a closed-system RDM, the change of residue concentration through time in isotope form
is:
𝐶𝑖 (𝑡) = 𝐶𝑖0𝑒−𝑘𝑡 (2.3)
where C(t) is concentration at time t, C0 is initial concentration, and i denotes different
isotopes.
For a RTM, assuming there is a sole input with concentration of C0, the ion or molecule of
interest diffuses out in one dimension from this point. Setting the depth of the input at 0 and
considering diffusion, advection, and reaction, we can describe the system by a mass
conservation equation (58):
𝜕𝛷𝐶
𝜕𝑡 =
𝜕
𝜕𝑧(𝐷𝛷
𝜕𝐶
𝜕𝑧) −
𝜕
𝜕𝑧(𝜔𝛷𝐶) + 𝛷 ∑ 𝑅𝑖 (2.4)
where D is the diffusion coefficient, z is depth, Φ is sediment porosity, ω is the
sedimentation rate, and ΣRi is the sum of the individual reaction rate for the production and
consumption of a given constituent or isotope. In a sediment column where a sedimentation rate
is negligible in the given time period, or in a still water column, the advection term (∂(ωΦC)/∂z)
in Eq. 2.4 can be omitted. Assuming D and Φ remain constant in the whole system during the
observation period, and the system has sufficient time to reach a isotope steady-state (∂C/∂t = 0)
(56), Eq. 2.4 can be simplified to the consumption term and the diffusion term, expressed as:
14
𝐷𝜕2𝐶
𝜕𝑧2− 𝑘𝐶 = 0 (2.5)
At the observation moment to, when the system reaches steady-state, with the boundary
condition C(0, to) = C0, C(∞,to) = 0, the concentration at different depths of this column can be
solved and expressed in isotope form as:
𝐶𝑖 (𝑧, 𝑡𝑜) = 𝐶𝑖0𝑒
−√𝑘𝑖
𝐷𝑖 ×𝑧
(2.6)
Diffusional isotope fractionation of aqueous oxyanions is ignored in our model, thus, we
have rD = cD, solving the diagnostic α by Eq. 2.6 and Eq. 2.2, we obtain:
𝑅𝑡 𝑅0⁄ = 𝑓√𝛼−1 (2.7)
from a single-source RTM. This equation is similar in form to Eq.1 from a closed-system
RDM, but “α” is replaced by “√𝛼”.
The slopes of trajectories on δ'-δ' space for a RDM (SRDM) and a single-source RTM (SRTM)
are:
𝑆𝑅𝐷𝑀 = 𝛼𝐵−1
𝛼𝐴−1 (2.8)
𝑆𝑅𝑇𝑀 = √𝛼𝐵−1
√𝛼𝐴−1 (2.9)
In δ'17O-δ'18O space, given a 17θ value of 0.528, we will get an apparent SRDM of 0.530
(Fig.1, C) and SRTM of 0.529 (Fig.1, A). In δ'18O-δ'15N space for nitrate, SRDM = SRTM = 1 if we
assign 18α = 15α (Fig.1 B, D). Our analysis indicates that for a defined reaction, the two
trajectories on δ'-δ' spaces from a closed-system, time-dependent RDM and from a one-
dimensional diffusional, steady-state, single-source RTM are numerically similar because the
diagnostic α’s for both isotope systems (e.g. δ'18O-δ'15N or δ'17O-δ'18O) in most natural isotope
systems are close to 1.0.
15
Using a Monte Carlo method, we tested the deviation of SRTM from SRDM (ΔS = SRTM - SRDM)
under different αA and SRDM values. We gave αA value in the range of 0.95 to 1.00 and SRDM
value 0 to 1 (Fig. 2.3., left) and 1 to 5 (Fig. 2.3., right) respectively. If αA value is close to 1.0, we
found the deviation is small in the entire range of SRDM (SRTM ≈ SRDM, Fig. 2.3., black). This is
because α-1 ≈ lnα and α0.5-1 ≈ 0.5×lnα, when α does not deviate far from unity or the degree of
fractionation is small.
Figure 2.3. Predicted trajectory slope difference between a RTM and a RDM (ΔS, y-axis) vs.
slopes predicted by a RDM (x-axis). SRDM is in range of 0 to 1 (left), and 1 to 5 (right). Gradient
scale indicates the α of species A. We set 𝐷𝑁𝑂2−/𝐷𝑁𝑂3
− = 1 for the RTM.
Larger isotope fractionation produces larger ΔS. When 0 < SRDM < 1, we found that the
maximum deviation ΔS of -0.003 occurs when αA = 0.95 and SRDM = 0.5. For mass-dependent
triple isotope relationship, such as 17θ of oxygen and 33θ of sulfur, where θ ≈ 0.5, the slope
obtained by a RTM model can have a maximum deviation of ~ 0.003 from that obtained from a
RDM model. When SRDM > 1, the maximum ΔS increases with the increase of SRDM. The ΔS can
be as large as 0.326 when αA = 0.95 and SRDM = 5. Therefore, this difference is negligible for
most natural processes.
16
In the case of denitrification, SRDM = 1 because 18α = 15α. The SRTM will also be equal to 1
because √𝛼𝐴 = √𝛼𝐵. Thus, if denitrification is the only process in a RTM system with diffusional
mass transfer, the observed slope of trajectory on δ'18O-δ'15N space should still be 1. Therefore,
an observed slope deviation of a nitrate δ'18O-δ'15N trajectory from 1 must be caused by other
processes. Now, let us examine one of the likely causes, complex nitrogen cycling processes.
2.3.2 N Cycling
For a system when NO3- is involved in a complex nitrogen cycling with NAR, NXR and NIR,
the predicted δ'18O-δ'15N trajectories for nitrate by a RTM or a RDM under different kNXR/kNAR
ratios and 𝛿′18𝑂𝐻2𝑂 are shown in Fig. 2.4. Here we plot only the trajectory with Δδ'15N < 20‰,
since it is mostly the observed range of Δδ'15N in the field Casciotti, 2007 #11;Casciotti, 2013 #4}
(Our predicted trajectories under different dynamics in a wider range of Δδ'15N < 40‰ can be
found in S1. Model results). The predicted trajectories for complex systems are not linear. Thus,
the slope we mentioned here is the regression slope of the predicted (δ'18O, δ'15N) points. The
slope of the RDM trajectories are always greater than that of the RTM trajectories, regardless of
the slope of the RDM trajectories being greater or less than 1 (Fig. 2.4. and Fig. 2.11.). In the
range of Δδ'15N < 20‰, the deviation of a RTM slope from a RDM slope ranges from 0.03 to 0.3.
For instance, larger fluxes of NXR (kNXR/kNAR = 2) with initial ∆𝛿′18𝑂𝑁𝑂2−−𝐻2𝑂 = 0‰ produces a
slope of 1.375 in a RDM (blue dash lines, Fig. 2.4. B), but the slope of a RTM is approximately
equal to 1 (1.012, blue solid lines, Fig. 2.4. B).
Although definitive data are lacking, the k for nitrite-water oxygen isotope exchange is
likely much faster than the k’s of either NAR or NXR in both freshwater and seawater systems
(11, 36, 40, 48, 57). Assuming kexch.f/kNAR = 10 (13, 43, 51), NO2- can quickly reach oxygen
isotope equilibrium with water. NXR produces NO3- by introducing one oxygen from water to
17
NO2- (38). Thus, eventually, the𝛿′18𝑂𝑁𝑂3
−will reach isotope steady-state, which is buffered by
the 𝛿′18𝑂𝐻2𝑂 . The isotope steady-state is displayed by a flattening-out NO3- trajectory of a
constant δ'18O but an increasing δ'15N. The regression slope is defined by the initial and the
isotope steady-state, as well as the rate of the system approaching isotope steady-state. The
regression slope of δ'18O-δ'15N trajectory can be negative if the initial ∆𝛿′18𝑂𝑁𝑂3−−𝐻2𝑂 is large,
since the isotope steady-state has a much lower δ'18O than the source 𝛿′18𝑂𝑁𝑂3−
𝑖𝑛𝑖. For instance,
in a RTM system where atmospheric NO3- (δ'15N ≈ 0‰ and δ'18O≈ 70‰ (59)) as the only NO3
-
source, and 𝛿′18𝑂𝐻2𝑂 = 0‰. Given the dynamics of kNXR/kNAR = 2, kexch/kNAR = 10, kNIR/kNAR =
1, and kAMO/kNAR = 0, the regression slope of δ'18O-δ'15N trajectory will be -0.987.
Figure 2.4. Predicted δ'18O-δ'15N trajectories for nitrate with kNXR/kNAR = 0.1 (A) and kNXR/kNAR
= 2 (B), by a RTM (solid lines) and a RDM (dash lines). The 𝛿′18𝑂𝐻2𝑂 was set at 0‰ (blue) or -
20‰ (pink). We set kexch.f/kNAR = 10, kNIR/kNAR = 1, kAMO/kNAR = 0, [NO2-]ini = 0 μmol/L for both
models, and 𝐷𝑁𝑂2−/𝐷𝑁𝑂3
− = 1 for the RTM. Black line is the reference line with a slope of 1.
The diffusion coefficient difference between NO2- and NO3
-, if any, has little influence on
the δ'18O-δ'15N trajectory. Therefore, it can be neglected in the discussion of diffusional mass
transfer (Fig. 2-1). Trajectories predicted by a RTM and a RDM respond to changing parameters
similarly, and are comparable to the results in GW model (see 2.6.1. Model Results). The δ'18O-
δ'15N trajectory of NO3- is mainly controlled by the initial ∆𝛿′18𝑂𝑁𝑂2
−−𝐻2𝑂 and the net NO3-
18
consumption flux (36). The greater the initial ∆𝛿′18𝑂𝑁𝑂2−−𝐻2𝑂 is, the smaller the slope will be.
The greater the net NO3- consumption flux is, the closer the trajectory is to the 1:1 ratio. Large
net NO3- consumption flux could be the results of small kNXR/kNAR ratio (Fig. 2-9) or large
kNIR/kNXR ratio (Fig. 2-10).
2.4. Implications
Our analysis demonstrates that an incorrect use of a RDM to spatial or mixed spatial and
temporal data would not give us the right diagnostic α values. In fact, we would always
underestimate the degree of diagnostic fractionation in such a case. However, the slope of a
trajectory on δ'18O-δ'15N space for spatial data, which should be treated by a single-source RTM,
would be identical to the trajectory predicted by a RDM. This “lucky” situation certainly
vanishes when the system has multiple NO3- sources. In other words, if the plotted data points
consist of NO3- of different origins, the slope of the trajectory can be any value. We must
understand the underlying physical processes of a set of data before we decide to plot them on a
δ-δ space. Otherwise, diagnostic isotope fractionation parameters cannot be deduced. NO3-
collected from different depths of water or sediment columns (13-14, 16-18), different sites at a
study area (19-33), or mixture of time and space (34-36), have been plotted, often
indiscriminately, on δ-δ space to explore nitrate-related reaction kinetics and dynamics. The
observed δ18O-δ15N regression slope varies from < 1 (16, 19-22, 24-25), ≈ 1 (13-14, 17, 34, 36), >
1 (18, 26, 28, 32-33), and to none (no significant correlation) (23, 29-31, 34-35). Some of these
cases, in which samples were collected from different depths of water or sediment columns (13-
14, 16-18), can be simplified to a diffusional RTM with a single NO3- source. The rest of them,
especially the temporal and spatial mixed data, cannot be easily described by a simple RTM or
RDM model.
19
One outstanding result of our modeling exercise is that the slope of δ'18O-δ'15N trajectories
predicted by the RTM model are rarely greater than 1, when nitrite-water oxygen isotope
exchange is faster than NXR in N cycling (kexch.f/kNAR > 10, Fig. 2-9). This is a result that is
different from previous studies (36) and warrants further investigation. Here we assumed
kexch.f/kNAR = 10, kNXR/kNAR = 0~2, kNIR/kNAR = 0.1~2, and 𝛿′18𝑂𝐻2𝑂 = 0~-20‰ with kAMO/kNAR =
0, and explored the δ'18O-δ'15N trajectories predicted by a RTM using a Monte Carlo method (Fig.
2.5.).
Figure 2.5. Predicted nitrate isotope compositions in profiles as discrete points simulated by a
Monte Carlo method on δ'18O-δ'15N space in a RTM. kNXR/kNAR = 0~2, kNIR/kNAR = 0.1~2,
𝛿′18𝑂𝐻2𝑂 = 0~-20‰, kexch.f/kNAR = 10, kAMO/kNAR = 0, [NO2-]ini = 0 μmol/L, and 𝐷𝑁𝑂2
−/𝐷𝑁𝑂3− = 1.
Black points are generated under the conditions of 𝛿′18𝑂𝐻2𝑂 < 4‰, kNxR/kNAR > 1.6 and
kNIR/kNAR < 0.5. Black lines are reference lines with slope of 1 (solid) and 1.2 (dashed)
respectively.
Our given conditions cover most of the natural settings on Earth surface. We found that the
slope of the δ'18O-δ'15N trajectories can only be slightly greater than 1 (< 1.2) when the initial
∆𝛿′18𝑂𝑁𝑂2−−𝐻2𝑂 is smaller than 4‰ and when NXR flux is large, i.e. kNxR/kNAR > 1.6 and
kNIR/kNAR < 0.5 (black points in Fig. 2.5.). The current observed range of ∆𝛿′18𝑂𝑁𝑂3−−𝐻2𝑂 in
20
freshwater systems is 9~30‰ (26, 29, 35), which should exclude the possibility of a δ'18O-δ'15N
trajectory with a regression slope greater than 1. In a seawater system where 𝛿′18𝑂𝐻2𝑂 is close to
0‰, if a NO3- source has 𝛿′18𝑂𝑁𝑂3
− < 4‰, the regression slope of observed trajectory can be
greater than 1, if NXR flux is large.
With this model prediction in mind, let us examine the reported slope > 1 cases in literature.
The NO3- profiles from Monterey Bay, California at different seasons display some regression
slopes greater than 1, for which the authors interpreted it as possibly the result of mixing of
upwelling NO3- from oxygen-deficient waters with NO3
- assimilated by phytoplankton in
euphotic zone (18). NO3- samples collected from columns at different sites and seasons of the
Yellow Sea have Δδ18O/Δδ15N = 1 to 3 (32-33). NO3- samples from different sites of Changjiang
(the Yangtze River) align on the δ18O-δ15N regression slope of 1.5 (stream (26)) and 1.29
(estuary (28)). We noticed that all these cases sample sets contain samples from different sites.
Thus, the >1 regression slope can readily be explained by the mixing different NO3- sources that
differ more in the δ18O than in the δ15N, same to the Monterey case (18). Even when a set of
NO3- samples collected from different depths, sites and/or seasons displays a good linear
relationship of regression slope less than 1 on δ18O-δ15N space (16, 19-22, 24-25), it does not
necessarily mean they were produced by different extents of NXR, if the physical processes are
not well constrained a priori.
Similarly, trajectories on δ-δ space exhibited by samples of unknown relationships have
been widely applied to deduce triple oxygen or quadruple sulfur isotope exponent θ’s. For
example, multiple-sulfur isotope compositions were measured for pyrite in sedimentary rock
profiles. Well-correlated linear relationships between δ33S and δ34S with various regression
slopes are observed in several profiles (60-61). However, pyrite grains precipitated at different
21
times can have different sources or are the integrated product of a long-term sulfate reduction
processes over different depths and conditions. Plotting and linking such a data set cannot be
simply treated by a closed-system RDM or a single-source, steady-state RTM to deduce the
underlying diagnostic isotope fractionation parameters. Again, to plot or process a set of data and
deduce diagnostic parameters of mechanistic implication, we must have a clear context of known
or hypothesized a priori the physical processes that link these data.
2.5. Conclusions
Assuming reactions are of first order kinetics and giving a set of diagnostic fractionations, we
compared a closed-system time-dependent Rayleigh Distillation model (RDM) with a one-
dimensional diffusion, steady-state, single-source Reaction-Transport-Model (RTM) in their
prediction of the trajectories on δ'18O-δ'15N space of nitrate that is involved in N cycling. When a
RDM is incorrectly applied to spatial data for which a RTM should have been applied to, the
degree of the diagnostic isotope fractionation is underestimated. However, the derived
trajectories on δ'18O-δ'15N space are unaffected in a system with only NAR processes. When
NO3- is involved in more complex N cycling processes than just a NAR, the slope of temporal
data trajectories on δ'18O-δ'15N space predicted by a RDM are slightly, but measurably greater
than that of the respective spatial data trajectories predicted by a RTM. Our RTM model shows
that a slope value > 1 in δ'18O-δ'15N space is unlikely in single-sourced N cycling processes in
nature, except for the rare condition when the initial δ'18O difference between water and NO3-
smaller than 4‰ and the reoxidation flux is exceptionally large (kNxR/kNAR > 1.6 and kNIR/kNAR <
0.5). In fact, we predicted that almost all the reported cases of slopes > 1 are the results of
plotting together multiple NO3- sources of different space and/or time. To deduce diagnostic
22
parameters of mechanistic implication, we must understand a priori the physical processes
underlying data points of interest.
2.6. Supporting Information
Using N cycling adapted from GW model (Fig. 2.1.), we predicted δ'18O-δ'15N trajectories for
nitrate with different parameters of 𝐷𝑁𝑂2− /𝐷𝑁𝑂3
− , kexch.f/kNAR, 𝛿′18𝑂𝐻2𝑂 , kNXR/kNAR kNIR/kNAR,
kAMO/kNAR and kAMX/kNAR in the range of Δδ15N < 40‰.
2.6.1. Model Results
2.6.1.1. Diffusion Coefficient Difference
Our models assumed the same diffusion coefficients for NO3- and NO2
- (𝐷𝑁𝑂2− /𝐷𝑁𝑂3
− = 1).
Currently, the exact NO2- diffusion coefficient is not well calibrated. Here we tested 𝐷𝑁𝑂2
−/𝐷𝑁𝑂3−
ratio in the range of 1 to 5 under different dynamics and 𝛿′18𝑂𝐻2𝑂 (Fig. 2.6.). The predicted
trajectories are unaffected. Therefore, the difference of diffusion coefficients between NO3- and
NO2- can be ignored in discussion.
2.6.1.2. Nitrite-Water Exchange Rate Constant
Nitrite-water oxygen exchange kexch.f is an important model parameter. Due to the lack of a
precise calibration, we need to explore its sensitivity. Since kexch.f is temperature and pH
dependent (23), we explored the trajectories under a wide range of kexch.f (Fig. 2.7.). The larger
the kexch.f is, the faster the δ'18ONO2 reaches equilibrium with water. kexch.f/kNAR = 0 when NO2-
does not exchange with water, and kexch.f/kNAR = 1000 when NO2- exchanges and reaches
equilibrium with water rapidly.
Under the condition that NO2- does not equilibrate with water (kexch.f/kNAR = 0), the isotope
compositions of NO2- will be defined by the NO3
- isotope compositions and the denitrification
α’s (18). NO3- reduction to NO2
- loses one oxygen and is associated with a branching oxygen
23
isotope effect (αNAR.BR = 0.975 (29)). This branching isotope effect produces NO2- with its δ'18O
~10‰ heavier than the δ'18O of NO3-. If the isotope compositions of NO2
- is defined by NAR,
such difference overwhelms the contribution of water oxygen during NXR, and causes nitrate
δ'18O increase faster than the δ'15N (18). Thus, the slope of δ'18O-δ'15N trajectory will be greater
than 1 at the early stage of N cycling in both a RTM and a RDM when NXR flux is large
(kNXR/kNAR = 2, Fig. 2.7., C, D, G, H, green lines). However, trajectories will be gradually
flattened out at later stage of N cycling and reaches isotope steady-state, when NO2- accumulated,
and its isotope composition is not completely defined by NAR.
Figure 2.6. Predicted δ'18O-δ'15N trajectories for nitrate with different 𝐷𝑁𝑂2−/𝐷𝑁𝑂3
− by a RTM
(top) and a RDM (bottom). The 𝛿′18𝑂𝐻2𝑂 was set at 0‰ or -20‰. Nitrite was set to not
exchange oxygen isotopes with water (kexch.f/kNAR = 0) or quickly exchange with water
(kexch.f/kNAR = 1000). Other parameters are set as kNIR/kNAR = 1, and kAMO/kNAR = 0. Black line is
reference line with slope of 1.
24
Figure 2.7. Predicted δ'18O-δ'15N trajectories for nitrate with different kexch.f/kNAR ratios in a RTM
(top) and a RDM (bottom). The kNXR/kNAR is set as 0.1 or 2 and the 𝛿′18𝑂𝐻2𝑂 was set at 0‰ or -
20‰. Other parameters are set as kNIR/kNAR = 1, kAMO/kNAR = 0, [NO2-]ini = 0 μmol/L and
𝐷𝑁𝑂2−/𝐷𝑁𝑂3
− = 1. Black line is reference line with slope of 1.
If NO2- equilibrates with water quickly (kexch.f/kNAR = 1000), the NO3
- produced by NXR
will not have a lower δ'18O, and the slope of the trajectory will not be greater than 1.
2.6.1.3. Water Isotope Compositions
During NXR, one oxygen will be added to NO2- from water to NO3
- (18). Thus, the lighter the
𝛿′18𝑂𝐻2𝑂 is, the lighter the 𝛿′18𝑂𝑁𝑂3− at isotope steady-state is. Thus, a larger net NXR flux will
result in a smaller slope of the δ'18O-δ'15N trajectory. In addition, the slope of the δ'18O-δ'15N
trajectory will be smaller when nitrite-water oxygen isotope exchange rate is fast (e.g. Fig. S4, C,
D, G, H, kexch/kNAR = 1000) than when nitrite-water do not exchange oxygen (e.g. Fig. S4, A, B,
E, F, kexch/kNAR = 0).
25
Figure 2.8. Predicted δ'18O-δ'15N trajectories for nitrate with different 𝛿′18𝑂𝐻2𝑂 by a RTM (top)
and a RDM (bottom). The kNXR/kNAR is set at 0.1 or 2. Nitrite was set to not exchange oxygen
isotopes with water (kexch.f/kNAR = 0) or exchange with water quickly (kexch.f/kNAR = 1000). Other
parameters are set as kNIR/kNAR = 1, kAMO/kNAR = 0, and 𝐷𝑁𝑂2−/𝐷𝑁𝑂3
− = 1. Black line is the
reference line with a slope of 1.
2.6.1.4. NXR Rate Constant
The kNXR/kNAR ratio characterizes the net input of NO3- from NO2
-. The larger the kNXR/kNAR ratio
is, the larger NXR flux is, and the more the trajectory deviates from the slope of 1 (Fig. 2.9.).
2.6.1.5. NIR Rate Constant
When kNIR is greater than kNXR, more NO2- will be further reduced to other nitrogen pools instead
of being re-oxidized to NO3-. Thus, when kNIR ≫ kNXR, even with large kNXR/kNAR ratio, the net
flux for NO3- is still consumption. Under such condition, the trajectory will be similar to the
NAR only trajectory, regardless the kNXR/kNAR ratio or 𝛿′18𝑂𝐻2𝑂 because NAR is the dominating
process controlling NO3- concentration (Fig. 2.10., pink lines).
26
Figure 2.9. Predicted δ'18O-δ'15N trajectories for nitrate with different kNXR/kNAR ratios by a RTM
(top) and a RDM (bottom). The 𝛿′18𝑂𝐻2𝑂 was set at 0‰ or -20‰. Nitrite was set to not
exchange oxygen isotopes with water (kexch.f/kNAR = 0) or exchange with water quickly
(kexch.f/kNAR = 1000). Other parameters are set as kNIR/kNAR = 1, kAMO/kNAR = 0, and 𝐷𝑁𝑂2−/𝐷𝑁𝑂3
− =
1. Black line is the reference line with a slope of 1.
2.6.1.6. AMO and AMX Rate Constants
The kAMO/kNAR and kAMX/kNAR ratios characterize the net input of NO2- from NH4
+. When kAMX
≫ kAMO, more NH4+ will be oxidized to other nitrogen pools instead of being oxidized to NO2
-.
Thus, the smaller the kAMO/kNAR is or the larger the kAMX/kNAR is, the closer the trajectory to
kAMO/kNAR = 0 is (Fig. 2.11.).
27
Figure 2.10. Predicted δ'18O-δ'15N trajectories for nitrate with different kNIR/kNAR ratios by a
RTM (top) and a RDM (bottom). The 𝛿′18𝑂𝐻2𝑂 was set at 0‰ or -20‰. Nitrite was set to not
exchange oxygen isotopes with water (kexch.f/kNAR = 0) or exchange with water quickly
(kexch.f/kNAR = 1000). Other parameters are set as kNXR/kNAR = 2, kAMO/kNAR = 0, and 𝐷𝑁𝑂2−/𝐷𝑁𝑂3
− =
1. Black line is the reference line with a slope of 1.
Figure 2.11. Predicted δ'18O-δ'15N trajectories for nitrate with different kAMO/kNAR (kAMX/kNAR =
1, A, B) or kAMX/kNAR (kAMO/kNAR = 1, C, D) ratios by a RTM (top) and a RDM (bottom). Other
parameters are set at kNXR/kNAR = 0.5, kNIR/kNAR = 1, kexch.f/kNAR = 10, 𝛿′18𝑂𝐻2𝑂 = 0‰, and
𝐷𝑁𝑂2−/𝐷𝑁𝑂3
− = 1. Black line is the reference line with a slope of 1.
28
2.6.2. Notations
To avoid messiness in illustration, × 1000‰ factor is omitted from any of the derivations as
suggested in Bao (62).
Isotope composition can be reported using the classical δ-notation:
𝛿 ≡ (𝑅𝑠𝑎𝑚𝑝𝑙𝑒
𝑅𝑟𝑒𝑓− 1) (S2.1)
where R denotes the ratio between the number of moles of a rare isotope and a common isotope
of an element (e.g. 18R = 18O/16O for oxygen).
To simplify mathematical treatments, in this paper, we adopt and recommend the δ'
notation:
𝛿′ ≡ ln (𝑅𝑠𝑎𝑚𝑝𝑙𝑒
𝑅𝑟𝑒𝑓) (S2.2)
Since δ can be measured to a higher precision than R, Rayleigh Distillation is expressed
as
(𝛿𝑡+1)
(𝛿0+1) = 𝑓𝛼−1 (S2.3)
In δ' notation, we have
𝛿′𝑡 − 𝛿′0 = (𝛼 − 1) × ln 𝑓 (S2.4)
A fractionation factor (α) is defined as:
𝛼𝐴−𝐵 ≡𝑅𝐴
𝑅𝐵 (S2.5)
where A and B denote different phases in a reaction.
Using δ notation, for a single reaction system, the slope of a trajectory on δ-δ space
equals approximately to 𝛼𝐵−1
𝛼𝐴−1 when δ0 ≪1, since (4-5):
∆𝛿 ≈ (𝛼 − 1) × ln 𝑓 (S2.6)
29
However, if the δ’-notation is applied, the slope equals to 𝛼𝐵−1
𝛼𝐴−1 without mathematical
approximation.
The high-dimensional isotope parameter θ describes the relationship between three
isotopes of the same element. For example, oxygen has isotopes 16O, 17O, and 18O. The triple
oxygen isotope exponent θ is expressed as:
𝜃17 ≡ ln 𝛼17
ln 𝛼18 (S2.7)
2.6.3. Equations
Every reaction consists of a forward and a backward reaction. In most cases, only the net
reaction can be observed, which is the combined result of the forward and backward reactions.
As are the case for most processes, the individual forward and backward k’s and KIEs are not
available for N cycling processes. Therefore, in our models, we considered net k’s and diagnostic
α’s for all reactions, except for the nitrite-water exchange reaction.
During nitrite-water oxygen isotope exchange reaction, the concentrations of NO2- and
H2O remain constant:
𝑑([𝑁𝑂2
−])
𝑑𝑡 = 𝑘𝑒𝑥𝑐ℎ.𝑓 × [𝑁𝑂2
−] = 𝑘𝑒𝑥𝑐ℎ.𝑏 × [𝐻2𝑂] = 𝑑([𝐻2𝑂])
𝑑𝑡 (S2.8)
If we assume kexch.f = kexch.b, a minor imbalance of concentrations will occur, i.e.
d([N16O2-]+[N17O2
-]+[N18O2-])/dt is not equal to d([H2
16O]+ [H217O]+[H2
18O])/dt. This small
deviation will gradually accumulate and turn significant after many simulation steps when NO3-
concentration approaches 0. Therefore, we only assign kexch.f from which the corresponding kexch.b
is calculated.
𝑘𝑒𝑥𝑐ℎ.𝑓 × [𝑁𝑂2−] = −𝑘𝑒𝑥𝑐ℎ.𝑓 × [ 𝑁𝑂2
−16 ] − 𝑘𝑒𝑥𝑐ℎ.𝑓 × 𝐾𝐼𝐸𝑒𝑥𝑐ℎ.𝑓17 × [ 𝑁𝑂2
−17 ] −
𝑘𝑒𝑥𝑐ℎ.𝑓 × 𝐾𝐼𝐸𝑒𝑥𝑐ℎ.𝑓18 × [ 𝑁𝑂2
−18 ] (S2.9)
30
𝑘𝑒𝑥𝑐ℎ.𝑏 × [𝐻2𝑂] = −𝑘𝑒𝑥𝑐ℎ.𝑏 × [ 𝐻2𝑂16 ] − 𝑘𝑒𝑥𝑐ℎ.𝑏 × 𝐾𝐼𝐸𝑒𝑥𝑐ℎ.𝑏17 × [ 𝐻2𝑂17 ] −
𝑘𝑒𝑥𝑐ℎ.𝑏 × 𝐾𝐼𝐸𝑒𝑥𝑐ℎ.𝑏18 × [ 𝐻2𝑂18 ] (S2.10)
Insert Eq. S3.2 and Eq. S3.3 to Eq. S3.1, we can solve:
𝑘𝑒𝑥𝑐ℎ.𝑏 = 𝑘𝑒𝑥𝑐ℎ.𝑓 × ([ 𝑁𝑂2−16 ] + 𝐾𝐼𝐸𝑒𝑥𝑐ℎ.𝑓
17 × [ 𝑁𝑂2−17 ] + 𝐾𝐼𝐸𝑒𝑥𝑐ℎ.𝑓
18 × [ 𝑁𝑂2−18 ])/
([ 𝐻2𝑂16 ] + 𝐾𝐼𝐸𝑒𝑥𝑐ℎ.𝑏17 × [ 𝐻2𝑂17 ] + 𝐾𝐼𝐸𝑒𝑥𝑐ℎ.𝑏
18 × [ 𝐻2𝑂18 ]) (S2.11)
For each isotope:
𝑑[ 𝐻2𝑂𝑖 ]
𝑑𝑡 𝑒𝑥𝑐ℎ =
𝑑( 𝐴𝐻2𝑂𝑖 ×[𝐻2𝑂])
𝑑𝑡 = 𝐴𝐻2𝑂
𝑖 ×𝑑([𝐻2𝑂])
𝑑𝑡 = 𝑘𝑒𝑥𝑐ℎ.𝑓 × 𝛼𝑒𝑞
𝑖 × 𝐴𝐻2𝑂𝑖 ×
([ 𝑁𝑂2−16 ] + 𝐾𝐼𝐸𝑒𝑥𝑐ℎ.𝑓
17 × [ 𝑁𝑂2−17 ] + 𝐾𝐼𝐸𝑒𝑥𝑐ℎ.𝑓
18 × [ 𝑁𝑂2−18 ])/( 𝐴𝐻2𝑂
16 + 𝐾𝐼𝐸𝑒𝑥𝑐ℎ.𝑏17 × 𝐴𝐻2𝑂
17 +
𝐾𝐼𝐸𝑒𝑥𝑐ℎ.𝑏18 × 𝐴𝐻2𝑂
18 ) (S2.12)
To separate the two reactions, we need the forward and backward KIEs (18KIEexch.f and
18KIEexch.b). Since only the 18αeq is known, we assign 18KIEexch.f = 1 and 18KIEexch.b is therefore
calculated using:
𝛼𝑁𝑂2−−𝐻2𝑂(𝑒𝑞) =
𝐾𝐼𝐸𝑒𝑥𝑐ℎ.𝑏
𝐾𝐼𝐸𝑒𝑥𝑐ℎ.𝑓 (S2.13)
Similarly, the k of oxygen of NO3- produced from H2O (kNXR.H2O) is calculated from kNXR:
𝑘𝑁𝑋𝑅.𝐻2𝑂 = 0.5 × 𝑘𝑁𝑋𝑅 × ([ 𝑁𝑂2−16 ] + 𝛼𝑁𝑋𝑅
17 × [ 𝑁𝑂2−17 ] + 𝛼𝑁𝑋𝑅
18 × [ 𝑁𝑂2−18 ])/
([ 𝐻2𝑂16 ] + 𝛼𝑁𝑋𝑅.𝐻2𝑂17 × [ 𝐻2𝑂17 ] + 𝛼𝑁𝑋𝑅.𝐻2𝑂
18 × [ 𝐻2𝑂18 ]) (S2.14)
For each isotope:
𝑑[ 𝐻2𝑂𝑖 ]
𝑑𝑡 𝑁𝑋𝑅 = 0.5 × 𝑘𝑁𝑋𝑅 × 𝛼𝑁𝑋𝑅.𝐻2𝑂
𝑖 × 𝐴𝐻2𝑂𝑖 × ([ 𝑁𝑂2
−16 ] + 𝛼𝑁𝑋𝑅17 × [ 𝑁𝑂2
−17 ] +
𝛼𝑁𝑋𝑅18 × [ 𝑁𝑂2
−18 ])/( 𝐴𝐻2𝑂16 + 𝛼𝑁𝑋𝑅.𝐻2𝑂
17 × 𝐴𝐻2𝑂17 + 𝛼𝑁𝑋𝑅.𝐻2𝑂
18 × 𝐴𝐻2𝑂18 ) (S2.15)
The k’s of oxygen of NO2- produced from O2 (kAMO.O2) and H2O (kAMO.H2O) is calculated
from kAMO:
31
𝑘𝐴𝑀𝑂.𝐻2𝑂 = 𝑘𝐴𝑀𝑂 × ([ 𝑁𝐻4+14 ] + 𝛼𝐴𝑀𝑂
15 × [ 𝑁𝐻4+15 ])/([ 𝐻2𝑂16 ] + 𝛼𝐴𝑀𝑂.𝐻2𝑂
17 ×
[ 𝐻2𝑂17 ] + 𝛼𝐴𝑀𝑂.𝐻2𝑂18 × [ 𝐻2𝑂18 ]) (S2.16)
𝑘𝐴𝑀𝑂.𝑂2 = 𝑘𝐴𝑀𝑂 × ([ 𝑁𝐻4
+14 ] + 𝛼𝐴𝑀𝑂15 × [ 𝑁𝐻4
+15 ])/([ 𝐻2𝑂16 ] + 𝛼𝐴𝑀𝑂.𝑂2
17 × [ 𝐻2𝑂17 ] +
𝛼𝐴𝑀𝑂.𝑂2
18 × [ 𝐻2𝑂18 ]) (S2.17)
For each isotope:
𝑑[ 𝐻2𝑂𝑖 ]
𝑑𝑡 𝑁𝑋𝑅 = 𝑘𝑁𝑋𝑅 × 𝛼𝐴𝑀𝑂.𝐻2𝑂
𝑖 × 𝐴𝐻2𝑂𝑖 × ([ 𝑁𝐻4
+14 ] + 𝛼𝐴𝑀𝑂15 × [ 𝑁𝐻4
+15 ])/(𝐴𝐻2𝑂16 +
𝛼𝐴𝑀𝑂.𝐻2𝑂17 × 𝐴𝐻2𝑂
17 + 𝛼𝐴𝑀𝑂.𝐻2𝑂18 × 𝐴𝐻2𝑂
18 ) (S2.18)
𝑑[ 𝑂2𝑖 ]
𝑑𝑡 𝑁𝑋𝑅 = 𝑘𝑁𝑋𝑅 × 𝛼𝐴𝑀𝑂.𝑂2
𝑖 × 𝐴𝐻2𝑂𝑖 × ([ 𝑁𝐻4
+14 ] + 𝛼𝐴𝑀𝑂15 × [ 𝑁𝐻4
+15 ])/(𝐴𝑂2
16 +
𝛼𝐴𝑀𝑂.𝑂2
17 × 𝐴𝐻2𝑂17 + 𝛼𝐴𝑀𝑂.𝑂2
18 × 𝐴𝑂2
18 ) (S2.19)
2.6.4. Model details and script in R language
The differential equation used in a RDM is dC/dt = ΣRi, and the partial differential equation used
in a RTM is ∂C/∂t = D∂2C/∂z2 + ΣRi = 0. Equations are solved numerically by functions ode
(RDM) (63) and steady.1D (RTM) (64) in R. Since all reactions are assumed to be first-order
kinetics, the concentrations approach 0 with the increasing of time (RDM) or distance (RTM).
Concentration lowers than 1 μmol/L is hardly measurable and has little environmental
significance. Therefore, we only plot the trajectories when [NO3-] > 1 μmol/L. For each
simulation, 1000 points are generated. The last 40% were omitted for purpose of accuracy, since
the algorithms of the functions (ode and steady.1D, which we used to solve the differential
equations) and log transformation causes non-linearity while the calculation approaching
boundary conditions (personal communication with Soetaert).
The followings are executable scripts in R language:
#=======================#
32
# RDM Model #
#=======================#
RDM.model <- function(t, y, pars = NULL) {
with(as.list(y),{
dC14.NO3 <- (k.NXR * C14.NO2 - k.NAR * C14.NO3)
dC15.NO3 <- (k.NXR * a15.NXR * C15.NO2 - k.NAR * a15.NAR * C15.NO3)
dC16.NO3 <- (k.NXR * C16.NO2 + (0.5 * k.NXR * A16. * (C16.NO2 + a17.NXR.NO2
* C17.NO2 + a18.NXR.NO2 * C18.NO2) / (A16.H2O + a17.NXR.H2O * A17.H2O +
a18.NXR.H2O * A18.H2O)) - k.NAR * C16.NO3)
dC17.NO3 <- (k.NXR * a17.NXR.NO2 * C17.NO2 + (0.5 * k.NXR * a17.NXR.H2O *
A17.H2O * (C16.NO2 + a17.NXR.NO2 * C17.NO2 + a18.NXR.NO2 * C18.NO2) / (A16.H2O
+ a17.NXR.H2O * A17.H2O + a18.NXR.H2O * A18.H2O)) - k.NAR * a17.NAR * C17.NO3)
dC18.NO3 <- (k.NXR * a18.NXR.NO2 * C18.NO2 + (0.5 * k.NXR * a18.NXR.H2O *
A18.H2O * (C16.NO2 + a17.NXR.NO2 * C17.NO2 + a18.NXR.NO2 * C18.NO2) / (A16.H2O
+ a17.NXR.H2O * A17.H2O + a18.NXR.H2O * A18.H2O)) - k.NAR * a18.NAR * C18.NO3)
dC14.NO2 <- (k.NAR * C14.NO3 + k.AMO * C14.NH4 - k.NXR * C14.NO2 - k.NIR *
C14.NO2)
dC15.NO2 <- (k.NAR * a15.NAR * C15.NO3 + k.AMO * a15.AMO * C15.NH4 -
k.NXR * a15.NXR * C15.NO2 - k.NIR * a15.NIR * C15.NO2)
dC16.NO2 <- (k.NAR * C16.NO3 * 2 / 3 + ( k.AMO * A16.O2 * (C14.NH4 +
a15.AMO * C15.NH4) / (A16.O2 + a17.AMO.O2 * A17.O2 + a18.AMO.O2 * A18.O2)) +
( k.AMO * A16.H2O * (C14.NH4 + a15.AMO * C15.NH4) / (A16.H2O + a17.AMO.H2O *
A17.H2O + a18.AMO.H2O * A18.H2O)) - k.NXR * C16.NO2 - k.NIR * C16.NO2 - k.exch.f *
C16.NO2 + ( k.exch.f * A16.H2O * (C16.NO2 + a17.f * C17.NO2 + a18.f * C18.NO2) /
(A16.H2O + a17.f * a17.eq * A17.H2O + a18.f * a18.eq * A18.H2O)))
dC17.NO2 <- (k.NAR * a17.NAR / a17.NAR.BR * C17.NO3 * 2 / 3 + ( k.AMO *
a17.AMO.O2 * A17.O2 * (C14.NH4 + a15.AMO * C15.NH4) / (A16.O2 + a17.AMO.O2 *
A17.O2 + a18.AMO.O2 * A18.O2)) + ( k.AMO * a17.AMO.H2O * A17.H2O * (C14.NH4
+ a15.AMO * C15.NH4) / (A16.H2O + a17.AMO.H2O * A17.H2O + a18.AMO.H2O *
A18.H2O)) - k.NXR * a17.NXR.NO2 * C17.NO2 - k.NIR * a17.NIR * C17.NO2 - k.exch.f *
a17.f * C17.NO2 + ( k.exch.f * a17.f * a17.eq * A17.H2O * (C16.NO2 + a17.f * C17.NO2 +
a18.f * C18.NO2) / (A16.H2O + a17.f * a17.eq * A17.H2O + a18.f * a18.eq * A18.H2O)))
dC18.NO2 <- (k.NAR * a18.NAR / a18.NAR.BR * C18.NO3 * 2 / 3 + ( k.AMO
* a18.AMO.O2 * A18.O2 * (C14.NH4 + a15.AMO * C15.NH4) / (A16.O2 + a17.AMO.O2 *
A17.O2 + a18.AMO.O2 * A18.O2)) + ( k.AMO * a18.AMO.H2O * A18.H2O * (C14.NH4 +
a15.AMO * C15.NH4) / (A16.H2O + a17.AMO.H2O * A17.H2O + a18.AMO.H2O * A18.H2O))
- k.NXR * a18.NXR.NO2 * C18.NO2 - k.NIR * a18.NIR * C18.NO2 - k.exch.f * a18.f *
C18.NO2 + (k.exch.f * a18.f * a18.eq * A18.H2O * (C16.NO2 + a17.f * C17.NO2 + a18.f *
C18.NO2) / (A16.H2O + a17.f * a17.eq * A17.H2O + a18.f * a18.eq * A18.H2O)))
dC14.NH4 <- (- k.AMO * C14.NH4 - k.AMX * C14.NH4 )
dC15.NH4 <- (- k.AMO * a15.AMO * C15.NH4 - k.AMX * a15.AMX * C15.NH4)
return(list(c(dC14.NO3 = dC14.NO3, dC15.NO3 = dC15.NO3, dC14.NO2 = dC14.NO2,
dC15.NO2 = dC15.NO2, dC14.NH4 = dC14.NH4, dC15.NH4 = dC15.NH4, dC16.NO3 =
dC16.NO3, dC18.NO3 = dC18.NO3, dC17.NO3 = dC17.NO3, dC16.NO2 = dC16.NO2,
dC18.NO2 = dC18.NO2, dC17.NO2 = dC17.NO2)))})}
#=======================#
33
# RTM Model #
#=======================#
RTM.model <- function(t = 0, state,pars = NULL) {
C14.NO3 <- state[( 0*N+1) : ( 1*N)]
C15.NO3 <- state[( 1*N+1) : ( 2*N)]
C14.NO2 <- state[( 2*N+1) : ( 3*N)]
C15.NO2 <- state[( 3*N+1) : ( 4*N)]
C14.NH4 <- state[( 4*N+1) : ( 5*N)]
C15.NH4 <- state[( 5*N+1) : ( 6*N)]
C16.NO3 <- state[( 6*N+1) : ( 7*N)]
C18.NO3 <- state[( 7*N+1) : ( 8*N)]
C17.NO3 <- state[( 8*N+1) : ( 9*N)]
C16.NO2 <- state[( 9*N+1) : (10*N)]
C18.NO2 <- state[(10*N+1) : (11*N)]
C17.NO2 <- state[(11*N+1) : (12*N)]
C14.NO3.bot <- C14.NO3[N]
tran14.NO3 <- D.NO3 * diff(diff(c(C14.NO3.0, C14.NO3, C14.NO3.bot))/dz)/dz
reac14.NO3 <- (k.NXR * C14.NO2 - k.NAR * C14.NO3)
dC14.NO3 <- tran14.NO3 + reac14.NO3
C15.NO3.bot <- C15.NO3[N]
tran15.NO3 <- D.NO3 * diff(diff(c(C15.NO3.0, C15.NO3, C15.NO3.bot))/dz)/dz
reac15.NO3 <- (k.NXR * a15.NXR * C15.NO2 - k.NAR * a15.NAR * C15.NO3)
dC15.NO3 <- tran15.NO3 + reac15.NO3
C16.NO3.bot <- C16.NO3[N]
tran16.NO3 <- D.NO3 * diff(diff(c(C16.NO3.0, C16.NO3, C16.NO3.bot))/dz)/dz
reac16.NO3 <- (k.NXR * C16.NO2 + (0.5 * k.NXR * A16. * (C16.NO2 +
a17.NXR.NO2 * C17.NO2 + a18.NXR.NO2 * C18.NO2) / (A16.H2O + a17.NXR.H2O *
A17.H2O + a18.NXR.H2O * A18.H2O)) - k.NAR * C16.NO3)
dC16.NO3 <- tran16.NO3 + reac16.NO3
C17.NO3.bot <- C17.NO3[N]
tran17.NO3 <- D.NO3 * diff(diff(c(C17.NO3.0, C17.NO3, C17.NO3.bot))/dz)/dz
reac17.NO3 <- (k.NXR * a17.NXR.NO2 * C17.NO2 + (0.5 * k.NXR * a17.NXR.H2O *
A17.H2O * (C16.NO2 + a17.NXR.NO2 * C17.NO2 + a18.NXR.NO2 * C18.NO2) / (A16.H2O
+ a17.NXR.H2O * A17.H2O + a18.NXR.H2O * A18.H2O)) - k.NAR * a17.NAR * C17.NO3)
dC17.NO3 <- tran17.NO3 + reac17.NO3
C18.NO3.bot <- C18.NO3[N]
tran18.NO3 <- D.NO3 * diff(diff(c(C18.NO3.0, C18.NO3, C18.NO3.bot))/dz)/dz
reac18.NO3 <- (k.NXR * a18.NXR.NO2 * C18.NO2 + (0.5 * k.NXR * a18.NXR.H2O *
A18.H2O * (C16.NO2 + a17.NXR.NO2 * C17.NO2 + a18.NXR.NO2 * C18.NO2) / (A16.H2O
+ a17.NXR.H2O * A17.H2O + a18.NXR.H2O * A18.H2O)) - k.NAR * a18.NAR * C18.NO3)
dC18.NO3 <- tran18.NO3 + reac18.NO3
C14.NO2.bot <- C14.NO2[N]
tran14.NO2 <- D.NO2 * diff(diff(c(C14.NO2.0, C14.NO2, C14.NO2.bot))/dz)/dz
reac14.NO2 <- (k.NAR * C14.NO3 + k.AMO * C14.NH4 - k.NXR * C14.NO2 - k.NIR *
C14.NO2)
dC14.NO2 <- tran14.NO2 + reac14.NO2
34
C15.NO2.bot <- C15.NO2[N]
tran15.NO2 <- D.NO2 * diff(diff(c(C15.NO2.0, C15.NO2, C15.NO2.bot))/dz)/dz
reac15.NO2 <- (k.NAR * a15.NAR * C15.NO3 + k.AMO * a15.AMO * C15.NH4 -
k.NXR * a15.NXR * C15.NO2 - k.NIR * a15.NIR * C15.NO2)
dC15.NO2 <- tran15.NO2 + reac15.NO2
C16.NO2.bot <- C16.NO2[N]
tran16.NO2 <- D.NO2 * diff(diff(c(C16.NO2.0, C16.NO2, C16.NO2.bot))/dz)/dz
reac16.NO2 <- (k.NAR * C16.NO3 * 2 / 3 + ( k.AMO * A16.O2 * (C14.NH4 +
a15.AMO * C15.NH4) / (A16.O2 + a17.AMO.O2 * A17.O2 + a18.AMO.O2 * A18.O2)) +
( k.AMO * A16.H2O * (C14.NH4 + a15.AMO * C15.NH4) / (A16.H2O + a17.AMO.H2O *
A17.H2O + a18.AMO.H2O * A18.H2O)) - k.NXR * C16.NO2 - k.NIR * C16.NO2 - k.exch.f *
C16.NO2 + ( k.exch.f * A16.H2O * (C16.NO2 + a17.f * C17.NO2 + a18.f * C18.NO2) /
(A16.H2O + a17.f * a17.eq * A17.H2O + a18.f * a18.eq * A18.H2O)))
dC16.NO2 <- tran16.NO2 + reac16.NO2
C17.NO2.bot <- C17.NO2[N]
tran17.NO2 <- D.NO2 * diff(diff(c(C17.NO2.0, C17.NO2, C17.NO2.bot))/dz)/dz
reac17.NO2 <- (k.NAR * a17.NAR / a17.NAR.BR * C17.NO3 * 2 / 3 + ( k.AMO
* a17.AMO.O2 * A17.O2 * (C14.NH4 + a15.AMO * C15.NH4) / (A16.O2 + a17.AMO.O2
* A17.O2 + a18.AMO.O2 * A18.O2)) + ( k.AMO * a17.AMO.H2O * A17.H2O * (C14.NH4
+ a15.AMO * C15.NH4) / (A16.H2O + a17.AMO.H2O * A17.H2O + a18.AMO.H2O *
A18.H2O)) - k.NXR * a17.NXR.NO2 * C17.NO2 - k.NIR * a17.NIR * C17.NO2 - k.exch.f *
a17.f * C17.NO2 + ( k.exch.f * a17.f * a17.eq * A17.H2O * (C16.NO2 + a17.f * C17.NO2 +
a18.f * C18.NO2) / (A16.H2O + a17.f * a17.eq * A17.H2O + a18.f * a18.eq * A18.H2O)))
dC17.NO2 <- tran17.NO2 + reac17.NO2
C18.NO2.bot <- C18.NO2[N]
tran18.NO2 <- D.NO2 * diff(diff(c(C18.NO2.0, C18.NO2, C18.NO2.bot))/dz)/dz
reac18.NO2 <- (k.NAR * a18.NAR / a18.NAR.BR * C18.NO3 * 2 / 3 + ( k.AMO
* a18.AMO.O2 * A18.O2 * (C14.NH4 + a15.AMO * C15.NH4) / (A16.O2 + a17.AMO.O2 *
A17.O2 + a18.AMO.O2 * A18.O2)) + ( k.AMO * a18.AMO.H2O * A18.H2O * (C14.NH4 +
a15.AMO * C15.NH4) / (A16.H2O + a17.AMO.H2O * A17.H2O + a18.AMO.H2O * A18.H2O))
- k.NXR * a18.NXR.NO2 * C18.NO2 - k.NIR * a18.NIR * C18.NO2 - k.exch.f * a18.f *
C18.NO2 + (k.exch.f * a18.f * a18.eq * A18.H2O * (C16.NO2 + a17.f * C17.NO2 + a18.f *
C18.NO2) / (A16.H2O + a17.f * a17.eq * A17.H2O + a18.f * a18.eq * A18.H2O)))
dC18.NO2 <- tran18.NO2 + reac18.NO2
C14.NH4.bot <- C14.NH4[N]
tran14.NH4 <- D.NH4 * diff(diff(c(C14.NH4.0, C14.NH4, C14.NH4.bot))/dz)/dz
reac14.NH4 <- (- k.AMO * C14.NH4 - k.AMX * C14.NH4 )
dC14.NH4 <- tran14.NH4 + reac14.NH4
C15.NH4.bot <- C15.NH4[N]
tran15.NH4 <- D.NH4 * diff(diff(c(C15.NH4.0, C15.NH4, C15.NH4.bot))/dz)/dz
reac15.NH4 <- (- k.AMO * a15.AMO * C15.NH4 - k.AMX * a15.AMX * C15.NH4)
dC15.NH4 <- tran15.NH4 + reac15.NH4
return(list(c(dC14.NO3 = dC14.NO3, dC15.NO3 = dC15.NO3, dC14.NO2 = dC14.NO2,
dC15.NO2 = dC15.NO2, dC14.NH4 = dC14.NH4, dC15.NH4 = dC15.NH4, dC16.NO3 =
35
dC16.NO3, dC18.NO3 = dC18.NO3, dC17.NO3 = dC17.NO3, dC16.NO2 = dC16.NO2,
dC18.NO2 = dC18.NO2, dC17.NO2 = dC17.NO2)))}
#======================#
# Parameter definition #
#======================#
C.NO3.0 <- 10 #initial concentration of NO3[umol L-1]
C.NO2.0 <- 0 #initial concentration of NO2, 0 to 20 [umol L-1]
C.NH4.0 <- 10 #initial concentration of NH4[umol L-1]
#diffusion coefficient
D.NO3 <- 0.1728 #[0.1728 m2 day-1](Kuypers,2003)
D.NO2 <- 0.1728 #= 1 to 5*D.NO3
D.NH4 <- 0.1728 #= D.NO3
#reaction constant [day-1], first order kinetics Rate=k[C]
k.NAR <- 1e-3 #denitrification (Casciotti, 2012)
k.NXR <- 0*k.NAR #nitrification, 0 to 2* k.NAR
k.NIR <- 1*k.NAR #nitrite reduction, 1 to 100* k.NAR
k.AMO <- 0*k.NAR #aerobic ammonia oxidation, 0 to 100*k.NAR
k.AMX <- 1*k.NAR #anammox, 1 to 100* k.NAR
k.exch.f <- 0*k.NAR #NO2 exchange with water, forward, 0 to 1000* k.NAR
#reference of isotope compositions
O18.VSMOW <- (2005.20/(10^6))
O17.VSMOW <- (379.9/(10^6))
N.air <- (0.00366)
# initial N and O isotope compositions
d15.NO3.0 <- 0
d15.NO2.0 <- -40 #not used in our model
d15.NH4.0 <- 0
d18.NO3.0 <- 0
d18.NO2.0 <- -40 #not used in our model
d18.H2O <- 0 # 0 to -20
d18.O2 <- 23
theta <-0.528
d17.NO3.0 <- theta * d18.NO3.0
d17.NO2.0 <- theta * d18.NO2.0
d17.H2O <- theta * d18.H2O
d17.O2 <- theta * d18.O2
#fractionation factors (Granger, 2016)
a15.NAR <- (-15 / 1000 + 1)
a18.NAR <- (a15.NAR)
a17.NAR <- 0.992
a18.NAR.BR <- (-25 / 1000 + 1)
a17.NAR.BR <- a18.NAR.BR^theta
a15.NXR <- (+15 / 1000 + 1)
a18.NXR.NO2 <- (+4 / 1000 + 1)
a17.NXR.NO2 <- a18.NXR.NO2^theta
a18.NXR.H2O <- (-14 / 1000 + 1)
36
a17.NXR.H2O <- a18.NXR.H2O^theta
a15.NIR <- (-5 / 1000 + 1)
a18.NIR <- (a15.NIR)
a17.NIR <- a18.NIR^theta
a15.AMX <- (-35 / 1000 + 1)
a15.AMO <- (-26 / 1000 + 1)
a18.AMO.H2O <- (-14 / 1000 + 1)
a17.AMO.H2O <- a18.AMO.H2O^theta
a18.AMO.O2 <- (-14 / 1000 + 1)
a17.AMO.O2 <- a18.AMO.O2^theta
a18.eq <- (13.5 / 1000 + 1) #equilibrium alpha between NO2 and H2O
a17.eq <- a18.eq^theta
a18.f <- (0 / 1000 + 1) #forward KIE NO2->H2O
a17.f <- a18.f^theta
# Calculate initial condition (R: isotope ratio, C: concentration, A: isotope abundance)
R15.NO3.0 <- (exp(d15.NO3.0 / 1000) * N.air)
C14.NO3.0 <- (C.NO3.0 / (1 + R15.NO3.0))
C15.NO3.0 <- (C.NO3.0 * (R15.NO3.0)/ (1 + R15.NO3.0))
R18.NO3.0 <- (exp(d18.NO3.0 / 1000) * O18.VSMOW)
R17.NO3.0 <- (exp(d17.NO3.0 / 1000) * O17.VSMOW)
C16.NO3.0 <- (3 * C.NO3.0 / (1 + R18.NO3.0 + R17.NO3.0))
C18.NO3.0 <- (3 * C.NO3.0 * R18.NO3.0 / (1 + R18.NO3.0 + R17.NO3.0))
C17.NO3.0 <- (3 * C.NO3.0 * R17.NO3.0 / (1 + R18.NO3.0 + R17.NO3.0))
R15.NO2.0 <- (exp(d15.NO2.0 / 1000) * N.air)
C14.NO2.0 <- (C.NO2.0 / (1 + R15.NO2.0))
C15.NO2.0 <- (C.NO2.0 * (R15.NO2.0)/ (1 + R15.NO2.0))
R18.NO2.0 <- (exp(d18.NO2.0 / 1000) * O18.VSMOW)
R17.NO2.0 <- (exp(d17.NO2.0 / 1000) * O17.VSMOW)
C16.NO2.0 <- (2 * C.NO2.0 / (1 + R18.NO2.0 + R17.NO2.0))
C18.NO2.0 <- (2 * C.NO2.0 * R18.NO2.0 / (1 + R18.NO2.0 + R17.NO2.0))
C17.NO2.0 <- (2 * C.NO2.0 * R17.NO2.0 / (1 + R18.NO2.0 + R17.NO2.0))
R15.NH4.0 <- (exp(d15.NH4.0 / 1000) * N.air)
C14.NH4.0 <- (C.NH4.0 / (1 + R15.NH4.0))
C15.NH4.0 <- (C.NH4.0 * (R15.NH4.0)/ (1 + R15.NH4.0))
R18.H2O <- (exp(d18.H2O / 1000) * O18.VSMOW)
R17.H2O <- (exp(d17.H2O / 1000) * O17.VSMOW)
A16.H2O <- (1 / (1 + R18.H2O + R17.H2O))
A18.H2O <- (R18.H2O / (1 + R18.H2O + R17.H2O))
A17.H2O <- (R17.H2O / (1 + R18.H2O + R17.H2O))
R18.O2 <- (exp(d18.O2 / 1000) * O18.VSMOW)
R17.O2 <- (exp(d17.O2 / 1000) * O17.VSMOW)
A16.O2 <- (1 / (1 + R18.O2 + R17.O2))
A18.O2 <- (R18.O2 / (1 + R18.O2 + R17.O2))
A17.O2 <- (R17.O2 / (1 + R18.O2 + R17.O2))
#====================#
# Solve RTM Model #
37
#====================#
#Load package
library(rootSolve)
#Set Grid
dz <- 0.1
z <-seq(0,100,by=dz)
N <- length(z)
#Solve RTM model
yini.rtm <- rep(0, length.out = 12*N)
rtmout <- steady.1D( y = yini.rtm, pos = TRUE, func = RTM.model, nspec = 12, names =
c("C14.NO3", "C15.NO3", "C14.NO2", "C15.NO2", "C14.NH4", "C15.NH4", "C16.NO3",
"C18.NO3", "C17.NO3", "C16.NO2", "C18.NO2", "C17.NO2"))
#Read isotope concentrations of nitrate
C14.NO3.rtm <- rtmout$y[ , "C14.NO3"]
C15.NO3.rtm <- rtmout$y[ , "C15.NO3"]
C16.NO3.rtm <- rtmout$y[ , "C16.NO3"]
C18.NO3.rtm <- rtmout$y[ , "C18.NO3"]
C17.NO3.rtm <- rtmout$y[ , "C17.NO3"]
#Omit last 40% data for accuracy
C14.NO3.rtm <- C14.NO3.rtm[-c((length(C14.NO3.rtm)-round((length(C14.NO3.rtm)/2.5),
digits = 0)+1):length(C14.NO3.rtm))]
C15.NO3.rtm <- C15.NO3.rtm[-c((length(C15.NO3.rtm)-round((length(C15.NO3.rtm)/2.5),
digits = 0)+1):length(C15.NO3.rtm))]
C16.NO3.rtm <- C16.NO3.rtm[-c((length(C16.NO3.rtm)-round((length(C16.NO3.rtm)/2.5),
digits = 0)+1):length(C16.NO3.rtm))]
C18.NO3.rtm <- C18.NO3.rtm[-c((length(C18.NO3.rtm)-round((length(C18.NO3.rtm)/2.5),
digits = 0)+1):length(C18.NO3.rtm))]
C17.NO3.rtm <- C17.NO3.rtm[-c((length(C17.NO3.rtm)-round((length(C17.NO3.rtm)/2.5),
digits = 0)+1):length(C17.NO3.rtm))]
#Calculate isotope compositions
R15.NO3.rtm <- C15.NO3.rtm / C14.NO3.rtm
d15.NO3.rtm <- 1000 * log((R15.NO3.rtm) / N.air)
R18.NO3.rtm <- C18.NO3.rtm / C16.NO3.rtm
d18.NO3.rtm <- 1000 * log((R18.NO3.rtm) / O18.VSMOW)
R17.NO3.rtm <- C17.NO3.rtm / C16.NO3.rtm
d17.NO3.rtm <- 1000 * log((R17.NO3.rtm) / O17.VSMOW)
#====================#
# Solve Batch Model #
#====================#
#Load package
library(deSolve)
#Set Grid
dt <- 4
times <-seq(0,4000,by=dt)
#Give initial condition
38
yini.rdm <- c( C14.NO3 = C14.NO3.0, C15.NO3 = C15.NO3.0, C14.NO2 = C14.NO2.0,
C15.NO2 = C15.NO2.0, C14.NH4 = C14.NH4.0, C15.NH4 = C15.NH4.0, C16.NO3 =
C16.NO3.0, C18.NO3 = C18.NO3.0, C17.NO3 = C17.NO3.0, C16.NO2 = C16.NO2.0,
C18.NO2 = C18.NO2.0, C17.NO2 = C17.NO2.0)
#Solve RDM model
rdmout <- ode(y = yini.rdm, time = times, func = RDM.model, parms = NULL)
#Read isotope concentrations of nitrate
C14.NO3.rdm <- rdmout[ , "C14.NO3"]
C15.NO3.rdm <- rdmout[ , "C15.NO3"]
C16.NO3.rdm <- rdmout[ , "C16.NO3"]
C18.NO3.rdm <- rdmout[ , "C18.NO3"]
C17.NO3.rdm <- rdmout[ , "C17.NO3"]
#Omit last 40% data for accuracy
C14.NO3.rdm <- C14.NO3.rdm[-c((length(C14.NO3.rdm)-round((length(C14.NO3.rdm)/2.5),
digits = 0)+1):length(C14.NO3.rdm))]
C15.NO3.rdm <- C15.NO3.rdm[-c((length(C15.NO3.rdm)-round((length(C15.NO3.rdm)/2.5),
digits = 0)+1):length(C15.NO3.rdm))]
C16.NO3.rdm <- C16.NO3.rdm[-c((length(C16.NO3.rdm)-round((length(C16.NO3.rdm)/2.5),
digits = 0)+1):length(C16.NO3.rdm))]
C18.NO3.rdm <- C18.NO3.rdm[-c((length(C18.NO3.rdm)-round((length(C18.NO3.rdm)/2.5),
digits = 0)+1):length(C18.NO3.rdm))]
C17.NO3.rdm <- C17.NO3.rdm[-c((length(C17.NO3.rdm)-round((length(C17.NO3.rdm)/2.5),
digits = 0)+1):length(C17.NO3.rdm))]
#Calculate isotope compositions
R15.NO3.rdm <- (C15.NO3.rdm / C14.NO3.rdm)
d15.NO3.rdm <- (1000 * log((R15.NO3.rdm) / N.air))
R18.NO3.rdm <- (C18.NO3.rdm / C16.NO3.rdm)
d18.NO3.rdm <- (1000 * log((R18.NO3.rdm) / O18.VSMOW))
R17.NO3.rdm <- (C17.NO3.rdm / C16.NO3.rdm)
d17.NO3.rdm <- (1000 * log((R17.NO3.rdm) / O17.VSMOW))
39
CHAPTER 3. IDENTIFYING APPARENT LOCAL STABLE ISOTOPE
EQUILIBRIUM IN A COMPLEX NON-EQUILIBRIUM SYSTEM1
3.1. Introduction
Living systems are open systems that exchange energy and matter with the environment. Life is
inherently out-of-equilibrium chemically in general and stable isotopically in particular among
its many constituents. However, reversibility of enzymatic reactions is well recognized (65-67).
It leads to the possibility that an organism can attain local isotope equilibrium among or within
certain biomolecules for some elements like carbon if a subset of biomolecules is involved in
fully reversible reactions (68-71). By inference, biomolecules involved in the least reversible
reactions should have isotope composition away from a thermodynamic equilibrium state (68-
70). Thus, the deviation of the stable isotope composition of a set of biomolecules from their
expected thermodynamic equilibrium state may be used as a measure of the degree of expressed
reversibility in enzymatic reactions.
It was observed that, among or within some biomolecules in organisms, good linear
correlation exists between carbon isotope composition (δ13C) and the corresponding 13β value
(68). Here, 13β is the reduced partition function ratio (RPFR), which is the predicted equilibrium
isotope fractionation factor between a carbon compound or a site-specific carbon and the atomic
carbon (i.e. the calculated equilibrium 13RA/13Ratomic value at a given temperature, where 13R =
[13C]/[12C]).The concept of β or RPFR was proposed independently by Urey (72) and Bigeleisen
and Mayer (73). It is a theoretical corner-stone of stable isotope geochemistry, which facilitates a
universal comparison of equilibrium stable isotope fractionation between any compounds of
interest.
1This chapter, previously published as He, Y., Cao, X., Wang, J. and Bao, H., 2018. Identifying apparent local stable
isotope equilibrium in a complex non‐equilibrium system. Rapid Communications in Mass Spectrometry, 32(4),
pp.306-310. is reprinted here by permission, according to the “COPYRIGHT TRANSFER AGREEMENT of Rapid
Communications in Mass Spectrometry Published by Wiley (the "Owner").”
40
The 13β-δ13C correlation is described as:
𝛿13𝐶 − 𝛿13𝐶𝑎𝑣𝑒 = 𝜒(13𝛽−13𝛽𝑎𝑣𝑒 ) × 103 (3.1)
where ave denotes the arithmetic average of δ or β values for all the components of
interest, and χ is the regression coefficient, reflecting the degree of “thermodynamic order” with
respect to a complete equilibrium state when χ = 1 (68). Some of the observed strong positive
13β-δ13C correlations led Galimov (68) to propose that thermodynamic order or certain degree of
isotope equilibrium is sometimes present among biomolecules, even though as a whole a living
system is not in thermodynamic equilibrium. He pointed out that the fundamental reason behind
the apparent thermodynamic order is the reversibility of enzymatic reactions.
The idea that reversibility of enzymatic reactions is commonly expressed in biological
systems was disputed (74). Also, the cited positive 13β-δ13C correlation as a valid evidence for
the existence of close-to-equilibrium isotope distribution has been argued, since 13β-δ13C
correlation is not commonly observed in living systems and, even if an apparent 13β-δ13C
correlation is observed, thermodynamic control is not the unique explanation (75-76). For
instance, a δ13C value distribution of a set of molecules produced by non-equilibrium
biochemical process (e.g. kinetic isotope effect) could also correlate with their corresponding 13β
values, or that the observed correlation could just be a systematic deviation of biomolecules
relative to the corresponding equilibrium state (75).
We noticed that the 13β-δ13C correlation (68-70) uses the average isotope composition of
a given system as the reference for its 13β-δ13C correlation. In a system consisting of multiple
components, if we choose a component as a reference for mutual comparison, even if the
reference is the average stable isotope composition of compounds of interest, we have effectively
assigned that reference to be at equilibrium. The use of such a reference is not mathematically
41
rigorous and can often be misleading when dealing with a complex non-equilibrium system. This
is simply because we do not know a priori which compound or set of compounds represents the
state of isotope equilibrium. In this contribution, we propose that the deviation of a complex
system from its equilibrium state can be rigorously described as a graph problem, as applied in
discrete mathematics. It can be represented by an apparent α difference matrix (|Δα|), i.e. the
difference matrix between the observed isotope difference matrix (|αob|) and its corresponding
equilibrium fractionation matrix (|αeq|). We show that the |Δα| matrix can be used to assess
apparent local equilibrium in an overall non-equilibrium complex system, both in general and in
specific cases.
3.2. Methods For Evaluating a System at Disequilibrium States
A system can have only one possible distribution, i.e. one set of α values, at isotope equilibrium
but numerous possible distributions at isotope disequilibrium.
If we treat each biomolecule as a node with one variable (δ’ or 1000lnβ values) and the
apparent difference in isotope composition as distance, the distribution of the isotope
composition of all biomolecules in question in a system is, mathematically, a graph problem.
Distance matrix is an exhaustive tabulation that consists of pairwise relationships between
components. It is an abstraction that represents the relative position information of a graph,
which can be used to describe the distribution of a graph and the similarities between graphs (77).
We have applied here the concept of distance matrix to describe the distribution of isotope
compositions of multiple components in a complex system.
For a system with K number of components, the isotope composition distribution can be
described by a K×K matrix:
42
|𝛼| = |
𝛼11 𝛼12 𝛼13 ⋯ 𝛼1𝐾𝛼21
⋮𝛼22
⋮𝛼23
⋮⋯⋱
𝛼2𝐾
⋮𝛼𝐾1 𝛼𝐾2 𝛼𝐾3 ⋯ 𝛼𝐾𝐾
| (3.2)
where αij (αij = δ’i-δ’j) denotes the difference in isotope compositions between component
i and component j. For predicted equilibrium state, αij = 1000lnβi-1000lnβj.
The difference between observed isotope difference |αob| and theoretical calculated
equilibrium isotope difference |αeq|:
|𝛥𝛼| = |𝛼𝑜𝑏| − |𝛼𝑒𝑞| (3.3)
describes the deviation of the measured isotope composition distribution from the
equilibrium state. |Δα| is a zero matrix when the system reaches equilibrium. Thus, the |Δα|
method removes the influence of isotope source and considers directly the distribution pattern of
isotope composition among components. Thus, different sets of isotope compositions, regardless
of the background isotope composition, can be compared directly.
|Δα| matrices can be visualized with a heat map, in which the absolute values of apparent
α difference (the deviation from equilibrium isotope fractionation) are illustrated in color. Our
heat map illustrates only the lower triangle of the matrix without a diagonal because the matrix is
symmetric with a diagonal of 0 (Figs. 3.1. and 3.2.).
In addition to the overall isotope distribution, each Δαij measures the degree of deviation
of the apparent α difference between component i and j from their respective equilibrium values.
Thus, apparent local equilibrium can also be identified using |Δα| matrices by looking for
components with |Δαij| ≈ 0. The heat map visually helps inspect the closeness of any paired
components to their isotope equilibrium state. In the next section, we re-process some of the
published data using the |Δα| method to examine if there is indeed local equilibrium among
amino acids (AAs) in living systems.
43
3.3. Applications
Using equilibrium β values of AAs calculated by Rustad (78), we evaluate two cases, potato and
a green alga, in the studies of thermodynamic order (68). AAs are grouped into different families
on the basis of their properties and biosynthetic pathways. The six families are: (i) α-
ketoglutarate family, including glutamic acid, glutamine, proline and arginine; (ii) 3-
phosphoglycerate family, including serine, glycine and cysteine; (iii) oxaloacetate family,
including aspartic acid, asparagine, methionine, threonine, lysine and isoleucine; (iv) pyruvate
family, including alanine, valine and leucine; (v) phosphoenol pyruvate and erythrose 4-
phosphate family, including tryptophan, phenylalanine and tyrosine; and (vi) ribose 5-phosphate
family, histidine.
3.3.1. Potato Leaves and Tubers
The very existence of “thermodynamic order” in AAs in potato leaves (79) has been heavily
debated (68-69, 75-76). Our constructed heat maps of |Δα| matrices (Fig. 3-1) show that local
equilibrium does exist among the synthetic families of AAs in potato leaves.
Fig. 3-1 shows that, in potato leaves, AAs in the same synthetic families are close to
apparent carbon isotope equilibrium with each other: (i) alanine and valine are in the pyruvate
family (Δαala-val 0.9‰); (ii) lysine is in the oxaloacetate family, synthesized from aspartic acid
(Δαasp-lys 0.4‰); (iii) proline is in the α-ketoglutarate family, synthesized from glutamic acid
(Δαglu-pro 0.4‰); and (iv) phenylalanine and tyrosine are both synthesized from phosphoenol
pyruvate and erythrose-4-phosphate (Δαphe-tyr 0.1‰). An apparent close-to-equilibrium
relationship also exists among proline, lysine, glutamic acid, and aspartic acid (oxaloacetate
family and α-ketoglutarate family), which may be caused by isotope exchange between their
precursors, α-ketoglutarate and oxaloacetate, in the citric acid cycle (80). In addition,
44
oxaloacetate can be synthesized from pyruvate, and the metabolic product of valine, succinyl-
CoA, can also enter the citric acid cycle, which may explain the apparently close-to-equilibrium
relationship between valine and aspartic acid (Δαasp-val 0.7‰).
Figure 3.1. |Δα| matrix heat maps for carbon isotope composition of AAs in potato leaves (left)
and tubers (right). Each cell corresponds to the apparent α difference (Δα) for correspondingly
paired AAs. The value of Δα is represented in a color scale, with deep blue for close to apparent
equilibrium and with light blue for the largest deviation. Black squares mark paired AAs of the
same synthetic family, ala, alanine; val, valine; asp, aspartic acid; lys, lysine; glu, glutamic acid;
pro, proline; phe, phenylalanine; tyr, tyrosine; ser, serine. 13β values were calculated by Rustad
(78) and δ13C are measured by Gleixner et al (79).
Interestingly, compared with potato leaves, the carbon isotope composition of AAs in the
tubers of the same potato displays a different distribution pattern. Specifically, the apparent
close-to-equilibrium isotope differences among synthetic families that are observed in leaves are
absent from the tubers. However, other apparent close-to-equilibrium isotope differences appear,
e.g. Δαlys-val 0.5‰, Δαglu-val 0.5‰, Δαglu-asp 0.2‰, Δαpro-tyr 0.2‰, and Δαphe-ser 0.0‰. The
difference in |Δα| matrices for potato leaves and tubers indicates that different degrees of
reversibility in AAs synthetic pathways or entirely different synthetic pathways may exist in the
two different potato parts. Conventionally, AAs synthesis is believed to take place entirely in
leaves (81). If AAs in tubers are synthesized in leaves and distributed to the tuber from leaves
45
alone, |Δα| for tubers should be the same as for leaves. Recent studies reveal, however, that some
AAs, e.g. asparagine, can be synthesized in situ in potato tubers (82). Our results support
Muttucumaru et al.’s suggestion that AAs synthesis may not occur entirely in leaves. In addition,
the degradation of AAs in tubers can alter the isotope distribution, which may lead to the
disappearance of apparent local equilibrium that was initially transferred from potato leaves.
3.3.2. The Green Alga Chlorella Pyrenoidosa
Besides the apparently well-correlated 13β-δ13C cases, some cases of poor 13β-δ13C correlations
are presented (68). The poor correlation was interpreted as an evidence for less expressed
reversibility in enzymatic reactions. For instance, a “thermodynamic order” was shown to exist
among different AAs in the autotrophic green alga, Chlorella pyrenoidosa, but was absent from a
glucose-cultured group. This difference was attributed to the disruption of “thermodynamic order”
when the alga was fed with external nutrients or another carbon source like glucose. We
reanalyzed the data from Abelson and Hoering (83) and the results are presented in Fig. 3.2.
The study of 13β-δ13C correlations led to a conclusion that the degree of reversibility of
enzymatic reactions in the glucose-cultured green alga was much poorer than in the autotrophic
one (68). However, our |Δα| matrix analysis shows that a different apparent local equilibrium
exists among different AAs regardless of the culture method or nutrition source of the alga.
Opposite to the prediction that external carbon source disturbs thermodynamic order in
organisms (68), the pyruvate family (alanine and leucine, Δαala-leu 0.3‰) and oxaloacetate family
(isoleucine and lysine, Δαile-lys 0.1‰) are closer to apparent equilibrium in the glucose-cultured
alga than in the autotrophic alga (Δαala-leu 7.1‰, Δαile-lys 1.8‰). It would be expected that the
isotope distribution would differ in the two differently cultured alga, since the external carbon
46
source would alter the metabolic kinetics. The current data cannot lead to the conclusion that one
of the systems is more in equilibrium than the other.
Figure 3.2. |Δα| matrix heat maps for carbon isotope composition of AAs in autotrophic (left)
and glucose-cultured (right) Chlorella pyrenoidosa. Each cell corresponds to the apparent α
difference (Δα) for a pair of AAs. The value of Δα is represented on a color scale, with deep blue
for close to apparent equilibrium and with light blue for the largest deviation. Black squares
mark paired AAs of the same synthetic family; for abbreviations see Figure 3.1. 13β values were
calculated by Rustad (78) and δ13C are measured by Abelson and Heoring (83).
3.4. Implications and Future Works
Our |Δα| matrix approach for analyzing intermolecular isotope distribution of a complex system
can be applied to those of intramolecular systems as well. The analysis can serve as forensic
identifiers for biosynthetic pathways and their reversibility, metabolic states, environmental
conditions, and other information. For instance, different mechanism of fatty acids production in
E. coli and S. cerevisiae are deciphered using intramolecular δ13C variations (84-85). Before we
understand intramolecular δ13C variations, we need to find an appropriate way to describe it. Our
|Δα| matrix serves to this purpose by looking at a system as a whole. Another important condition
in applying |Δα| matrix approach is to know the equilibrium state, from which the disequilibrium
state can be evaluate. Some average and site-specific 13β values for organic molecules are
47
calculated (68, 78, 86-87). The ab initio calculation can be used to estimate 13β values
theoretically.
3.5. Conclusions
We have demonstrated that the |Δα| matrix describes rigorously the stable isotope distributions in
a system that has many interacting components. The previous simple linear δ-β correlation
method is inadequate mainly because we are dealing with complex, multi-component systems in
which each compound or site has its own predicted difference between measured and
equilibrium values. The |Δα| approach can help identify apparent local equilibrium in an
inherently disequilibrium system. The applications of |Δα| matrix to the carbon isotope
composition of different AAs from earlier published data, e.g. potato leaves and tubers, as well
as the autotrophic vs. glucose-cultured green alga, Chlorella pyrenoidosa, shows that remarkable
apparent local equilibrium does exist in potato leaves and also to a certain degree in both
example of the green alga.
48
CHAPTER 4. EQUILIBRIUM INTRAMOLECULAR ISOTOPE
DISTRIBUTION IN LARGE ORGANIC MOLECULES
4.1. Introduction
Large molecules have different positions for carbon, oxygen, or hydrogen. Benefiting
from the development of position-specific (PS) isotope measurement techniques,
intramolecular isotope distribution (Intra-ID) has become a promising tool for identifying
and quantifying the production and consumption pathways of large molecules. For
example, Intra-ID has been used to study metabolic processes in organisms or
biodegradation processes of large organic molecules. Albertson and Hoering (83) first
isolated and measured δ13C values of carboxyl carbons in amino acids (AAs). Their
results suggested that CO2 fixation could occur via citric acid cycle other than the
ribulose diphosphate pathway in photosynthesizing algae. Monson and Hayes (84-85)
revealed different fatty acid (FA) metabolic and catabolic processes in E. coli and S.
cerevisiae from intramolecular δ13C variations. An increasing number of PS δ13C has
been measured using analytical approaches such as chemical degradation (e.g. acetoin
(88), glucose (89-90), chlorophyll and malonic acid (69 and reference therein), and AAs
(91-92)) or pyrolysis (e.g. methyl palmitate (93) and aromatic carboxylic acids (94)). In
recent years, research groups at University of Nantes (95-105), Tokyo Institute of
Technology (106-109), California Institute of Technology (86, 110-111), and Guangzhou
Institute of Geochemistry, Chinese Academy of Sciences (Li, 2018) are actively pursuing
technique innovation for PS δ13C analysis. However, along with the rapid development of
analytical technique, the information of reaction processes and their associated isotope
effects contained in Intra-ID, concealed or lost in compound-specific isotope composition
analysis, has not received its deserved attention.
49
Nitrogen isotope site preference (SP) of nitrous oxide (NNO) is one of the most
extensively studied cases of PS isotope compositions. The SP is defined by the difference
between δ15Nα (center nitrogen) and δ15Nβ (terminal nitrogen). Predicted equilibrium SP
of N2O is 45 ‰ (112). Although most observations fit the prediction that 15N
preferentially enriches in α-N position for 30-40‰ (113-115), negative SP were also
observed (116-117). It was later explained by different synthetic pathways with
symmetrical or unsymmetrical precursors. If the precursor of N2O is symmetric (e.g.
ONNO), the two nitrogens are at the same position. Any prior pathway and source
differences are eliminated by the symmetrical structures. Therefore, N2O has positive SP
that agrees with predicted equilibrium SP, since two nitrogens in the symmetrical
precursor undergo same isotope effect that leads to 15N depletion on the bond cleavage
position. If the precursor is unsymmetrical, the two nitrogens are not positionally
equivalent. Therefore, it is expected that two nitrogens have different equilibrium or
kinetic isotope effects (EIE or KIE) because they go through different reaction pathways
and may have different sources (115, 118-119).
Intra-ID has been applied more generally in biochemistry. Predicted reduced
partition function ratio (RPFR or β factor), which gives the equilibrium isotope
fractionation factor (αeq) between a PS carbon in a molecule and its atomic form.
Galimov used the similarity between measured δ13C and his predicted 13β as evidence for
biomolecules approaching intramolecular isotope equilibrium and the existence of
various degrees of reversibility in metabolism (68-70). Based on Galimov’s (68)
predicted 13β factors, Hayes (120) illustrated the equilibrium Intra-IDs of chlorophyll, and
proposed an example of using Intra-ID to interpret KIE of FA metabolic pathways. The
50
structure of FA is built by carbons alternating from methyl and carboxyl positions in
acetate. If the FA production reactions are in equilibrium, the FA molecule should have
equilibrium Intra-ID, which all the methylene carbons should have same abundance of
13C that depleted than the carboxyl carbon and enriched than the methyl carbon. If the FA
production reactions are kinetically controlled, FA should have a repeated alternating
isotope pattern, which inherited from its precursor, acetate. Based on this prediction,
Hayes interpreted different FA production mechanisms in E. coli and S. cerevisiae, as
well as kinetic isotope effects during FA biosynthesis (120).
Like cases above, isotope chemists consider the equilibrium isotope state of an
organic molecule as the state with equilibrium Intra-ID. However, unlike inorganic
reactions, reactants and products in an organic reaction commonly have different
positions for the same element. If the reaction is fully reversible, the corresponding
positions between reactants and products will reach equilibrium with each other.
However, if there is no intramolecular exchanging mechanism, different positions in
reactants or products cannot reach equilibrium. Therefore, such reaction with EIE can
also produce disequilibrium Intra-ID, if the PS carbon has disequilirbium sources.
While all these factors can result in various disequilibrium states of Intra-ID, to
quantify these disequilibrium states, we need a reference, and the end-member reference
should be the equilibrium Intra-ID state regardless whether it can be ultimately reached
or not. The information of production pathways cannot be extracted from the SP of N2O
or the variations of δ13C in FA without knowing the predicted equilibrium states. The
equilibrium state is a necessary reference for differentiating disequilibrium Intra-IDs
51
produced by a disequilibrium process from produced by an equilibrium process with
disequilibrium sources.
Using second-order Møller-Plesset (MP2) perturbation theory and Density
Functional Theory (DFT) with the B3LYP exchange-correlation functional, Rustad
calculated the average and site-specific 13β values for carbons in 20 standard amino acids
(78). The results showed that the 13β values of amine carbons can differ by as large as
15.8‰ between different AAs, which in Galimov’s method are considered the same. Ab
initio calculation offers a reliable approach to estimating β factors theoretically. Wang et
al. (121) calculated hydrogen PS 2β factors for organic molecules. Since Rustad’s and
Wang et al’s (78, 121-122), effort of calculating the equilibrium baselines for Intra-ID in
organic molecules has only resumed recently on simpler molecules such as ethane and
propane (87, 123). One of the important reasons may be that organic molecules are
usually large molecules, and the calculations are time consuming or beyond current
computational resources. All those calculations, including a RPFR calculation for Mg
isotopes in chlorophyll (124), calculated the whole organic molecule, which is time-
consuming. The PS isotope measurement is available for some large molecules like
sucrose (98), long chain n-alkane (125), nicotine (105), and predictably will be available
for even more complex biomolecules like proteins and DNAs in the coming years. Our
current limitation in PS RPFR computation for large molecules has hindered our
understanding and therefore a broader application of Intra-ID.
Quantum mechanical/molecular mechanical (QM/MM) calculations use a lower
theoretical level for outer layers to simplify the calculations for potential enzymatic
reaction mechanisms. An active site in an enzyme is bonded immediately to a few
52
functional groups in its inner layers and subsequently surrounded by out-layer functional
groups. The outer-layers exert much less influence to the active site than the inner ones.
Based on force constants and molecular geometry obtained from the QM/MM approach,
compound specific equilibrium or kinetic isotope effects for enzyme catalytic
biochemical reactions are calculated using, for example, ISOEFF program (126-129).
Besides QM/MM approach, cluster model has also been used to simplify optimization
and energy calculations of organic molecules. For example, Cu-
(imidazole)2(SCH3)(S(CH3)2)+ has been used to represent plastocyanin and obtained
consistent optical spectrum (130-131).
We argue that the calculation can be further simplified to obtain vibrational
frequencies for each position. Since the vibrational frequencies of an atom can only be
influenced by its close surroundings, a variety of cluster models have been applied to
represent minerals or solutions (132-138). In these cluster models, isotope effect of the
target atom is affected only by the nearest three molecules, and the atoms out of this
cluster size can be ignored. Cluster models can also be applicable to the calculation of
RPFRs in large organic molecules, where a set of organic molecule clusters is used to
represent different positions in a large organic molecule. Such method can drastically
simplify the calculation of PS RPFRs in a large organic molecule.
Since data from small organic molecule can be used construct PS β factors in
large organic molecules, we started to build a PS 13β database for all organic molecules.
In this study, we use density functional theory (DFT) and Urey-Bigeleisen-Mayer model
to calculate the PS 13β factors for a set of organic molecules including Pentane, Hexane,
Heptane, 3-methylpentane , 3-methylhexane, Acetic acid, Propionic acid, Butanoic acid,
53
Pentanoic Acid, hexanoic acid, 2-Hexanone, Pentanoyl chloride, n-butanol, Butylamine,
n-Butyl chloride, 1-Butanethiol, Butyl methyl ether, Pentanenitrile, n-Butylphosphate, 2-
Aminopentanoic acid, n-butyl cyclopentadiene, Butylbenzene, n-butylpyrrole, 2-
butyltetrahydrofuran and Coenzyme A (CoA). Furthermore, to display the utility, we
analyze a case of PS 13β factors in C16 FA which we constructed from the results of
pentane and pentanoic acid. Using acetate PS δ13C values from literature, and the
calculated equilibrium Intra-ID of acetate and C16 FA, we calculated the Intra-IDs of C16
FA produced by equilibrium isotope fractionation processes.
4.2. Method
13β factors can be estimated in harmonic approximation using Urey- Bigeleisen-Mayer
model (72-73). For an isotope exchange reaction between species A and B:
A+B* = A*+B (4.1)
where the superscript “*” indicates rare isotope (e.g. 13C) substituted molecules and the
one without superscript is the reference isotope (e.g. 12C).
The equilibrium constant Keq can be expressed by the ratio of partition functions:
𝐾𝑒𝑞 =[𝐴∗]
[𝐴]
[𝐵∗]
[𝐵]⁄ =
𝑄𝐴∗
𝑄𝐴
𝑄𝐵∗
𝑄𝐵⁄ =
(𝑄𝑡𝑟𝑎𝑛𝑠𝑄𝑟𝑜𝑡𝑄𝑣𝑖𝑏𝑄𝑒𝑙𝑒𝑐)𝐴∗
(𝑄𝑡𝑟𝑎𝑛𝑠𝑄𝑟𝑜𝑡𝑄𝑣𝑖𝑏𝑄𝑒𝑙𝑒𝑐)𝐴
(𝑄𝑡𝑟𝑎𝑛𝑠𝑄𝑟𝑜𝑡𝑄𝑣𝑖𝑏𝑄𝑒𝑙𝑒𝑐)𝐵∗
(𝑄𝑡𝑟𝑎𝑛𝑠𝑄𝑟𝑜𝑡𝑄𝑣𝑖𝑏𝑄𝑒𝑙𝑒𝑐)𝐵⁄ (4.2)
where Q denotes partition function of a given species. Qtrans, Qrot, Qvib and Qelec
are the translational, rotational, vibrational and electronic partition functions.
Urey- Bigeleisen-Mayer model ignored the electronic energy difference and other
quantum effects for simplicity. Using the harmonic oscillator and rigid rotator
approximations to describe the motion of a molecule, the Qtrans, Qrot, Qvib can be
expressed as (take a non-linear polyatomic molecule contains N atoms as example):
𝑄𝑡𝑟𝑎𝑛𝑠 = 𝑉 (2𝜋𝑀𝑘𝐵𝑇
ℎ2)
3/2
(4.3)
54
𝑄𝑟𝑜𝑡 =𝜋1/2(8𝜋2𝑘𝐵𝑇)
3/2(𝐼𝐴𝐼𝐵𝐼𝐶)1/2
𝜎ℎ3 (4.4)
𝑄𝑣𝑖𝑏 = ∏𝑒−𝑢𝑖/2
1−𝑒−𝑢𝑖
3𝑁−6𝑖=1 (4.5)
𝑢 =ℎ𝜐
𝑘𝐵𝑇 (4.6)
where V is volume, M is the molecular mass, kB is the Boltzmann constant, T is
temperature in K, h is the Planck constant, IA, IB, IC is the moment of inertia around axis
A, B, C of rotation, σ is the symmetry number of the molecule, and ν is the harmonic
vibration frequency.
Teller-Redlich product rule (139) was employed to simplify Urey- Bigeleisen-
Mayer model. The Teller-Redlich product rule is described as:
(𝐼𝐴𝐼𝐵𝐼𝐶)𝐴∗1/2
(𝐼𝐴𝐼𝐵𝐼𝐶)𝐴1/2 (
𝑀𝐴∗
𝑀𝐴)
3/2
= (𝑚∗
𝑚)
3𝑛/2∏
𝑢𝑖,𝐴∗
𝑢𝑖,𝐴
3𝑁−6𝑖=1 (4.7)
where n is the number of atoms been substituted (in this study, n = 1) and m*, m is
the mass of the rare and common atoms.
𝑄𝐴∗
𝑄𝐴=
𝜎𝐴
𝜎𝐴∗(
𝑚∗
𝑚)
3/2∏ (
𝑢𝑖∗
𝑢𝑖)3𝑛−6
𝑖 (𝑒−𝑢𝑖
∗ 2⁄
𝑒−𝑢𝑖 2⁄ ) (1−𝑒−𝑢𝑖
1−𝑒−𝑢𝑖∗) (4.8)
where 𝑄𝐴∗
𝑄𝐴
𝜎𝐴∗
𝜎𝐴/ (
𝑚∗
𝑚)
3/2
was called the reduced partition function ratio (RPFR) of
the isotopologue pair A*/A . Theorists often use the β factor to describe the enrichment of
rare isotope in a certain compound:
β𝑏 = RPFR = ∏ (𝑢𝑖
∗
𝑢𝑖)3𝑛−6
𝑖 (𝑒−𝑢𝑖
∗ 2⁄
𝑒−𝑢𝑖 2⁄ ) (1−𝑒−𝑢𝑖
1−𝑒−𝑢𝑖∗) (4.9)
where b is the number of X atoms substituted in the exchange reaction.
The equilibrium fractionation factor α is equal to Keq, if it is for a single-isotope-
substitution reaction:
55
𝐾𝑒𝑞 = 𝛼𝐴−𝐵 =𝑅𝑃𝐹𝑅𝐴
𝑅𝑃𝐹𝑅𝐵=
𝛽𝐴
𝛽𝐵 (4.10)
All optimization and harmonic vibrational frequencies calculations for the ground
states were performed using density functional theory (DFT) with the B3LYP exchange-
correlation functional (140-141) with the 6-311G++/d,p basis set in software Gaussian 09.
4.3. Results
The relative equilibrium PS 13C enrichments at 25 ℃ of position i are given as the
equilibrium fractionation values relative to CO2(g) (lnαeq=ln(βi/βco2)).
The molecules we chose are mostly a carbon chain with a functional group.
Similar carbons have similar lnαeq value. The stronger the bond environment is, the
heavier the lnαeq value is. Methyl carbon only has one C-C bond and three C-H bonds, its
lnαeq value is the lightest (~ -50‰). Methylene carbon has two C-C bonds and two C-H
bonds, which has a stronger bond environment than methyl carbon. Thus, methylene
carbon is heavier (lnαeq = ~ -36‰) than methyl carbon. Carboxyl carbon has one C-C
bond, one C=O bond and one C-O bond, which is in a very strong bond environment.
Thus, its lnαeq value is much heavier than other carbon types (lnαeq = -0.2 ~-2.5 ‰).
Methyl group (C1 for all molecules except for Coenzyme-A) has similar lnαeq values (~-
50‰), when the functional group is two or three carbons away from it. Therefore, if
methyl carbon is in a carbon chain, and a functional group is 3 carbons away from it, its
lnαeq values can be estimated from pentane (see 4.6.1. Validation of Cluster Model).
56
Table 4.1. Calculated position-specific lnαeq values (‰), indicating enrichments in 13C relative to CO2(g) at 25 ℃
Molecule C1 C2 C3 C4 C5 C6 C7 C8 C9 C10
Pentane
-50.3 -36.0 -35.0 -36.0 -50.3
Hexane
-50.4 -36.0 -35.0 -35.0 -35.9 -50.4
Heptane -50.4 -36.1 -35.1 -35.1 -35.1 -36.0 -50.4
3-methylpentane
-50.3 -35.5 -24.7 -35.5 -50.3 -48.9
3-methylhexane
-50.4 -36.1 -34.4 -24.9 -34.4 -36.1 -50.3 -49.0
Acetic acid -47.7 -0.2
Propionic acid
-48.9 -37.4 -2.2
Butanoic acid
-49.8 -34.8 -36.6 -2.4
Pentanoic Acid
-50.0 -35.7 -33.9 -36.7 -2.5
hexanoic acid
-50.2 -35.7 -34.7 -33.9 -36.6 -2.4
2-Hexanone
-50.2 -35.9 -34.0 -37.9 -9.2 -52.0
Pentanoyl chloride
-50.0 -35.8 -34.2 -37.1 -27.3
n-butanol
-50.2 -36.1 -32.5 -33.8
Butylamine
-50.4 -36.3 -34.1 -34.7
n-Butyl chloride
-50.1 -36.8 -34.3 -52.5
1-Butanethiol
-50.1 -36.3 -35.2 -52.1
Butyl methyl ether
-50.2 -36.0 -32.5 -31.6 -46.4
Pentanenitrile
-49.9 -35.7 -35.8 -38.3 -33.8
n-Butylphosphate
-50.0 -36.4 -32.5 -37.0
2- Aminopentanoic acid
-50.1 -35.9 -33.8 -25.9 0.4
n-butyl cyclopentadiene
-50.3 -36.0 -34.9 -38.7 -30.2 -33.6 -33.5 -33.5 -33.6
Butylbenzene
-50.5 -36.2 -36.5 -35.2 -20.3 -31.8 -32.0 -32.4 -32.0 -31.8
(table cont’d.)
57
Molecule C1 C2 C3 C4 C5 C6 C7 C8 C9 C10
n-butylpyrrole
-50.3 -36.3 -34.7 -30.7 -29.3 -33.2 -33.2 -29.2
2-butyltetrahydrofuran
-50.2 -35.8 -35.1 -31.6 -21.6 -37.6 -38.9 -34.0
Coenzyme-A C1 C2 C3 C4 C5 C6 C7 C8 C9 C10
-4.6 -21.1 -4.2 -10.1 -22.0 -17.5 -14.6 -20.9 -21.2 -35.3
C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 C21
-35.9 -16.4 -46.8 -46.3 -26.7 1.1 -31.9 -35.5 0.4 -31.5 -51.7
Table 4.2. lnαeq value and equilibrium fractionation factors between acetate and C16 FA
C# C1 C2 C3 C4 … C11 C12 C13 C14 C15 C16
CH3 CH2 CH2 CH2 … CH2 CH2 CH2 CH2 CH2 COOH
lnαeq -50.3 -36.0 -35.0 -35.0 … -35.0 -35.0 -35.0 -33.9 -36.7 -2.5
αacetate-
C16FA -2.6 -35.8 12.7 -34.8
…
12.7 -34.8 12.7 -33.7 11 -2.3
58
4.4. Discussions and Applications
In discussions, we started from the application of Intracrystalline isotope effect of minerals, and
expanded to Intra-ID of large organic molecules. Using position-specific lnαeq values of acetate
and C16 FA as an example, we showed the application of using equilibrium Intra-IDs to study
equilibrium isotope fractionation process.
Intracrystalline isotope effect has been brought out as a potential single mineral
geothermometer. For example, the crystal structure of biogenic apatite contains different
position-specific oxygens: phosphate/ carbonate sites and hydroxyl sites. Each of the sites have
O in distinct bonding environments. Assuming phosphate/carbonate ion in solution reaches
equilibrate with water, the isotope composition difference between phosphate/carbonate and
water (Δδ18O) is temperature dependent and can be used to reconstruct paleotemperatures (142-
144). Similar approaches had been proposed for CuSO4•5H2O (145), illite (146), Quartz (147)
and alunite (148), assuming source δ18O for the different positions are in equilibrium, or the
Δδ18O between the source does not change through time. However, this approach has not been
validated, since the assumption of equilibrated source δ18O or constant Δδ18O is in question.
Thus, it is questionable that the diagenetic environment is recorded in the observed
intracrystalline isotope effect because both source δ18O and isotope fractionation factor (α) of
each oxygen positions are unknown. If the PS oxygens in apatite or other minerals have
independent sources and reaction paths and there is no equilibration among them, we cannot treat
them as the same element in the molecule. In fact, the two oxygen in the mineral is more like two
different elements, just like as if they were sulfur and oxygen in a sulfate ion. Often, one of the
positions is more susceptible to later burial alteration than the other, which would create new
59
disequilibrium. For example, crystallization water oxygen of gypsum is more easily being altered
than the gypsum sulfate oxygen by diagenesis and weathering.
Equilibrium Intra-ID can only be reached, if an exchanging mechanism between
different positions exist, which is not the case for majority of large biochemical molecules. This
fact seems to be conveniently ignored by most practitioners in the field so far. Therefore, instead
of equilibrium Intra-ID, equilibrium among corresponding positions in reactant and product is
more informative.
Here, based on the lnαeq values of pentanoic acid and pentane, we estimated lnαeq values
of C16 FA by cluster model. Using lnαeq values of acetate and C16 FA, we calculated the
equilibrium isotope fractionation effects of C16 FA produced from acetate. Acetate is made of a
carboxyl carbon and a methyl carbon, which are named Cak and Cam, respectively. The δ13C
difference between them are defined by the site preference of acetate (SPa = δ13Cam- δ13Cak). It is
assumed that odd carbons of C16 are produced in equilibrium from Cam, and even carbons are
produced from Cak, alternatingly. The methyl carbon in C16 FA is named Cm_end, carboxyl carbon
is named Ck_end, and rest methylene carbons are named Ck, if it is produced from Cak, and Cm, if
it is produced from Cam.
PS lnαeq factors for C16 FA are estimated by cluster model. The lnαeq value and
equilibrium fractionation factors between acetate and C16 FA (αacetate-C16FA) for each carbon are
shown in Table 4.2. Cm_end is labeled C1 and Ck_end is labeled C16. lnαeq value of C1 is -50.3‰
(=lnαeq value of pentane C1, Table 4.1.), of C2 is -36.0‰ (=lnαeq value of pentane C2, Table
4.1.), of C3-C13 is -35.0‰ (=lnαeq value of pentane C3, Table 4.1.), of C14 is -33.9‰ (=lnαeq
value of pentanoic acid C3, Table 4.1.), of C14 is -36.7‰ (=lnαeq value of pentanoic acid C4,
Table 4.1.), of C14 is -2.5‰ (=lnαeq value of pentanoic acid C5, Table 4.1.).
60
Figure 4.1. Intra-ID of C16 FA produced by equilibrium isotope effect with different sources.
Diameter of each circle is proportional to the abundance of 13C expected at isotopic equilibrium.
The smallest circle (rmin) represents the lightest δ’13C, and the largest circle (rmax) represents the
heaviest δ’13C. (B. 1974-Meinschein; C. 2008-Thomas; D. 2014-Yamada)
Using calculated αacetate-C16FA, we illustrated Intra-ID produced by EIE with different
acetate sources. The observed SPa are in the range of -23.7‰ to 29.3‰ (108, 149-152). The
equilibrium SPa-eq = -47.5‰. The Intra-ID of C16 FA produced by equilibrium process are shown
in Fig. 4.1. To illustrate the chemical structure of C16 FA with isotope dimension, we adopted the
visualization invented by Galimov (68), where the variation of 13C abundance of each carbon are
represented by different-sized circles. As we can see from Fig. 4-1, the C16 FA will have
equilibrium Intra-ID, when the source acetate is in equilibrium. Intra-ID of C16 FA will always
be an alternating pattern, where Cm is enriched and Ck is depleted in 13C. It is results in the
difference of αacetate-C16FA between Cm and Ck. Cam fractionates 12.7‰ to reach equilibrium with
61
Cm, and Cak fractionates -34.8‰ to reach equilibrium with Ck. The equilibrium fractionation
difference overwrites SPa information, and the C16 FA Intra-ID appears to be heavier at Cm
position and lighter at Ck position.
The Intra-ID of a molecule contains information of both source and reaction pathways.
Equilibrium isotope fractionation process can produce C16 FA with equilibrium Intra-ID, if the
source carbons are in equilibrium (SPa = SPa-eq). However, equilibrium isotope fractionation
cannot produce C16 FA Intra-ID with lighter Cm and heavier Ck, if SPa > SPa-eq. In other words, if
C16 FA, which is in equilibrium with acetate, has δ13Ck > δ13Cm, the source acetate must have SPa
< -47.5 ‰. Based on the current observed SPa, we can confidently say that the observed Intra-ID
of FA or normal alkane, which is produced from FA, with δ13Ck > δ13Cm cannot produced by
equilibrium isotope fractionation processes from acetate. However, if δ13Ck < δ13Cm, the FA or
normal alkane can be produced from either equilibrium isotope effect or kinetic isotope effect.
Even if we observed equilibrium Intra-ID in FA or normal alkane, it could be produced from
equilibrium isotope effect with equilibrium sources, or KIE with disequilibrium sources. Thus,
the reaction process or formation environment cannot be determined, only knowing the Intra-ID
of product.
4.5. Conclusions
In this paper, we provided 13β factors for a set of organic molecules. It contributes to the database
which can be used construct PS 13β factors in large organic molecules. We further explored the
utility of the 13β factors by calculating equilibrium isotope effect between acetate and C16 FA. If
the C16 FA production process is in equilibrium, the equilibrium Intra-ID of C16 FA can only be
produced from acetate with equilibrium SPa. Based on the current observed acetate SPa range,
acetate-C16 FA equilibrium isotope effect can only produce C16 FA with 13C enriched in Cm
62
positions. The case illustrates that without knowing the source and pathway information, the site
preference of Intra-ID of a molecule is not informative to interpret its formation history.
4.6. Support Information
Figure 4.2. Influence of functional group to 13β values of methyl. Molecule name see Table 4.1.
Each cell represents the 13βCH3 values difference between two molecules (ln αij). The value of ln
αij is represented on a color scale, with red for the largest deviation (ln αij = 2‰) and with gray
for the no differences (ln αij = 0‰).
4.6.1. Validation of Cluster model
Using methyl (CH3) as example, we calculated the 13β(CH3) value in different clusters. The 13β(CH3)
value differences between clusters are illustrated as distance matrix, where each component
represents the difference between the 13β(CH3) value of two molecules (ln αij = ln (βi/βj)). The
distance matrix is visualized by heat map (Fig. 4.2.). The value of ln αij is represented in color
scale, where red indicates large difference (ln αij = 2‰) and gray indicates no difference (ln αij =
0‰). Using carboxyl group as example, with the increasing of cluster size (from acetic acid,
propionic acid, butanoic acid, pentanoic acid to hexanoic acid), the influence of carboxyl to
63
13β(CH3) value decreases. The ln α value difference of methyl group between molecule and
pentane 13β(CH3) value is <0.2‰, when the carboxyl group is three or more methylenes away
from the methyl. Similarly, when the functional group, ketone, acyl chlorid , amine, amine and
carboxyl, alcohol, phosphate, ether, thiol, nitrile, chloride, benzene, cyclopentadiene,
tetrahydrofuran or pyrrole, is three methylenes away from the methyl, their influences on the
13β(CH3) value are all smaller than 0.4‰. If we want to estimate the 13β(CH3) value of a methyl
group, which is connected with two methylene group and another functional group, we can use a
propane methyl 13β(CH3) value to represent it.
4.6.2. Applications of Cluster Model
Here we use CoA to illustrate the application of cluster model to estimate 13β values of a large
molecule. The structure of the CoA is shown as cluster 0 in Fig. 4.3. We reduced CoA to small
clusters, where each target atom has at least three adjacent functional groups, to represent the
whole structure. We built cluster 1 from C0/1-C0/14. In cluster 1, C1/1-C1/10 are in the same
positions as C0/1-C0/10 with at least 3 neighboring functional groups. Similarly, C2/2-C2/6, C3/6, C4/3,
and C5/3-C5/6 are representative of C0/11-C0/15, C0/16, C0/17, and C0/18-C0/21, respectively (Fig. 4.3.).
As we can see in Table 4.3, the 1000lnαeq values estimated by cluster model are in good
agreement with those calculated by calculating entire molecule. Furthermore, the computer time
taken by the cluster calculations is only 0.2% of that for the whole molecule (Table 4.4.). It can
save even more for larger molecules or when using higher theoretical levels.
64
Figure 4.3. Structures of Coenzyme A for whole molecule (cluster 0) calculation and five
clusters build for cluster model calculation. Gray sphere represents carbon atoms, white sphere
represents hydrogen atoms, red sphere represents oxygen atoms, blue sphere represents nitrogen
atoms, yellow sphere represents sulfur atoms and orange sphere represents phosphorus atoms.
Cluster model calculation breaks a large organic molecule into small clusters, where
target atoms are surrounded by two to three adjacent functional groups. The method provides
accurate PS β values for large organic molecules and takes at least 5 orders of magnitudes less
time than calculating entire molecule. It is especially useful for the KIE/EIE calculation for
enzymatic reactions. The previous calculation for structure and activation energies of enzymatic
reactions uses QM/MM to optimize whole molecules. For RPFR calculation, building a large
65
enzyme molecule is unnecessary. Instead, we can use a small cluster to represent the enzyme.
Compare to use a free molecule as the precursor (e.g. considering acetate as the precursor for
fatty acid production), we can get a better prediction by considering the direct precursor (e.g.
Acetyl-CoA carboxylase) by using cluster model to represent a large enzyme. In addition,
instead of calculate molecules as gas phase, we can easily add solvent into the model, since
molecule size is small.
Table 4.3. Comparison of 1000lnαeq values (‰) calculated using cluster model and WMM
Whole molecule Cluster model Diff Whole molecule Cluster model Diff
Atom C0/1 C1/1
C0/12 C2/3
lnαeq -4.6 -4.5 0.1 -16.4 -16.6 -0.2
Atom C0/2 C1/2
C0/13 C2/4
lnαeq -21.1 -21.2 -0.1 -46.8 -47.0 -0.2
Atom C0/3 C1/3
C0/14 C2/5
lnαeq -4.2 -4.3 -0.1 -46.3 -46.2 0.1
Atom C0/4 C1/4
C0/15 C2/6
lnαeq -10.1 -10.1 0.0 -26.7 -26.7 0.0
Atom C0/5 C1/5
C0/16 C3/6
lnαeq -22 -22 0.0 1.1 1.4 0.3
Atom C0/6 C1/6
C0/17 C4/3
lnαeq -17.5 -17.5 0.0 -31.9 -31.6 0.3
Atom C0/7 C1/7
C0/18 C5/3
lnαeq -14.6 -14.8 -0.2 -35.5 -35.2 0.3
Atom C0/8 C1/8
C0/19 C5/4
lnαeq -20.9 -20.9 0.0 0.4 0.4 0.0
Atom C0/9 C1/9
C0/20 C5/5
lnαeq -21.2 -21.1 0.1 -31.5 -31.5 0.0
Atom C0/10 C1/10
C0/21 C5/6
lnαeq -35.3 -35.3 0.0 -51.7 -51.7 0.0
Atom C0/11 C2/2
Average Average
lnαeq -35.9 -35.8 0.1 -23.5 -23.4 0.1
66
Table 4.4. Comparison of computer time for whole fragment and clusters
Cluster Computer time (s)
1 25975135.6
2 39528.1
3 5968.6
4 2530.8
5 3803.3
total 26026966.4
0 1106913651286.5
67
CHAPTER 5. OVERALL CONCLUSION
Relative to a chemical concentration, stable isotope ratio or a δ’ value is a high dimensional
parameter that offers new insights into processes involved. Along the same line of thinking,
higher dimensional information can further be uncovered in the relationships among the δ's.
Atmospheric chemists, geochemists, cosmochemists, ecologists, and environmental chemists
have made numerous important and often ground-breaking discoveries in recent years using
high-dimensional stable isotopes as a tool.
In my dissertation, I built a closed-system time-dependent Rayleigh Distillation Model
and a one-dimensional diffusion, steady-state, single-source Reaction Transport Model to discuss
the multiple isotope relationships of different element in the same molecule or ion in chapter 2.
In chapter 3 and 4, position-specific isotope effect and the equilibrium state of inter- or intra-
molecular isotope distributions are discussed.
This dissertation provide tools for the development of high-dimensional isotope
relationships, which can be applied in the future field works.
68
REFERENCES
1. Farquhar, J.; Bao, H.; Thiemens, M., Atmospheric influence of Earth's earliest sulfur
cycle. Science. 2000, 289 (5480), 756-758.
2. Craig, H.; Gordon, L., Deuterium and oxygen-18 variations in the ocean and the marine
atmosphere, Stable Isotopes in Oceanographic Studies and Paleotemperatures E. Toniori, 9–131,
Cons. Nat. Rich. Lab. Geol. Nucl., Pisa, Italy. 1965.
3. Granger, J.; Sigman, D. M.; Lehmann, M. F.; Tortell, P. D., Nitrogen and oxygen isotope
fractionation during dissimilatory nitrate reduction by denitrifying bacteria. Limnology and
Oceanography. 2008, 53 (6), 2533-2545.
4. Sturchio, N. C.; Böhlke, J. K.; Beloso, A. D.; Streger, S. H.; Heraty, L. J.; Hatzinger, P.
B., Oxygen and chlorine isotopic fractionation during perchlorate biodegradation: Laboratory
results and implications for forensics and natural attenuation studies. Environmental science &
technology. 2007, 41 (8), 2796-2802.
5. Palau, J.; Shouakar-Stash, O.; Hatijah Mortan, S.; Yu, R.; Rosell, M.; Marco-Urrea, E.;
Freedman, D. L.; Aravena, R.; Soler, A.; Hunkeler, D., Hydrogen Isotope Fractionation during
the Biodegradation of 1, 2-Dichloroethane: Potential for Pathway Identification Using a Multi-
element (C, Cl, and H) Isotope Approach. Environmental Science & Technology. 2017, 51 (18),
10526-10535.
6. Elsner, M.; McKelvie, J.; Lacrampe Couloume, G.; Sherwood Lollar, B., Insight into
methyl tert-butyl ether (MTBE) stable isotope fractionation from abiotic reference experiments.
Environmental science & technology. 2007, 41 (16), 5693-5700.
7. Fischer, A.; Herklotz, I.; Herrmann, S.; Thullner, M.; Weelink, S. A.; Stams, A. J.;
Schlömann, M.; Richnow, H.-H.; Vogt, C., Combined carbon and hydrogen isotope fractionation
investigations for elucidating benzene biodegradation pathways. Environmental science &
technology. 2008, 42 (12), 4356-4363.
8. Fischer, A.; Theuerkorn, K.; Stelzer, N.; Gehre, M.; Thullner, M.; Richnow, H. H.,
Applicability of stable isotope fractionation analysis for the characterization of benzene
biodegradation in a BTEX-contaminated aquifer. Environmental science & technology. 2007, 41
(10), 3689-3696.
9. Elsner, M., Stable isotope fractionation to investigate natural transformation mechanisms
of organic contaminants: principles, prospects and limitations. Journal of Environmental
Monitoring. 2010, 12 (11), 2005-2031.
10. Abe, Y.; Hunkeler, D., Does the Rayleigh equation apply to evaluate field isotope data in
contaminant hydrogeology? Environmental science & technology. 2006, 40 (5), 1588-1596.
69
11. Hatzinger, P. B.; Böhlke, J. K.; Sturchio, N. C.; Gu, B.; Heraty, L. J.; Borden, R. C.,
Fractionation of stable isotopes in perchlorate and nitrate during in situ biodegradation in a sandy
aquifer. Environmental Chemistry. 2009, 6 (1), 44-52.
12. Thullner, M.; Fischer, A.; Richnow, H.-H.; Wick, L. Y., Influence of mass transfer on
stable isotope fractionation. Applied microbiology and biotechnology. 2013, 97 (2), 441-452.
13. Casciotti, K.; McIlvin, M., Isotopic analyses of nitrate and nitrite from reference mixtures
and application to Eastern Tropical North Pacific waters. Marine Chemistry. 2007, 107 (2), 184-
201.
14. Casciotti, K. L.; Buchwald, C.; McIlvin, M., Implications of nitrate and nitrite isotopic
measurements for the mechanisms of nitrogen cycling in the Peru oxygen deficient zone. Deep
Sea Research Part I: Oceanographic Research Papers. 2013, 80, 78-93.
15. Thullner, M.; Centler, F.; Richnow, H.-H.; Fischer, A., Quantification of organic
pollutant degradation in contaminated aquifers using compound specific stable isotope analysis–
review of recent developments. Organic geochemistry. 2012, 42 (12), 1440-1460.
16. Fukada, T.; Hiscock, K. M.; Dennis, P. F., A dual-isotope approach to the nitrogen
hydrochemistry of an urban aquifer. Applied Geochemistry. 2004, 19 (5), 709-719.
17. Sigman, D. M.; Granger, J.; DiFiore, P. J.; Lehmann, M. M.; Ho, R.; Cane, G.; van Geen,
A., Coupled nitrogen and oxygen isotope measurements of nitrate along the eastern North Pacific
margin. Global Biogeochemical Cycles. 2005, 19 (4).
18. Wankel, S. D.; Kendall, C.; Pennington, J. T.; Chavez, F. P.; Paytan, A., Nitrification in
the euphotic zone as evidenced by nitrate dual isotopic composition: Observations from
Monterey Bay, California. Global Biogeochemical Cycles. 2007, 21 (2).
19. Cey, E. E.; Rudolph, D. L.; Aravena, R.; Parkin, G., Role of the riparian zone in
controlling the distribution and fate of agricultural nitrogen near a small stream in southern
Ontario. Journal of Contaminant Hydrology. 1999, 37 (1-2), 45-67.
20. Mengis, M.; Schif, S.; Harris, M.; English, M.; Aravena, R.; Elgood, R.; MacLean, A.,
Multiple geochemical and isotopic approaches for assessing ground water NO3− elimination in a
riparian zone. Groundwater. 1999, 37 (3), 448-457.
21. Fukada, T.; Hiscock, K. M.; Dennis, P. F.; Grischek, T., A dual isotope approach to
identify denitrification in groundwater at a river-bank infiltration site. Water Research. 2003, 37
(13), 3070-3078.
22. Aravena, R.; Robertson, W. D., Use of multiple isotope tracers to evaluate denitrification
in ground water: study of nitrate from a large‐flux septic system plume. Groundwater. 1998, 36
(6), 975-982.
70
23. Liu, C.-Q.; Li, S.-L.; Lang, Y.-C.; Xiao, H.-Y., Using δ15N-and δ18O-values to identify
nitrate sources in karst ground water, Guiyang, Southwest China. Environmental science &
technology. 2006, 40 (22), 6928-6933.
24. Chen, F.; Jia, G.; Chen, J., Nitrate sources and watershed denitrification inferred from
nitrate dual isotopes in the Beijiang River, south China. Biogeochemistry. 2009, 94 (2), 163-174.
25. Otero, N.; Torrentó, C.; Soler, A.; Menció, A.; Mas-Pla, J., Monitoring groundwater
nitrate attenuation in a regional system coupling hydrogeology with multi-isotopic methods: the
case of Plana de Vic (Osona, Spain). Agriculture, ecosystems & environment. 2009, 133 (1-2),
103-113.
26. Li, S.-L.; Liu, C.-Q.; Li, J.; Liu, X.; Chetelat, B.; Wang, B.; Wang, F., Assessment of the
sources of nitrate in the Changjiang River, China using a nitrogen and oxygen isotopic approach.
Environmental Science & Technology. 2010, 44 (5), 1573-1578.
27. Itoh, M.; Takemon, Y.; Makabe, A.; Yoshimizu, C.; Kohzu, A.; Ohte, N.; Tumurskh, D.;
Tayasu, I.; Yoshida, N.; Nagata, T., Evaluation of wastewater nitrogen transformation in a
natural wetland (Ulaanbaatar, Mongolia) using dual-isotope analysis of nitrate. Science of the
Total Environment. 2011, 409 (8), 1530-1538.
28. Chen, F.; Chen, J.; Jia, G.; Jin, H.; Xu, J.; Yang, Z.; Zhuang, Y.; Liu, X.; Zhang, H.,
Nitrate δ 15 N and δ 18 O evidence for active biological transformation in the Changjiang
Estuary and the adjacent East China Sea. Acta Oceanologica Sinica. 2013, 32 (4), 11-17.
29. Li, S.-L.; Liu, C.-Q.; Li, J.; Xue, Z.; Guan, J.; Lang, Y.; Ding, H.; Li, L., Evaluation of
nitrate source in surface water of southwestern China based on stable isotopes. Environmental
earth sciences. 2013, 68 (1), 219-228.
30. Yue, F.-J.; Li, S.-L.; Liu, C.-Q.; Zhao, Z.-Q.; Hu, J., Using dual isotopes to evaluate
sources and transformation of nitrogen in the Liao River, northeast China. Applied geochemistry.
2013, 36, 1-9.
31. Ding, J.; Xi, B.; Gao, R.; He, L.; Liu, H.; Dai, X.; Yu, Y., Identifying diffused nitrate
sources in a stream in an agricultural field using a dual isotopic approach. Science of The Total
Environment. 2014, 484, 10-18.
32. Umezawa, Y.; Yamaguchi, A.; Ishizaka, J.; Hasegawa, T.; Yoshimizu, C.; Tayasu, I.;
Yoshimura, H.; Morii, Y.; Aoshima, T.; Yamawaki, N., Seasonal shifts in the contributions of
the Changjiang River and the Kuroshio Current to nitrate dynamics at the continental shelf of the
northern East China Sea based on a nitrate dual isotopic composition approach. Biogeosciences
Discussions. 2013, 10, 10143-10188.
33. Liu, S. M.; Altabet, M. A.; Zhao, L.; Larkum, J.; Song, G. D.; Zhang, G. L.; Jin, H.; Han,
L. J., Tracing Nitrogen Biogeochemistry During the Beginning of a Spring Phytoplankton Bloom
in the Yellow Sea Using Coupled Nitrate Nitrogen and Oxygen Isotope Ratios. Journal of
Geophysical Research: Biogeosciences. 2017, 122 (10), 2490-2508.
71
34. Hosono, T.; Tokunaga, T.; Kagabu, M.; Nakata, H.; Orishikida, T.; Lin, I.-T.; Shimada, J.,
The use of δ15N and δ18O tracers with an understanding of groundwater flow dynamics for
evaluating the origins and attenuation mechanisms of nitrate pollution. Water research. 2013, 47
(8), 2661-2675.
35. Zhang, Y.; Li, F.; Zhang, Q.; Li, J.; Liu, Q., Tracing nitrate pollution sources and
transformation in surface-and ground-waters using environmental isotopes. Science of the Total
Environment. 2014, 490, 213-222.
36. Kim, H.; Kaown, D.; Mayer, B.; Lee, J.-Y.; Hyun, Y.; Lee, K.-K., Identifying the sources
of nitrate contamination of groundwater in an agricultural area (Haean basin, Korea) using
isotope and microbial community analyses. Science of the Total environment. 2015, 533, 566-
575.
37. Bourbonnais, A.; Altabet, M. A.; Charoenpong, C. N.; Larkum, J.; Hu, H.; Bange, H. W.;
Stramma, L., N‐loss isotope effects in the Peru oxygen minimum zone studied using a mesoscale
eddy as a natural tracer experiment. Global Biogeochemical Cycles. 2015, 29 (6), 793-811.
38. Buchwald, C.; Santoro, A. E.; Stanley, R. H.; Casciotti, K. L., Nitrogen cycling in the
secondary nitrite maximum of the eastern tropical North Pacific off Costa Rica. Global
Biogeochemical Cycles. 2015, 29 (12), 2061-2081.
39. Granger, J.; Wankel, S. D., Isotopic overprinting of nitrification on denitrification as a
ubiquitous and unifying feature of environmental nitrogen cycling. Proceedings of the National
Academy of Sciences. 2016, 113 (42), E6391-E6400.
40. Casciotti, K. L., Inverse kinetic isotope fractionation during bacterial nitrite oxidation.
Geochimica et Cosmochimica Acta. 2009, 73 (7), 2061-2076.
41. Buchwald, C.; Casciotti, K. L., Oxygen isotopic fractionation and exchange during
bacterial nitrite oxidation. Limnology and Oceanography. 2010, 55 (3), 1064-1074.
42. Casciotti, K. L.; McIlvin, M.; Buchwald, C., Oxygen isotopic exchange and fractionation
during bacterial ammonia oxidation. Limnology and Oceanography. 2010, 55 (2), 753-762.
43. Buchwald, C.; Casciotti, K. L., Isotopic ratios of nitrite as tracers of the sources and age
of oceanic nitrite. Nature Geoscience. 2013, 6 (4), 308.
44. Casciotti, K. L.; Sigman, D. M.; Ward, B. B., Linking diversity and stable isotope
fractionation in ammonia-oxidizing bacteria. Geomicrobiology Journal. 2003, 20 (4), 335-353.
45. Brunner, B.; Contreras, S.; Lehmann, M. F.; Matantseva, O.; Rollog, M.; Kalvelage, T.;
Klockgether, G.; Lavik, G.; Jetten, M. S.; Kartal, B., Nitrogen isotope effects induced by
anammox bacteria. Proceedings of the National Academy of Sciences. 2013, 110 (47), 18994-
18999.
72
46. Robertson, W.; Moore, T.; Spoelstra, J.; Li, L.; Elgood, R.; Clark, I.; Schiff, S.; Aravena,
R.; Neufeld, J., Natural attenuation of septic system nitrogen by anammox. Groundwater. 2012,
50 (4), 541-553.
47. Bryan, B. A.; Shearer, G.; Skeeters, J. L.; Kohl, D. H., Variable expression of the
nitrogen isotope effect associated with denitrification of nitrite. Journal of Biological Chemistry.
1983, 258 (14), 8613-8617.
48. Casciotti, K. L.; Böhlke, J. K.; McIlvin, M. R.; Mroczkowski, S. J.; Hannon, J. E.,
Oxygen isotopes in nitrite: analysis, calibration, and equilibration. Analytical Chemistry. 2007,
79 (6), 2427-2436.
49. Casciotti, K.; Sigman, D.; Hastings, M. G.; Böhlke, J.; Hilkert, A., Measurement of the
oxygen isotopic composition of nitrate in seawater and freshwater using the denitrifier method.
Analytical Chemistry. 2002, 74 (19), 4905-4912.
50. Granger, J.; Sigman, D. M.; Needoba, J. A.; Harrison, P. J., Coupled nitrogen and oxygen
isotope fractionation of nitrate during assimilation by cultures of marine phytoplankton.
Limnology and Oceanography. 2004, 49 (5), 1763-1773.
51. Casciotti, K. L.; Buchwald, C., Insights on the marine microbial nitrogen cycle from
isotopic approaches to nitrification. Frontiers in microbiology. 2012, 3, 356.
52. Chernyavsky, B. M.; Wortmann, U. G., REMAP: A reaction transport model for isotope
ratio calculations in porous media. Geochemistry, Geophysics, Geosystems. 2007, 8 (2).
53. Wortmann, U. G.; Chernyavsky, B. M., The significance of isotope specific diffusion
coefficients for reaction-transport models of sulfate reduction in marine sediments. Geochimica
et Cosmochimica Acta. 2011, 75 (11), 3046-3056.
54. Wortmann, U. G.; Chernyavsky, B.; Bernasconi, S. M.; Brunner, B.; Böttcher, M. E.;
Swart, P. K., Oxygen isotope biogeochemistry of pore water sulfate in the deep biosphere:
dominance of isotope exchange reactions with ambient water during microbial sulfate reduction
(ODP Site 1130). Geochimica et Cosmochimica Acta. 2007, 71 (17), 4221-4232.
55. Antler, G.; Turchyn, A. V.; Ono, S.; Sivan, O.; Bosak, T., Combined 34S, 33S and 18O
isotope fractionations record different intracellular steps of microbial sulfate reduction.
Geochimica et Cosmochimica Acta. 2017, 203, 364-380.
56. Knowles, R., Denitrification. Microbiological reviews. 1982, 46 (1), 43.
57. Kuypers, M. M.; Sliekers, A. O.; Lavik, G.; Schmid, M.; Jørgensen, B. B.; Kuenen, J. G.;
Damsté, J. S. S.; Strous, M.; Jetten, M. S., Anaerobic ammonium oxidation by anammox bacteria
in the Black Sea. Nature. 2003, 422 (6932), 608.
58. Berner, R. A., An idealized model of dissolved sulfate distribution in recent sediments.
Geochimica et Cosmochimica Acta. 1964, 28 (9), 1497-1503.
73
59. Jackson, W. A.; Böhlke, J. K.; Andraski, B. J.; Fahlquist, L.; Bexfield, L.; Eckardt, F. D.;
Gates, J. B.; Davila, A. F.; McKay, C. P.; Rao, B., Global patterns and environmental controls of
perchlorate and nitrate co-occurrence in arid and semi-arid environments. Geochimica et
Cosmochimica Acta. 2015, 164, 502-522.
60. Kaufman, A. J.; Johnston, D. T.; Farquhar, J.; Masterson, A. L.; Lyons, T. W.; Bates, S.;
Anbar, A. D.; Arnold, G. L.; Garvin, J.; Buick, R., Late Archean biospheric oxygenation and
atmospheric evolution. Science. 2007, 317 (5846), 1900-1903.
61. Ono, S.; Kaufman, A. J.; Farquhar, J.; Sumner, D. Y.; Beukes, N. J., Lithofacies control
on multiple-sulfur isotope records and Neoarchean sulfur cycles. Precambrian Research. 2009,
169 (1-4), 58-67.
62. Bao, H.; Cao, X.; Hayles, J. A., Triple oxygen isotopes: fundamental relationships and
applications. Annual Review of Earth and Planetary Sciences. 2016, 44, 463-492.
63. Soetaert, K.; Petzoldt, T.; Setzer, R. W., Solving differential equations in R: package
deSolve. Journal of Statistical Software. 2010, 33.
64. Soetaert, K.; Herman, P. M., A practical guide to ecological modelling: using R as a
simulation platform. Springer Science & Business Media: 2009.
65. Igamberdiev, A. U.; Kleczkowski, L. A., Metabolic systems maintain stable non‐
equilibrium via thermodynamic buffering. Bioessays. 2009, 31 (10), 1091-1099.
66. Igamberdiev, A. U.; Kleczkowski, L. A., Optimization of CO 2 fixation in photosynthetic
cells via thermodynamic buffering. Bio Systems. 2011, 103 (2), 224-229.
67. Morandini, P., Rethinking metabolic control. Plant science : an international journal of
experimental plant biology. 2009, 176 (4), 441-451.
68. Galimov, E. M., The Biological Fractionation of Isotopes. 1985.
69. Galimov, E. M., Phenomenon of life: between equilibrium and non-linearity. Origins of
Life and Evolution of Biospheres. 2004, 34 (6), 599-613.
70. Galimov, E. M., Phenomenon of life: between equilibrium and non-linearity. Origin and
principles of evolution. Geochemistry International. 2006, 44 (1), S1-S95.
71. Hayes, J. M., Fractionation of carbon and hydrogen isotopes in biosynthetic processes.
Reviews in mineralogy and geochemistry. 2001, 43 (1), 225-277.
72. Urey, H. C., The thermodynamic properties of isotopic substances. Journal of the
Chemical Society (Resumed). 1947, 562-581.
73. Bigeleisen, J.; Mayer, M. G., Calculation of equilibrium constants for isotopic exchange
reactions. The Journal of Chemical Physics. 1947, 15 (5), 261-267.
74
74. Buchachenko, A. L., On isotope fractionation in enzymatic reaction. Russian Journal of
Physical Chemistry A. 2007, 81 (5), 836-836.
75. Schmidt, H. L., Fundamentals and systematics of the non-statistical distributions of
isotopes in natural compounds. Die Naturwissenschaften. 2003, 90 (12), 537-552.
76. Schmidt, H. L.; Robins, R. J.; Werner, R. A., Multi-factorial in vivo stable isotope
fractionation: causes, correlations, consequences and applications. Isotopes in environmental and
health studies. 2015, 51 (1), 155-199.
77. Dattorro, J., Convex optimization & Euclidean distance geometry. Lulu. com: 2010.
78. Rustad, J. R., Ab initio calculation of the carbon isotope signatures of amino acids.
Organic Geochemistry. 2009, 40 (6), 720-723.
79. Gleixner, G.; Scrimgeour, C.; Schmidt, H.-L.; Viola, R., Stable isotope distribution in the
major metabolites of source and sink organs of Solanum tuberosum L.: a powerful tool in the
study of metabolic partitioning in intact plants. Planta. 1998, 207 (2), 241-245.
80. Bender, D. A., Amino acid metabolism. John Wiley & Sons: 2012.
81. Foyer, C. H.; Parry, M.; Noctor, G., Markers and signals associated with nitrogen
assimilation in higher plants. Journal of experimental botany. 2003, 54 (382), 585-593.
82. Muttucumaru, N.; Keys, A. J.; Parry, M. A.; Powers, S. J.; Halford, N. G., Photosynthetic
assimilation of 14C into amino acids in potato (Solanumtuberosum) and asparagine in the tubers.
Planta. 2014, 239 (1), 161-170.
83. Abelson, P. H.; Hoering, T. C., Carbon isotope fractionation in formation of amino acids
by photosynthetic organisms. Proceedings of the National Academy of Sciences. 1961, 47 (5),
623-632.
84. Monson, K.; Hayes, J., Biosynthetic control of the natural abundance of carbon 13 at
specific positions within fatty acids in Saccharomyces cerevisiae. Isotopic fractionation in lipid
synthesis as evidence for peroxisomal regulation. Journal of Biological Chemistry. 1982, 257
(10), 5568-5575.
85. Monson, K. D.; Hayes, J., Biosynthetic control of the natural abundance of Carbon 13 at
specific positions within fatty acids in Escherichia coli. Evidence regarding the coupling of fatty
acid and phospholipid synthesis. Journal of Biological Chemistry. 1980, 255 (23), 11435-11441.
86. Piasecki, A.; Sessions, A.; Lawson, M.; Ferreira, A.; Neto, E. S.; Ellis, G. S.; Lewan, M.
D.; Eiler, J. M., Position-specific 13 C distributions within propane from experiments and natural
gas samples. Geochimica et Cosmochimica Acta. 2017.
87. Webb, M. A.; Miller III, T. F., Position-specific and clumped stable isotope studies:
Comparison of the Urey and path-integral approaches for carbon dioxide, nitrous oxide, methane,
and propane. The Journal of Physical Chemistry A. 2014, 118 (2), 467-474.
75
88. Rinaldi, G.; Meinschein, W.; Hayes, J., Intramolecular carbon isotopic distribution in
biologically produced acetoin. Biological Mass Spectrometry. 1974, 1 (6), 415-417.
89. Rossmann, A.; Butzenlechner, M.; Schmidt, H. L., Evidence for a nonstatistical carbon
isotope distribution in natural glucose. Plant Physiology. 1991, 96 (2), 609-614.
90. Gleixner, G.; Schmidt, H.-L., Carbon isotope effects on the fructose-1, 6-bisphosphate
aldolase reaction, origin for non-statistical 13C distributions in carbohydrates. Journal of
Biological Chemistry. 1997, 272 (9), 5382-5387.
91. Savidge, W. B.; Blair, N. E., Patterns of intramolecular carbon isotopic heterogeneity
within amino acids of autotrophs and heterotrophs. Oecologia. 2004, 139 (2), 178-189.
92. Savidge, W. B.; Blair, N. E., Intramolecular carbon isotopic composition of monosodium
glutamate: biochemical pathways and product source identification. Journal of agricultural and
food chemistry. 2005, 53 (2), 197-201.
93. Corso, T. N.; Brenna, J. T., High-precision position-specific isotope analysis.
Proceedings of the National Academy of Sciences. 1997, 94 (4), 1049-1053.
94. Oba, Y.; Naraoka, H., Site‐specific carbon isotope analysis of aromatic carboxylic acids
by elemental analysis/pyrolysis/isotope ratio mass spectrometry. Rapid communications in mass
spectrometry. 2006, 20 (24), 3649-3653.
95. Bayle, K.; Gilbert, A.; Julien, M.; Yamada, K.; Silvestre, V.; Robins, R. J.; Akoka, S.;
Yoshida, N.; Remaud, G. S., Conditions to obtain precise and true measurements of the
intramolecular 13 C distribution in organic molecules by isotopic 13 C nuclear magnetic
resonance spectrometry. Analytica chimica acta. 2014, 846, 1-7.
96. Bayle, K.; Grand, M.; Chaintreau, A.; Robins, R. J.; Fieber, W.; Sommer, H.; Akoka, S.;
Remaud, G. r. S., Internal Referencing for 13C Position-Specific Isotope Analysis Measured by
NMR Spectrometry. Analytical chemistry. 2015, 87 (15), 7550-7554.
97. Diomande, D. G.; Martineau, E.; Gilbert, A.; Nun, P.; Murata, A.; Yamada, K.; Watanabe,
N.; Tea, I.; Robins, R. J.; Yoshida, N., Position-specific isotope analysis of xanthines: a 13C
Nuclear Magnetic Resonance method to determine the 13C intramolecular composition at natural
abundance. Analytical chemistry. 2015, 87 (13), 6600-6606.
98. Gilbert, A.; Robins, R. J.; Remaud, G. S.; Tcherkez, G. G., Intramolecular 13C pattern in
hexoses from autotrophic and heterotrophic C3 plant tissues. Proceedings of the National
Academy of Sciences. 2012, 109 (44), 18204-18209.
99. Gilbert, A.; Silvestre, V.; Robins, R. J.; Remaud, G. S., Impact of the deuterium isotope
effect on the accuracy of 13C NMR measurements of site-specific isotope ratios at natural
abundance in glucose. Analytical and bioanalytical chemistry. 2010, 398 (5), 1979-1984.
100. Gilbert, A.; Silvestre, V.; Robins, R. J.; Remaud, G. S.; Tcherkez, G., Biochemical and
physiological determinants of intramolecular isotope patterns in sucrose from C 3, C 4 and CAM
76
plants accessed by isotopic 13C NMR spectrometry: a viewpoint. Natural product reports. 2012,
29 (4), 476-486.
101. Gilbert, A.; Silvestre, V.; Robins, R. J.; Tcherkez, G.; Remaud, G. S., A 13C NMR
spectrometric method for the determination of intramolecular δ13C values in fructose from plant
sucrose samples. New Phytologist. 2011, 191 (2), 579-588.
102. Gilbert, A.; Silvestre, V.; Segebarth, N.; Tcherkez, G.; Guillou, C.; Robins, R. J.; Akoka,
S.; Remaud, G. S., The intramolecular 13C‐distribution in ethanol reveals the influence of the
CO2‐fixation pathway and environmental conditions on the site‐specific 13C variation in glucose.
Plant, cell & environment. 2011, 34 (7), 1104-1112.
103. Remaud, G. S.; Akoka, S., A review of flavors authentication by position‐specific isotope
analysis by nuclear magnetic resonance spectrometry: the example of vanillin. Flavour and
Fragrance Journal. 2017, 32 (2), 77-84.
104. Robins, R. J.; Romek, K. M.; Remaud, G. S.; Paneth, P., Non-statistical isotope
fractionation as a novel “retro-biosynthetic” approach to understanding alkaloid metabolic
pathways. Phytochemistry Letters. 2016.
105. Romek, K. M.; Remaud, G. S.; Silvestre, V.; Paneth, P.; Robins, R. J., Non-statistical
13C fractionation distinguishes co-incident and divergent steps in the biosynthesis of the
alkaloids nicotine and tropine. Journal of Biological Chemistry. 2016, 291 (32), 16620-16629.
106. Gilbert, A.; Yamada, K.; Suda, K.; Ueno, Y.; Yoshida, N., Measurement of position-
specific 13 C isotopic composition of propane at the nanomole level. Geochimica et
Cosmochimica Acta. 2016, 177, 205-216.
107. Gilbert, A.; Yamada, K.; Yoshida, N., Evaluation of on-line pyrolysis coupled to isotope
ratio mass spectrometry for the determination of position-specific 13 C isotope composition of
short chain n-alkanes (C 6–C 12). Talanta. 2016, 153, 158-162.
108. Nimmanwudipong, T.; Gilbert, A.; Yamada, K.; Yoshida, N., Analytical method for
simultaneous determination of bulk and intramolecular 13C‐isotope compositions of acetic acid.
Rapid Communications in Mass Spectrometry. 2015, 29 (24), 2337-2340.
109. Suda, K.; Gilbert, A.; Yamada, K.; Yoshida, N.; Ueno, Y., Compound–and position–
specific carbon isotopic signatures of abiogenic hydrocarbons from on–land serpentinite–hosted
Hakuba Happo hot spring in Japan. Geochimica et Cosmochimica Acta. 2017, 206, 201-215.
110. Piasecki, A. M. Site-specific isotopes in small organic molecules. California Institute of
Technology, 2015.
111. Piasecki, A.; Sessions, A.; Lawson, M.; Ferreira, A.; Neto, E. S.; Eiler, J. M., Analysis of
the site-specific carbon isotope composition of propane by gas source isotope ratio mass
spectrometer. Geochimica et Cosmochimica Acta. 2016, 188, 58-72.
77
112. Yung, Y. L.; Miller, C. E., Isotopic fractionation of stratospheric nitrous oxide. Science.
1997, 278 (5344), 1778-1780.
113. Yoshida, N.; Toyoda, S., Constraining the atmospheric N2O budget from intramolecular
site preference in N2O isotopomers. Nature. 2000, 405 (6784), 330.
114. Toyoda, S.; Yoshida, N.; Miwa, T.; Matsui, Y.; Yamagishi, H.; Tsunogai, U.; Nojiri, Y.;
Tsurushima, N., Production mechanism and global budget of N2O inferred from its isotopomers
in the western North Pacific. Geophysical research letters. 2002, 29 (3).
115. Sutka, R. L.; Ostrom, N.; Ostrom, P.; Breznak, J.; Gandhi, H.; Pitt, A.; Li, F.,
Distinguishing nitrous oxide production from nitrification and denitrification on the basis of
isotopomer abundances. Applied and environmental microbiology. 2006, 72 (1), 638-644.
116. Sutka, R.; Ostrom, N.; Ostrom, P.; Gandhi, H.; Breznak, J., Nitrogen isotopomer site
preference of N2O produced by Nitrosomonas europaea and Methylococcus capsulatus Bath.
Rapid Communications in Mass Spectrometry. 2003, 17 (7), 738-745.
117. Yamulki, S.; Toyoda, S.; Yoshida, N.; Veldkamp, E.; Grant, B.; Bol, R., Diurnal fluxes
and the isotopomer ratios of N2O in a temperate grassland following urine amendment. Rapid
Communications in Mass Spectrometry. 2001, 15 (15), 1263-1269.
118. Schmidt, H. L.; Werner, R. A.; Yoshida, N.; Well, R., Is the isotopic composition of
nitrous oxide an indicator for its origin from nitrification or denitrification? A theoretical
approach from referred data and microbiological and enzyme kinetic aspects. Rapid
Communications in Mass Spectrometry. 2004, 18 (18), 2036-2040.
119. Toyoda, S.; Mutobe, H.; Yamagishi, H.; Yoshida, N.; Tanji, Y., Fractionation of N 2 O
isotopomers during production by denitrifier. Soil Biology and Biochemistry. 2005, 37 (8), 1535-
1545.
120. Hayes, J. M., Isotopic order, biogeochemical processes, and earth history - Goldschmidt
Lecture, Davos, Switzerland, August 2002. Geochimica Et Cosmochimica Acta. 2004, 68 (8),
1691-1700.
121. Wang, Y.; Sessions, A. L.; Nielsen, R. J.; Goddard, W. A., Equilibrium 2 H/1 H
fractionations in organic molecules: I. Experimental calibration of ab initio calculations.
Geochimica et Cosmochimica Acta. 2009, 73 (23), 7060-7075.
122. Wang, Y.; Sessions, A. L.; Nielsen, R. J.; Goddard, W. A., Equilibrium 2 H/1 H
fractionations in organic molecules. II: Linear alkanes, alkenes, ketones, carboxylic acids, esters,
alcohols and ethers. Geochimica et Cosmochimica Acta. 2009, 73 (23), 7076-7086.
123. Piasecki, A.; Sessions, A.; Peterson, B.; Eiler, J., Prediction of equilibrium distributions
of isotopologues for methane, ethane and propane using density functional theory. Geochimica et
Cosmochimica Acta. 2016, 190, 1-12.
78
124. Black, J. R.; Yin, Q.-z.; Rustad, J. R.; Casey, W. H., Magnesium isotopic equilibrium in
chlorophylls. Journal of the American Chemical Society. 2007, 129 (28), 8690-8691.
125. Gilbert, A.; Yamada, K.; Yoshida, N., Exploration of intramolecular 13 C isotope
distribution in long chain n-alkanes (C 11–C 31) using isotopic 13 C NMR. Organic
geochemistry. 2013, 62, 56-61.
126. Anisimov, V.; Paneth, P., ISOEFF98. A program for studies of isotope effects using
Hessian modifications. Journal of Mathematical Chemistry. 1999, 26 (1), 75-86.
127. Swiderek, K.; Paneth, P., Extending limits of chlorine kinetic isotope effects. The Journal
of organic chemistry. 2012, 77 (11), 5120-5124.
128. Swiderek, K.; Paneth, P., Binding ligands and cofactor to L-lactate dehydrogenase from
human skeletal and heart muscles. The Journal of Physical Chemistry B. 2011, 115 (19), 6366-
6376.
129. Swiderek, K.; Paneth, P., Binding isotope effects. Chemical reviews. 2013, 113 (10),
7851-7879.
130. Siegbahn, P. E.; Blomberg, M. R., Transition-metal systems in biochemistry studied by
high-accuracy quantum chemical methods. Chemical Reviews. 2000, 100 (2), 421-438.
131. Siegbahn, P. E.; Blomberg, M. R., Density functional theory of biologically relevant
metal centers. Annual review of physical chemistry. 1999, 50 (1), 221-249.
132. Rustad, J. R.; Nelmes, S. L.; Jackson, V. E.; Dixon, D. A., Quantum-chemical
calculations of carbon-isotope fractionation in CO2 (g), aqueous carbonate species, and
carbonate minerals. The Journal of Physical Chemistry A. 2008, 112 (3), 542-555.
133. Rustad, J. R.; Casey, W. H.; Yin, Q.-Z.; Bylaska, E. J.; Felmy, A. R.; Bogatko, S. A.;
Jackson, V. E.; Dixon, D. A., Isotopic fractionation of Mg 2+(aq), Ca 2+(aq), and Fe 2+(aq) with
carbonate minerals. Geochimica et Cosmochimica Acta. 2010, 74 (22), 6301-6323.
134. He, H.-t.; Liu, Y., Silicon isotope fractionation during the precipitation of quartz and the
adsorption of H4SiO4 (aq) on Fe (III)-oxyhydroxide surfaces. Chinese Journal of Geochemistry.
2015, 34 (4), 459-468.
135. Li, X.; Liu, Y., Equilibrium Se isotope fractionation parameters: a first-principles study.
Earth and Planetary Science Letters. 2011, 304 (1), 113-120.
136. Li, X.; Liu, Y., A theoretical model of isotopic fractionation by thermal diffusion and its
implementation on silicate melts. Geochimica et Cosmochimica Acta. 2015, 154, 18-27.
137. Liu, Y.; Tossell, J. A., Ab initio molecular orbital calculations for boron isotope
fractionations on boric acids and borates. Geochimica et Cosmochimica Acta. 2005, 69 (16),
3995-4006.
79
138. Zhang, S.; Liu, Y., Molecular-level mechanisms of quartz dissolution under neutral and
alkaline conditions in the presence of electrolytes. Geochemical Journal. 2014, 48 (2), 189-205.
139. Wilson, E. B.; Decius, J. C.; Cross, P. C., Molecular vibrations: the theory of infrared
and Raman vibrational spectra. Courier Corporation: 1955.
140. Becke, A. D., Density‐functional thermochemistry. III. The role of exact exchange. The
Journal of chemical physics. 1993, 98 (7), 5648-5652.
141. Lee, C.; Yang, W.; Parr, R. G., Development of the Colle-Salvetti correlation-energy
formula into a functional of the electron density. Physical review B. 1988, 37 (2), 785.
142. Aufort, J.; Ségalen, L.; Gervais, C.; Paulatto, L.; Blanchard, M.; Balan, E., Site-specific
equilibrium isotopic fractionation of oxygen, carbon and calcium in apatite. Geochimica et
Cosmochimica Acta. 2017, 219, 57-73.
143. Blake, R. E.; O’Neil, J. R.; Surkov, A. V., Biogeochemical cycling of phosphorus:
insights from oxygen isotope effects of phosphoenzymes. American Journal of Science. 2005,
305 (6-8), 596-620.
144. Chang, S. J.; Blake, R. E., Precise calibration of equilibrium oxygen isotope
fractionations between dissolved phosphate and water from 3 to 37 C. Geochimica et
Cosmochimica Acta. 2015, 150, 314-329.
145. Götz, D.; Heinzinger, K.; Klemm, A., The Dependence of the Oxygen Isotope
Fractionation in the Hydration Water of CuSO4· 5 H2O on the Crystallization Temperature.
Zeitschrift für Naturforschung A. 1975, 30 (12), 1667-1674.
146. Bechtel, A.; Hoernes, S., Oxygen isotope fractionation between oxygen of different sites
in illite minerals: a potential single-mineral thermometer. Contributions to Mineralogy and
Petrology. 1990, 104 (4), 463-470.
147. Klemm, A.; Banerjee, A.; Hoernes, S., A new intracrystalline isotope effect: 18O under
the faces of amethyst. Zeitschrift für Naturforschung A. 1990, 45 (11-12), 1374-1376.
148. Arehart, G. B.; Kesler, S. E.; O'Neil, J. R.; Foland, K., Evidence for the supergene origin
of alunite in sediment-hosted micron gold deposits, Nevada. Economic Geology. 1992, 87 (2),
263-270.
149. Meinschein, W.; Rinaldi, G.; Hayes, J.; Schoeller, D., Intramolecular isotopic order in
biologically produced acetic acid. Biological Mass Spectrometry. 1974, 1 (3), 172-174.
150. Thomas, R. B. Intramolecular isotopic variation in acetate in sediments and wetland soils.
The Pennsylvania State University, 2008.
151. Hattori, R.; Yamada, K.; Kikuchi, M.; Hirano, S.; Yoshida, N., Intramolecular carbon
isotope distribution of acetic acid in vinegar. Journal of agricultural and food chemistry. 2011,
59 (17), 9049-9053.
80
152. Yamada, K.; Kikuchi, M.; Gilbert, A.; Yoshida, N.; Wasano, N.; Hattori, R.; Hirano, S.,
Evaluation of commercially available reagents as a reference material for intramolecular carbon
isotopic measurements of acetic acid. Rapid Communications in Mass Spectrometry. 2014, 28
(16), 1821-1828.
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APPENDIX; COPYRIGHT INFORMATION
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VITA
Yuyang He, born in Sichuan, China, received her bachelor’s and master’s degree from the Ocean
University of China, majored in geology and marine geology, respectively. Geoscience is about
exploration, adventure, and discovery of the unknown, about which she is passionate. After
finished her master’s degree, she decided to enter the Department of Geology & Geophysics at
Louisiana State University.
Upon completion of her PhD program, she plan to remain in geoscience academia to
continue her research while sharing her knowledge and my enthusiasm of science with people.
She is always thankful for that she have the opportunity to be well educated, and she believe that
it is her responsibility now to protect the educational rights of the next generation; to help them
achieve something better than she could. In her understanding, the value of education is more
than getting a better job or earning more money, it is the infinite potential of science, art and
technology for our future.