High-Dimensional Isotope Relationships

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High-Dimensional Isotope RelationshipsYuyang HeLouisiana State University and Agricultural and Mechanical College, [email protected]

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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:


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:


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 *



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



reac15.NO3 <- (k.NXR * a15.NXR * C15.NO2 - k.NAR * a15.NAR * C15.NO3)




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)




reac17.NO3 <- (k.NXR * a17.NXR.NO2 * C17.NO2 + (0.5 * k.NXR * a17.NXR.H2O *






reac18.NO3 <- (k.NXR * a18.NXR.NO2 * C18.NO2 + (0.5 * k.NXR * a18.NXR.H2O *






reac14.NO2 <- (k.NAR * C14.NO3 + k.AMO * C14.NH4 - k.NXR * C14.NO2 - k.NIR *

C14.NO2)


34



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)




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)))




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)))




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)))


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)


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))


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))



C16.NO2.0 <- (2 * C.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"]





#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))]









#Calculate isotope compositions

R15.NO3.rtm <- C15.NO3.rtm / C14.NO3.rtm

d15.NO3.rtm <- 1000 * log((R15.NO3.rtm) / N.air)


d18.NO3.rtm <- 1000 * log((R18.NO3.rtm) / O18.VSMOW)


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"]





#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))]









#Calculate isotope compositions

R15.NO3.rdm <- (C15.NO3.rdm / C14.NO3.rdm)

d15.NO3.rdm <- (1000 * log((R15.NO3.rdm) / N.air))


d18.NO3.rdm <- (1000 * log((R18.NO3.rdm) / O18.VSMOW))


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

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APPENDIX; COPYRIGHT INFORMATION

88

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.

High-Dimensional Isotope Relationships

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