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Geochimica et Cosmochimica Acta 172 (2016) 22–38 www.elsevier.com/locate/gca

Combinatorial effects on clumped isotopes and their significance in biogeochemistry

Laurence Y. Yeung

Department of Earth Science, Rice University, Houston, TX 77005, USA

Received 28 May 2015; accepted in revised form 17 September 2015; Available online 30 September 2015

Abstract

The arrangement of isotopes within a collection of molecules records their physical and chemical histories. Clumped- isotope analysis interrogates these arrangements, i.e., how often rare isotopes are bound together, which in many cases can be explained by equilibrium and/or kinetic isotope fractionation. However, purely combinatorial effects, rooted in the statistics of pairing atoms in a closed system, are also relevant, and not well understood. Here, I show that combinatorial isotope effects are most important when two identical atoms are neighbors on the same molecule (e.g., O2,N2, and D–D clumping in CH4). When the two halves of an atom pair are either assembled with different isotopic preferences or drawn from different reservoirs, combinatorial effects cause depletions in clumped-isotope abundance that are most likely between zero and 1‰, although they could potentially be 10‰ or larger for D–D pairs. These depletions are of similar magnitude, but of opposite sign, to low-temperature equilibrium clumped-isotope effects for many small molecules. Enzymatic isotope- pairing reactions, which can have site-specific isotopic fractionation factors and atom reservoirs, should express this class of combinatorial isotope effect, although it is not limited to biological reactions. Chemical-kinetic isotope effects, which are related to a bond-forming transition state, arise independently and express second-order combinatorial effects related to the abundance of the rare isotope. Heteronuclear moeties (e.g., CAO and CAH), are insensitive to direct combinatorial influences, but secondary combinatorial influences are evident. In general, both combinatorial and chemical-kinetic factors are important for calculating and interpreting clumped-isotope signatures of kinetically controlled reactions. I apply this analytical framework to isotope-pairing reactions relevant to geochemical oxygen, carbon, and nitrogen cycling that may be influenced by combinatorial clumped-isotope effects. These isotopic signatures, manifest as either directly bound isotope ‘‘clumps” or as features of a molecule’s isotopic anatomy, are linked to molecular mechanisms and may eventually provide additional information about biogeochemical cycling on environmentally relevant spatial scales. Ó 2015 Elsevier Ltd. All rights reserved.

1. INTRODUCTION quent chemical and transport history. For example, for many carbonates (Ghosh et al., 2006, 2007; Schauble Studies of molecules containing more than one rare iso- et al., 2006; Dennis and Schrag, 2010; Eagle et al., 2010, tope indicate that isotopes are seldom distributed randomly 2013; Tripati et al., 2010; Dennis et al., 2011; Eiler, 2011; among the isotopic variants of a molecule (Eiler, 2013). Thiagarajan et al., 2011; Passey and Henkes, 2012; These isotope distributions in principle contain information Henkes et al., 2013; Zaarur et al., 2013; Came et al., how a molecule was formed and, to some extent, its subse- 2014; Fernandez et al., 2014; Petrizzo et al., 2014; Tang et al., 2014; Kluge et al., 2014a, 2015; Kele et al., 2015; Kluge and John, 2015a,b), atmospheric CO2 and O2 E-mail address: [email protected] (Eiler and Schauble, 2004; Affek and Eiler, 2006; Affek http://dx.doi.org/10.1016/j.gca.2015.09.020 0016-7037/Ó 2015 Elsevier Ltd. All rights reserved. L.Y. Yeung / Geochimica et Cosmochimica Acta 172 (2016) 22–38 23 et al., 2007; Yeung et al., 2012, 2014; Affek, 2013; Clog relative bulk isotopic composition of each atom within et al., 2014), and CH4 in certain environments (Stolper the isotope pair. Traditional kinetic isotope effects can yield et al., 2014a, 2015; Wang et al., 2015), isotope-exchange clumped-isotope deficits or excesses in the products, but reactions distribute rare isotopes at or near internal isotopic they operate in conjunction with these combinatorial equilibrium. Under those conditions, the relative abun- effects. The influence of combinatorics is therefore likely dance of singly- and multiply-substituted isotopic versions widespread, introducing challenges to existing clumped- of a molecule—its ‘‘clumped” isotope distribution—reflects isotope approaches, while opening up possibilities for the isotopic equilibration temperatures. Often, however, application of clumped-isotope analysis to new problems isotope-exchange reactions are disfavored or absent, and in biogeochemistry. kinetic isotope fractionation governs the distribution of iso- topes within a molecular species (Affek et al., 2008, 2014; 2. THEORY Guo et al., 2009; Yeung et al., 2009; Kluge and Affek, 2012; Saenger et al., 2012; Affek and Zaarur, 2014; 2.1. The stochastic reference frame Joelsson et al., 2014; Kluge et al., 2014b; Schmidt and Johnson, 2015; Wang et al., 2015). Clumping of rare The clumped-isotope composition of any given isotope isotopes in those cases likely depends in part on extrinsic pair is reported relative to the stochastic distribution of iso- factors (e.g., enzyme-substrate binding) in addition to fac- topes, i.e., tors intrinsic to the reaction in question (e.g., vibrational n Rsample Dn 1 1 frequencies of reactants and transition states). Therefore, ¼ nR ð Þ the range of clumped-isotope variations expected from stochastic kinetically controlled reactions, particularly those relevant where R is the ratio of the multiply-substituted analyte with to biogeochemistry, is not obvious. Rapidly improving cardinal mass n to its most abundant isotopologue (e.g., 47 47 44 analytical capabilities for isotope and clumped-isotope geo- R = CO2/ CO2), with the Dn value reported in per mil n chemistry (Young et al., 2011; Eiler et al., 2013; Ono et al., (‰). Importantly, Rstochastic is a calculated isotopologue 2014) will place our understanding of these isotope effects abundance that is defined internally; it depends on the bulk under intense scrutiny. isotope ratios (e.g., 13C/12C) in the sample itself and not on Indeed, recent studies indicate that clumped-isotope an arbitrary external standard. This internal comparison geochemistry is more diverse than previously recognized. between singly- and multiply-substituted isotopic species Equilibrium thermodynamic partitioning yields slight results in some non-intuitive clumped-isotope systematics. excesses in multiply-substituted species relative to a random Mixing, for example, does not conserve Dn values because n n distribution of isotopes, with the magnitude of those Rsample and Rstochastic do not covary linearly (Eiler and excesses varying with the relative zero-point energies of a Schauble, 2004; Yeung et al., 2009; Stolper et al., 2014b; molecule’s isotopic variants (Wang et al., 2004; Schauble Defliese and Lohmann, 2015). At the same time, the inter- et al., 2006; Cao and Liu, 2012; Webb and Miller, 2014). nal comparison makes Dn values insensitive to reservoir However, laboratory experiments show that enzymatic effects when the only relevant process is bond alteration reactions can produce an isotopic distribution that is (Ghosh et al., 2006; Passey and Henkes, 2012; Yeung depleted relative to the random distribution; biologically et al., 2014, 2015): in these cases, the arrangement of iso- mediated formation of gases can lead to products with an topes does not depend on the source of atoms. ” ‘‘anticlumped isotope distribution (Stolper et al., 2015; This sensitivity of Dn values to chemistry has been inves- Wang et al., 2015; Yeung et al., 2015). The origins of these tigated for isotopic equilibrium (Wang et al., 2004; Ghosh anticlumped distributions are clearly kinetic, but the preva- et al., 2006; Schauble et al., 2006; Cao and Liu, 2012; Hill lence and range of these signatures in nature is not yet et al., 2014; Webb and Miller, 2014) and in specific cases known. where kinetics are important (Eiler and Schauble, 2004; In this manuscript, I examine the systematics of the Affek et al., 2007, 2014; Guo et al., 2009; Yeung et al., clumped-isotope distribution within simple molecules 2009, 2015; Dae¨ron et al., 2011; Kluge and Affek, 2012; formed through kinetically controlled reactions. I analyze Saenger et al., 2012; Joelsson et al., 2014; Stolper et al., how isotopic preferences are propagated through elemen- 2015; Wang et al., 2015). Here, I approach bond formation tary chemical mechanisms, and I find that the clumped- systematically: Beginning with a conceptual model of iso- isotope distribution of products depends in part on the tope pairing by irreversible reactions, I build toward an sampling statistics of the component atoms and not simply understanding of clumped-isotope distributions in poly- on bond-making chemistry. These effects, which are only atomic molecules that are formed with some degree of evident when two atoms are combined, can dominate the reversibility. I focus on the systematics of bond formation, product’s clumped isotope distribution when other isotope as bond scission can lead to isotopomer-dependent isotope effects are small. For example, when rare isotopes are effects that are not as easily generalized (Johnston et al., paired, e.g., making 18O–18Oor15N–15N, these combinato- 1995; Rahn et al., 1998; Farquhar et al., 2001; Miller rial effects can lead to clumped-isotope distributions that et al., 2005; Lyons, 2007; Ostrom et al., 2007; are depleted relative to the stochastic distribution of iso- Chakraborty et al., 2008; Danielache et al., 2008; Hattori topes even in the absence of kinetic isotope effects during et al., 2011; Ono et al., 2013; Whitehill et al., 2013; bond formation. The magnitude of these deficits depends Schmidt and Johnson, 2015). The simplified reaction on the isotope-pair system being studied, as well as the schemes described herein have analogs in gas-phase, 24 L.Y. Yeung / Geochimica et Cosmochimica Acta 172 (2016) 22–38 biological, and geological chemistry. They provide useful in which E1 and E2 are substrate binding sites 1 and 2, starting points for predicting and interpreting natural iso- respectively. Similar reactions can be written for each topologue variations. isotopic substitution. The initial steps can be reversible, Throughout this manuscript I will use the term ‘‘irre- and are associated with an isotopic fractionation factor at versible reaction” as shorthand for all reactions in which each binding site. For an infinite source reservoir and a the bond-forming step does not fractionate the isotopes two-isotope system (i.e., A and A0, with a constant ratio that are clumped together. It is an approximation meant of [A0]/[A]), the rate of product formation is described for- to simplify the conception of combinatorial clumped- mally by the following differential equations: isotope effects. When the atoms are directly bound together, d ½ðAAÞ k bond formation does not fractionate isotopes only in spe- dt ¼ 3½AE1½AE2ð2Þ cial irreversible cases. It may be a sufficient approximation d 0 ½ðAA Þ 0 0 when the atom pairs being considered are not directly k 0 k 0 dt ¼ 31 ½A E1½AE2þ 32 ½AE1½A E2ð3Þ bound together or when the transition state for bond for- d 0 0 mation is sufficiently loose, e.g., when the rate-limiting step ½ðA A Þ 0 0 ¼ k 00 ½A E1½A E2ð4Þ of a multi-step reaction is not the bond-making step of dt 3 interest. The latter scenario may be most relevant to in which k3, k310, k320, and k300 are the forward rates for A–A multi-step biochemical redox reactions, which can be association (reaction R3), and the primes denote the 0 limited by the rate of electron transfer instead of bond number of isotopic substitutions to A (e.g., k310 is a single formation (Mathai et al., 1993). substitution at site 1, and k320 is a single substitution at site 2). The bound AE1 and AE2 species are described by d 2.2. Irreversible isotope pairing ½ðAE1Þ k k dt ¼ 1½A½E1 1½AE1ð5Þ

To understand the combinatorial clumped-isotope effect d½ðAE2Þ ¼ k ½A½E k ½AE ð6Þ qualitatively, consider, as an analog for bond-forming reac- dt 2 2 2 2 tions, the act of choosing two apples from a bag. In this with analogous equations for reactions involving A0. In Eqs. bag, there are apples of two colors, red and green. One (5) and (6), k1 and k1 represent the forward and backward chooses an apple with each hand and joins the two to make rate constants, respectively, for ‘‘enzyme-substrate”-type a pair. After a large number of pairs have been assembled, binding (reaction R1), with the same convention describing the color distribution of the pairs is analyzed. If neither reaction R2. At steady state for substrate binding, i.e., d hand had a preference for a color, then the number of [(AE )]/dt = 0 and d[(AE )]/dt = 0, etc., 1 2 red–red or green–green pairs will appear random, i.e., dic- d½ðAAÞ k1k2 2 tated by chance, based on the total number of apples ¼ k3 ½A ½E1½E2ð7Þ dt k1k2 picked. If, however, the left hand had only picked red d 0 k k k k apples while the right hand had only picked green apples, ½ðAA Þ k 10 2 k 1 20 0 dt ¼ 310 k k þ 320 k k ½A ½A½E1½E2ð8Þ there would be no color-matched pairs, clearly fewer 10 2 1 20 ” color-matched ‘‘clumps than if the apples had been paired d 0 0 k k ½ðA A Þ k 10 20 0 2 by chance alone. These two endmember cases suggest that dt ¼ 300 k k ½A ½E1½E2ð9Þ deficits in color-matched pairs, relative to chance statistics, 10 20 can also arise when the right and left hands have different The primes denote rate constants for reactions involving color preferences. Below, I will derive this idea formally isotopically substituted species. When A–A bond formation as it applies to the pairing of rare isotopes: Deficits in has no kinetic isotope effect, i.e., k3 = k310 = k320 = k300 (see rare-isotope pairs, relative to the stochastic distribution of Section 2.2.3 for discussion of the more general case), the isotopes, will be apparent whenever the two halves of a composition of each product isotopologue is: 0 0 0 symmetric dimer are assembled with unequal isotopic A–A0 ½ðAA Þ ½A ½A Rsample ¼ ¼ a þ a ð10Þ preferences. ½ðAAÞ ½A 1 ½A 2 0 0 0 0 A0–A0 ½ðA A Þ ½A ½A 2.2.1. Two-isotope systems Rsample ¼ ¼ a1 a2 ð11Þ A simple isotope-pairing reaction to investigate is one in ½ðAAÞ ½A ½A a a which two substrates are bound together by an irreversible where 1 =(k10/k10)/(k1/k1) and 2 =(k20/k20)/(k2/k2) bond-forming reaction. This class of reaction is ubiquitous are the isotopic fractionation factors for reactions R1 and A–A0 in biogeochemistry and relevant to biogeochemical pro- R2, respectively. Note that the expression for Rsample 0 0 cesses such as O2 evolution from photosynthesis (see (Eq. (10)) explicitly accounts for both A–A and A –A Section 3). The simplest form of this reaction pairs two isotopologues. The stochastic distribution of isotopes in atoms, here represented as a pair of arbitrary atoms A, the product is thus: ! 2 through the elementary steps: 0 0 0 0 2 0 0 ½A product AA 2 A A A –A R ½ð Þ þ ½ð Þ stochastic ¼ ¼ 0 A þ E1 AE1 ðR1Þ ½Aproduct 2½ðAAÞ þ ½ðAA Þ A–A0 A0–A0 2 A þ E2 AE2 ðR2Þ Rsample þ 2 Rsample 12 ¼ A–A0 ð Þ AE1 þ AE2 ! A–A þ E1 þ E2 ðR3Þ 2 þ Rsample L.Y. Yeung / Geochimica et Cosmochimica Acta 172 (2016) 22–38 25 and the clumped-isotope composition of the products is characterizes the number of singly-substituted isotopo- therefore logues. Because the geometric mean of two numbers is 2 3 0 2 always equal to or less than their arithmetic mean, purely a a f2 þ ½A ða þ a Þg 4 1 2 ½A 1 2 5 combinatorial isotope effects almost always yield an D 0 0 A –A ¼ 2 1 ð13Þ ½A0 ‘‘anticlumped” isotope distribution (i.e., D 0 0 6 0 with a fa þ a þ 2 a a g A –A 1 2 ½A 1 2 couple exceptions; see Fig. 1 and discussion below). One

This exact expression reveals a dependence of DA0–A0 on can also consider this phenomenon as symmetry-based: the fractionation factors in reactions R1 and R2 (i.e., a1 and When the two halves of a symmetric dimer are assembled a2, respectively) as well as the rare-isotope abundance of the through reactions R1, R2, R3, the abundances of singly- source reservoir (i.e., [A0]/[A]). The clumped-isotope distri- substituted product isotopomers may not be equal (i.e., bution of the A–A pair therefore depends on two factors [A0–A] – [A–A0]). Yet, the arithmetic mean of the that are unrelated to A–A bond-forming chemistry. The isotopomer abundances is still nearly equal to the bulk iso- consequences of these combinatorial systematics are plotted topic composition of A0 (for systems in which [A0]/[A] ? 0). in Fig. 1 for several isotope pairs; in general, they lead to The geometric mean of the isotopomer abundances, how- products with fewer rare-isotope pairs than a stochastic dis- ever, will be smaller than the arithmetic mean, resulting in D tribution would predict. A0–A0 < 0. In contrast, for a symmetric mechanism that To understand the origin of these nonstochastic produces [A0–A] and [A–A0] in equal abundance (i.e., a a clumped-isotope deficits, it is useful to examine equation 1 = 2), the arithmetic and geometric means of isotopomer 0 D (13) in the low-abundance limit, i.e., [A ]/[A] ? 0. In that abundances are equal, yielding A0–A0 0. limit, the clumped-isotope composition can be approxi- Many isotopologue systems, such as those for H2 and mated as: N2, are close to the low-abundance limit described by "#"# Eq. (14) (see Fig. 1). For example, when a1 = 1 and 1 2 2 a1a2 ða1a2Þ a2 = 1.05, DA0–A0 = 0.59‰ in the low-abundance limit, D 0– 0 1 ¼ 1 ð14Þ A A 1 2 1 ‰ a a ða1 þ a2Þ and the deviation from that is +0.0002 for D–D pairing 4 ð 1 þ 2Þ 2 and + 0.01‰ for a 13C–13C pairing. Species with more The combinatorial effect for isotope pairing is thus fun- abundant rare isotopes have larger nominal departures damentally related to the ratio of the geometric mean, from the low-abundance limit, such as +0.3‰ for 37Cl–37Cl a a 1/2 1 a a ‰ 81 81 ( 1 2) , and the arithmetic mean, 2( 1 + 2), of the bound and +0.59 for Br– Br. In the endmember case in which species’ isotopic composition. The geometric mean charac- [A0]/[A] = 1, the isotopes in the product are in slight excess terizes the number of doubly-substituted (isotopically relative to the stochastic distribution (DA0–A0 = +0.007‰, clumped) isotopologues, while the arithmetic mean or +0.6‰ relative to that expected in the low-abundance limit). This excess arises from the [A0]/[A]-dependent terms in Eq. (13), and is likely near the upper limit for DA0–A0 val- ues caused by biological enzyme-substrate binding. For iso- topic fractionation factors a = 0.95 1.05, DA0–A0 values do not exceed 0.01‰ when [A0]/[A] = 1. Another consequence of the [A0]/[A]-dependent terms in Eq. (13) is a slight asym- metry in for DA0–A0 values a < 1 vs. a >1. In principle, DA0–A0 also depends on the isotopic compo- sition of the source reservoir. However, the dependence of DA0–A0 on isotopic composition is generally negligible if the molecule is built from a single source of constant iso- topic composition. For example, hydrogen isotope pairing from an infinite source with an extreme dDSMOW value of +500‰ (SMOW = Standard Mean Ocean Water) produces D4 that is only 0.0001‰ (0.1 ppm) higher than from a source with dDSMOW = 0. Similarly, chlorine isotope pair- 37 ing from a source with high d ClSMOC = +10‰ (SMOC = Standard Mean Ocean Chloride) produces D74 that is 0.002‰ higher than from a source with 37 d ClSMOC = 0. If each atom is drawn from a different reservoir, however (e.g., dDSMOW = +500‰ at one site Fig. 1. Calculated clumped-isotope signatures for irreversible d isotope-pairing reactions R1, R2, and R3 for a = 1. Combinato- and DSMOW = 0 at the second), then the characteristic 1 D rial effects, driven here by a difference in isotopic fractionation for dependence of n values on binding fractionation factors ” the two halves of a symmetric dimer, cause Dn values to be less than is offset; an ‘‘anticlumped isotope distribution could be zero. As the isotopic contrast deviates from unity (plotted here as produced even when a1 = a2, and a stochastic distribution a a 2/ 1 = 1 for no isotopic contrast), the magnitude of the combi- could be produced when a1 – a2. In those cases, the gener- D natorial effect increases quadratically. The sensitivity of n values alized expression for DA0–A0 produced, in terms of site- to site-specific isotope contrast decreases as rare-isotope abundance 0 0 0 0 specific isotope ratios R1 = a1[A ]/[A] and R2 = a2[A ]/ 15 ? 13 ? 37 increases ( N C Cl at natural abundances covers two [A], is calculated from Eq. (13) to be: orders of magnitude). 26 L.Y. Yeung / Geochimica et Cosmochimica Acta 172 (2016) 22–38 "# 0R 0R 0R 0R 2 which yields D 1 2ð2 þ 1 þ 2Þ A0–A0 ¼ 1 ð15Þ 2 no3 0 0 0 0 2 2 ð R1 þ R2 þ 2 R1 R2Þ ½A ½A0 a a a a a a 6 1 2 2 þ ½A ð 1 þ 2 Þþ ½A ð 1 þ 2Þ 7 D 0 0 4no5 A contrast in isotopic abundances at each site, irrespec- A –A ¼ 2 1 ½A0 ½A a a 2 a a a a a a tive of its origin, will lead to a nonstochastic deficit in rare- 1 þ 2 þ ½A 2 1 þ ½A ð 1 2 þ 1 2 Þ isotope pairs in the product molecules. Note, however, that ð20Þ Eqs. (13) and (15) describe instantaneous DA0–A0 values, i.e., D 0 for a common and infinite reservoir, and the A0–A0 values for a given set of conditions. Changing R1 "# 0 and R2 values, e.g., in systems with finite atom reservoirs 0R 0R R R 0R 0R 2 D 1 2ð2 þ 1 þ 2 þ 1 þ 2Þ subject to Rayleigh fractionation, could result in instanta- A0–A0 ¼ 1 0 0 0 0 0 0 2 ðÞR1 þ R2 þ 2 R1 R2 þ R1 R2 þ R1 R2 neous Dn values that vary over time. Finally, the sensitivity of DA0–A0 values to bulk isotopic ð21Þ composition increases with the abundance of the rare- when generalized to the individual isotope ratios at each isotope pair. Concomitantly, however, the sensitivity of site. Similar expressions can be derived for DA–A . D 0 0 values to a and a values decreases (Fig. 1). In this A –A 1 2 Eqs. (20) and (21) resemble Eqs. (13) and (15), except with sense, stable-isotope systems with a rare minor isotope have additional terms involving the third isotope. These terms larger combinatorial effects, all else equal. Isotope-labeling represent the sequestration of A and A0 in molecular bonds experiments, for which combinatorial clumped-isotope with A*, which affects the stochastic distribution of effects will be less important, may allow one to disentangle isotopes. In molecular oxygen, this sequestration leads to intrinsic kinetic isotope effects from combinatorial effects negligible effects on the clumped-isotope distribution: on the clumped-isotope distributions observed in natural Ignoring the contribution from 17O when calculating D systems. 36 for 18O–18O isotope pairing only results in an error of 0.0003‰ when a = 1 and a = 1.05. 2.2.2. Three-isotope systems 1 2 The presence of a third isotope also allows for the for- To describe species that contain three isotopes, e.g., 16O, mation of a third clumped-isotope composition, D – 0 . 17O, and 18O, I introduce the isotope A*, which can pair A A The stochastic abundance for A*–A0 is defined as: with A and A0 to form the A–A*,A*–A*, and A*–A0 ! ! isotopologues with abundances described by 0 A–A0 ½A product ½A product Rstochastic ¼ 2 ½Aproduct ½Aproduct A–A ½ðAA Þ ½A ½A Rsample ¼ ¼ a1 þ a2 ð16Þ ½ðAAÞ ½A ½A ½ðAAÞþ½ðAA0Þþ2½ðAAÞ ¼ 2 2½ðAAÞþ½ðAA0Þþ½ðAAÞ A–A ½ðA A Þ ½A ½A Rsample ¼ ¼ a a ð17Þ ½ðAAÞ ½A 1 ½A 2 ½ðAA0Þþ½ðAA0Þþ2½ðA0A0Þ 0 0 0 2½ðAAÞþ½ðAA Þþ½ðAA Þ – 0 ½ðA A Þ ½A ½A A A 0 Rsample ¼ ¼ a1 a2 AA R A –A R A –A R ½ðAAÞ ½A ½A sample þ sample þ2 sample ¼ 2 0 – 0 2þ AA Rsample þ A A Rsample ½A ½A þ a1 a2 ð18Þ A–A0 A–A0 A0–A0 ½A ½A Rsample þ Rsample þ2 Rsample – 0 – ð22Þ 2þ A A Rsample þ A A Rsample The stochastic distribution of A0 in A0–A0 becomes: making the clumped-isotope distribution:

2 hi3 2 ½A ½A0 a a a a a a a a 6 ð 1 2 þ 1 2 Þ 2 þ A ð 1 þ 2 Þþ A ð 1 þ 2Þ 7 D hi½ hi½ A–A0 ¼ 4 15 ð23Þ ½A0 ½A ½A ½A0 a a a a a a a a a a a a a a a a 22½A 1 2 þ ½A ð 1 2 þ 2 1Þþ 1 þ 2 2 ½A 1 2 þ ½A ð 1 2 þ 1 2 Þþ 1 þ 2 " # ðR 0R þ 0R R Þð2 þ R þ R þ 0R þ 0R Þ2 1 2 1 2 1 2 1 2 1 24 ¼ 0 0 0 R 0 0 0 0 ð Þ 2ð R1 þ R2 þ R1 R2 þ R1 R2 þ 2 R1 R2Þð R1 þ 2 þ R1 R2 þ R1 R2 þ 2 R1 R2Þ

AA0 A–A0 A0–A0 2 0 0 Rsample Rsample 2 Rsample In the low-abundance limit, expression (23) simplifies to A –A R þ þ "# stochastic ¼ – 0 – 2 þ A A Rsample þ A A Rsample a a a a D 1 2 þ 1 2 A–A0 1 ð25Þ ð19Þ 1 a a a a 2 ð 1 þ 2Þð 1 þ 2 Þ L.Y. Yeung / Geochimica et Cosmochimica Acta 172 (2016) 22–38 27

17 17 17 18 18 18 Fig. 2. Calculated combinatorial effects for rare-isotope pairing in O2. (Left) Dependence of D34 ( O O), D35 ( O O), and D36 ( O O) on site-specific isotope contrast, where the mass-dependent relationship 17R =(18R)0.528 was used. The smaller fractionation of 17O vs. 18O at each 18 18 binding site results in D34 and D35 values being less sensitive to R2/ R1 compared to D36. (Right) Covariation in D34, D35, and D36 values for combinatorial isotope pairing (black and white points) and isotopic equilibrium (grey points). Combinatorial effects result in clumped-isotope compositions in quadrant III, while isotopic equilibrium yields clumped-isotope compositions in quadrant I. The slopes of the trends in D–D space are also unique to each process within each quadrant.

Combinatorial effects for isotope pairing in the oxygen d½ðAAÞ ¼ k ½ðE A–AE Þzð26Þ system are shown in Fig. 2. Qualitatively, they resemble dt 4 1 2 those in Fig. 1: nonstochastic deficits in rare-isotope pairs d – z ½ðE1A AE2Þ k k – z are expected, and they increase in magnitude as the two dt ¼ 3½AE1½AE2 3½ðE1A AE2Þ ð27Þ sites’ isotopic preferences diverge. However, mass- If the reaction is in steady state with respect to dependent relationships between the three multiply- à à substituted isotopologues are apparent in D –D and (E1A–AE2) , i.e., d[(E1A–AE2) ]/dt = 0, then 36 34 D –D plots. They correspond to mass-dependent rela- 36 35 d½ðAAÞ k3k4 k1k2 2 tionships between 16O, 17O and 18O. ¼ ½A ½E1½E2ð28Þ dt k3 k1k2 The expression for combinatorial effects on Dn increases in complexity as the number of relevant isotopes increases. with similar expressions for the production of the singly- Yet, the qualitative dependence of Dn on fractionation fac- and doubly-substituted isotopologues of the product: tors remains the same: Combinatorial factors almost always 0 d½ðAA Þ k 0 k 0 k 0 k k 0 k 0 k k 0 drive D values toward ‘‘anticlumped” values less than zero 31 41 1 2 32 42 1 2 n dt ¼ k k k þ k k k for irreversible isotope-pairing processes. 310 10 2 320 1 20 0 ½A ½A½E1½E2ð29Þ 2.2.3. Isotope pairing controlled by the A–A transition state d 0 0 k k k k ½ðA A Þ 300 400 10 20 0 2 Consider the case when two atoms from the same infi- ¼ ½A ½E1½E2ð30Þ dt k 00 k 0 k 0 nite reservoir (A or A0) are joined through a transition state 3 1 2 to make a diatomic molecule (A–A): Integrating with respect to time and assuming that the source of A and A0 is infinite, the isotopologue ratios are: AE þ AE ðE A–AE Þz ! A–A þ E þ E ðR4Þ 1 2 1 2 1 2 0 0 0 AA A In classical transition state theory (Eyring, 1935; A–A R ½ð Þ ½ a a a a sample ¼ ¼ ð 1 k10 þ 2 k20 Þð31Þ Bigeleisen and Wolfsberg, 1958), the abundance of A–A ½ðAAÞ ½A products depends on (1) the reversible association of 0 0 0 2 0 0 A A A A –A R ½ð Þ a ½ a a reactants AE1 and AE2 that forms the transition state sample ¼ ¼ k00 1 2 ð32Þ à ½ðAAÞ ½A (E1A–AE2) and (2) the rate of A–A formation from that a a transition state. This schematic example is analogous to in which k10 =(k310k410/k310)/(k3k4/k3), k20 =(k320k420/ the case in which an isotopomer-specific fractionation exists k320)/(k3k4/k3), and ak00 =(k300k400/k300)/(k3k4/k3) are during A–A isotope pairing (i.e., k3 – k310 – k320 – k300 in the kinetic fractionation factors for reaction R4. Eqs. 2–4). Isotopomer-dependent transmission factors, e.g., k400, are Let k3 and k3 represent the forward and backward rate uncommon because of the nearly identical potential energy constants, respectively, for the association of the transition surfaces for isotopically substituted reactions (Bigeleisen state, and let k4 represent the rate constant (i.e., transmis- and Wolfsberg, 1958); they are included here for complete- sion coefficient) for A–A production from that transition ness. The clumped-isotope composition of the product is state. The relevant reaction rate expressions are: therefore 28 L.Y. Yeung / Geochimica et Cosmochimica Acta 172 (2016) 22–38 2 no3 2 ½A0 (or weaker) dependence on the other isotopic fractionation a a a 00 a a 0 a a 0 6 1 2 k 2 þ ½A ð 1 k1 þ 2 k2 Þ 7 D 0 0 4 no5 factors. The magnitude of kinetic isotope effects (i.e., the A –A ¼ 2 1 ð33Þ a a a a ½A0 a a a tightness of the transition state) controls the importance 1 k10 þ 2 k20 þ 2 1 2 k00 "#½A of combinatorial effects in reactions like R4. 0R 0R a 0R a 0R a 2 2 1 2 k00 ð2 þ 1 k10 þ 2 k20 Þ When kinetic isotope effects are small [i.e., ak00/(ak0) ? ¼ 2 1 ð34Þ 0R a 0R a 0R 0R a 1], D 0 0 values are dominated by combinatorial ‘‘anti- ð 1 k10 þ 2 k20 þ 2 1 2 k00 Þ A –A clumping,” likely of order 0–1‰. Even when kinetic iso- 2 which are generalizations of Eqs. (13) and (15). tope effects are large [i.e., ak00/(ak0) – 1], combinatorial While the dependence of DA0–A0 on kinetic fractionation effects tend to reduce the effective preference for rare- factors is not straightforward, the basic form of isotope pairs; they generally decrease DA0–A0 values (see Eqs. (33) and (34) can be understood more clearly when Fig. 3). For example, a transition state that pairs 18O a a a k0 = k10 = k20, i.e.: atoms by 5‰ moreso than what chance dictates, i.e., ak00/ a 2 D 2 no3 ( k0) = 1.005, will express a 18O–18O value in its products 2 ½A0 that is slightly less than 5‰. An asymmetry in substrate- a a a a ak0 6 1 2 2 þ A ð 1 þ 2Þ a 00 7 D ½ k D A0–A0 4no 15 ð35Þ binding fractionation reduces the 18O–18O value further: a 0 2 2 18 18 k00 ½A ðak0 Þ a a a1 þ a2 þ 2 a1a2 For 2/ 1 values between 0.95 and 1.05, combinatorial ak0 ½A effects reduce the product’s D18O–18O values to between 2 and the rare-isotope abundance is vanishingly low (i.e., 4.32‰ and 4.98‰. If, instead, ak00/(ak0) = 0.995 for the 0 [A ]/[A] ? 0): transition state, then D18O–18O values would be between "# 5.63‰ and 4.98‰ instead of 5.00‰ in the absence

a1a2 ak00 of combinatorial effects. Furthermore, increasing rare- D 0– 0 1 ð36Þ A A 1 2 2 isotope abundance decreases the sensitivity of DA0–A0 values ða þ a Þ ðak0 Þ 4 1 2 to chemical-kinetic effects, rendering combinatorial effects Similar expressions can be derived for triple-isotope more important (see Fig. 3). However, rare-isotope abun- systems. The first term in Eqs. (35) and (36) is the pure com- dance, i.e., [A0]/[A], can modulate the magnitude of the binatorial effect, analogous to the first term in Eqs. (13) and combinatorial term in Eq. (35). Similar to the case for irre- (14), respectively. The second term, determined by the phys- versible reactions (Section 2.2.1), DA0–A0 values here become à 0 ical chemistry of the (E1A–AE2) transition state, is the less sensitive to site-specific isotope contrast as [A ]/[A] 2 chemical preference for rare-isotope clumping in excess of increases. For example, when when ak00/(ak0) = 1.005, the stochastic distribution. It can be extracted from quan- pairing of 18O atoms at natural abundances results in a tum–mechanical calculations of transition-state properties maximum D18O–18O value of 4.98‰, which is reduced to 18 18 (Guo et al., 2009; Joelsson et al., 2014). 4.38‰ (i.e., 0.60‰) when a2/ a1 = 1.05; a hypothetical 2 37 When ak00/(ak0) > 1, kinetic isotope fractionation leads transition state that pairs Cl atoms, which have a relative 2 D directly to DA0–A0 > 0, while if ak00/(ak0) < 1, it leads to abundance 100 times higher, has a maximum 37Cl–37Cl ‰ ‰ DA0–A0 < 0. As illustrated in Fig. 3, the DA0–A0 value has a value of 2.57 , which is only reduced to 2.22 ‰ 37a 37a first-order dependence on ak00 and a second-order (i.e., 0.35 ) when 2/ 1 = 1.05.

Fig. 3. Calculated combinatorial effects for reversible (but kinetically controlled) rare-isotope pairing. (Left) Rare-isotope paring is strongly 2 dependent on intrinsic kinetic clumping/anticlumping preferences, i.e., ak00 /(ak0) in Eq. (35), but depletions in Dn values caused by combinatorial effects are still relevant. (Left and right) Increasing abundance of the rare isotope reduces the sensitivity of Dn values to clumping/anticlumping preferences; 15N ? 13C ? 37Cl at natural abundances covers two orders of magnitude. This secondary combinatorial effect arises from the [A0]/[A]-dependent terms in Eqs. 33–35. L.Y. Yeung / Geochimica et Cosmochimica Acta 172 (2016) 22–38 29

0 0 Eqs. 33–35 also apply to reactions in which the first sub- A–B0 ½ðAB Þ ½B Rsample ¼ ¼ a ð38Þ strate is bound and the second reacts directly by making an ½ðABÞ ½B B A–A bond, i.e., 0 0 0 0 A0–B0 ½ðA B Þ ½A ½B Rsample ¼ ¼ a a ð39Þ z AB A A B B AE1 þ A ðE1A–AÞ ! A–A þ E1 ðR5Þ ½ð Þ ½ ½ This reaction mechanism applies to a variety of pro- For two-isotope systems such as carbon and hydrogen, cesses such as denitrification [(Riplinger and Neese, 2011; the stochastic distribution of isotopes in the product is: Blomberg and Siegbahn, 2012); see Section 3.4]. The 0 0 A0–B0 ½A p½B p Rstochastic absence of the second binding site E2 is represented by ¼ ½Ap½Bp a2 = 1, as it represents no isotopic fractionation associated ½ðA0BÞ þ ½ðA0B0Þ ½ðAB0Þ þ ½ðA0B0Þ with substrate binding. ¼ ½ðABÞ þ ½ðAB0Þ ½ðABÞ þ ½ðA0BÞ The preceding derivations show that clumped-isotope A0 –B A0–B0 A–B0 A0 –B0 distributions for kinetically controlled isotope-pairing reac- Rsample þ Rsample Rsample þ Rsample ¼ – 0 0– 1 þ A B Rsample 1 þ A BRsample tions are determined by (1) the bulk isotopic abundances at each atom position and (2) the isotopomer-specific fraction- A0–B A–B0 ¼ Rsample Rsample ation factors for forward reaction. The former is controlled by properties extrinsic to the bond-making chemistry, while ð40Þ the latter is controlled by intrinsic isotope-pairing chem- The expression for DA0–B0 then yields istry. While the sensitivity of clumped-isotope distributions "# A0–B A–B0 to atomic source might be viewed as a confounding compli- Rsample Rsample D 0 A –B0 ¼ 0– – 0 1 ¼ 0 ð41Þ cation, note that many isotope-pairing reactions draw A BRsample A B Rsample atoms from a common reservoir. For example, in photo- synthetic O formation from water, the source reservoir With only one atom and two isotopes of each type in the 2 D for both oxygen atoms is identical, well-mixed, and effec- molecule, the above expression is trivial in hindsight: A0–B0 tively infinite. In that case, the clumped-isotope signature values from irreversible heteronuclear association are primarily reflects the biochemistry of photosynthesis and always zero, i.e., there is no combinatorial effect. not the isotopic composition of the source water (Yeung The expression also holds for multi-isotope systems. For example, when A* and B* isotopes are also relevant, then et al., 2015). For N2 formation from nitric oxide in denitri- fication, however, the source reservoir is finite and poten- the calculated stochastic distribution does not change: tially sensitive to environmental conditions [(Yang et al., A0–BR A0–B0 R A0–B R A0–B0 R sample þ sample þ sample 2014); see Section 3.4]. Nevertheless, Dn values are sensitive stochastic ¼ – 0 – 1 þ A B Rsample þ A B Rsample to biochemistry at the molecular level. They provide A–B0 R A0–B0 R AB0 R additional information to bulk isotope measurements, sample þ sample þ sample 0– – 1 A BRsample A BRsample potentially offering new ways to quantify the imprints of þþ biogeochemical processes on environmentally meaningful A0–B A–B0 ¼ Rsample Rsample ð42Þ spatial scales.

Indeed, one can show that Dn = 0 for all possible 2.3. Heteronuclear atom association multiply-substituted isotopologues in this heteronuclear diatomic system. These systematics are a manifestation of 2.3.1. Irreversible atom association the lack of symmetry across the A–B bond: the ” 0 Are combinatorial effects important when atoms from ‘‘lop-sidedness that can arise during asymmetric A –A iso- different isotope systems are bound irreversibly? The irre- topomer formation cannot develop during A–B formation. versible assembly of a heteronuclear diatomic molecule, The B atom does not affect the stochastic distribution of the A–B, expresses combinatorial effects differently from the A atom, nor vice versa. In the language of the example used preceding examples utilizing isotope pairs. To derive a gen- at the beginning of Section 2, the left hand might choose eralized expression, I use a reaction scheme involving A and between red and green apples, but the right hand chooses B atoms, drawn from an infinite reservoir, which bind to between red and green tomatoes; there is always only one of each fruit in a pair, no matter the color. substrate-specific sites E1 and E2: Thus, the first-order combinatorial effect for irreversible bond formation between two different species is to yield A þ E1 AE1 ðR6Þ D 0 0 = 0. Chemical effects are important, but they depend A –B B þ E2 BE2 ðR7Þ on the system (see below). AE1 þ BE2 ! A–B þ E1 þ E2 ðR8Þ 2.3.2. Atom association controlled by the A–B transition with similar reactions for each possible isotopic substitu- state tion. The isotopic composition of the products is Finally, I examine the association of a heteronuclear diatomic molecule through a transition state, i.e., 0 0 A0–B ½ðA BÞ ½A z Rsample ¼ ¼ a ð37Þ AE1 þ BE2 ðE1A–BE2Þ ! A–B þ E1 þ E2 ðR9Þ ½ðABÞ ½A A 30 L.Y. Yeung / Geochimica et Cosmochimica Acta 172 (2016) 22–38 where the isotopologue abundances are analogous to those isotope-pairing systematics described in Section 2.2.3, in Eqs. (31) and (32): DA0–B0 values depend on the bulk abundance of both A 0 0 and B in a nonlinear way. They are shown for several moi- A0–B ½ðA BÞ A ½A Rsample ¼ ¼ ak0 a ð43Þ eties of carbon species in Fig. 4 (left panel). As the bulk ½ðAAÞ ½A A abundance of the rare isotopes increases (i.e., increasing 0 0 A0 B0 D A–B0 ½ðAB Þ B ½B magnitude of R and/or R values), A0–B0 values become Rsample ¼ ¼ ak0 a ð44Þ ½ðAAÞ ½B B less sensitive to properties of the transition state. 0 0 0 0 This tempering of intrinsic kinetic fractionation factors A0–B0 ½ðA B Þ ½A ½B Rsample ¼ ¼ ak00 aAaB ð45Þ is one of two combinatorial effects for isotope clumping ½ðAAÞ ½A½B between two different species. The second effect is a depen- D In Eqs. 43–45, the subscripts and superscripts A and B dence of A0–B0 values on the fractionation factors relevant Aa Ba denote the atom to which the isotopic fractionation factor to the singly-substituted products ( k0 and k0 values). A a applies. For example, ak0 is the isotopomer fractiona- For a given isotope-clumping tendency expressed by the 0 a Aa Ba tion factor for producing A –B in reaction R9, whereas transition state, i.e., constant k00/( k0 k0) ratios, different A B B 0 a 0 a 0 D 0 0 ak0 applies to A–B . The stochastic distribution is k and k values result in offsets in A –B values. These therefore: offsets depend on the bulk abundance of each atom: they increase in magnitude from 0.01‰ for the relatively rare A0B A0B0 A–B0 A0–B0 13 18 0 0 R R R R ‰ A –B R sample þ sample sample þ sample C– O moiety to 0.1 for the more abundant stochastic ¼ 0 0– 13 37 1 þ AB Rsample 1 þ A BRsample C– Cl moiety when the bulk-isotope fraction factors 23 0 a 0 Aa ½A a k00 ½B a are between 0.995 and 1.005 (see Fig. 4, right panel). k0 1þ A ½A A a 0 ½B B ¼ 4 k 5 ½B0 In the low-abundance limit, Eq. (48) becomes simply Ba 0 a 1 þ k ½B B 23 related to the properties of the transition state: 0 a 0 Ba ½B a k00 ½A a k0 B 1 þ Ba A a 4 ½B k0 ½A 5 k00 ð46Þ D 0– 0 1 49 ½A0 A B A B ð Þ Aa 0 a ak0 ak0 1þ k ½A A And the clumped isotope distribution is One must exercise caution in applying this approxima- tion when making ab initio predictions for heteronuclear 2 3 B ½B0 A ½A0 isotope combinations, however: While Eqs. (47) and (48) ak0 a ak0 a a 00 1 þ ½B B 1 þ ½A A 4 k 5 D 0 0 D 0 0 show that A –B is largely determined by the chemistry of A –B ¼ 0 0 1 ð47Þ Aa 0 Ba 0 ak00 ½B ak00 ½A k k a a D 0 0 1 þ Aa B 1 þ Ba A the reactive transition state, A –B values are much more 2 k0 ½B k0 ½A 3 D sensitive to rare-isotope abundance than A0–A0 values are B B0 A A0 a a 0 R a 0 R 4 k00 1þ k 1 þ k 5 (Section 2.2.3). ¼ A B a a 1 ð48Þ ak0 ak0 k00 B0 R k00 A0 R 1 þ Aa 1þ Ba k0 k0 3. BIOGEOCHEMICAL APPLICATIONS A0 0 B0 0 with R = aA[A ]/[A] and R = aB[B ]/[B] for the reac- tants. Systems of three isotopes or more have similar, but The generalized derivations in the preceding section analogous expressions not shown here. Similar to the reveal a fundamental feature of clumped-isotope

Fig. 4. Influence of rare-isotope abundance on product Dn values of heteronuclear moieties. Similar to the isotope-pairing case, rare-isotope A B clumping in heteronuclear moieties is strongly dependent on intrinsic kinetic clumping/anticlumping preferences, i.e., aki/( ak0 ak0)in Eqs. (47) and (48). (Left) Increasing abundance of the doubly-substituted moiety reduces the sensitivity of Dn values to kinetic clumping/ A B anticlumping preferences. (Right) Isotopic preferences relevant to the singly-substituted species will offset Dn vs. ak00 /( ak0 ak0) curves, as shown for 13C18O moieties. Note the difference in scale on the right-hand plot. L.Y. Yeung / Geochimica et Cosmochimica Acta 172 (2016) 22–38 31 geochemistry: kinetic signatures depend both combinatorial reactants, (2) combinatorial factors, and (3) properties of and chemical-kinetic factors. For isotope pairing, a the bond-forming transition state. Consider the irreversible ‘‘clumped” isotope distribution (i.e., Dn > 0) is produced and non-fractionating addition (after Section 2.3.1)ofan when the chemistry of the transition state favors the forma- O-atom to a CO fragment of arbitrary D13C–18O =+1‰ tion of rare-atom pairs enough to overcome the combinato- to form CO2: rial tendencies favoring the opposite outcome. Unequal O þ COðD – ¼ 1%Þ!CO ðD – 0:5%ÞðR10Þ abundances of rare isotopes at different molecular positions 13C 18O 2 13C 18O D tend to reduce Dn values from the transition-state’s prefer- Under these assumptions, the 13C–18O value for CO2 ence for rare-isotope clumping. The abundance of the rare would be approximately half that of the CO fragment, isotope has an influence as well. i.e., 0.5‰, because the inherited CO fragment with In heteronuclear systems with a unique atom B (e.g., ‘‘clumped” composition constitutes half the total number 2 A A CO, NO3 ,SO4 ), the lack of symmetry-based peculiarities of C O bonds; the other C O bonds are added according on A0–B0 clumping reduces the variability of combinatorial to chance alone (Eq. (41)), with secondary isotope effects factors. Yet, isotope pairs within those molecules (e.g., O–O here assumed to be negligible. The D18O–18O value, in 2 pairs in NO3 ,SO4 ) will still be subject to the combinato- contrast, is determined combinatorially, as the reaction rial factors described above. The expressions for Dn values chemistry imposed here resembles the irreversible isotope- derived in Section 2 are summarized in Table 1. pairing case outlined in Section 2.2. Relative-rate predic- How will Dn values vary in nature? Isotope pairs in dia- tions for reaction (R10) indicate that its chemical isotope tomic molecules produced biochemically, such as O2 from effects are quite large (Gao and Marcus, 2007), so the com- photosynthesis, H2 from nitrogen fixation (Hadfield and binatorial effects discussed in this simplified example should Bulen, 1969; Scranton et al., 1987), and N2 from denitrifica- be considered endmember effects. tion are sensitive to combinatorial effects in their clumped I reiterate here that this paper is not meant to be an isotope distributions, but the expression of combinatorial exhaustive account of all possible clumped-isotope distribu- effects in these systems will depend on the magnitude of tions arising from kinetically controlled reactions. It is intrinsic kinetic isotope effects. In polyatomic molecules, meant instead to provide a framework in which one can combinatorial effects are not as obvious: While analytical examine the effects of these reactions on the clumped- expressions exist for the sequential molecular assembly of isotope composition of geochemically interesting molecules. polyatomic molecules, they are quite complex (even for These concepts are, in some sense, already incorporated A–B–A molecules) because of the number of possible iso- into more sophisticated formulations of biosynthesis topomer products. However, combinatorial effects apply (Hayes, 2001; Valentine et al., 2004; Brunner and during each atomic addition step, so the insights gained Bernasconi, 2005; Stolper et al., 2015; Wang et al., 2015), from examining diatomic systems apply generally to the but the concepts described here offer a more general under- synthesis of polyatomic molecules. Furthermore, the occur- standing of how to interpret and envision clumped-isotope rence of isotope pairs within any polyatomic molecule can distributions in nature. be described by the systematics outlined in Section 2, regardless of their bonding partners (e.g., two hydrogen 3.1. Marine oxygen cycling atoms bound to two different carbon atoms). For the general case in which the clumped-isotope distri- The oxidation of two water molecules by plants and bution in polyatomic molecules is inherited partially from algae to form molecular oxygen occurs at the other processes or reservoirs, the Dn value of the molecule oxygen-evolving complex of the photosystem II enzyme. depends on (1) the inherited isotopic composition of the It catalyzes the following four-electron oxidation:

Table 1 Summary of expressions for isotope pairing and heteronuclear isotope clumping for two-isotope systems.

Reaction Dn + 1 (single, infinite atom Dn + 1 (arbitrary atom Section reservoir) reservoirs) hi 2 0 Irreversible bond formation a a 2þ½A ða þa Þ 1 2 ½A 1 2 0R 0R 0R 0R 2 A0E +A0E ? A0–A0 +E +E (two-isotope hi 1 2ð2þ 1þ 2Þ 2.2.1 1 2 1 2 2 ðÞ0R þ0R þ20R 0R 2 a a ½A0a a 1 2 1 2 system) 1þ 2þ2 2 1 hi½A 2 ½A ½A0 a a 2þ ða þa Þþ ða þa Þ 1 2 ½A 1 2 ½A 1 2 0R 0R R R 0R 0R 2 A0E +A0E ? A0–A0 +E +E (three-isotope hi1 2ð2þ 1þ 2þ 1þ 2Þ 2.2.2 1 2 1 2 2 0R 0R 0R 0R R 0R 0R R 2 ½A0 ½A ðÞ1þ 2þ2 1 2þ 1 2þ 1 2 a a a a a a a a system) 1þ 2þ2 ½A 2 1þ ½A ð 1 2þ 1 2 Þ

0 0 0 0 A E1 +BE2 ? A –B +E1 +E2 112.3.1 no 2 Bond formation controlled by transition state ½A0 a a a 00 2þ ða a 0 þa a 0 Þ à 1 2 k ½A 1 k1 2 k2 0R 0R a 0R a 0R a 2 0 0 0 0 ? 0 0 no1 2 k00 ð2þ 1 k10 þ 2 k20 Þ A E1 +AE2 (E1A –A E2) A –A +E1 +E2 2 2 2.2.3 0R a 0R a 0R 0R a ½A0 ðÞ1 k10 þ 2 k20 þ2 1 2 k00 a a 0 þa a 0 þ2 a a ak00 1 k1 2 k2 ½A 1 2 B0 A0 1 Ba ½ a 1 Aa ½ a 0 0 a þ k0 ½B B þ k0 ½A A a 1 Ba B R 1 Aa A R 0 0 à ? 0 0 k00 k00 ðÞþ k0 ðÞþ k0 A E1 +BE2 (E1A–BE2) A –B +E1 +E2 A B A B 2.3.2 ak0 ak0 a 0 a 0 ak0 ak0 a a k00 ½B a k00 ½A a k00 B0 R k00 A0 R 1þAa ½B B 1þBa ½A A 1þAa 1þB a k0 k0 k0 k0 32 L.Y. Yeung / Geochimica et Cosmochimica Acta 172 (2016) 22–38

þ 2H2O þ 4hm ! O2 þ 4H þ 4e ðR11Þ yield insights into the oxygen budget of the deep ocean (Bender, 1990; Levine et al., 2009). The reaction is a multi-step redox mechanism (McEvoy 17 18 and Brudvig, 2006) that does not fractionate Oor O 3.2. Methanogenesis during production of O2 from the water substrate (Guy et al., 1993; Helman et al., 2005). This absence of bulk iso- Methanogenesis from CO and acetate progresses a a 2 topic fractionation suggests that k10 1 and k20 1in through reversible two-electron reductions coupled with A Eq. (33), i.e., photosynthetic O O bond formation has a hydrogen-atom addition (Rouvie`re and Wolfe, 1988). These A loose transition state. If O O bond formation is a mass- hydrogen atoms are correlated only through secondary iso- a a a a dependent process, then k00 k10 k20, resulting in k00/ tope effects, resulting in a strong combinatorial effect when a a ( k10 k20) values that are close to unity as well (Young the reversibility is low and methane production is strongly et al., 2002), and no preference for rare-isotope pairing favored. If H2O ultimately provides all of the hydrogen beyond chance statistics; OAO isotope pairing may there- atoms on methane (Whiticar et al., 1986; Valentine et al., fore be described well by Eqs. (20) and (23). 2004), one might expect the combinatorial effects on 12 Isotope-labeling experiments have shown that water D CH2D2 values to be dominated by the intrinsic deuterium 12 binds to two non-equivalent binding sites on the enzyme kinetic fractionation factors, rendering D CH2D2 values (Hillier and Wydrzynski, 2008; Rapatskiy et al., 2012). insensitive to the isotopic composition of the source water. D Eqs. (20) and (23) therefore predict that 35 < 0 and In contrast, if one or more of the hydrogen atoms on D 36 < 0 for the evolved O2. Yeung et al. (2015) presented methane comes from a different source, as may be the case data from a closed terrarium experiment utilizing water for acetoclastic methanogenesis (CH4 production from acet- hyacinths (Eichhorniae crassipes on 12 h/12 h light/dark ate substrates), then both enzymatic and inherited fractiona- D D 12 cycles) that showed 35 and 36 values evolving toward tion factors will influence D CH2D2 values of the methane a steady-state at the stochastic distribution of isotopes evolved. Reversibility in methane production may also drive D D 12 ( 35 = 0 and 36 = 0). They observed that dark incuba- D CH2D2 values toward equilibrium values, which are D D tions increased 35 and 36 values and consequently greater than zero (Cao and Liu, 2012; Stolper et al., 2014a, inferred that photosynthesis must generate O2 with b, 2015; Webb and Miller, 2014; Wang et al., 2015). D D 13 35 < 0 and 36 < 0 to compensate. Whether this The CH3D clumped-isotope composition of methane 13 clumped-isotope deficit is purely combinatorial in origin (D CH3D) has been applied to distinguish thermogenic or also influenced by subtle kinetic isotope effects remains and biogenic sources of methane (Stolper et al., 2014a, to be tested; combinatorial effects are unlikely to yield O2 2015; Wang et al., 2015). It can express both equilibrium D ‰ D ‰ with 35 < 0.5 and 36 < 1 , however, because of and non-equilibrium signatures, depending on hydrogen the typical magnitude of oxygen-isotope effects associated availability; H2-replete conditions seem to promote kineti- with enzyme-substrate binding [i.e., a = 0.97 1.03; cally constrained reactions, resulting in methane formed 13 (Angeles-Boza and Roth, 2012; Angeles-Boza et al., with D CH3D values as low as 5‰ (Stolper et al., 2014)]. 2015; Wang et al., 2015). The equations derived in Sec- While the clumped-isotope signatures of photosynthesis tion 2.3 suggest that the source of these large anticlumped 13 have not yet been determined unequivocally (values as low CH3D signatures is unlikely to be combinatorial; in D ‰ D ‰ as 35 = 5 and 36 = 10 cannot yet be ruled out), clumps of two dissimilar atomic species, anticlumped iso- the clumped-isotope composition of O2 can nonetheless tope distributions arise only from kinetic isotope effects. 13 12 be further developed as a tracer of marine oxygen cycling. The D CH3D and D CH2D2 values in methane can A clumped-isotope approach would complement oxygen- therefore complement each other in biogeochemical studies. isotope methods for determining primary productivity Clumped-isotope disequilibria due to kinetic effects will be 13 12 (Bender and Grande, 1987; Quay et al., 1993; Luz and correlated on a plot of D CH3D vs. D CH2D2 (Young D Barkan, 2000; Juranek and Quay, 2013) because 35 and et al., 2011), while D–D combinatorial effects will only D 12 36, unlike other oxygen-isotope tracers, would be insensi- affect D CH2D2 and other multiply-deuterated methanes. tive to the isotopic composition of the photosynthetic Using Da values between 0.9 and 1.1, Eq. (13) suggests that source water. They would allow one to solve for the end- the magnitude of combinatorial effects can be as large as 12 member isotopic composition of photosynthetic O2 in dif- 10‰ in D CH2D2, comparable in magnitude to 12 ferent environments, which can be uncertain (Eisenstadt D CH2D2 values expected at isotopic equilibrium (Young et al., 2010; Kaiser, 2011; Luz and Barkan, 2011). Further- et al., 2011). Combinatorial effects may therefore be impor- more, it may provide more constraints on the fate of pho- tant to the relative abundances of multiply-deuterated tosynthetically produced oxygen (Kaiser, 2011; Nicholson methanes under kinetically controlled conditions. Applica- D et al., 2014). Using an endmember composition of 36,p = tion of such a multiple-isotopologue technique may help ‰ 0.4 – 10 for photosynthetic O2, a 10% mixture of pho- disentangle the ubiquitous chemical and reservoir effects tosynthetic O2 with atmospheric O2 in the upper ocean relevant to biogeochemical methane cycling. [D36,air =2‰ (Yeung et al., 2014)] would yield signals between 0.2‰ and 1‰ in magnitude, within detection limits 3.3. Biological nitrogen fixation of current methods (Yeung et al., 2014). Respiration and gas exchange can cause additional isotopologue fractiona- Molecular hydrogen is a byproduct of biological nitro- tion, however. Future development of this approach may gen fixation (Hadfield and Bulen, 1969; Scranton et al., L.Y. Yeung / Geochimica et Cosmochimica Acta 172 (2016) 22–38 33

1987). In the molybdenum-containing variant of nitroge- 3.4. Conversion of fixed nitrogen to N2 and N2O nase, hydrogen is believed to be evolved from hydrides (H) bound to the catalytic iron-molybdenum cofactor Nitrogen-nitrogen bonds are formed during denitrifica- (FeMo-co) after N2 binds to one of the Fe metal centers tion, nitrification, and anaerobic ammonia oxidation of the cluster (Hoffman et al., 2014). While the reaction is (anammox). During biological denitrification, nitrate part of a multi-step sequence, its essential elements with (NO3 ) is reduced to dinitrogen (N2) through a series of respect to isotope pairing are: intermediate steps involving nitrite (NO2 ), nitric oxide (NO), and nitrous oxide (N2O), i.e., H –½FeMo-co–H þ N2 H2 þ½FeMo-co–N2 ðR12Þ NO ! NO ! NO ! N O ! N ðR13Þ Hydrogen evolution is reversible (Yang et al., 2013; 3 2 2 2 + Lukoyanov et al., 2015), but both combinatorial and The reduction 2 NO + 2H +2e ? N2O+H2O is the chemical-kinetic factors could be important for the relevant reaction that pairs nitrogen atoms; it sets the rela- 15 15 clumped-isotope distribution of H2. In particular, variabil- tive abundance of NA N bonds (i.e., the D30 value) in ity in H2 partial pressure may modulate the reversibility of N2O, which is then inherited by N2 (Ostrom et al., 2007). reaction R12 and its attendant isotopic fractionation, such The reaction is catalyzed by the enzyme nitric oxide reduc- as in soils where partial pressures of H2 can be higher than tase (NOR), which has different variants in bacteria and those in the free atmosphere (Conrad and Seiler, 1979). fungi. Each variant utilizes a different mechanism (Fig. 5), Consequently, clumped-isotope compositions of H2 evolved which could impart its signature on the D30 values of from high- and low-H2 environments may differ. In addi- N2O and/or N2 produced (Shiro et al., 1995, 2012; tion, hydrogenase and nitrogenase enzymes can catalyze Riplinger and Neese, 2011; Blomberg and Siegbahn, 2012). isotope exchange between H2 and H2O(Jackson et al., These D30 values and the intramolecular distribution of 15 1968; Whiticar et al., 1986; Valentine et al., 2004; Vignais, NinN2O(Toyoda et al., 2005) share a common origin, 2005). If these isotope-exchange reactions are important, i.e., enzymatic and chemical fractionation factors for then combinatorial signatures will be negligible compared NAN bond formation, so the theory and observations 15 to those caused by equilibrium and kinetic processes [e.g., regarding the site preference of NinN2O are also ger- D ‰ ° 4 200 at 25 C equilibrium; (Richet et al., 1977; mane to D30 values in N2O and N2. I will examine them Yang et al., 2012)]. If isotope exchange is rapid and com- below. plete, then H2 from nitrogen fixation may express a A variety of denitrifiers evolve N2O with asymmetric 15 15 15 clumped-isotope composition more consistent with H2 for- N site preferences (SP = d Ncentral d Nterminal). For mation temperatures and the environments where nitrogen bacterial denitrification, measurements of pure cultures is fixed. range from SP = 10‰ to 0‰ (Toyoda et al., 2005; Nevertheless, the marine nitrogen budget remains uncer- Frame and Casciotti, 2010; Mothet et al., 2013; Yamazaki tain (Mahaffey et al., 2005; Deutsch et al., 2007; Galloway et al., 2014). Experiments on fungal denitrification show a et al., 2008; Eugster and Gruber, 2012) in part because past larger and more consistent site preference [e.g., measurements of marine nitrogen fixation may have be SP = 37‰;(Sutka et al., 2008)], although its magnitude is inaccurate (Grosskopf et al., 2012); additional in situ esti- species- and culture-dependent (Rohe et al., 2014). Nitrifi- mates would be useful. Observations of H2 supersaturation cation by ammonia- and methane-oxidizing bacteria and in the mixed layer (Moore et al., 2009, 2014; Wilson et al., archaea can also produce N2O through hydroxylamine 2013) suggest that the isotopic composition of H2 may (NH2OH) and NO intermediates. The NH2OH + NO path- record information about nitrogen fixation. way [‘‘nitrifier-denitrification” (Ritchie and Nicholas,

15 15 Fig. 5. Candidate mechanisms for N2O production by bacterial and fungal NOR, which likely lead to characteristic NA N clumping in the products. The bacterial NOR mechanism was proposed by Blomberg and Siegbahn (2012) while the fungal (P450nor) mechanism was proposed by Riplinger and Neese (2011). The iron atoms represent two sites on the enzyme that are involved in NAN bond formation. Atoms that comprise the product N2O are highlighted in red. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 34 L.Y. Yeung / Geochimica et Cosmochimica Acta 172 (2016) 22–38

1972)] results in a 15N site preference similar to that of fun- Seitzinger, 2006), and clumped-isotope measurements may gal denitrification (Sutka et al., 2003; Santoro et al., 2011; yield new insights. The magnitude of expected equilibrium Jung et al., 2014; Yamazaki et al., 2014), while the NO and combinatorial clumped-isotope signatures is similar + NO pathway results in a 15N site preference similar to [of order ±1‰;(Wang et al., 2004)], and kinetic isotope that for bacterial denitrification (Sutka et al., 2008; Frame effects may be larger in magnitude, so new constraints on and Casciotti, 2010). nitrogen cycling may emerge if D30 values can be measured These site preferences are mechanism-specific, but they with analytical precision of ±0.1‰. can be evaluated within the framework of reaction R5 in Section 2.2.3. In general, D30 values in the N2O and N2 4. CONCLUSIONS products are a convolution of site-specific 15N/14N contrast and kinetic isotope effects for NAN bond formation. Using Clumped-isotope distributions are sensitive to physical 15N fractionation factors of order 0.99–1.04 [e.g., (Frame (e.g., diffusive), chemical (e.g., transition-state), and combi- and Casciotti, 2010; Yamazaki et al., 2014; Yang et al., natorial isotope effects. Chemical and combinatorial effects, 2014)], combinatorial clumped-isotope effects are predicted in particular, are intertwined during bond formation: 15 15 D to deplete NA N isotopomers of N2O and N2 by up to Combinatorial effects tend to decrease n values of the 0.6‰ relative to the kinetic isotope effects associated with products in isotope pairing reactions. These effects arise 15 15 NA N pairing. Future measurements of D30 values will from unequal isotopic preferences during substrate selec- 15 reveal whether the N site preference in N2O is caused tion. A secondary combinatorial effect is related to rare- by binding interactions between NO and NOR, kinetic iso- isotope abundance: The sequestration of rare isotopes into tope effects in NAN bond formation, or both. multiply-substituted moieties reduces the sensitivity of the Anaerobic ammonium oxidation is a third important clumped-isotope distribution to both chemical and combi- pathway through which NAN bonds are formed in nature. natorial factors. Explicit accounting of these combinatorial Unlike the previous mechanisms, the anammox pathway effects is required for high-precision comparison of pre- + combines NO and ammonium (NH4 ) to form hydrazine dicted clumped-isotope signatures (e.g., from ab initio cal- (N2H4) before being oxidized to N2 (Kartal et al., 2011): culations) and those observed in nature. þ Furthermore, I find that recent observations of non- NO ! NO þ NH ! N2H4 ! N2 ðR14Þ 2 4 equilibrium clumped-isotope signatures in CH4 and O2 Little is known about the enzymatic steps leading to (Stolper et al., 2014a, 2015; Wang et al., 2015; Yeung N2H4 formation, but the two different nitrogen pools for et al., 2015) originate from two different phenomena. + 15 13 NO2 and NH4 have different d N compositions. In Anticlumped distributions of CH3DinCH4 are minimally oxygen-deficient areas where anammox is particularly affected by purely combinatorial clumped-isotope effects; + active, NO2 and NH4 could differ in composition by many chemical-kinetic isotope effects control the number of 13 tens of per mil, because enzyme-catalyzed nitrogen-isotope C–D isotope clumps relative to the stochastic distribu- 15 exchange may deplete d NinNO2 by as much as 60‰ tion. In contrast, photosynthetic O2 is most likely affected relative to NO3 (Brunner et al., 2013). Furthermore, exper- by combinatorial clumped-isotope effects, resulting in anti- 18 18 17 18 imental cultures of anammox bacteria were observed to clumped O O and O O distributions. Mechanisms of + A have isotopic fractionation factors for NO2 and NH4 N N bond formation relevant to the global nitrogen cycle assimilation that differed by 10‰ (Brunner et al., 2013). suggest that clumped-isotope distributions in N2O and N2 Therefore, combinatorial depletions in D30 are expected will carry diverse biochemical signatures. While not for anammox-derived N2: Using a 10–60‰ difference in discussed in detail here, non-equilibrium distributions of 15 + 15 15 13 13 d N between the NO2 and NH4 in R14, Eq. (33) predicts D–D, N– N, and C– C isotope pairs in biogenic that combinatorial effects will reduce the D30 value of N2 by organic molecules could conceivably be explored as 0.02–0.9‰ relative to chemical-kinetic preferences. The process-specific isotope signatures in the fossil record, pro- precise D30 value relative to that produced by other N2 vided the molecules are produced from kinetically con- sources is difficult to estimate, however, because of a pau- trolled biosynthetic pathways and the moieties of interest 15 + city of open-ocean d N–NH4 measurements (Altabet, are resistant to diagenetic alteration. 2006), nutrient cycling mechanisms in unique microenvi- Finally, combinatorial effects in isotope pairing are not ronments (Prokopenko et al., 2013), and poorly known unique to bond-forming chemistry. While bond-rupturing enzyme-15N fractionation factors. chemistry and bond-preserving (e.g., crystallization) chem- In all of these mechanisms for nitrogen loss, molecular istry cannot distinguish symmetrically equivalent molecular details about both the NAN bond-forming step and the isotopomers, they may express preferences for position- substrate-determining steps will likely influence D30 values specific isotopic substitutions within larger, less symmetric in N2O and N2 products. Clumped-isotope measurements molecules. of natural N2O and N2 may clarify the role of bacterial ACKNOWLEDGEMENTS and fungal nitrogen-loss mechanisms in soils (Seitzinger et al., 2006; Galloway et al., 2008), as well as the role of I thank Jeanine Ash, Edwin Schauble, and Ed Young for stim- anammox in the global nitrogen cycle (Galloway et al., ulating discussions that led to this manuscript being conceived, 2004; Kuypers et al., 2005; Gruber and Galloway, 2008). Shuning Li and Eric Hegg for comments on the manuscript, and Quantification of these nitrogen-loss mechanisms has been Ed Young for coming up with the term ‘‘combinatorial” to describe challenging (Risgaard-Petersen et al., 2003; Davidson and the clumped-isotope effects described herein. I also thank Shuhei L.Y. Yeung / Geochimica et Cosmochimica Acta 172 (2016) 22–38 35

Ono and two anonymous reviewers for their detailed comments. Chakraborty S., Ahmed M., Jackson T. L. and Thiemens M. H. This research was supported in part by the National Science Foun- (2008) Experimental test of self-shielding in vacuum ultraviolet dation Low-Temperature Geochemistry program (EAR-1349182). photodissociation of CO. Science 321, 1328–1331.

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