Abstract
High precision measurements of molecules containing more than one heavy isotope may provide novel constraints on element cycles in nature. These socalled clumped isotope signatures are reported relative to the random (stochastic) distribution of heavy isotopes over all available isotopocules of a molecule, which is the conventional reference. When multiple indistinguishable atoms of the same element are present in a molecule, this reference is calculated from the bulk (≈average) isotopic composition of the involved atoms. We show here that this referencing convention leads to apparent negative clumped isotope anomalies (anticlumping) when the indistinguishable atoms originate from isotopically different populations. Such statistical clumped isotope anomalies must occur in any system where two or more indistinguishable atoms of the same element, but with different isotopic composition, combine in a molecule. The size of the anticlumping signal is closely related to the difference of the initial isotope ratios of the indistinguishable atoms that have combined. Therefore, a measured statistical clumped isotope anomaly, relative to an expected (e.g. thermodynamical) clumped isotope composition, may allow assessment of the heterogeneity of the isotopic pools of atoms that are the substrate for formation of molecules.
Introduction
Analysis of the isotopic composition of molecules is one of the key tools for studying element cycles on earth. For the light elements H, C, N and O with relatively small heavytolight isotope ratios at natural abundance, the standard analytical instruments have largely limited isotope analysis to singlesubstituted isotopocules (isotopically substituted molecules). Studies of multiply substituted isotopocules, referred to as clumped isotopes, were only occasionally carried out, often using isotopically enriched substrates or labeling experiments^{1,2,3,4,5}. Recent analytical advancements using isotope ratio mass spectrometry^{6,7,8} or laser spectroscopy^{9} have enabled high precision measurements of clumped isotopes in several molecules such as CO_{2}, CH_{4}, O_{2} and N_{2}O^{8,10,11,12,13,14,15} and the field is rapidly expanding.
Since multiply substituted isotopocules are thermodynamically more stable than single substituted ones, classical isotope theory predicts small but measurable positive clumped isotope anomalies for most molecules under natural conditions^{16,17,18,19}. These clumped isotope signatures depend on temperature, which is the basis of the new field of clumped isotope thermometry^{20}.
Yeung et al.^{14} and Wang et al.^{13} reported negative heavy isotope clumping in photosynthetic O_{2} formation and in biogenic CH_{4}, respectively. Yeung et al.^{14} attributed the negative Δ values (see equation 3 for definition) in photosynthetic O_{2} to different isotopic composition of the two O atoms originating from different sites in the oxygen evolving complex of photosystem II. Triggered by this observation we investigated this further and show here that negative clumping anomalies are necessarily expected whenever two or more indistinguishable atoms of the same element but with different isotopic composition combine in a molecule. The atoms do not need to share a common bond but can be at distant places in a molecule.
Yeung^{21} recently presented an analysis of such apparent statistical clumped isotope effects in combination with other isotope effects. In our paper, we restrict the analysis to statistical clumped isotope effects and phrase the calculations exclusively in terms of isotope ratios in order to elucidate the underlying general nature of these apparent isotope signatures. The fundamental origin of the apparent statistical clumped isotope effect is thoroughly presented and visualized geometrically. We then provide a general mathematical formalism for apparent statistical clumped isotope signatures in any multiple isotope system. Finally we demonstrate quantitatively how a certain measured statistical anticlumping signal can be used to determine the isotopic heterogeneity of indistinguishable atoms in a molecule.
Origin of the statistical negative clumped isotope signatures
We describe and calculate the statistical clumped isotope effects in terms of heavytolight isotope ratios ^{i}R of individual atoms and molecules, where the index i indicates the mass of the atom or molecule. The same letter R is used for both atomic isotope ratios and molecular isotopocule ratios. In particular, for molecules with multiple heavy isotopes (clumped isotopes), the heavyto light isotopocule ratio is defined as
When two atoms with atomic heavytolight isotope ratios R_{1} and R_{2} (where R_{i} can be, e.g., ^{2}H/^{1}H, ^{13}C/^{12}C, ^{18}O/^{16}O, etc.) combine in a molecule in a purely random manner, i.e., without any isotope effect, the ratio of molecules that include the heavy isotopes of both of these atoms relative to molecules including only light isotopes is simply the product of the atomic isotope ratios of the two atoms
Clumped isotope signatures Δ_{i} are then by convention calculated as the relative difference between a certain (measured) clumped isotopocule ratio and the random clumped isotope ratio, usually reported in per mill (‰).
The apparent negative statistical clumped isotope signatures that we describe in this paper are fundamentally related to this referencing convention, in particular the choice of the reference ratio ^{i}R_{cl,random} that is required to calculate Δ_{i} in Eq. 3. When the isotope ratios R_{1} and R_{2} of the involved atoms are individually known, ^{i}R_{cl,random} = R_{1} · R_{2} can be precisely calculated. This is always the case when heavy isotopes of different elements clump together (e.g. ^{13}C and ^{18}O in CO or in CO_{2}). It also holds when molecules of the same element, for which the individual isotope ratios are known, clump together (e.g. ^{15}N^{α} and ^{15}N^{β} in N_{2}O, where α and β indicate the central and terminal position of the N atom in the linear NNO molecule, which can be determined independently^{22,23}). This is graphically illustrated in Fig. 1, where R_{1} and R_{2} are plotted on the x and yaxis and their product, ^{i}R_{cl,random}, is shown as the blue area.
When a molecule contains indistinguishable atoms of the same element, it is impossible to determine the individual atomic isotope ratios of these atoms. Nevertheless, the atoms may originate from isotopically distinct populations with isotope ratios R_{1} and R_{2}. To calculate the correct value of ^{i}R_{cl,random} we would therefore need to know the individual isotope ratios R_{1} and R_{2}. However, since these isotope ratios cannot be retrieved for indistinguishable atoms, it is common (and reasonable) to assign the bulk (≈average, see below) isotopic composition of the atoms. Through this choice, the real stochastic clumped isotope ratio R_{1} ⋅ R_{2} (blue area in Fig. 1) is substituted by the approximated value R_{av} ⋅ R_{av} (red area in Fig. 1). Both areas have the same perimeter 2 ⋅ (R_{1} + R_{2}) =2 ⋅ (R_{av} + R_{av}), but the red square has a larger area than the blue rectangle, i.e., R_{av} ⋅ R_{av} > R_{1} ⋅ R_{2}. Replacing the area of the blue rectangle by the one of the red square in the denominator of Eq. 3 causes a systematic negative artifact. This produces the apparent negative statistical clumped isotope effect. The anticlumping signal is the larger the more different the individual isotope ratios of the indistinguishable atoms are. In the following chapters we derive the general mathematical formalism to calculate these apparent statistical clumped isotope effects in any multiisotope system. We also show that the measurement of statistical anticlumping in principle allows quantifying the heterogeneity of the isotopic composition of indistinguishable atoms in a molecule, i.e. to reconstructing the blue area from the red area in Fig. 1.
We emphasize that the apparent anticlumping signature is not related to a physical isotope effect, but is a mathematical artifact that originates from the referencing convention. It will never occur when the contributing atoms are distinguishable (thus never for atoms from different elements, e.g. for ^{13}C^{18}O clumping in CO_{2}), but it will always occur when two indistinguishable atoms of the same element combine in a molecule (e.g. for ^{18}O^{18}O clumping in CO_{2}). Table 1 shows a selection of common atmospheric molecules and specific clumping signatures for which statistical anticlumping will occur or not occur, respectively.
Clumping of indistinguishable atoms in one molecule
Molecules with two atoms of the same element (e.g. N_{2}, O_{2}, H_{2})
As mentioned above, two indistinguishable atoms of the same element in one molecule may generally originate from different reservoirs or involve different fractionation effects such that their isotope ratios R_{1} and R_{2} represented two distinct pools when the molecule formed. Since the two atoms are now indistinguishable, we cannot independently measure the isotope ratios R_{1} and R_{2}. In fact, in conventional isotope ratio measurements of single substituted isotopocules the arithmetic average ratio of the two ratios (e.g. ^{29}R = 2 ^{15}R_{av} for ^{15}N measurements in N_{2}) is determined. For rare heavy isotopes (R_{1}, R_{2} ≪1), this average ratio ^{15}R_{av} is generally similar to the bulk isotope ratio R_{bulk} of the sample (see Supplementary Information). In this case, it is common and reasonable to assign R_{bulk} ≈ R_{av} to each of the indistinguishable atoms for further calculations. For the remainder of this paper, we use R_{bulk} = R_{av}, which considerably simplifies the formulas and removes the dependency of the apparent clumped isotope signal on the isotope ratio. The differences between using R_{bulk} and R_{av} are discussed in detail in the Supplementary Information.
The stochastically expected (random) ratio of isotopocules with two heavy atoms relative to the light isotopocules, from a population of atoms with average heavy isotope ratio of , is
However, when the atoms represent different isotopic pools with possibly different isotope ratios, the real clumped isotope ratio, R_{cl}, of doubly substituted isotopocules relative to the light isotopocules is
The apparent statistical clumped isotope Δ is the relative difference between the real and the stochastically expected clumped isotope ratio
The clumped isotope composition is always negative, except for the case R_{1}= R_{2}, for which Δ = 0. Thus, when two atoms of the same element with different isotopic composition combine in a molecule, the resulting molecule will always have an apparent negative clumping signature. The black curve in Fig. 2a shows the size of this quadratic statistical negative isotope clumping according to Eq. (6). The negative clumping signal Δ does not depend on the absolute value of the underlying isotope ratios, but only on the relative difference of the isotope ratios. In the following we refer to this effect as “statistical clumped isotope signature”. Eq. 6 was first derived for the case of formation of molecular O_{2} in photosynthesis by Yeung et al.^{14}, who indeed observed negative clumped isotope signals relative to the thermodynamically expected values for photosynthetic O_{2}.
Generalization: Molecules with three or more atoms of the same element – complete substitution
When a molecule contains three or more atoms of the same element, these atoms can generally represent populations with different isotope ratios R_{1}, R_{2}, … R_{n}. We first consider the case of full heavyisotope substitution, which is the generalization of the twoatom case presented above. As it is not possible to independently measure the individual isotope ratios R_{i} of the indistinguishable atoms, the arithmetic average isotope ratio ^{18}R_{av}(≈^{18}R_{bulk}, see Supplementary Information) is assigned to each of the indistinguishable atoms for further calculations
As the atoms are all assigned the same atomic isotope ratio R_{av}, the stochastically expected ratio of fully substituted isotopocules relative to nonsubstituted isotopocules from this population of atoms is the nth power of R_{av}.
This is the ndimensional equivalent of replacing blue rectangle in Fig. 1 by the red square. The real (=observed) ratio of fullysubstituted isotopocules is the product of all isotope ratios involved, which is identical to the nth power of the geometric mean of the isotope ratios
Thus, the statistical clumped isotope signature for fully substituted isotopocules is
This equation applies to any set of indistinguishable atoms in a molecule. Since the arithmetic mean is always larger or equal than the geometric mean, the statistical clumped isotope signal is always negative, except for the case where all ratios R_{i} are identical, in which case the arithmetic and geometric means are equal and thus Δ = 0. Figure 2 shows the variation of Δ with the relative difference of the isotope ratios in the 2, 3, 4 and 10atom systems. For the cases illustrated in Fig. 2a, only one isotope ratio is varied and the isotope ratios of all other atoms are kept constant and identical. For the same relative difference in isotope ratio of a single atom, the clumping signal decreases with increasing number of atoms. Although the 10atom case may not be of much practical use, it is included to emphasize the point that the statistical negative heavy isotope clumping does not require the heavy isotopes to be linked directly by a common chemical bond. For example, statistical negative DD clumping in ethane (C_{2}H_{6}, Table 1) may involve pairs of hydrogen atoms at any of the 6 positions.
The isotope combinations presented in Fig. 2a all include the point where all isotope ratios are equal, which corresponds to Δ = 0‰. However, in general, the isotope ratios of the atoms at different positions are not equal. Figures 2b,c show the clumping signal for the 3 and 4atom cases when one ratio is varied again and the other ratios are held constant, but at different values for the individual atoms. Now the situation that all isotope ratios are equal cannot occur and the parabolashaped curves are shifted towards negative Δ values. The yaxis offset increases with increasing difference of the individual isotope ratios, thus with the heterogeneity of the isotopic composition of the individual indistinguishable atoms. The curves in Fig. 2b,c are selected 2dimensional crosssections of a multidimensional space, which illustrate the effect of varying one of the multiple isotope ratios relative to a fixed set of other ratios. In practice, a certain combination of isotope ratios among indistinguishable atoms will correspond to one single value of Δ and we will show below that the statistical clumped isotope signal Δ is a measure for the heterogeneity of the isotopic composition of indistinguishable atoms in a molecule.
Molecules with three or more atoms of the same element – incomplete substitution
Clumping of two heavy isotopes in molecules with three indistinguishable atoms (e.g. ^{18}O^{18}O or ^{17}O^{17}O clumping in O_{3} or NO_{3})
As a first example we consider molecules with three indistinguishable atoms and calculate the clumping signature of double substituted isotopocules. The three atoms generally represent three isotopically different populations with isotope ratios R_{1}, R_{2} and R_{3}. However, as the atoms are indistinguishable, the individual ratios cannot be determined and for further calculations they are assigned the average isotope ratio, which can be determined from measurement of the single substituted isotopocules
The stochastically expected isotope ratio of isotopocules with exactly two out of three possible heavy isotopes relative to the light isotopocules from a population of indistinguishable atoms with assigned isotope ratio R_{av} is
The factor 3 gives the number of possible permutations of two heavy isotopes over three atom positions. The real probability for finding a molecule with exactly two heavy atoms from the three atoms with isotope ratios R_{1}, R_{2} and R_{3} is
Thus, the statistical clumped isotope signal for clumping of two out of 3 possible heavy isotopes, Δ_{2/3}, is
Since all terms in the numerator and denominator are squares and thus positive, Δ is always negative except for the case R_{1} = R_{2} = R_{3} when Δ_{2/3} = 0. Some examples for the clumping of two out of three heavy isotopes in a molecule are shown in Fig. 3a. The solid line again includes the case where all ratios are identical and Δ = 0. The dashed and dotted lines show examples where one isotope ratio varies and the other ones are constant but different, so that the case that all are equal does not occur. Again, the dashed and dotted curves are shifted to more negative Δ values.
Clumping of two heavy isotopes in molecules with four indistinguishable atoms (e.g. DD clumping in CH_{4})
We now consider molecules with four indistinguishable atoms and calculate the clumping signature of doublysubstituted isotopocules. The four atoms generally represent isotopically distinct pools with isotope ratios R_{1}, R_{2}, R_{3} and R_{4}. As the ratios cannot be determined individually, they are assigned the average atomic isotope ratio
The stochastically expected probability for forming a molecule with exactly two out of four possible heavy isotopes from a population of indistinguishable atoms with assigned isotope ratio R_{av} is
The factor 6 again gives the number of possible permutations of 2 heavy isotopes over 4 atom positions . However, the real probability to form an isotopocule with exactly two heavy atoms from the four atoms with different isotope ratios R_{1}, R_{2}, R_{3} and R_{4} is
Thus, the clumped isotope signal Δ is
Since Δ can again be expressed as a negative sum of squares, it is always negative, except for R_{1} = R_{2} = R_{3} = R_{4} where Δ_{2/4} = 0. Some examples for the statistical clumped isotope effect of two out of four heavy isotopes in a molecule are shown in Fig. 3b. An important example for this case is the DD clumping in methane. The reservoirs that supply the different hydrogen atoms in the formation of methane can vary considerably and significant apparent statistical anticlumping is expected.
Generalization: Clumping of m heavy isotopes in molecules with n indistinguishable atoms
For the general case of n indistinguishable atoms that represent isotopic pools with isotope ratios R_{1}, R_{2}, … R_{n}, we assign again the arithmetic mean isotope ratio R_{av} to each of the atoms.
The stochastically expected ratio of isotopocules with exactly m out of a possible n heavy atoms relative to the light isotopocules from a population of indistinguishable atoms with this average heavy isotope ratio R_{av} is
The real ratio of isotopocules with m heavy isotopes relative to nonsubstituted isotopocules (R_{cl}) from n atoms with isotope ratios R_{1}, R_{2}, … R_{n} is
Thus, the clumped isotope signal Δ is
Eq. 22 is the most general equation to calculate statistical negative isotope clumping in any multiisotope system. In the case where all ratios R_{i} are identical, R_{ji} = R_{av}, the numerator becomes identical to the denominator because the number of possible subsets {j_{1}, … j_{m}} out of {1, … n} is equal to the binomial coefficient and the products reduce to the factors (R_{av})^{m}. In this case, Δ = 0.
More than 2 stable isotopes: ^{17}O^{18}O clumping for oxygen
So far we have formally only treated molecules with two stable isotopes. Oxygen has three stable isotopes ^{16}O, ^{17}O and ^{18}O. Clumping of heavy isotopes of the same sort (i.e., clumping of multiple ^{17}O atoms or multiple ^{18}O atoms in one molecule) follows the examples and general rules outlined above (see Table 1). However, it is also possible that the heavy ^{17}O and ^{18}O atoms clump together in one molecule.
O_{2} and other molecules with two indistinguishable O atoms
In the case of molecular O_{2}, the apparent statistical clumping signal for ^{17}O^{18}O clumping was derived in Yeung et al.^{14}. Since the isotope ratios of the individual O atoms cannot be determined individually, they are assigned the average heavy isotope ratios
The stochastically expected (random) ratio of molecules with one ^{17}O and one ^{18}O isotope relative to ^{16}O^{16}O from this population of O atoms is
The real probability for forming ^{17}O^{18}O molecules from two atoms with isotope ratio ^{17}R_{1}, ^{18}R_{1} and ^{17}R_{2,} ^{18}R_{2} is
Therefore,
For normal mass dependent fractionation, where ref. 24, with threeisotope exponent β ≈ 0.53 ref. 25, ^{17}R_{1} and ^{18}R_{1} are either both smaller or both larger than ^{17}R_{2} and ^{18}R_{2}, thus again the clumped isotope signature ^{35}Δ is always <0 ( = 0 if ^{i}R_{1} and ^{i}R_{2} are identical). This can also be shown by inserting the mass dependent fractionation relation as follows:
As the atoms do not need to share the same bond to generate statistical isotope clumping, the equation also applies to ^{17}O^{18}O clumping of other molecules with 2 O atoms (importantly ^{17}O^{18}O clumping in CO_{2}). The resulting negative clumping signature is shown in Fig. 4.
Heavy isotope clumping for three indistinguishable oxygen atoms (e.g. O_{3})
When a molecule has three indistinguishable (or not distinguished) oxygen atoms, we assign the average ratios and to each of the atoms. The statistical clumping signatures for complete ^{17}O or ^{18}O substituted isotopocules and for ^{17}O^{17}O and ^{18}O^{18}O clumping can be derived according to the formalism derived above. Here we also consider the remaining combinations where ^{17}O and ^{18}O clump together in one molecule.
Since there are 3! = 6 different possibilities to distribute the three distinguishable atoms (^{16}O, ^{17}O and ^{18}O) over the three positions, the stochastically expected clumping is
The real clumping is calculated by considering explicitly all 6 configurations of ^{16}O, ^{17}O and ^{18}O
and thus
As argued above for O_{2}, for normal mass dependent fractionation, ^{17}R_{i} and ^{18}R_{i} are either both smaller or both larger than ^{17}R_{j} and ^{18}R_{j}, thus Δ is always < 0 (=0 if all ratios are identical). Inserting the mass dependent fractionation relation again, Eq. 30 can be transformed to
The corresponding values of Δ are shown in Fig. 5 together with all other possible isotope clumping combinations for ^{17}O and ^{18}O isotopes in O_{3}. These are always calculated as the relative difference of the explicitly calculated clumped isotope ratio ^{i}R_{cl} from the stochastically expected heavy isotope ratio ^{i}R_{cl,random} (Equation 30). For example, for ^{17}O^{17}O^{18}O clumping there are three possible configurations to distribute the atoms (^{17}O, ^{17}O and ^{18}O) over the three positions, so the stochastically expected clumping is
The real clumping is calculated by considering explicitly all 3 configurations
and thus
The corresponding equation for ^{17}O^{18}O^{18}O clumping is easily derived by exchanging ^{17}O and ^{18}O.
Important examples of molecules with three oxygen atoms are O_{3} and the nitrate ion NO_{3}^{−}. For O_{3}, the central and terminal O atoms can be distinguished with suitable techniques^{2,26,27,28,29}. In these cases, statistical anticlumping only occurs when the indistinguishable terminal O atoms are involved (Table 1). In many cases, however, the isotopic composition of O_{3} is determined without position information^{30,31,32,33} and in this case the atoms can be treated as effectively indistinguishable and need to be treated according to the formalism developed here.
Statistical clumped isotope signals and the heterogeneity of the isotope ratios of indistinguishable atoms
The derivations above show that statistical combination of indistinguishable atoms in a molecule leads to apparent negative clumped isotope signals. The size of the apparent negative clumping in the molecule increases with increasing difference in isotopic composition between the individual atoms, so it may actually contain scientifically relevant information. In order to investigate this further, we created random (within a certain range) sets of isotope ratios for all multiisotope systems with up to 5 indistinguishable atoms and calculated the apparent negative multiisotope clumping signature Δ. Figure 6 shows that for each multiisotope set all these random sets of isotope ratios yield Δ values that fall on distinct curves when Δ is plotted versus the relative standard deviation of the individual isotope ratios. For each of the multiisotope clumping combinations with m out of n heavy isotopes, the curves can be parameterized as:
Note that the Δ values in Fig. 6 are plotted as ‰, so the factor 1/2 in Eq. 35 corresponds to 500‰ in the fit curves.
In the case of m = 2 (2 heavy isotopes in any multiisotope system) the fit is perfect, but for more heavy atoms clumping in one molecule there is some scatter around these fit lines. This originates from the fact that there is no fixed analytical relation between the arithmetic and geometric means. Figure 7 shows the relative deviation of the explicitly calculated Δ values and the approximation using Eq. 35 for the randomly chosen sets of isotope ratios with known standard deviation and average values. The outer envelopes of the point clouds for each multiisotope system define the error with which the statistical clumped isotope signal Δ in a molecule can be predicted from Eq. 35 when the standard deviation and the mean of the individual isotope ratios are known.
Scientifically, the opposite relation
is more attractive, since the isotopic variability among indistinguishable atoms in a molecule is usually not known, but Δ may be measurable^{14}. This means that measurement of the statistical clumped isotope anomaly Δ could provide a novel tracer to determine the heterogeneity (quantified by the standard deviation) of the isotopic pools of indistinguishable atoms in a molecule. For example, in the 2isotope system O_{2}, a Δ value of 1.5‰ below the thermodynamically expected value as measured by Yeung et al.^{14} would indicate a relative standard deviation in the isotope ratios of the two O atoms of about 5.5% (Fig. 6, blue curve).
Figure 8 shows the relative error that is made when Eq. 36 is used to calculate the relative standard deviation of the isotope ratios from the Δ value for randomly chosen sets of (known) isotope ratios. The outer envelope for each isotope system quantifies the error with which for a group of indistinguishable atoms can be derived from Δ. Above Δ values of −1.5‰ the relative error in is generally below 1% and above Δ values of −5‰ the relative error is still only 2%. Thus, whereas we are not able to measure the isotopic composition of individual indistinguishable atoms, the apparent statistical isotope clumping provides a means to obtain information about the heterogeneity of the isotope ratios with quite good precision.
Conclusions
The statistical combination of indistinguishable atoms with different isotope ratios in a molecule always leads to apparent negative clumped isotope signals. We emphasize the term apparent, because this signal does not relate to a physical negative clumping process. The underlying reason is that in the calculation of the stochastic reference value for calculating Δ, the actual isotopic composition at each individual atom position is replaced by the average of the isotopic composition of all indistinguishable atoms (which is similar bulk isotopic composition of the molecule for small isotope ratios). Thus the apparent statistical Δ is by nature an artifact originating from our limitation to measure the isotope ratios of indistinguishable atoms. Using the formalism presented in this paper, this apparent statistical clumping signal can be calculated for any multiisotope system.
We have calculated here the pure apparent statistical heavy isotope clumping values. In nature these apparent clumping signals will always occur in combination with thermodynamic heavy isotope clumping and possible other kinetic isotope effects that lead to clumped isotope anomalies (Yeung^{21}. The statistical clumped isotope signatures will always occur whenever two or more indistinguishable atoms clump together in a molecule. For isotope heterogeneities of a few percent, the effect is of the same order of magnitude as the thermodynamic effects in many molecules. Thus, it is important to take these effects into consideration when interpreting isotopic clumping of indistinguishable atoms in nature. Furthermore, when the statistical clumping signature can be separated from other contributions, its magnitude provides quantitative information on the heterogeneity of the isotopic composition of the indistinguishable atoms in a molecule.
Additional Information
How to cite this article: Röckmann, T. et al. Statistical clumped isotope signatures. Sci. Rep. 6, 31947; doi: 10.1038/srep31947 (2016).
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Acknowledgements
This work was carried out as part of the program of the Netherlands Earth System Science Centre (NESSC), financially supported by the Dutch Ministry of Education, Culture and Science (OCW).
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T.R., M.E.P. and M.E.G.H. developed the conceptual framework for the manuscript and discussed the content. M.C.K. contributed code to produce the figures. T.R. developed the mathematical formalism and wrote the manuscript with input from all authors.
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Röckmann, T., Popa, M., Krol, M. et al. Statistical clumped isotope signatures. Sci Rep 6, 31947 (2016). https://doi.org/10.1038/srep31947
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