# Low 13C-13C abundances in abiotic ethane

Koudai Taguchi, Alexis Gilbert, Barbara Sherwood Lollar, Thomas Giunta,
Christopher J. Boreham, Qi Liu, Juske Horita, Yuichiro Ueno

► To cite this version:

Koudai Taguchi, Alexis Gilbert, Barbara Sherwood Lollar, Thomas Giunta, Christopher J. Boreham, et al.. Low 13C-13C abundances in abiotic ethane. Nature Communications, 2022, 13 (1), 5790 (10p.). 10.1038/s41467-022-33538-9. hal-04203890

HAL Id: hal-04203890

https://hal.science/hal-04203890

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# Low 13C-13C abundances in abiotic ethane

Koudai Taguchi, Alexis Gilbert, Barbara Sherwood Lollar, Thomas Giunta,
Christopher J. Boreham, Qi Liu, Juske Horita, Yuichiro Ueno

► To cite this version:

Koudai Taguchi, Alexis Gilbert, Barbara Sherwood Lollar, Thomas Giunta, Christopher J. Boreham, et al.. Low 13C-13C abundances in abiotic ethane. Nature Communications, 2022, 13 (1), 5790 (10p.). 10.1038/s41467-022-33538-9. hal-04203890

HAL Id: hal-04203890

https://hal.science/hal-04203890v1

Submitted on 17 May 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

# Low ¹³C-¹³C abundances in abiotic ethane

Received: 24 April 2022

Accepted: 22 September 2022

Published online: 02 October 2022

<figure><img src="image_3.png" /></figure>

Check for updates

Koudai Taguchi Ⓔ<fnref n="1" />, Alexis Gilbert Ⓔ<fnref n="1" />,², Barbara Sherwood Lollar³,⁴, Thomas Giunta³,⁵, Christopher J. Boreham⁶, Qi Liu Ⓔ⁷, Juske Horita⁸ & Yuichiro Ueno Ⓔ<fnref n="1" />,²,⁹

Distinguishing biotic compounds from abiotic ones is important in resource geology, biogeochemistry, and the search for life in the universe. Stable isotopes have traditionally been used to discriminate the origins of organic materials, with particular focus on hydrocarbons. However, despite extensive efforts, unequivocal distinction of abiotic hydrocarbons remains challenging. Recent development of clumped-isotope analysis provides more robust information because it is independent of the stable isotopic composition of the starting material. Here, we report data from a ¹³C-¹³C clumped-isotope analysis of ethane and demonstrate that the abiotically-synthesized ethane shows distinctively low ¹³C-¹³C abundances compared to thermogenic ethane. A collision frequency model predicts the observed low ¹³C-¹³C abundances (anti-clumping) in ethane produced from methyl radical recombination. In contrast, thermogenic ethane presumably exhibits near stochastic ¹³C-¹³C distribution inherited from the biological precursor, which undergoes C-C bond cleavage/recombination during metabolism. Further, we find an exceptionally high ¹³C-¹³C signature in ethane remaining after microbial oxidation. In summary, the approach distinguishes between thermogenic, microbially altered, and abiotic hydrocarbons. The ¹³C-¹³C signature can provide an important step forward for discrimination of the origin of organic molecules on Earth and in extra-terrestrial environments.

Detecting organic molecules synthesised via biological processes and distinguishing them from those synthesised via abiotic processes is critical to the search for life elsewhere in the universe<fnref n="1" />⁻³. Hydrocarbons have been detected on Mars⁴, Enceladus⁵, and certain meteorites⁶, although their origins remain a matter of debate. On Earth, natural hydrocarbons are mainly biotic in origin, produced either by thermal decomposition of sedimentary organic matter or microbial production including methanogenesis⁷,⁸. Contrastingly, some abiotic hydrocarbons are produced by a variety of reactions (including free-radical, the Sabatier, and Fischer-Tropsch-type reactions⁹⁻¹²) in both deep

crustal fluids, hydrothermal systems, and sites of low-temperature water-rock reaction such as serpentinization.

Stable isotopes of carbon and hydrogen have long been used to discriminate hydrocarbon sources⁷,⁸. Thermogenic and abiotic hydrocarbons can sometimes be distinguished using compound-specific isotopic analysis (CSIA)¹⁰,¹³; namely, the relationship between the ¹³C/¹²C and ²H/¹H ratios of individual hydrocarbons (methane, ethane, propane, and *n*-butane) (Fig. 1b). However, isotopic identification of abiotic hydrocarbons is often challenging (see ref. 11 and references therein), partly because the CSIA requires a set of

<footnote n="1">¹Department of Earth and Planetary Sciences, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan. ²Earth-Life Science Institute (WPI-ELSI), Tokyo Institute of Technology, Meguro, Tokyo 152-8550, Japan. ³Department of Earth Sciences, University of Toronto, Toronto, ON M5S 3B1, Canada. ⁴Institut de physique du globe de Paris (IPGP), Université Paris Cité, Paris, France. ⁵Univ Brest, CNRS, Ifremer, Geo-Ocean, F-29280 Plouzané, France. ⁶Geoscience Australia, Canberra, ACT, PO Box 378, 2601, Australia. ⁷State Key Laboratory of Ore Deposit Geochemistry, Institute of Geochemistry, Chinese Academy of Sciences, Guiyang 550081, China. ⁸Department of Geosciences, Texas Tech University, Lubbock, TX 79409, USA. ⁹Institute for Extra-cutting-edge Science and Technology Avant-garde Research (X-star), Japan Agency for Marine-Earth Science and Technology (JAMSTEC), Natsushima-cho, Yokosuka 237-0061, Japan. ✉e-mail: taguchi.k.ab@m.titech.ac.jp; gilbert.a.aa@m.titech.ac.jp; ueno.y.ac@m.titech.ac.jp</footnote>
<figure><img src="image_3.png" /></figure>

# Low <sup>13</sup>C-<sup>13</sup>C abundances in abiotic ethane

Received: 24 April 2022

Accepted: 22 September 2022

Published online: 02 October 2022

<figure><img src="image_4.png" /></figure>

Check for updates

Koudai Taguchi Ⓔ<fnref n="1" />, Alexis Gilbert Ⓔ<fnref n="1" />,², Barbara Sherwood Lollar³,⁴, Thomas Giunta³,⁵, Christopher J. Boreham⁶, Qi Liu Ⓔ⁷, Juske Horita⁸ & Yuichiro Ueno Ⓔ<fnref n="1" />,²,⁹

Distinguishing biotic compounds from abiotic ones is important in resource geology, biogeochemistry, and the search for life in the universe. Stable isotopes have traditionally been used to discriminate the origins of organic materials, with particular focus on hydrocarbons. However, despite extensive efforts, unequivocal distinction of abiotic hydrocarbons remains challenging. Recent development of clumped-isotope analysis provides more robust information because it is independent of the stable isotopic composition of the starting material. Here, we report data from a <sup>13</sup>C-<sup>13</sup>C clumped-isotope analysis of ethane and demonstrate that the abiotically-synthesized ethane shows distinctively low <sup>13</sup>C-<sup>13</sup>C abundances compared to thermogenic ethane. A collision frequency model predicts the observed low <sup>13</sup>C-<sup>13</sup>C abundances (anti-clumping) in ethane produced from methyl radical recombination. In contrast, thermogenic ethane presumably exhibits near stochastic <sup>13</sup>C-<sup>13</sup>C distribution inherited from the biological precursor, which undergoes C-C bond cleavage/recombination during metabolism. Further, we find an exceptionally high <sup>13</sup>C-<sup>13</sup>C signature in ethane remaining after microbial oxidation. In summary, the approach distinguishes between thermogenic, microbially altered, and abiotic hydrocarbons. The <sup>13</sup>C-<sup>13</sup>C signature can provide an important step forward for discrimination of the origin of organic molecules on Earth and in extra-terrestrial environments.

Detecting organic molecules synthesised via biological processes and distinguishing them from those synthesised via abiotic processes is critical to the search for life elsewhere in the universe<fnref n="1" />⁻³. Hydrocarbons have been detected on Mars⁴, Enceladus⁵, and certain meteorites⁶, although their origins remain a matter of debate. On Earth, natural hydrocarbons are mainly biotic in origin, produced either by thermal decomposition of sedimentary organic matter or microbial production including methanogenesis⁷,⁸. Contrastingly, some abiotic hydrocarbons are produced by a variety of reactions (including free-radical, the Sabatier, and Fischer-Tropsch-type reactions⁹⁻¹²) in both deep

crustal fluids, hydrothermal systems, and sites of low-temperature water-rock reaction such as serpentinization.

Stable isotopes of carbon and hydrogen have long been used to discriminate hydrocarbon sources⁷,⁸. Thermogenic and abiotic hydrocarbons can sometimes be distinguished using compound-specific isotopic analysis (CSIA)¹⁰,¹³; namely, the relationship between the <sup>13</sup>C/<sup>12</sup>C and <sup>2</sup>H/<sup>1</sup>H ratios of individual hydrocarbons (methane, ethane, propane, and *n*-butane) (Fig. 1b). However, isotopic identification of abiotic hydrocarbons is often challenging (see ref. 11 and references therein), partly because the CSIA requires a set of

<footnote n="1">¹Department of Earth and Planetary Sciences, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan. ²Earth-Life Science Institute (WPI-ELSI), Tokyo Institute of Technology, Meguro, Tokyo 152-8550, Japan. ³Department of Earth Sciences, University of Toronto, Toronto, ON M5S 3B1, Canada. ⁴Institut de physique du globe de Paris (IPGP), Université Paris Cité, Paris, France. ⁵Univ Brest, CNRS, Ifremer, Geo-Ocean, F-29280 Plouzané, France. ⁶Geoscience Australia, Canberra, ACT, PO Box 378, 2601, Australia. ⁷State Key Laboratory of Ore Deposit Geochemistry, Institute of Geochemistry, Chinese Academy of Sciences, Guiyang 550081, China. ⁸Department of Geosciences, Texas Tech University, Lubbock, TX 79409, USA. ⁹Institute for Extra-cutting-edge Science and Technology Avant-garde Research (X-star), Japan Agency for Marine-Earth Science and Technology (JAMSTEC), Natsushima-cho, Yokosuka 237-0061, Japan. ✉e-mail: taguchi.k.ab@m.titech.ac.jp; gilbert.a.aa@m.titech.ac.jp; ueno.y.ac@m.titech.ac.jp</footnote>
<figure><img src="image_4.png" /><figcaption>**Fig. 1 |** Results of Δ¹³C¹³C and compound-specific isotopic analysis (CSIA) of hydrocarbons (δ¹³C). **a** Relationship between Δ¹³C¹³C~Ethane~ and δ¹³C~Ethane~ value normalised against Vienna Pee Dee Belemnite (VPDB). Symbols are the same as shown in the legend of panel **b**. Grey circles denote bio-ethanol from three plants: the C3-type, the C4-type, and the Crassulacean Acid Metabolism (CAM)¹⁹. Light blue triangles represent ethene (C₂H₄) produced by propane pyrolysis (see Methods).</figcaption></figure>

<figure><img src="image_5.png" /></figure>

Blue symbols represent proposed thermogenic gases, and orange represent pro-
posed abiotic gases. The error bars are the standard error of the mean. The ana-
lytical uncertainty for the δ¹³C value is within the symbol. **b** Inverse of carbon
number (n_c) of individual hydrocarbon versus its carbon isotope composition
relative to methane (δ¹³C~CH₄~). The analytical uncertainty for the δ¹³C value is within
the symbol.

molecules, all of which are not always available to sample. The recent
development of the clumped-isotope analysis allows the collation of
information preserved within a single molecule without a need for
analysing other related molecules¹⁴⁻¹⁶. For example, the abundance of
clumped isotopes of methane (¹³CH₃D and CH₂D₂) is now routinely
used as a geothermometer¹⁷, although the C-H bonding is susceptible
to isotopic exchange which can lead, in some cases, to the reset of
¹³CH₃D and CH₂D₂ abundances¹⁸. More robust information may come
from ¹³C-¹³C clumping in organic molecules, because carbon in ethane
is less readily exchanged than hydrogen, which is exchanged with
surrounding water¹⁸.

We have developed a method to determine the relative abun-
dances of the three isotopologues in ethane (¹²CH₃¹²CH₃, ¹²CH₃¹³CH₃,
and ¹³CH₃¹³CH₃) reported as Δ¹³C¹³C values¹⁹,²⁰ (see Methods). Here, we
show that this tool can be used to distinguish between abiotic and
biotic hydrocarbons. We have examined natural gas ethane from var-
ious geological settings and compared them with abiotic ethane syn-
thesised from methane in the laboratory. Based on experimental
results and observations, we present a mechanistic understanding of
¹³C-¹³C abundances in hydrocarbons to account for the results found on
abiotic and thermogenic ethane.

**Results and discussion**

General trends

Results of our clumped-isotope analysis show that thermogenic nat-
ural gas exhibits relatively high Δ¹³C¹³C values (Fig. 1a and Supple-
mentary Table 1) and have a typical δ¹³C distribution pattern classically
observed for thermogenic gas¹³,²¹, in which each longer-chain alkane is
more enriched in ¹³C than the previous alkane (Fig. 1b and Supple-
mentary Table 2). In contrast, abiotic ethane synthesised from CH₄
exhibit distinctively low Δ¹³C¹³C values irrespective of the energy
source used (i.e., ultraviolet [UV] light, spark discharge, and gamma-
ray irradiation: see Methods). The low Δ¹³C¹³C values are also seen in
hydrocarbons from deep fracture fluids in Kidd Creek (Canada) and
the Dingo gas field in the Amadeus Basin (Australia) (see Methods for
geologic settings), also proposed to have an abiotic origin.

A previous study using high-resolution mass spectrometry
exhibited 4‰ variation of Δ¹³C¹³C values of thermogenic ethane²². This

variation has been suggested to arise from the pyrolysis of ethane, which leads to a decrease in Δ¹³C¹³C values²²,²³. The present study using a conventional isotope ratio mass spectrometry after conversion of C₂H₆ to C₂F₆ shows a narrower range of the Δ¹³C¹³C values (0.57‰). Our pyrolysis experiment conducted at the same temperature as in ref. 22 (600 °C) and using a similar quartz vessel showed no change in the Δ¹³C¹³C values in contrast to ref. 22 (Supplementary Fig. 1 and Supplementary Table 3). These observations point to a potential discrepancy between the two methods for isotopologues analysis. Further interlaboratory comparisons will be necessary to calibrate the data from the two methods. The data presented here will be obtained solely by the method presented in ref. 19 that gives reproducible Δ¹³C¹³C values with no scale compression²⁰ (see Methods).

**Thermogenic ethane**

The Δ¹³C¹³C value of thermogenic ethane could be attributed to C-C
bonding in precursor molecules and the kinetic isotope effect during
thermal cracking, as discussed in previous studies²²,²³ (Fig. 2a). In an
ideal case where ethane is produced by breaking at least one C-C bond
in an organic precursor, the kinetic isotope effect is relevant only to
one carbon in ethane resulting in preferential ¹³C enrichment in one of
the two carbons. In this case, the intramolecular bias in ¹³C lowers the
Δ¹³C¹³C value owing to the combinatorial effect²⁴,²⁵, even though the
two carbons in ethane are symmetrically equivalent. Combinatorial
isotope effects are statistical clumped-isotope anomalies that occur
when two atoms at two different positions in a single molecule iso-
topically differ²⁴,²⁵. In other words, when ¹³C is not evenly distributed at
the two positions in an ethane molecule, the abundances of ¹³C-¹³C are
lower than the stochastic (random) distribution. This does not apply to
molecules with non-equivalent atomic sites, typically ethanol, for
which an accurate stochastic distribution can be calculated based on
the ¹³C/¹²C ratio of both sites.

We estimated the intramolecular bias of thermogenic ethane
based on a simple model in which hydrocarbons are formed by the
cleavage of at least one C-C bond in an *n*-alkyl chain, as the simplest
case considered in previous model<sup>21-23</sup>. For ethane, the δ<sup>13</sup>C (=[(<sup>13</sup>C/<sup>12</sup>C)<sub>sample</sub>/(<sup>13</sup>C/<sup>12</sup>C)<sub>standard</sub>] - 1) of one C-atom (C<sub>1</sub>) is that of the original
precursor, whereas that of the other C-atom (C<sub>2</sub>) is altered by the
<figure><img src="image_5.png" /><figcaption>**Fig. 1 |** Results of Δ¹³C¹³C and compound-specific isotopic analysis (CSIA) of hydrocarbons (δ¹³C). **a** Relationship between Δ¹³C¹³C~Ethane~ and δ¹³C~Ethane~ value normalised against Vienna Pee Dee Belemnite (VPDB). Symbols are the same as shown in the legend of panel **b**. Grey circles denote bio-ethanol from three plants: the C3-type, the C4-type, and the Crassulacean Acid Metabolism (CAM)¹⁹. Light blue triangles represent ethene (C₂H₄) produced by propane pyrolysis (see Methods).</figcaption></figure>

<figure><img src="image_6.png" /></figure>

Blue symbols represent proposed thermogenic gases, and orange represent pro-
posed abiotic gases. The error bars are the standard error of the mean. The ana-
lytical uncertainty for the δ¹³C value is within the symbol. **b** Inverse of carbon
number (n~C~) of individual hydrocarbon versus its carbon isotope composition
relative to methane (δ¹³C~CH₄~). The analytical uncertainty for the δ¹³C value is within
the symbol.

molecules, all of which are not always available to sample. The recent
development of the clumped-isotope analysis allows the collation of
information preserved within a single molecule without a need for
analysing other related molecules¹⁴⁻¹⁶. For example, the abundance of
clumped isotopes of methane (¹³CH₃D and CH₂D₂) is now routinely
used as a geothermometer¹⁷, although the C-H bonding is susceptible
to isotopic exchange which can lead, in some cases, to the reset of
¹³CH₃D and CH₂D₂ abundances¹⁸. More robust information may come
from ¹³C-¹³C clumping in organic molecules, because carbon in ethane
is less readily exchanged than hydrogen, which is exchanged with
surrounding water¹⁸.

We have developed a method to determine the relative abun-
dances of the three isotopologues in ethane (¹²CH₃¹²CH₃, ¹²CH₃¹³CH₃,
and ¹³CH₃¹³CH₃) reported as Δ¹³C¹³C values¹⁹,²⁰ (see Methods). Here, we
show that this tool can be used to distinguish between abiotic and
biotic hydrocarbons. We have examined natural gas ethane from var-
ious geological settings and compared them with abiotic ethane syn-
thesised from methane in the laboratory. Based on experimental
results and observations, we present a mechanistic understanding of
¹³C-¹³C abundances in hydrocarbons to account for the results found on
abiotic and thermogenic ethane.

**Results and discussion**

General trends

Results of our clumped-isotope analysis show that thermogenic nat-
ural gas exhibits relatively high Δ¹³C¹³C values (Fig. 1a and Supple-
mentary Table 1) and have a typical δ¹³C distribution pattern classically
observed for thermogenic gas¹³,²¹, in which each longer-chain alkane is
more enriched in ¹³C than the previous alkane (Fig. 1b and Supple-
mentary Table 2). In contrast, abiotic ethane synthesised from CH₄
exhibit distinctively low Δ¹³C¹³C values irrespective of the energy
source used (i.e., ultraviolet [UV] light, spark discharge, and gamma-
ray irradiation: see Methods). The low Δ¹³C¹³C values are also seen in
hydrocarbons from deep fracture fluids in Kidd Creek (Canada) and
the Dingo gas field in the Amadeus Basin (Australia) (see Methods for
geologic settings), also proposed to have an abiotic origin.

A previous study using high-resolution mass spectrometry
exhibited 4‰ variation of Δ¹³C¹³C values of thermogenic ethane²². This

variation has been suggested to arise from the pyrolysis of ethane, which leads to a decrease in Δ¹³C¹³C values²²,²³. The present study using a conventional isotope ratio mass spectrometry after conversion of C₂H₆ to C₂F₆ shows a narrower range of the Δ¹³C¹³C values (0.57‰). Our pyrolysis experiment conducted at the same temperature as in ref. 22 (600 °C) and using a similar quartz vessel showed no change in the Δ¹³C¹³C values in contrast to ref. 22 (Supplementary Fig. 1 and Supplementary Table 3). These observations point to a potential discrepancy between the two methods for isotopologues analysis. Further interlaboratory comparisons will be necessary to calibrate the data from the two methods. The data presented here will be obtained solely by the method presented in ref. 19 that gives reproducible Δ¹³C¹³C values with no scale compression²⁰ (see Methods).

**Thermogenic ethane**

The Δ¹³C¹³C value of thermogenic ethane could be attributed to C-C
bonding in precursor molecules and the kinetic isotope effect during
thermal cracking, as discussed in previous studies²²,²³ (Fig. 2a). In an
ideal case where ethane is produced by breaking at least one C-C bond
in an organic precursor, the kinetic isotope effect is relevant only to
one carbon in ethane resulting in preferential ¹³C enrichment in one of
the two carbons. In this case, the intramolecular bias in ¹³C lowers the
Δ¹³C¹³C value owing to the combinatorial effect²⁴,²⁵, even though the
two carbons in ethane are symmetrically equivalent. Combinatorial
isotope effects are statistical clumped-isotope anomalies that occur
when two atoms at two different positions in a single molecule iso-
topically differ²⁴,²⁵. In other words, when ¹³C is not evenly distributed at
the two positions in an ethane molecule, the abundances of ¹³C-¹³C are
lower than the stochastic (random) distribution. This does not apply to
molecules with non-equivalent atomic sites, typically ethanol, for
which an accurate stochastic distribution can be calculated based on
the ¹³C/¹²C ratio of both sites.

We estimated the intramolecular bias of thermogenic ethane
based on a simple model in which hydrocarbons are formed by the
cleavage of at least one C-C bond in an *n*-alkyl chain, as the simplest
case considered in previous model²¹⁻²³. For ethane, the δ¹³C (=[(¹³C/
¹²C)sample/(¹³C/¹²C)standard] - 1) of one C-atom (C₁) is that of the original
precursor, whereas that of the other C-atom (C₂) is altered by the
<figure><img src="image_6.png" /><figcaption>Fig. 2 | Δ¹³C¹³C vs. slope of compound-specific isotopic analysis (CSIA) trend.</figcaption></figure>

The horizontal axis shows the slope of the relationship between δ¹³C and 1/n_c (derived from Fig. 1b), which also assumed to be an intrinsic δ¹³C bias between the two positions of carbon in a molecule (see text). The Δ¹³C¹³C scale to the stochastic distribution was estimated by assuming that the C-C bonds of biological glucose are under homogeneous isotopic equilibrium (see Methods). The error bars are the standard error of the mean. a The curved black line shows the predicted Δ¹³C¹³C value of ethane produced by C-C bond cleavage from an organic precursor (denoted as 'P') at each temperature, considering the combinatorial effect (see Methods). The CSIA slope at each temperature is also calculated using the same

<figure><img src="image_7.png" /></figure>

thermal cracking model<sup>21-23,26</sup> (see Methods). The grey shaded area shows the uncertainty of the calculation, mainly derived from the possible range in Δ¹³C¹³C and differences in δ¹³C values between two adjacent positions of precursor molecules for the thermogenic hydrocarbons (see Methods). The dotted arrow shows the expected change due to microbial oxidation of ethane for Tokamachi mud volcano (Supplementary Fig. 2). b The curved black line shows the theoretically calculated Δ¹³C¹³C value of ethane at each temperature (Supplementary Fig. 3 and Supplementary Table 5) and δ¹³C values of hydrocarbons<sup>67,68</sup> (see Methods). The dotted arrows show the notional changes due to isotope exchange among hydrocarbons after formation.

cracking process. This creates an intramolecular bias within the ethane molecule, denoted as Δ¹³C<sub>Ethane</sub> (= δ¹³C<sub>C1</sub> - δ¹³C<sub>C2</sub>). The Δ¹³C<sub>Ethane</sub> value can be obtained through the slope of the relationship between the δ¹³C values of individual hydrocarbons and the inverse of their carbon number (i.e., 1/n_c) (see Methods) (Fig. 1b)<sup>21-23</sup>. The combinatorial effect (Δ¹³C¹³C<sub>Comb</sub>) can therefore be calculated as follows (see Methods for detailed calculation):

$$ \Delta^{13}C^{13}C_{\text{Comb}} = -(\Delta^{13}C_{\text{Ethane}}/2(\delta^{13}C_{\text{Ethane}} + 1000))^{2} \times 1000, \quad (1) $$

where δ¹³C<sub>Ethane</sub> represents the carbon isotope composition of ethane. Assuming a typical δ¹³C<sub>Ethane</sub> value of -40‰ and a kinetic isotope effect (¹³k/¹²k) from 0.958 to 0.986 (i.e., δ¹³C<sub>C1</sub> > δ¹³C<sub>C2</sub>), corresponding to temperatures ranging from 25 to 300 °C<sup>26</sup>, the Δ¹³C¹³C<sub>Comb</sub> is predicted to range from -0.04 to -0.45‰ (Fig. 2a). Assuming a cracking temperature of 100 °C, the Δ¹³C¹³C value of ethane would be lower than -0.25‰ relative to the Δ¹³C¹³C value of precursor hydrocarbons. Estimating the Δ¹³C¹³C value of the original precursors is difficult, since measurement of ¹³C¹³C clumped isotopes has not been applied to natural gas precursors, such as alkanes and kerogen. The only available data for biological molecules to date are derived from bio-ethanol (Fig. 1a)<sup>19</sup>. Despite the different types of photosynthetic pathway (the C3-type, the C4-type, and the Crassulacean Acid Metabolism [CAM]), the bio-ethanol show a narrow range of Δ¹³C¹³C values from +0.90 to +0.98‰, suggesting bio-ethanol is a good representative of biological molecules<sup>19</sup>. Using the Δ¹³C¹³C value of the ethanol as a starting estimate for an organic precursor (denoted as 'P' in Fig. 2a), the Δ¹³C¹³C value of thermogenic ethane can be calculated (Fig. 2a). Remarkably, the observed Δ¹³C¹³C values of the majority of the thermogenic natural gas ethanes in this study fall within the range predicted by this model at about 150 °C (Fig. 2a). The thermogenic ethane seems aligned perpendicular to the equilibrium temperature curve (Fig. 2b), which may potentially reflect the variation of Δ¹³C¹³C of the organic precursor, though, at present, the available Δ¹³C¹³C data of the organic precursor is only limited to bio-ethanol. Future studies should pursue Δ¹³C¹³C of organic molecules such as n-alkanes, fatty acids, and lignin to evaluate the Δ¹³C¹³C variations in the organic precursor.

## Microbial oxidation of ethane

Three natural gas samples from Tokamachi show Δ¹³C¹³C values exceeding the predicted range of thermogenic ethane from this model (Fig. 2a). The high Δ¹³C¹³C in these samples could be due to microbial oxidation of ethane. Microbial strains capable of metabolising non-methane hydrocarbons (NMHCs) reportedly inhabit various geological settings under anoxic conditions<sup>27-29</sup>. The samples showing the high Δ¹³C¹³C values also have high C₁/(C₂ + C₃) ratios, consistent with the anaerobic oxidation of NMHCs (Supplementary Fig. 2)<sup>30</sup>. Furthermore, at each of these localities, the high Δ¹³C¹³C in ethane correlated with the ¹³C enrichment of the central carbon in propane, strongly indicating microbial degradation of NMHCs (Supplementary Fig. 2 and Supplementary Table 4)<sup>31</sup>. Microbial oxidation of NMHCs is known to yield increased ¹³C enrichment in higher hydrocarbons<sup>31</sup> (i.e., the steeper slope of the CSIA trend seen in Fig. 1b), consistent with the higher Δ¹³C¹³C values of ethane indicative of anaerobic oxidation origin.

Although the Δ¹³C¹³C analysis of pure culture anaerobic oxidation remains unreported, a conjecture can nevertheless be made based on recent findings on the anaerobic oxidation of ethane by archaea<sup>28,29</sup>. Previous culture experiments on ethane-oxidising archaea have shown that the enzymatic steps from ethane to CO₂ by the archaea 'Candidatus Ethanoperedens' can be reversible within the cell<sup>29</sup>. In such case, a reverse reaction in which some CO₂ is converted back into ethane may yield isotopic bond re-ordering towards thermodynamic equilibrium. Note that a similar mechanism has been proposed for the anaerobic oxidation of methane and is believed to promote isotopic re-equilibration in the residual methane<sup>32</sup>. Accordingly, it is conceivable that residual ethane undergoing microbial oxidation would approach isotopic equilibrium (Δ¹³C¹³C ≥ + 0.10‰ relative to the stochastic distribution of ethane; Fig. 2b, Supplementary Fig. 3 and Supplementary Table 5) at the microbiologically functional temperature range (<122 °C<sup>33</sup>) based on the assumption of stochastic reference frame (see Methods). A more precise estimate of the Δ¹³C¹³C change in microbial degradation is not possible at this stage, owing to the uncertainty in the growth temperature and degree of reversibility in natural populations. Nonetheless, the observed Δ¹³C¹³C increase seen in the three samples here is consistent with microbial degradation of
<figure><img src="image_7.png" /><figcaption>Fig. 2 | Δ¹³C¹³C vs. slope of compound-specific isotopic analysis (CSIA) trend.</figcaption></figure>

The horizontal axis shows the slope of the relationship between δ¹³C and 1/n_c (derived from Fig. 1b), which also assumed to be an intrinsic δ¹³C bias between the two positions of carbon in a molecule (see text). The Δ¹³C¹³C scale to the stochastic distribution was estimated by assuming that the C-C bonds of biological glucose are under homogeneous isotopic equilibrium (see Methods). The error bars are the standard error of the mean. a The curved black line shows the predicted Δ¹³C¹³C value of ethane produced by C-C bond cleavage from an organic precursor (denoted as 'P') at each temperature, considering the combinatorial effect (see Methods). The CSIA slope at each temperature is also calculated using the same

<figure><img src="image_8.png" /></figure>

thermal cracking model<sup>21-23,26</sup> (see Methods). The grey shaded area shows the uncertainty of the calculation, mainly derived from the possible range in Δ¹³C¹³C and differences in δ¹³C values between two adjacent positions of precursor molecules for the thermogenic hydrocarbons (see Methods). The dotted arrow shows the expected change due to microbial oxidation of ethane for Tokamachi mud volcano (Supplementary Fig. 2). b The curved black line shows the theoretically calculated Δ¹³C¹³C value of ethane at each temperature (Supplementary Fig. 3 and Supplementary Table 5) and δ¹³C values of hydrocarbons<sup>67,68</sup> (see Methods). The dotted arrows show the notional changes due to isotope exchange among hydrocarbons after formation.

cracking process. This creates an intramolecular bias within the ethane molecule, denoted as Δ¹³C<sub>Ethane</sub> (= δ¹³C<sub>C1</sub> - δ¹³C<sub>C2</sub>). The Δ¹³C<sub>Ethane</sub> value can be obtained through the slope of the relationship between the δ¹³C values of individual hydrocarbons and the inverse of their carbon number (i.e., 1/n_c) (see Methods) (Fig. 1b)<sup>21-23</sup>. The combinatorial effect (Δ¹³C¹³C<sub>Comb</sub>) can therefore be calculated as follows (see Methods for detailed calculation):

$$ \Delta^{13}C^{13}C_{\text{Comb}} = - \left( \frac{\Delta^{13}C_{\text{Ethane}}}{2(\delta^{13}C_{\text{Ethane}} + 1000)} \right)^2 \times 1000, \quad (1) $$

where δ¹³C<sub>Ethane</sub> represents the carbon isotope composition of ethane. Assuming a typical δ¹³C<sub>Ethane</sub> value of -40‰ and a kinetic isotope effect (¹³k/¹²k) from 0.958 to 0.986 (i.e., δ¹³C<sub>C1</sub> > δ¹³C<sub>C2</sub>), corresponding to temperatures ranging from 25 to 300 °C<sup>26</sup>, the Δ¹³C¹³C<sub>Comb</sub> is predicted to range from -0.04 to -0.45‰ (Fig. 2a). Assuming a cracking temperature of 100 °C, the Δ¹³C¹³C value of ethane would be lower than -0.25‰ relative to the Δ¹³C¹³C value of precursor hydrocarbons. Estimating the Δ¹³C¹³C value of the original precursors is difficult, since measurement of ¹³C¹³C clumped isotopes has not been applied to natural gas precursors, such as alkanes and kerogen. The only available data for biological molecules to date are derived from bio-ethanol (Fig. 1a)<sup>19</sup>. Despite the different types of photosynthetic pathway (the C3-type, the C4-type, and the Crassulacean Acid Metabolism [CAM]), the bio-ethanol show a narrow range of Δ¹³C¹³C values from +0.90 to +0.98‰, suggesting bio-ethanol is a good representative of biological molecules<sup>19</sup>. Using the Δ¹³C¹³C value of the ethanol as a starting estimate for an organic precursor (denoted as 'P' in Fig. 2a), the Δ¹³C¹³C value of thermogenic ethane can be calculated (Fig. 2a). Remarkably, the observed Δ¹³C¹³C values of the majority of the thermogenic natural gas ethanes in this study fall within the range predicted by this model at about 150 °C (Fig. 2a). The thermogenic ethane seems aligned perpendicular to the equilibrium temperature curve (Fig. 2b), which may potentially reflect the variation of Δ¹³C¹³C of the organic precursor, though, at present, the available Δ¹³C¹³C data of the organic precursor is only limited to bio-ethanol. Future studies should pursue Δ¹³C¹³C of organic molecules such as n-alkanes, fatty acids, and lignin to evaluate the Δ¹³C¹³C variations in the organic precursor.

## Microbial oxidation of ethane

Three natural gas samples from Tokamachi show Δ¹³C¹³C values exceeding the predicted range of thermogenic ethane from this model (Fig. 2a). The high Δ¹³C¹³C in these samples could be due to microbial oxidation of ethane. Microbial strains capable of metabolising non-methane hydrocarbons (NMHCs) reportedly inhabit various geological settings under anoxic conditions<sup>27-29</sup>. The samples showing the high Δ¹³C¹³C values also have high C₁/(C₂ + C₃) ratios, consistent with the anaerobic oxidation of NMHCs (Supplementary Fig. 2)<sup>30</sup>. Furthermore, at each of these localities, the high Δ¹³C¹³C in ethane correlated with the ¹³C enrichment of the central carbon in propane, strongly indicating microbial degradation of NMHCs (Supplementary Fig. 2 and Supplementary Table 4)<sup>31</sup>. Microbial oxidation of NMHCs is known to yield increased ¹³C enrichment in higher hydrocarbons<sup>31</sup> (i.e., the steeper slope of the CSIA trend seen in Fig. 1b), consistent with the higher Δ¹³C¹³C values of ethane indicative of anaerobic oxidation origin.

Although the Δ¹³C¹³C analysis of pure culture anaerobic oxidation remains unreported, a conjecture can nevertheless be made based on recent findings on the anaerobic oxidation of ethane by archaea<sup>28,29</sup>. Previous culture experiments on ethane-oxidising archaea have shown that the enzymatic steps from ethane to CO₂ by the archaea 'Candidatus Ethanoperedens' can be reversible within the cell<sup>29</sup>. In such case, a reverse reaction in which some CO₂ is converted back into ethane may yield isotopic bond re-ordering towards thermodynamic equilibrium. Note that a similar mechanism has been proposed for the anaerobic oxidation of methane and is believed to promote isotopic re-equilibration in the residual methane<sup>32</sup>. Accordingly, it is conceivable that residual ethane undergoing microbial oxidation would approach isotopic equilibrium (Δ¹³C¹³C ≥ +0.10‰ relative to the stochastic distribution of ethane; Fig. 2b, Supplementary Fig. 3 and Supplementary Table 5) at the microbiologically functional temperature range (<122 °C<sup>33</sup>) based on the assumption of stochastic reference frame (see Methods). A more precise estimate of the Δ¹³C¹³C change in microbial degradation is not possible at this stage, owing to the uncertainty in the growth temperature and degree of reversibility in natural populations. Nonetheless, the observed Δ¹³C¹³C increase seen in the three samples here is consistent with microbial degradation of
ethane (Fig. 2b). Future studies should pursue Δ¹³C¹³C analysis to evaluate microbial activity pertaining to the anaerobic oxidation of ethane. Because the anaerobic oxidation of ethane increases Δ¹³C¹³C and its direction of the trend is away from those of abiotic ethane, the ability of Δ¹³C¹³C to distinguish between abiotic and thermogenic ethane is not impaired. Other processes potentially alter the Δ¹³C¹³C value in ethane and are compatible with the observed variation of thermogenic ethane. These include diffusion (an increase of Δ¹³C¹³C value by 0.3‰ in the case where molecular collision is important), mixing with different sources (an increase of Δ¹³C¹³C value by up to 0.13‰ in the case of mixing samples with the same Δ¹³C¹³C values but with different δ¹³C values of -20‰ and -45‰) and secondary cracking of ethane itself (no Δ¹³C¹³C variations at 600 °C in this study; see Supplementary Fig. 1)²²,²³. However, again, the discrimination potential of the Δ¹³C¹³C value of ethane is not weakened, because all these factors tend to increase the Δ¹³C¹³C value.

## Abiotic hydrocarbon synthesis

In the stochastic reference frame assumed here (see Methods), the observed negative Δ¹³C¹³C values (i.e., anti-clumping²⁵) of abiotic ethane can be explained by kinetic isotope effects governed by collision frequency between CH₃ radicals (Fig. 2b). A radical–radical combination such as CH₃ + CH₃ is typically a barrierless reaction³⁴, associated with mild kinetic isotope effects, depending on the ratio of collision frequencies³⁵. These collision frequencies are scaled exactly by the inverse square root of the reduced mass (μ) of the collision pair (m₁ and m₂, respectively), i.e., μ = 1/m₁ + 1/m₂. Thus, the kinetic isotope effect due to the difference in collision frequency can be obtained from the relative reaction rate, calculated from the reduced mass ratio. In the case of a collision between CH₃ radicals, the kinetic isotope effects are calculated to be k/k = 0.9842 and k''/k = 0.9682, where the prime and double prime refers to singly and doubly substituted isop topologues, respectively. If kinetic isotope effects follow a stochastic distribution, (k''/k)/(k/k)² must be equal to 1 (see Supplementary Note 1). Any deviation from unity leads to Δ¹³C¹³C values different from 0. The intrinsic kinetic clumped-isotope effect calculated from the reduced mass of methyl radicals corresponds to a Δ¹³C¹³C value of -0.52‰. A more accurate quantitative estimate requires calculations that include a configurational effect, as isotopic substitutions can affect the minimum energy configuration of the transition state by changing its centre of mass³⁶. However, the simple collision frequency calculation shows that Δ¹³C¹³C should be negative (anti-clumping) compared to a stochastic distribution during the methyl radical recombination reaction. Notably, a combinatorial isotope effect is not expected in the case of a methyl radical recombination since the two methyl radicals arise from the same reservoir. However, if one considers surface reactions such as Fischer–Tropsch-type reactions, the mechanism itself may lead to an intramolecular bias in ethane³⁷, resulting in even lower Δ¹³C¹³C values. The data presented here provide, for the first time, a strong indicator of abiogenicity for ethane, based on the observed pattern of ¹³C-¹³C anti-clumping.

## Isotope exchange in abiotic synthesis

After hydrocarbons are produced, their destruction enhances carbon exchange among individual hydrocarbons, which may potentially lead partial isotopic equilibrium through repeated production and destruction cycling. In our UV irradiation experiments of methane, C₂H₆ was produced from the combination of CH₃ radicals (CH₃ being generated through the reaction of CH₄ with OH radicals derived from the photodissociation of water³⁸) by the following reaction³⁹:

$$CH_3 + CH_3 + M \rightarrow C_2H_6 + M \quad (2)$$

where M represents any third-body collision partner. In the wavelength range of UV light used in this study, neither CH₄, C₂H₆, nor C₃H₈

photodissociates³⁸. In addition, the C-C bonds in C₂H₆ and C₃H₈ are not decomposed by reactions with OH, H, O, and CH₃ radicals, all of which should be present during UV experiments³⁸. Hence, the C₂H₆ and C₃H₈ produced by UV irradiation of methane are unlikely to undergo C-C bond decomposition because of the lack of high-energy photon below 150 nm in our experimental setting. Conversely, for spark discharge and gamma-rays experiment, the C-C bonds of C₂+ hydrocarbons frequently cleave after their formation as follows⁴⁰ (Supplementary Table 6):

$$C_2H_6 \rightarrow 2CH_3 \quad (3)$$

$$C_3H_8 \rightarrow CH_3 + C_2H_5 \quad (4)$$

$$C_3H_8 \rightarrow CH_4 + C_2H_4 \quad (5)$$

In the case of gamma-ray irradiation of methane, the calculated dosage was sufficient to decompose the C₂+ hydrocarbons formed from methane, although C₂H₄ was not detected¹². Ethane in the spark discharge and gamma-ray irradiation experiments is produced not only by the CH₃ radical polymerisation but also by the C₃+ hydrocarbon decomposition. After production, ethane decomposes to CH₃, implying that the C₁ polymerisation is not unidirectional. The C-C chain elongation and shortening may alter the Δ¹³C¹³C values of ethane to an extent depending on the degree of reversibility. Fully reversible reactions may yield equilibrium isotope composition (the curved black line in Fig. 2b), whereas irreversible reactions tend to be governed by kinetic isotope effect as represented in the ethane synthesised by UV experiment (Fig. 2b). We suggest that cleavage of C-C bonds in hydrocarbons may enhance the reversibility and leads to an isotopic exchange, where Δ¹³C¹³C of abiotic ethane shifts toward the homogeneous isotopic equilibrium (Δ¹³C¹³C = +0.22‰ at 25 °C; Supplementary Fig. 3) (Fig. 2b).

## Δ¹³C¹³C systematics of abiotic hydrocarbons

In summary, the observed low Δ¹³C¹³C values in abiotic ethane can be explained by anti-clumping due to the kinetic isotope effect at the C-C bond formation (negative Δ¹³C¹³C). Subsequent isotope exchange facilitated through the backreaction from the higher hydrocarbons during the polymerisation sequence may cause the increase in Δ¹³C¹³C value (Fig. 2b). The two processes likely occur naturally and seem applicable to the abiotic hydrocarbons from Kidd Creek fracture fluids and the Dingo gas field (see Methods), both of which exhibit low Δ¹³C¹³C values within the expected abiotic range (Fig. 2b). The similarity of ethane from the gamma radiolysis experiments to the Kidd Creek samples is notable given the proposed role of radiolysis in producing acetate and formate at that site⁴¹. If isotope exchange continues, leading to homogeneous isotopic equilibrium, the Δ¹³C¹³C of abiotic ethane may eventually become indistinguishable from that of thermogenic hydrocarbons. However, the ¹³C-¹³C anti-clumping observed in the natural gases from the two sites (Kidd Creek and the Dingo gas field) demonstrates that the distinctively low abiotic Δ¹³C¹³C signature survives in nature. Based on these findings, ¹³C-¹³C anti-clumping in ethane can be a valuable approach to distinguish abiotic hydrocarbons from thermogenic, and potentially from microbial sources. The ¹³C-¹³C signature may thus be applied in investigations of the origin of ethane in terrestrial and extra-terrestrial settings. Moreover, not only ethane but a variety of organic molecules containing C-C bonds can be subjected to this analytical approach to distinguish abiotic formation pathways, in geological and even extra-terrestrial settings, such as Mars, Titan, and Enceladaus²⁵,⁴².
ethane (Fig. 2b). Future studies should pursue Δ¹³C¹³C analysis to evaluate microbial activity pertaining to the anaerobic oxidation of ethane. Because the anaerobic oxidation of ethane increases Δ¹³C¹³C and its direction of the trend is away from those of abiotic ethane, the ability of Δ¹³C¹³C to distinguish between abiotic and thermogenic ethane is not impaired. Other processes potentially alter the Δ¹³C¹³C value in ethane and are compatible with the observed variation of thermogenic ethane. These include diffusion (an increase of Δ¹³C¹³C value by 0.3‰ in the case where molecular collision is important), mixing with different sources (an increase of Δ¹³C¹³C value by up to 0.13‰ in the case of mixing samples with the same Δ¹³C¹³C values but with different δ¹³C values of -20‰ and -45‰) and secondary cracking of ethane itself (no Δ¹³C¹³C variations at 600 °C in this study; see Supplementary Fig. 1)²²,²³. However, again, the discrimination potential of the Δ¹³C¹³C value of ethane is not weakened, because all these factors tend to increase the Δ¹³C¹³C value.

## Abiotic hydrocarbon synthesis

In the stochastic reference frame assumed here (see Methods), the observed negative Δ¹³C¹³C values (i.e., anti-clumping²⁵) of abiotic ethane can be explained by kinetic isotope effects governed by collision frequency between CH₃ radicals (Fig. 2b). A radical–radical combination such as CH₃ + CH₃ is typically a barrierless reaction³⁴, associated with mild kinetic isotope effects, depending on the ratio of collision frequencies³⁵. These collision frequencies are scaled exactly by the inverse square root of the reduced mass (μ) of the collision pair (m₁ and m₂, respectively), i.e., μ = 1/m₁ + 1/m₂. Thus, the kinetic isotope effect due to the difference in collision frequency can be obtained from the relative reaction rate, calculated from the reduced mass ratio. In the case of a collision between CH₃ radicals, the kinetic isotope effects are calculated to be k/k = 0.9842 and k''/k = 0.9682, where the prime and double prime refers to singly and doubly substituted isotopologues, respectively. If kinetic isotope effects follow a stochastic distribution, (k''/k)/(k/k)² must be equal to 1 (see Supplementary Note 1). Any deviation from unity leads to Δ¹³C¹³C values different from 0. The intrinsic kinetic clumped-isotope effect calculated from the reduced mass of methyl radicals corresponds to a Δ¹³C¹³C value of -0.52‰. A more accurate quantitative estimate requires calculations that include a configurational effect, as isotopic substitutions can affect the minimum energy configuration of the transition state by changing its centre of mass³⁶. However, the simple collision frequency calculation shows that Δ¹³C¹³C should be negative (anti-clumping) compared to a stochastic distribution during the methyl radical recombination reaction. Notably, a combinatorial isotope effect is not expected in the case of a methyl radical recombination since the two methyl radicals arise from the same reservoir. However, if one considers surface reactions such as Fischer–Tropsch-type reactions, the mechanism itself may lead to an intramolecular bias in ethane³⁷, resulting in even lower Δ¹³C¹³C values. The data presented here provide, for the first time, a strong indicator of abiogenicity for ethane, based on the observed pattern of ¹³C-¹³C anti-clumping.

## Isotope exchange in abiotic synthesis

After hydrocarbons are produced, their destruction enhances carbon exchange among individual hydrocarbons, which may potentially lead partial isotopic equilibrium through repeated production and destruction cycling. In our UV irradiation experiments of methane, C₂H₆ was produced from the combination of CH₃ radicals (CH₃ being generated through the reaction of CH₄ with OH radicals derived from the photodissociation of water³⁸) by the following reaction³⁹:

$$CH_3 + CH_3 + M \rightarrow C_2H_6 + M \quad (2)$$

where M represents any third-body collision partner. In the wavelength range of UV light used in this study, neither CH₄, C₂H₆, nor C₃H₈

photodissociates³⁸. In addition, the C-C bonds in C₂H₆ and C₃H₈ are not decomposed by reactions with OH, H, O, and CH₃ radicals, all of which should be present during UV experiments³⁸. Hence, the C₂H₆ and C₃H₈ produced by UV irradiation of methane are unlikely to undergo C-C bond decomposition because of the lack of high-energy photon below 150 nm in our experimental setting. Conversely, for spark discharge and gamma-rays experiment, the C-C bonds of C₂+ hydrocarbons frequently cleave after their formation as follows⁴⁰ (Supplementary Table 6):

$$C_2H_6 \rightarrow 2CH_3 \quad (3)$$

$$C_3H_8 \rightarrow CH_3 + C_2H_5 \quad (4)$$

$$C_3H_8 \rightarrow CH_4 + C_2H_4 \quad (5)$$

In the case of gamma-ray irradiation of methane, the calculated dosage was sufficient to decompose the C₂+ hydrocarbons formed from methane, although C₂H₄ was not detected¹². Ethane in the spark discharge and gamma-ray irradiation experiments is produced not only by the CH₃ radical polymerisation but also by the C₃+ hydrocarbon decomposition. After production, ethane decomposes to CH₃, implying that the C₁ polymerisation is not unidirectional. The C-C chain elongation and shortening may alter the Δ¹³C¹³C values of ethane to an extent depending on the degree of reversibility. Fully reversible reactions may yield equilibrium isotope composition (the curved black line in Fig. 2b), whereas irreversible reactions tend to be governed by kinetic isotope effect as represented in the ethane synthesised by UV experiment (Fig. 2b). We suggest that cleavage of C-C bonds in hydrocarbons may enhance the reversibility and leads to an isotopic exchange, where Δ¹³C¹³C of abiotic ethane shifts toward the homogeneous isotopic equilibrium (Δ¹³C¹³C = +0.22‰ at 25 °C; Supplementary Fig. 3) (Fig. 2b).

## Δ¹³C¹³C systematics of abiotic hydrocarbons

In summary, the observed low Δ¹³C¹³C values in abiotic ethane can be explained by anti-clumping due to the kinetic isotope effect at the C-C bond formation (negative Δ¹³C¹³C). Subsequent isotope exchange facilitated through the backreaction from the higher hydrocarbons during the polymerisation sequence may cause the increase in Δ¹³C¹³C value (Fig. 2b). The two processes likely occur naturally and seem applicable to the abiotic hydrocarbons from Kidd Creek fracture fluids and the Dingo gas field (see Methods), both of which exhibit low Δ¹³C¹³C values within the expected abiotic range (Fig. 2b). The similarity of ethane from the gamma radiolysis experiments to the Kidd Creek samples is notable given the proposed role of radiolysis in producing acetate and formate at that site⁴¹. If isotope exchange continues, leading to homogeneous isotopic equilibrium, the Δ¹³C¹³C of abiotic ethane may eventually become indistinguishable from that of thermogenic hydrocarbons. However, the ¹³C-¹³C anti-clumping observed in the natural gases from the two sites (Kidd Creek and the Dingo gas field) demonstrates that the distinctively low abiotic Δ¹³C¹³C signature survives in nature. Based on these findings, ¹³C-¹³C anti-clumping in ethane can be a valuable approach to distinguish abiotic hydrocarbons from thermogenic, and potentially from microbial sources. The ¹³C-¹³C signature may thus be applied in investigations of the origin of ethane in terrestrial and extra-terrestrial settings. Moreover, not only ethane but a variety of organic molecules containing C-C bonds can be subjected to this analytical approach to distinguish abiotic formation pathways, in geological and even extra-terrestrial settings, such as Mars, Titan, and Enceladaus²⁵,⁴².
Methods

$\Delta^{13}C^{13}C$ notation

The abundance of $^{13}C-^{13}C$ isotopologues was conventionally reported as a deviation from the stochastic abundance of the isotopologues:

$$ \Delta^{13}C^{13}C \equiv \frac{^{1313}\text{R}_{\text{sample}}}{^{1313}\text{R}_{\text{stochastic}}} - 1 \quad (6) $$

where $^{1313}\text{R}$ is defined as the abundance of $^{13}C-^{13}C$ isotopologues compared to $^{12}C-^{12}C$ isotopologues, and $\text{R}_{\text{stochastic}}$ refers to the abundance ratio in a random distribution. The stochastic distribution of $C_2$ isotopologues is calculated as follows:

$$ ^{1313}\text{R}_{\text{stochastic}} = ^{13}\text{R} \times ^{13}\text{R} \quad (7) $$

where $^{13}\text{R}$ indicates the $^{13}C/^{12}C$ ratio in all $C_2$ molecules. In this study, we report $\Delta^{13}C^{13}C'$ as the natural logarithm of $\alpha$:

$$ \Delta^{13}C^{13}C' \equiv \ln(\alpha) \approx (\alpha - 1); \quad (\text{since } \alpha \approx 1) \quad (8) $$

where $\alpha$ represents the equilibrium constant of the homogeneous isotope exchange reaction:

$$ 2^{12}\text{C}^{13}\text{C} \rightleftharpoons 12^{12}\text{C}^{12}\text{C} + 13^{13}\text{C}^{13}\text{C} \quad (9) $$

$$ \alpha \equiv [^{12}\text{C}^{12}\text{C}][^{13}\text{C}^{13}\text{C}]/[^{12}\text{C}^{13}\text{C}]^2 \approx ^{1313}\text{R}/(2 \times ^{13}\text{R})^2 \quad (10) $$

where $^{13}\text{R}$ is calculated from the isotopologue ratio of $[^{12}\text{C}^{13}\text{C}]/[^{12}\text{C}^{12}\text{C}]$ divided by 2, reflecting the symmetry of two carbon atoms in ethane. The $\Delta^{13}C^{13}C'$ value is approximately equal to that of the conventional $\Delta^{13}C^{13}C$ (Eq. (6)). In the case of $C_2$ compounds, however, it is difficult to determine the $^{1313}\text{R}_{\text{stochastic}}$ accurately because C-C bond breaking and recombination do not usually occur reversibly. Thus, calibrating the value with experiments at different temperatures is not feasible. Therefore, $\Delta^{13}C^{13}C^*$ value is calculated as follows:

$$ \begin{aligned} \Delta^{13}C^{13}C^* &= \Delta^{13}C^{13}C'_{\text{sample}} - \Delta^{13}C^{13}C'_{\text{reference}} \\ &= \ln(\frac{^{1313}\text{R}_{\text{sample}}}{^{13}\text{R}^2_{\text{sample}}}) - \ln(\frac{^{1313}\text{R}_{\text{reference}}}{^{13}\text{R}^2_{\text{reference}}}) \\ &= \ln(\frac{^{1313}\text{R}_{\text{sample}}}{^{1313}\text{R}_{\text{reference}}}) - 2 \times \ln(\frac{^{1313}\text{R}_{\text{sample}}}{^{13}\text{R}_{\text{reference}}}) \\ &= \delta^{13}C^{13}C' - 2 \times \delta^{13}C' \end{aligned} \quad (11) $$

where $\delta^{13}C^{13}C'$ and $\delta^{13}C'$ represent the ratio of $^{1313}\text{R}$ and $^{13}\text{R}$ among sample and reference gases as the natural logarithm. Note that all isotope values ($\Delta^{13}C^{13}C^*$, $\delta^{13}C^{13}C'$ and $\delta^{13}C'$) are expressed in $\%$.

Measurement of $\Delta^{13}C^{13}C$

We used a fluorination method for the measurement of $^{13}C-^{13}C$ species$^{19,20}$, which is based on the fluorination of $C_2$ compounds to hexafluoroethane ($C_2F_6$) and subsequent measurement of its $^{13}C$ isotopologues with a conventional isotope ratio mass spectrometer. The purified $C_2F_6$ was introduced into a mass spectrometer (Thermo Fischer MAT253), used in the conventional dual inlet mode. The typical standard deviation of the mean for $\delta^{13}C'$, $\delta^{13}C^{13}C'$, and $\Delta^{13}C^{13}C^*$ values was $\pm 0.01\%$, $\pm 0.09\%$, and $\pm 0.09\%$, respectively ($n = 6$).

Consequently, we define the empirical transfer function as follows:

$$ \Delta^{13}C^{13}C^*_{\text{CSC}} = \lambda \times \Delta^{13}C^{13}C^* \quad (12) $$

where $\Delta^{13}C^{13}C^*_{\text{CSC}}$ value refers to the 'true scale' value, corrected for scale compression. The $\lambda$ value is 1.2541 $\pm$ 0.0101 for ethanol and ethene, whereas 1 for ethane since the latter is not prone to scrambling which may have occurred during the fluorination of ethene but not in the ion source of the mass spectrometer$^{20}$. For simplicity, we will

describe the corrected $\Delta^{13}C^{13}C^*_{\text{CSC}}$ as $\Delta^{13}C^{13}C$ in the following and main text.

Reference frame for $\Delta^{13}C^{13}C$ measurement

The $\Delta^{13}C^{13}C$ values obtained as described above are relative and are not referred to against the stochastic distribution but a working standard ($C_2F_6$)$^{19,20}$. To obtain values reported against a stochastic distribution, we constructed a reference frame for $^{13}C-^{13}C$ isotopologues analysis by estimating that C-C bonds in biological glucose could be under thermodynamic isotope equilibrium. The $\Delta^{13}C^{13}C$ of ethanol produced by the fermentation of sugars from different plants ($C_3$, $C_4$, and CAM) is reportedly uniform, whereas the $^{13}C/^{12}C$ ratio and position-specific isotope composition vary$^{19}$. The latter has been explained by differences in $CO_2$ assimilation$^{43}$ and internal metabolic fluxes$^{44}$; the narrow range of $\Delta^{13}C^{13}C$ values can be explained by the metabolic origin of sugars, i.e., the Calvin-Benson-Bassham (CBB) cycle$^{19}$. The C-C bonds of glucose are derived from the carboxylation, aldolisation, and transketolisation reactions in the CBB cycle. Enzymes facilitate these reactions, and while it is known that the carboxylation reaction is irreversible, the aldolisation and ketolisation reactions are at equilibrium$^{45}$. The measured ethanol$^{19}$ is derived from the carbons in the $C_1-C_2$ and $C_5-C_6$ positions of glucose which are produced in an aldolisation reaction$^{45}$. In this scenario, the $\Delta^{13}C^{13}C$ values of ethanol could be controlled by thermodynamic equilibrium, which enriches the doubly substituted isotopologue, leading to a positive $\Delta^{13}C^{13}C$ value compared to the stochastic distribution.

Because glucose is a complex molecule, we performed theoretical calculations to estimate equilibrium distributions of isotopologues of $CH_2=CH_2$, $CH_3=CH_3$, $CH_3=CH_2OH$, and $CH_2OH-CH_2OH$ as model molecules with C-C bonds (Supplementary Fig. 3 and Supplementary Table 5). The clumped-isotope signature at an isotopic equilibrium was calculated by applying the Bigeleisen-Mayer equation$^{46,47}$. The molecular constants within the Bigeleisen-Mayer equation were obtained through quantum chemical calculations. We used B3LYP/6-311+G (d, p) level$^{48-50}$ for geometry optimisations, single-point energy calculations, and harmonic frequency generations. All calculations were performed using the Gaussian 09 package$^{51}$. 'Very tight' geometry convergence criteria and 'superfine' grids built in Gaussian 09 were applied for geometry optimisation procedures and further computations.

The calculated $\Delta^{13}C^{13}C$ values for ethane were in good agreement with the previous studies$^{52}$. The obtained $\Delta^{13}C^{13}C$ differed by up to 0.02‰ among $CH_3=CH_3$, $CH_3=CH_2OH$, and $CH_2OH-CH_2OH$ molecules; this difference is sufficiently low compared to the analytical accuracy of $\Delta^{13}C^{13}C$ (Supplementary Fig. 3 and Supplementary Table 5). Thus, we used data obtained from $CH_2OH-CH_2OH$ to predict $\Delta^{13}C^{13}C$ of the $C_1-C_2$ and $C_5-C_6$ bonds in glucose under thermodynamic equilibrium, because the carbons in the $C_1-C_2$ and $C_5-C_6$ positions of glucose are composed as $CH_2OH-CHOH$ and $CHO-CHOH$, respectively. The $\Delta^{13}C^{13}C$ of the used $C_2F_6$ standard gas in this study was $-0.72\%$ compared to the stochastic distribution by assuming that the average of $\Delta^{13}C^{13}C$ of bio-ethanol (+0.93‰) is in thermodynamic equilibrium and using the results of the theoretical calculations for $CH_2OH-CH_2OH$ (+0.2‰ at 25 °C). In natural ethanol, the $\delta^{13}C$ values in positions $CH_3$ and $CH_2OH$ can be different by up to 11.4‰, which would result in lower $\Delta^{13}C^{13}C$ values compared with the stochastic distribution owing to combinatorial effects$^{25}$. However, the combinatorial effect in ethanol measured in this study is up to $-0.03\%$ considering a site-specific $^{13}C$ distribution in ethanol of 11.4‰ at maximum$^{53}$. The combinatorial effects calculated here are much lower than the analytical precision of $\pm 0.09\%$ and can thus be quantitatively neglected here.

Measurement of $\delta^{13}C$ values of hydrocarbons

The $\delta^{13}C$ values were determined using gas chromatography coupled with isotope ratio mass spectrometry (DeltaplusXP, Thermo Fisher Scientific K.K., Tokyo, Japan) via a combustion furnace and a conflow
## Methods

### Δ¹³C¹³C notation

The abundance of ¹³C-¹³C isotopologues was conventionally reported as a deviation from the stochastic abundance of the isotopologues:

$$ \Delta^{13}C^{13}C \equiv \frac{^{1313}\text{R}_{\text{sample}}}{^{1313}\text{R}_{\text{stochastic}}} - 1 \quad (6) $$

where ¹³¹R is defined as the abundance of ¹³C-¹³C isotopologues compared to ¹²C-¹²C isotopologues, and R<sub>stochastic</sub> refers to the abundance ratio in a random distribution. The stochastic distribution of C₂ isotopologues is calculated as follows:

$$ ^{1313}\text{R}_{\text{stochastic}} = ^{13}\text{R} \times ^{13}\text{R} \quad (7) $$

where ¹³R indicates the ¹³C/¹²C ratio in all C₂ molecules. In this study, we report Δ¹³C¹³C' as the natural logarithm of α:

$$ \Delta^{13}C^{13}C' \equiv \ln(\alpha) \approx (\alpha - 1); (\text{since } \alpha \approx 1) \quad (8) $$

where α represents the equilibrium constant of the homogeneous isotope exchange reaction:

$$ 2^{12}\text{C}^{13}\text{C} \rightleftharpoons 12^{12}\text{C}^{12}\text{C} + 13^{13}\text{C}^{13}\text{C} \quad (9) $$

$$ \alpha \equiv [^{12}\text{C}^{12}\text{C}][^{13}\text{C}^{13}\text{C}]/[^{12}\text{C}^{13}\text{C}]^2 \approx ^{1313}\text{R}/(2 \times ^{13}\text{R})^2 \quad (10) $$

where ¹³R is calculated from the isotopologue ratio of [¹²C¹³C]/[¹²C¹²C] divided by 2, reflecting the symmetry of two carbon atoms in ethane. The Δ¹³C¹³C' value is approximately equal to that of the conventional Δ¹³C¹³C (Eq. (6)). In the case of C₂ compounds, however, it is difficult to determine the ¹³¹R<sub>stochastic</sub> accurately because C-C bond breaking and recombination do not usually occur reversibly. Thus, calibrating the value with experiments at different temperatures is not feasible. Therefore, Δ¹³C¹³C* value is calculated as follows:

$$ \begin{aligned} \Delta^{13}C^{13}C^{*} &= \Delta^{13}C^{13}C'_{\text{sample}} - \Delta^{13}C^{13}C'_{\text{reference}} \\ &= \ln(\frac{^{1313}\text{R}_{\text{sample}}}{^{13}\text{R}^2_{\text{sample}}}) - \ln(\frac{^{1313}\text{R}_{\text{reference}}}{^{13}\text{R}^2_{\text{reference}}}) \\ &= \ln(\frac{^{1313}\text{R}_{\text{sample}}}{^{1313}\text{R}_{\text{reference}}}) - 2 \times \ln(\frac{^{13}\text{R}_{\text{sample}}}{^{13}\text{R}_{\text{reference}}}) \\ &= \delta^{13}C^{13}C' - 2 \times \delta^{13}C' \end{aligned} \quad (11) $$

where δ¹³C¹³C' and δ¹³C' represent the ratio of ¹³¹R and ¹³R among sample and reference gases as the natural logarithm. Note that all isotope values (Δ¹³C¹³C*, δ¹³C¹³C' and δ¹³C') are expressed in ‰.

### Measurement of Δ¹³C¹³C

We used a fluorination method for the measurement of ¹³C-¹³C species¹⁹,²⁰, which is based on the fluorination of C₂ compounds to hexafluoroethane (C₂F₆) and subsequent measurement of its ¹³C isotopologues with a conventional isotope ratio mass spectrometer. The purified C₂F₆ was introduced into a mass spectrometer (Thermo Fischer MAT253), used in the conventional dual inlet mode. The typical standard deviation of the mean for δ¹³C', δ¹³C¹³C', and Δ¹³C¹³C* values was ±0.01‰, ±0.09‰, and ±0.09‰, respectively (n = 6).

Consequently, we define the empirical transfer function as follows:

$$ \Delta^{13}C^{13}C^{*}_{\text{CSC}} = \lambda \times \Delta^{13}C^{13}C^{*} \quad (12) $$

where Δ¹³C¹³C*<sub>CSC</sub> value refers to the 'true scale' value, corrected for scale compression. The λ value is 1.2541 ± 0.0101 for ethanol and ethene, whereas 1 for ethane since the latter is not prone to scrambling which may have occurred during the fluorination of ethene but not in the ion source of the mass spectrometer²⁰. For simplicity, we will

describe the corrected Δ¹³C¹³C*<sub>CSC</sub> as Δ¹³C¹³C in the following and main text.

### Reference frame for Δ¹³C¹³C measurement

The Δ¹³C¹³C values obtained as described above are relative and are not referred to against the stochastic distribution but a working standard (C₂F₆)¹⁹,²⁰. To obtain values reported against a stochastic distribution, we constructed a reference frame for ¹³C-¹³C isotopologues analysis by estimating that C-C bonds in biological glucose could be under thermodynamic isotope equilibrium. The Δ¹³C¹³C of ethanol produced by the fermentation of sugars from different plants (C₃, C₄, and CAM) is reportedly uniform, whereas the ¹³C/¹²C ratio and position-specific isotope composition vary¹⁹. The latter has been explained by differences in CO₂ assimilation⁴³ and internal metabolic fluxes⁴⁴; the narrow range of Δ¹³C¹³C values can be explained by the metabolic origin of sugars, i.e., the Calvin-Benson-Bassham (CBB) cycle¹⁹. The C-C bonds of glucose are derived from the carboxylation, aldolisation, and transketolisation reactions in the CBB cycle. Enzymes facilitate these reactions, and while it is known that the carboxylation reaction is irreversible, the aldolisation and ketolisation reactions are at equilibrium⁴⁵. The measured ethanol¹⁹ is derived from the carbons in the C₁-C₂ and C₅-C₆ positions of glucose which are produced in an aldolisation reaction⁴⁵. In this scenario, the Δ¹³C¹³C values of ethanol could be controlled by thermodynamic equilibrium, which enriches the doubly substituted isotopologue, leading to a positive Δ¹³C¹³C value compared to the stochastic distribution.

Because glucose is a complex molecule, we performed theoretical calculations to estimate equilibrium distributions of isotopologues of CH₂=CH₂, CH₃-CH₃, CH₃-CH₂OH, and CH₂OH-CH₂OH as model molecules with C-C bonds (Supplementary Fig. 3 and Supplementary Table 5). The clumped-isotope signature at an isotopic equilibrium was calculated by applying the Bigeleisen-Mayer equation⁴⁶,⁴⁷. The molecular constants within the Bigeleisen-Mayer equation were obtained through quantum chemical calculations. We used B3LYP/6-311+G (d, p) level⁴⁸⁻⁵⁰ for geometry optimisations, single-point energy calculations, and harmonic frequency generations. All calculations were performed using the Gaussian 09 package⁵³. 'Very tight' geometry convergence criteria and 'superfine' grids built in Gaussian 09 were applied for geometry optimisation procedures and further computations.

The calculated Δ¹³C¹³C values for ethane were in good agreement with the previous studies⁵². The obtained Δ¹³C¹³C differed by up to 0.02‰ among CH₃-CH₃, CH₃-CH₂OH, and CH₂OH-CH₂OH molecules; this difference is sufficiently low compared to the analytical accuracy of Δ¹³C¹³C (Supplementary Fig. 3 and Supplementary Table 5). Thus, we used data obtained from CH₂OH-CH₂OH to predict Δ¹³C¹³C of the C₁-C₂ and C₅-C₆ bonds in glucose under thermodynamic equilibrium, because the carbons in the C₁-C₂ and C₅-C₆ positions of glucose are composed as CH₂OH-CHOH and CHO-CHOH, respectively. The Δ¹³C¹³C of the used C₂F₆ standard gas in this study was -0.72‰ compared to the stochastic distribution by assuming that the average of Δ¹³C¹³C of bio-ethanol (+0.93‰) is in thermodynamic equilibrium and using the results of the theoretical calculations for CH₂OH-CH₂OH (+0.2‰ at 25 °C). In natural ethanol, the δ¹³C values in positions CH₃ and CH₂OH can be different by up to 11.4‰, which would result in lower Δ¹³C¹³C values compared with the stochastic distribution owing to combinatorial effects²⁵. However, the combinatorial effect in ethanol measured in this study is up to -0.03‰ considering a site-specific ¹³C distribution in ethanol of 11.4‰ at maximum⁵³. The combinatorial effects calculated here are much lower than the analytical precision of ±0.09‰ and can thus be quantitatively neglected here.

### Measurement of δ¹³C values of hydrocarbons

The δ¹³C values were determined using gas chromatography coupled with isotope ratio mass spectrometry (DeltaplusXP, Thermo Fisher Scientific K.K., Tokyo, Japan) via a combustion furnace and a conflow
interface (GC Combustion III, Thermo Fisher Scientific K.K., Tokyo, Japan) (GC-C-IRMS). The gas chromatography column used was HP-PLOT-Q (30 m × 0.32 mm i.d., 10 µm film thickness; GL Sciences Inc., Tokyo, Japan), and the carrier gas was high-purity helium (99.999%; Fujii Co.). The conditions of the GC oven were as follows: injector temperature 250 °C; split mode (variable split ratio); flow rate 1.5 mL/min; oven temperature programme 50 °C (maintained for 5 min) raised to 200 °C (maintained for 10 min) at a rate of 10 °C/min. The combustion furnace consisted of a ceramic tube packed with CuO, NiO, and Pt wires, operating at 960 °C. Isotopic standardisation was made by CO₂ injections calibrated against the natural gas standard NGS-2 provided by the National Institute of Standards and Technology (NIST), Gaithersburg, MD, USA.

## Measurement of intramolecular δ¹³C composition in propane

Intramolecular δ¹³C bias of propane was determined by an online pyrolysis system coupled with GC-C-IRMS⁵⁴. Isotopic standardisation was made by CO₂ injections calibrated against the NIST natural gas standard NGS-2. The relative δ¹³C enrichment in a given position is defined as the difference in isotopic composition between central and terminal carbon positions. Three fragments are used for its calculation: CH₄, C₂H₄, and C₂H₆. CH₄ and C₂H₆ arise from the terminal position only, while C₂H₄ arises from an equal contribution of terminal and central positions. The relative δ¹³C enrichment in the central position (=Δ¹³C<sub>propane</sub>, expressed in ‰) is defined as follows:

$$ \Delta^{13}C_{\text{propane}} = \delta^{13}C_{\text{central}} - \delta^{13}C_{\text{terminal}} \quad (13) $$

where δ¹³C<sub>central</sub> and δ¹³C<sub>terminal</sub> refer to the carbon isotope composition of central and terminal positions, respectively.

## Purification of ethane and ethene

Ethene and ethane must be purified before the fluorination reaction¹⁹,²⁰. These gases were purified using a gas chromatograph GC-4000 plus (GL Sciences Inc., Tokyo, Japan) equipped with Hayesep Q column (1/8" od., 60/80 mesh, 4 m; GL Sciences Inc., Tokyo, Japan) connected to a vacuum line. The sample was directly introduced into the system using a gas-tight syringe through a rubber septum. In both cases, condensable products were trapped at -196 °C (liquid nitrogen), and the remaining non-condensable gases were evacuated under a vacuum. The condensed products were then released with a water bath at room temperature (25 °C) before being introduced into the gas chromatograph. High-purity helium (99.999%; Fujii Co.) was used as the carrier gas. Ethane and ethene could be separated and collected through the 6-port switching valve due to their different retention times. Other impurities were discarded. The conditions of the GC oven were as follows: injector temperature 120 °C; column pressure1200 kPa column; pressure2 160 kPa at 80 °C; oven temperature programme 35 °C (maintained for 15 min) raised to 200 °C (maintained for 16 min) at a rate of 60 °C/min.

## UV irradiation experiment

UV irradiation of methane was conducted in a glass flask (457 mL) used in a previous study³⁸. The top of the flask is made of UV-grade synthetic quartz window, which is transparent for >175 nm photon. A high-pressure xenon arc lamp (Xe lamp: Cermax, CX-04E, output setting 20 A) was used as the UV source, with a solar-like UV spectrum used in a previous report³⁸. Before UV irradiation, 50 mL of doubly distilled water was injected through a syringe port. The water was frozen using liquid nitrogen, and the remaining gas was evacuated from the vacuum line to remove the CO₂ or O₂ trapped in the water. Then, methane (purity 99.9%, GL Science Inc.) was introduced without purification into the flask from the vacuum line at 25 °C to a pressure of about 12 kPa. After introducing methane gas, the flask was kept at 25 °C using a water bath (MC-1, ASONE). An aliquot of gas phase was sampled from

the vacuum line to measure the chemical and carbon isotope composition before the UV irradiation (0 h). In this experiment, UV light was irradiated vertically from the top to the liquid water surface. Methane was exposed under UV light for 3 and 16 h. After the irradiation, the gas sample was collected from the vacuum line to a stainless-steel finger.

## Spark discharge experiment

C₂, hydrocarbons were produced by spark discharge of methane gas. This experiment was conducted in a glass flask (457 mL) previously evacuated through a vacuum line. Methane (purity 99.9%, GL Science Inc.) was introduced without purification into the flask using a gas-tight syringe to a pressure of 12 kPa. The glass flask was connected with a tungsten pole through Swagelok, and the pole was connected with a spark discharger (BD-50E Heavy Duty Generator). Spark discharge of methane was conducted cyclically for 15 min and then stopped for 15 min to avoid elevating the temperature of the reaction vessel. The two experiments were conducted at room temperature (25 °C) with a total duration of 15 min and 5 h. Output adjustment of spark discharger was controlled at eight levels. After exposing methane under spark discharge, gas samples were collected through the vacuum line to a stainless-steel finger.

## Gamma-ray irradiation experiment

The gamma-ray irradiation experiment was undertaken at ANSTO, Australia¹²,⁵⁵, where C₂-C₅ light hydrocarbons were synthesised via the ⁶⁰Co gamma-ray radiolysis of methane. The sample used in this study was irradiated for 650 h, and the temperature was maintained at 21 °C.

## Ethane pyrolysis

Ethane (purity 99.9%, GL Science Inc.) was decomposed by heating at 600 °C in a muffle furnace. Ethane was introduced into quartz tubes through a vacuum line using liquid nitrogen. Once sealed with a gas burner supplied with oxygen, the tubes were heated at 600 °C in a muffle furnace. The remaining ethane in the tubes was isolated from the other reaction products and measured by manometry to calculate the percentage of gas remaining and measured for its carbon isotope and clumped-isotope composition (Supplementary Fig. 1 and Supplementary Table 3).

## Pyrolysis of propane

Propane pyrolysis was conducted to produce hydrocarbons. Propane gas (purity 99.9%, GL Science Inc.) was introduced into empty Pyrex tubes through a vacuum line using liquid nitrogen. Once sealed with a gas burner, the tubes were heated at 500 °C in a muffle furnace. The chemical and isotopic compositions of the reaction products were characterised and measured using the methods presented above (Supplementary Table 7). Then, the produced ethene in the tubes was purified from the other reaction products to measure the Δ¹³C¹³C. Ethene was measured instead of ethane because the latter potentially arises from the recombination of CH₃ fragments (2 CH₃ → C₂H₆), not directly from propane cracking, contrary to ethene⁵⁴.

## Natural gas samples

We analysed samples from different natural gas reservoirs: Southwest Ontario Basin (Canada)³¹, Appalachian Basin (United States)⁵⁶,⁵⁷, and Arkoma Basin (United States)³⁸. Based on isotope composition, hydrocarbons in these basins are suggested to be mainly of thermogenic origin (Fig. 1b). We also analysed gases from the Kidd Creek and the Dingo gas field (Amadeus Basin). CSIA suggests that these hydrocarbons are of abiotic origin (Fig. 1b). In addition, we collected natural gas samples from the Tokamachi mud volcano, which is situated in the Tertiary sedimentary basin in Niigata Prefecture, Central Japan. Niigata Basin is part of the wider Green Tuff belt of Honshu, one of the most important petroleum (oil and gas) producing areas in Japan⁵⁹.
interface (GC Combustion III, Thermo Fisher Scientific K.K., Tokyo, Japan) (GC-C-IRMS). The gas chromatography column used was HP-PLOT-Q (30 m × 0.32 mm i.d., 10 µm film thickness; GL Sciences Inc., Tokyo, Japan), and the carrier gas was high-purity helium (99.999%; Fujii Co.). The conditions of the GC oven were as follows: injector temperature 250 °C; split mode (variable split ratio); flow rate 1.5 mL/min; oven temperature programme 50 °C (maintained for 5 min) raised to 200 °C (maintained for 10 min) at a rate of 10 °C/min. The combustion furnace consisted of a ceramic tube packed with CuO, NiO, and Pt wires, operating at 960 °C. Isotopic standardisation was made by CO₂ injections calibrated against the natural gas standard NGS-2 provided by the National Institute of Standards and Technology (NIST), Gaithersburg, MD, USA.

## Measurement of intramolecular δ¹³C composition in propane

Intramolecular δ¹³C bias of propane was determined by an online pyrolysis system coupled with GC-C-IRMS⁵⁴. Isotopic standardisation was made by CO₂ injections calibrated against the NIST natural gas standard NGS-2. The relative δ¹³C enrichment in a given position is defined as the difference in isotopic composition between central and terminal carbon positions. Three fragments are used for its calculation: CH₄, C₂H₄, and C₂H₆. CH₄ and C₂H₆ arise from the terminal position only, while C₂H₄ arises from an equal contribution of terminal and central positions. The relative δ¹³C enrichment in the central position (=Δ¹³C<sub>propane</sub>, expressed in ‰) is defined as follows:

$$ \Delta^{13}C_{\text{propane}} = \delta^{13}C_{\text{central}} - \delta^{13}C_{\text{terminal}} \quad (13) $$

where δ¹³C<sub>central</sub> and δ¹³C<sub>terminal</sub> refer to the carbon isotope composition of central and terminal positions, respectively.

## Purification of ethane and ethene

Ethene and ethane must be purified before the fluorination reaction¹⁹,²⁰. These gases were purified using a gas chromatograph GC-4000 plus (GL Sciences Inc., Tokyo, Japan) equipped with Hayesep Q column (1/8" od., 60/80 mesh, 4 m; GL Sciences Inc., Tokyo, Japan) connected to a vacuum line. The sample was directly introduced into the system using a gas-tight syringe through a rubber septum. In both cases, condensable products were trapped at -196 °C (liquid nitrogen), and the remaining non-condensable gases were evacuated under a vacuum. The condensed products were then released with a water bath at room temperature (25 °C) before being introduced into the gas chromatograph. High-purity helium (99.999%; Fujii Co.) was used as the carrier gas. Ethane and ethene could be separated and collected through the 6-port switching valve due to their different retention times. Other impurities were discarded. The conditions of the GC oven were as follows: injector temperature 120 °C; column pressure1 200 kPa column; pressure2 160 kPa at 80 °C; oven temperature programme 35 °C (maintained for 15 min) raised to 200 °C (maintained for 16 min) at a rate of 60 °C/min.

## UV irradiation experiment

UV irradiation of methane was conducted in a glass flask (457 mL) used in a previous study³⁸. The top of the flask is made of UV-grade synthetic quartz window, which is transparent for >175 nm photon. A high-pressure xenon arc lamp (Xe lamp: Cermax, CX-04E, output setting 20 A) was used as the UV source, with a solar-like UV spectrum used in a previous report³⁸. Before UV irradiation, 50 mL of doubly distilled water was injected through a syringe port. The water was frozen using liquid nitrogen, and the remaining gas was evacuated from the vacuum line to remove the CO₂ or O₂ trapped in the water. Then, methane (purity 99.9%, GL Science Inc.) was introduced without purification into the flask from the vacuum line at 25 °C to a pressure of about 12 kPa. After introducing methane gas, the flask was kept at 25 °C using a water bath (MC-1, ASONE). An aliquot of gas phase was sampled from

the vacuum line to measure the chemical and carbon isotope composition before the UV irradiation (0 h). In this experiment, UV light was irradiated vertically from the top to the liquid water surface. Methane was exposed under UV light for 3 and 16 h. After the irradiation, the gas sample was collected from the vacuum line to a stainless-steel finger.

## Spark discharge experiment

C₂, hydrocarbons were produced by spark discharge of methane gas. This experiment was conducted in a glass flask (457 mL) previously evacuated through a vacuum line. Methane (purity 99.9%, GL Science Inc.) was introduced without purification into the flask using a gas-tight syringe to a pressure of 12 kPa. The glass flask was connected with a tungsten pole through Swagelok, and the pole was connected with a spark discharger (BD-50E Heavy Duty Generator). Spark discharge of methane was conducted cyclically for 15 min and then stopped for 15 min to avoid elevating the temperature of the reaction vessel. The two experiments were conducted at room temperature (25 °C) with a total duration of 15 min and 5 h. Output adjustment of spark discharger was controlled at eight levels. After exposing methane under spark discharge, gas samples were collected through the vacuum line to a stainless-steel finger.

## Gamma-ray irradiation experiment

The gamma-ray irradiation experiment was undertaken at ANSTO, Australia¹²,⁵⁵, where C₂-C₅ light hydrocarbons were synthesised via the ⁶⁰Co gamma-ray radiolysis of methane. The sample used in this study was irradiated for 650 h, and the temperature was maintained at 21 °C.

## Ethane pyrolysis

Ethane (purity 99.9%, GL Science Inc.) was decomposed by heating at 600 °C in a muffle furnace. Ethane was introduced into quartz tubes through a vacuum line using liquid nitrogen. Once sealed with a gas burner supplied with oxygen, the tubes were heated at 600 °C in a muffle furnace. The remaining ethane in the tubes was isolated from the other reaction products and measured by manometry to calculate the percentage of gas remaining and measured for its carbon isotope and clumped-isotope composition (Supplementary Fig. 1 and Supplementary Table 3).

## Pyrolysis of propane

Propane pyrolysis was conducted to produce hydrocarbons. Propane gas (purity 99.9%, GL Science Inc.) was introduced into empty Pyrex tubes through a vacuum line using liquid nitrogen. Once sealed with a gas burner, the tubes were heated at 500 °C in a muffle furnace. The chemical and isotopic compositions of the reaction products were characterised and measured using the methods presented above (Supplementary Table 7). Then, the produced ethene in the tubes was purified from the other reaction products to measure the Δ¹³C¹³C. Ethene was measured instead of ethane because the latter potentially arises from the recombination of CH₃ fragments (2 CH₃ → C₂H₆), not directly from propane cracking, contrary to ethene⁵⁴.

## Natural gas samples

We analysed samples from different natural gas reservoirs: Southwest Ontario Basin (Canada)³¹, Appalachian Basin (United States)⁵⁶,⁵⁷, and Arkoma Basin (United States)³⁸. Based on isotope composition, hydrocarbons in these basins are suggested to be mainly of thermogenic origin (Fig. 1b). We also analysed gases from the Kidd Creek and the Dingo gas field (Amadeus Basin). CSIA suggests that these hydrocarbons are of abiotic origin (Fig. 1b). In addition, we collected natural gas samples from the Tokamachi mud volcano, which is situated in the Tertiary sedimentary basin in Niigata Prefecture, Central Japan. Niigata Basin is part of the wider Green Tuff belt of Honshu, one of the most important petroleum (oil and gas) producing areas in Japan⁵⁹.
**Southwest Ontario Basin.** The sedimentary strata of Southwestern Ontario consist of Late Cambrian to Devonian sediments. The samples analysed in this study are of Silurian to Middle Ordovician age and were collected between June 2012 and December 2013. For more details on the basin and sampling method, please refer to a previous study³¹.

**Appalachian Basin.** Samples from the Appalachian Basin are gas seeps collected south of Lake Erie in upstate New York in June 2018. In this region, natural gas seeps are abundant and generally found bubbling in small water ponds or riverbeds on fractured shales of Ordovician and Devonian ages³⁶,⁵⁷. Specifically, samples collected and measured here are those defined in ref. 56 as: Amherst State Park, Barcelona Gas Spring, Chestnut Ridge Eternal Flame, Gasport, and from Pipe Creek. These samples all have signatures of thermogenic generation with methane δ¹³C ranging between -42 and -52‰, low C₁/C₂+ ranging from 1.5 to 10, and ethane concentrations of 7–24 vol.%. The samples were collected by setting on the gas seeps an inverted funnel connected to Tygon tubing and a flow-through gas chamber to avoid air contamination. Gases are then sampled from the gas chamber with a gas-tight syringe and added to a pre-evacuated 60 cc serum vial pre-poisoned with HgCl₂.

**Arkoma Basin, Oklahoma, USA.** Hydrocarbons, the main material for shale gas, have been developed from late Devonian and early Mississippian formations, such as the Woodford Shale in Oklahoma and the Chattanooga Shale in Arkansas. Gas was collected from gas wells using standard well-sampling techniques⁵⁸.

**Tokamachi mud volcano, Niigata, Japan.** Tokamachi area consists of two active mud volcanoes, Murono and Gamo, located 10 km west of Tokamachi village. At Murono, groundwater, mud, and gases erupt at the bubbling crater; natural gas also seeps from cracks formed in the asphalt pavement along the road of a car test track built around the mud volcano; the crack-seepage mainly occurs in at least two sites. At Gamo, only two small mud craters were found during the survey performed in this study. The gas samples were sampled using the water-displacement method and stored in a glass vial sealed with a butyl rubber septum.

**Kidd Creek, Timmins, Ontario, Canada.** Sampling and characterisation of fracture fluids located at 2.4 km below the surface in a mine operating to 3 km depth in the 2.7-billion-year-old rocks of the Canadian Shield in a Cu-Ag-Zn deposit (stratiform volcanogenic massive sulfide) hosted in interlayered felsic, mafic, ultramafic and metasedimentary deposits that form part of Abitibi greenstone belt as described in previous studies¹⁰,⁴¹,⁶⁰⁻⁶⁴. We analysed ethane samples from a borehole at 7850' level in 2014 and at 9500' level in 2012 (Supplementary Table 2). These samples were stored in a glass vial sealed with a butyl rubber septum. Evidence from bulk carbon, hydrogen isotope signatures, and clumped methane has demonstrated that hydrocarbons of abiotic origin are predominant in Kidd Creek¹⁰,⁴¹,⁶².

**Amadeus Basin, Australia.** Gas samples from Dingo gas field in Amadeus Basin were collected as described in previous study⁶⁵. Both carbon and hydrogen signatures imply that hydrocarbons from Dingo gas field are derived from abiotic origins¹².

### Evaluation of the combinatorial effect of thermogenic ethane

We used the theoretical model to account for the carbon isotope composition of natural hydrocarbon gases produced by thermal cracking²¹⁻²³. Various alkyl groups attached to a large kerogen molecule are assumed to produce hydrocarbon gases. In this model, the carbon atoms of any individual natural gas hydrocarbon molecule are defined as 'Cn'. The carbon atom arising from the C-C bond breaking of

the alkyl chain is defined as 'Cn'. The remaining C-atoms in the hydrocarbons are defined as 'Cp'. Considering only primary isotope effects, the carbon atom of Cn is enriched in ¹²C because of the C-C bond breaking. However, the C-atoms (Cp) are unaffected by any isotope fractionation, thus recording the original isotope composition of the alkyl chain in kerogen. A given hydrocarbon with n atoms has one Cn atom and (n-1) Cp atoms. Therefore, its carbon isotope composition is:

$$ \delta^{13}C_n = - \Delta_q/n + \delta^{13}C_p \quad (14) $$

If the δ¹³Cn values are plotted as a function of 1/n ('natural gas plot' of ref. 21), the slope and the intercept of this plot represent the Δq (= δ¹³Cₚ - δ¹³Cₘ) and the δ¹³Cₚ values, respectively, based on Eq. (14). The slope of the plot represents the isotope fractionation associated with C-C bond breaking, leading to the difference in the carbon isotope composition between two carbon atoms of ethane ('intramolecular bias'). Based on this model, the ethane produced contains two symmetrically equivalent carbon atoms, but they originate from precursor sites with different kinetic isotope effects during ethane production. The ¹³C/¹²C ratio of ethane (RAVE) is the average of the two carbon positions, and because of the symmetry of ethane, the probability of ¹³C₂H₆ formation is expected to be proportional to the square of that average ratio (Rstochastic). In practice, however, the probability of the formation of ¹³C₂H₆ (R26/24) is proportional to the product of the ¹³C/¹²C ratio of the two different carbon positions. The deviation of the ¹³C-¹³C isotopologues abundance ratio of ethane from stochastic (Δ¹³C¹³CComb) is expressed as follows:

$$ \Delta^{13}C^{13}C_{\text{Comb}} = 1000(R_{26/24}/R_{\text{AVE}}^2 - 1) \quad (15) $$

$$ R_{AVE} = (R_A + R_B)/2 \quad (16) $$

$$ R_{26/24} = R_A \times R_B \quad (17) $$

where Rₐ and Rₑ represent the ¹³C/¹²C ratio for the two carbon positions Cₐ and Cₑ of ethane, respectively. The solution to the simultaneous equations of Eqs. (16) and (17) can be expressed as:

$$ R_A = R_{AVE} + (R_{AVE}^2 - R_{26/24})^{0.5} \quad (18) $$

$$ R_B = R_{AVE} - (R_{AVE}^2 - R_{26/24})^{0.5} \quad (19) $$

Therefore, the difference in the ¹³C/¹²C ratio between the two carbon positions of ethane can be expressed as:

$$ R_A - R_B = 2R_{AVE}(1 - R_{26/24}/R_{AVE}^2)^{0.5} \quad (20) $$

Converting Eq. (20) to δ values with respect to the standard (Rstd) is expressed as:

$$ \Delta^{13}C_{\text{Ethane}} = 2 \times (\delta^{13}C_{\text{Ethane}} + 1000) \times (-\Delta^{13}C^{13}C_{\text{Comb}}/1000)^{0.5} \quad (21) $$

where Δ¹³Cₚ and δ¹³Cₚ represent the difference and average of the two carbon positions in δ¹³C between the two carbon positions of ethane. Equation (21) can be rearranged as:

$$ \Delta^{13}C^{13}C_{\text{Comb}} = - (\Delta^{13}C_{\text{Ethane}}/2(\delta^{13}C_{\text{Ethane}} + 1000))^{2} \times 1000 \quad (22) $$

Combinatorial isotope effects associated with intramolecular bias in the organic precursor can be estimated through the position-specific isotope composition of long-chain alkanes measured by nuclear magnetic resonance⁶⁶. The latter shows differences in δ¹³C
**Southwest Ontario Basin.** The sedimentary strata of Southwestern Ontario consist of Late Cambrian to Devonian sediments. The samples analysed in this study are of Silurian to Middle Ordovician age and were collected between June 2012 and December 2013. For more details on the basin and sampling method, please refer to a previous study³¹.

**Appalachian Basin.** Samples from the Appalachian Basin are gas seeps collected south of Lake Erie in upstate New York in June 2018. In this region, natural gas seeps are abundant and generally found bubbling in small water ponds or riverbeds on fractured shales of Ordovician and Devonian ages³⁶,⁵⁷. Specifically, samples collected and measured here are those defined in ref. 56 as: Amherst State Park, Barcelona Gas Spring, Chestnut Ridge Eternal Flame, Gasport, and from Pipe Creek. These samples all have signatures of thermogenic generation with methane δ¹³C ranging between -42 and -52‰, low C₁/C₂+ ranging from 1.5 to 10, and ethane concentrations of 7-24 vol.%. The samples were collected by setting on the gas seeps an inverted funnel connected to Tygon tubing and a flow-through gas chamber to avoid air contamination. Gases are then sampled from the gas chamber with a gas-tight syringe and added to a pre-evacuated 60 cc serum vial pre-poisoned with HgCl₂.

**Arkoma Basin, Oklahoma, USA.** Hydrocarbons, the main material for shale gas, have been developed from late Devonian and early Mississippian formations, such as the Woodford Shale in Oklahoma and the Chattanooga Shale in Arkansas. Gas was collected from gas wells using standard well-sampling techniques⁵⁸.

**Tokamachi mud volcano, Niigata, Japan.** Tokamachi area consists of two active mud volcanoes, Murono and Gamo, located 10 km west of Tokamachi village. At Murono, groundwater, mud, and gases erupt at the bubbling crater; natural gas also seeps from cracks formed in the asphalt pavement along the road of a car test track built around the mud volcano; the crack-seepage mainly occurs in at least two sites. At Gamo, only two small mud craters were found during the survey performed in this study. The gas samples were sampled using the water-displacement method and stored in a glass vial sealed with a butyl rubber septum.

**Kidd Creek, Timmins, Ontario, Canada.** Sampling and characterisation of fracture fluids located at 2.4 km below the surface in a mine operating to 3 km depth in the 2.7-billion-year-old rocks of the Canadian Shield in a Cu-Ag-Zn deposit (stratiform volcanogenic massive sulfide) hosted in interlayered felsic, mafic, ultramafic and metasedimentary deposits that form part of Abitibi greenstone belt as described in previous studies¹⁰,⁴¹,⁶⁰⁻⁶⁴. We analysed ethane samples from a borehole at 7850' level in 2014 and at 9500' level in 2012 (Supplementary Table 2). These samples were stored in a glass vial sealed with a butyl rubber septum. Evidence from bulk carbon, hydrogen isotope signatures, and clumped methane has demonstrated that hydrocarbons of abiotic origin are predominant in Kidd Creek¹⁰,⁴¹,⁶².

**Amadeus Basin, Australia.** Gas samples from Dingo gas field in Amadeus Basin were collected as described in previous study⁶⁵. Both carbon and hydrogen signatures imply that hydrocarbons from Dingo gas field are derived from abiotic origins¹².

### Evaluation of the combinatorial effect of thermogenic ethane

We used the theoretical model to account for the carbon isotope composition of natural hydrocarbon gases produced by thermal cracking²¹⁻²³. Various alkyl groups attached to a large kerogen molecule are assumed to produce hydrocarbon gases. In this model, the carbon atoms of any individual natural gas hydrocarbon molecule are defined as 'Cn'. The carbon atom arising from the C-C bond breaking of

the alkyl chain is defined as 'Cn'. The remaining C-atoms in the hydrocarbons are defined as 'Cp'. Considering only primary isotope effects, the carbon atom of Cn is enriched in ¹²C because of the C-C bond breaking. However, the C-atoms (Cp) are unaffected by any isotope fractionation, thus recording the original isotope composition of the alkyl chain in kerogen. A given hydrocarbon with n atoms has one Cn atom and (n-1) Cp atoms. Therefore, its carbon isotope composition is:

$$ \delta^{13}C_n = - \Delta_q/n + \delta^{13}C_p \quad (14) $$

If the δ¹³Cn values are plotted as a function of 1/n ('natural gas plot' of ref. 21), the slope and the intercept of this plot represent the Δq (= δ¹³Cp - δ¹³Cm) and the δ¹³Cp values, respectively, based on Eq. (14). The slope of the plot represents the isotope fractionation associated with C-C bond breaking, leading to the difference in the carbon isotope composition between two carbon atoms of ethane ('intramolecular bias'). Based on this model, the ethane produced contains two symmetrically equivalent carbon atoms, but they originate from precursor sites with different kinetic isotope effects during ethane production. The ¹³C/¹²C ratio of ethane (RAVE) is the average of the two carbon positions, and because of the symmetry of ethane, the probability of ¹³C₂H₆ formation is expected to be proportional to the square of that average ratio (Rstochastic). In practice, however, the probability of the formation of ¹³C₂H₆ (R26/24) is proportional to the product of the ¹³C/¹²C ratio of the two different carbon positions. The deviation of the ¹³C-¹³C isotopologues abundance ratio of ethane from stochastic (Δ¹³C¹³CComb) is expressed as follows:

$$ \Delta^{13}C^{13}C_{\text{Comb}} = 1000(R_{26/24}/R_{\text{AVE}}^2 - 1) \quad (15) $$

$$ R_{AVE} = (R_A + R_B)/2 \quad (16) $$

$$ R_{26/24} = R_A \times R_B \quad (17) $$

where R_A and R_B represent the ¹³C/¹²C ratio for the two carbon positions C_A and C_B of ethane, respectively. The solution to the simultaneous equations of Eqs. (16) and (17) can be expressed as:

$$ R_A = R_{AVE} + (R_{AVE}^2 - R_{26/24})^{0.5} \quad (18) $$

$$ R_B = R_{AVE} - (R_{AVE}^2 - R_{26/24})^{0.5} \quad (19) $$

Therefore, the difference in the ¹³C/¹²C ratio between the two carbon positions of ethane can be expressed as:

$$ R_A - R_B = 2R_{AVE}(1 - R_{26/24}/R_{AVE}^2)^{0.5} \quad (20) $$

Converting Eq. (20) to δ values with respect to the standard (Rstd) is expressed as:

$$ \Delta^{13}C_{\text{Ethane}} = 2 \times (\delta^{13}C_{\text{Ethane}} + 1000) \times (-\Delta^{13}C^{13}C_{\text{Comb}}/1000)^{0.5} \quad (21) $$

where Δ¹³CEthane and δ¹³CEthane represent the difference and average of the two carbon positions in δ¹³C between the two carbon positions of ethane. Equation (21) can be rearranged as:

$$ \Delta^{13}C^{13}C_{\text{Comb}} = - (\Delta^{13}C_{\text{Ethane}}/2(\delta^{13}C_{\text{Ethane}} + 1000))^{2} \times 1000 \quad (22) $$

Combinatorial isotope effects associated with intramolecular bias in the organic precursor can be estimated through the position-specific isotope composition of long-chain alkanes measured by nuclear magnetic resonance⁶⁶. The latter shows differences in δ¹³C
values between two adjacent positions (=$\delta^{13}C_{CH3} - \delta^{13}C_{CH2}$) of ca. -3.9‰ (the C₁₆-C₃₁ range with odd carbon number), 10.4‰ (the C₁₆-C₃₁ range with even carbon number), and -12.5‰ (the C₁₁-C₁₅ range with odd and even carbon number)⁶⁶, which corresponds to depletion of Δ¹³C¹³C values of -0.004‰, -0.03‰, and -0.04‰, respectively. The combinatorial effects calculated here are much lower than the analytical precision of ±0.09‰ and can thus be quantitatively neglected.

## Data availability

All data generated or analysed during this study are included in this published article (and its supplementary information files).

## References

1. Pasteur, L. Recherches sur les relations qui peuvent exister entre la forme crystalline, la composition chimique et le sens de la polarisation rotatoire. *Ann. Chim. Phys.* **24**, 442-459 (1848).

2. House, C. H. et al. Depleted carbon isotope compositions observed at Gale crater, Mars. Proc. Natl Acad. Sci. USA. **119**, e2115651119 (2022).

3. Marshall, S. M. et al. Identifying molecules as biosignatures with assembly theory and mass spectrometry. *Nat. Commun.* **12**, 3033 (2021).

4. Webster, C. R. et al. Mars methane detection and variability at Gale crater. *Science* **347**, 415-417 (2015).

5. Waite, J. H. et al. Cassini finds molecular hydrogen in the Enceladus plume: evidence for hydrothermal processes. *Science* **356**, 155-159 (2017).

6. Hayes, J. M. Organic constituents of meteorites - a review. *Geochim. Cosmochim. Acta* **31**, 1395-1440 (1967).

7. Schoell, M. Genetic characterization of natural gases. *Am. Assoc. Petrol. Geol. Bull.* **67**, 2225-2238 (1983).

8. Whiticar, M. J., Faber, E. & Schoell, M. Biogenic methane formation in marine and freshwater environments: CO₂ reduction vs. acetate fermentation - Isotope evidence. *Geochim. Cosmochim. Acta* **50**, 693-709 (1986).

9. Kutcherov, V. G. & Krayushkin, V. A. Deep-seated abiogenic origin of petroleum: from geological assessment to physical theory. *Rev. Geophys.* **48**, RG1001 (2010).

10. Sherwood Lollar, B., Westgate, T. D., Ward, J. A., Slater, G. F. & Lacrampe-Couloume, G. Abiogenic formation of alkanes in the earth's crust as a minor source for global hydrocarbon reservoirs. *Nature* **416**, 522-524 (2002).

11. Etiope, G. & Sherwood Lollar, B. Abiotic methane on Earth. *Rev. Geophys.* **51**, 276-299 (2013).

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29. Hahn, C. J. et al. "Candidatus Ethanoperedens," a thermophilic genus of archaea mediating the anaerobic oxidation of ethane. *mBio* **11**, e00600-20 (2020).

30. Milkov, A. V. & Etiope, G. Revised genetic diagrams for natural gases based on a global dataset of >20,000 samples. *Org. Geochem.* **125**, 109-120 (2018).

31. Gilbert, A. et al. Intramolecular isotopic evidence for bacterial oxidation of propane in subsurface natural gas reservoirs. *Proc. Natl Acad. Sci. USA.* **116**, 6653-6658 (2019).

32. Ash, J. et al. Exchange catalysis during anaerobic methanotrophy revealed by ¹²CH₂D₂ and ¹³CH₃D in methane. *Geochem. Perspect. Lett.* **10**, 26-30 (2019).

33. Takai, K. et al. Cell proliferation at 122 °C and isotopically heavy CH₄ production by a hyperthermophilic methanogen under high-pressure cultivation. *Proc. Natl Acad. Sci. USA.* **105**, 10949-10954 (2008).

34. Yao, Q. et al. Potential energy surface for large barrierless reaction systems: application to the kinetic calculations of the dissociation of alkanes and the reverse recombination reactions. *J. Phys. Chem. A* **122**, 4869-4881 (2018).

35. Tsai, W. C. & Hu, W. P. Theoretical analysis on the kinetic isotope effects of bimolecular nucleophilic substitution (SN₂) reactions and their temperature dependence. *Molecules* **18**, 4816-4843 (2013).

36. Taatjes, C. A. & Klippenstein, S. J. Kinetic isotope effects and variable reaction coordinates in barrierless recombination reactions. *J. Phys. Chem. A* **105**, 8567-8578 (2001).

37. Suda, K., Gilbert, A., Yamada, K., Yoshida, N. & Ueno, Y. Compound-and position-specific carbon isotopic signatures of abiogenic hydrocarbons from on-land serpentinite-hosted Hakuba Happo hot spring in Japan. *Geochim. Cosmochim. Acta* **206**, 201-215 (2017).

38. Zang, X., Ueno, Y. & Kitadai, N. Photochemical synthesis of ammonia and amino acids from nitrous oxide. *Astrobiology* **22**, 387-398 (2022).

39. Wagner, A. F. & Wardlaw, D. M. Study of the recombination reaction CH₃+CH₃→C₂H₆. II: theory. *J. Phys. Chem.* **92**, 2462-2471 (1988).

40. Kado, S. et al. Reaction mechanism of methane activation using non-equilibrium pulsed discharge at room temperature. *Fuel* **82**, 2291-2297 (2003).

41. Sherwood Lollar, B. et al. A window into the abiotic carbon cycle - acetate and formate in fracture waters in 2.7 billion year-old host rocks of the Canadian Shield. *Geochim. Cosmochim. Acta* **294**, 295-314 (2021).
values between two adjacent positions (=$\delta^{13}C_{CH3} - \delta^{13}C_{CH2}$) of ca. -3.9‰ (the C₁₆-C₃₁ range with odd carbon number), 10.4‰ (the C₁₆-C₃₁ range with even carbon number), and -12.5‰ (the C₁₁-C₁₅ range with odd and even carbon number)⁶⁶, which corresponds to depletion of Δ¹³C¹³C values of -0.004‰, -0.03‰, and -0.04‰, respectively. The combinatorial effects calculated here are much lower than the analytical precision of ±0.09‰ and can thus be quantitatively neglected.

## Data availability

All data generated or analysed during this study are included in this published article (and its supplementary information files).

## References

1. Pasteur, L. Recherches sur les relations qui peuvent exister entre la forme crystalline, la composition chimique et le sens de la polarisation rotatoire. *Ann. Chim. Phys.* **24**, 442-459 (1848).

2. House, C. H. et al. Depleted carbon isotope compositions observed at Gale crater, Mars. Proc. Natl Acad. Sci. USA. **119**, e2115651119 (2022).

3. Marshall, S. M. et al. Identifying molecules as biosignatures with assembly theory and mass spectrometry. *Nat. Commun.* **12**, 3033 (2021).

4. Webster, C. R. et al. Mars methane detection and variability at Gale crater. *Science* **347**, 415-417 (2015).

5. Waite, J. H. et al. Cassini finds molecular hydrogen in the Enceladus plume: evidence for hydrothermal processes. *Science* **356**, 155-159 (2017).

6. Hayes, J. M. Organic constituents of meteorites - a review. *Geochim. Cosmochim. Acta* **31**, 1395-1440 (1967).

7. Schoell, M. Genetic characterization of natural gases. *Am. Assoc. Petrol. Geol. Bull.* **67**, 2225-2238 (1983).

8. Whiticar, M. J., Faber, E. & Schoell, M. Biogenic methane formation in marine and freshwater environments: CO₂ reduction vs. acetate fermentation - Isotope evidence. *Geochim. Cosmochim. Acta* **50**, 693-709 (1986).

9. Kutcherov, V. G. & Krayushkin, V. A. Deep-seated abiogenic origin of petroleum: from geological assessment to physical theory. *Rev. Geophys.* **48**, RG1001 (2010).

10. Sherwood Lollar, B., Westgate, T. D., Ward, J. A., Slater, G. F. & Lacrampe-Couloume, G. Abiogenic formation of alkanes in the earth's crust as a minor source for global hydrocarbon reservoirs. *Nature* **416**, 522-524 (2002).

11. Etiope, G. & Sherwood Lollar, B. Abiotic methane on Earth. *Rev. Geophys.* **51**, 276-299 (2013).

12. Boreham, C. J. & Davies, J. B. Carbon and hydrogen isotopes of the wet gases produced by gamma-ray-induced polymerisation of methane: insights into radiogenic mechanism and natural gas formation. *Radiat. Phys. Chem.* **168**, 108546 (2020).

13. Des Marais, D. J., Donchin, J. H., Nehring, N. L. & Truesdell, A. H. Molecular carbon isotopic evidence for the origin of geothermal hydrocarbons. *Nature* **292**, 826-828 (1981).

14. Eiler, J. M. The isotopic anatomies of molecules and minerals. *Annu. Rev. Earth Planet. Sci.* **41**, 411-441 (2013).

15. Wang, D. T. et al. Nonequilibrium clumped isotope signals in microbial methane. *Science* **348**, 428-431 (2015).

16. Yeung, L. Y., Ash, J. L. & Young, E. D. Biological signatures in clumped isotopes of O₂. *Science* **348**, 431-434 (2015).

17. Stolper, D. A. et al. Formation temperatures of thermogenic and biogenic methane. *Science* **344**, 1500-1503 (2014).

18. Giunta, T. et al. Evidence for methane isotopic re-ordering in gas reservoirs sourcing cold seeps from the Sea of Marmara. *Earth Planet. Sci. Lett.* **553**, 116619 (2021).

19. Taguchi, K., Yamamoto, T., Nakagawa, M., Gilbert, A. & Ueno, Y. A fluorination method for measuring the ¹³C-¹³C isotopologue of C₂ molecules. *Rapid Commun. Mass Spectrom.* **34**, 1-10 (2020).

20. Taguchi, K., Gilbert, A. & Ueno, Y. Standardization for ¹³C-¹³C clumped isotope analysis by the fluorination method. *Rapid Commun. Mass Spectrom.* **35**, 1-12 (2021).

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34. Yao, Q. et al. Potential energy surface for large barrierless reaction systems: application to the kinetic calculations of the dissociation of alkanes and the reverse recombination reactions. *J. Phys. Chem. A* **122**, 4869-4881 (2018).

35. Tsai, W. C. & Hu, W. P. Theoretical analysis on the kinetic isotope effects of bimolecular nucleophilic substitution (SN₂) reactions and their temperature dependence. *Molecules* **18**, 4816-4843 (2013).

36. Taatjes, C. A. & Klippenstein, S. J. Kinetic isotope effects and variable reaction coordinates in barrierless recombination reactions. *J. Phys. Chem. A* **105**, 8567-8578 (2001).

37. Suda, K., Gilbert, A., Yamada, K., Yoshida, N. & Ueno, Y. Compound-and position-specific carbon isotopic signatures of abiogenic hydrocarbons from on-land serpentinite-hosted Hakuba Happo hot spring in Japan. *Geochim. Cosmochim. Acta* **206**, 201-215 (2017).

38. Zang, X., Ueno, Y. & Kitadai, N. Photochemical synthesis of ammonia and amino acids from nitrous oxide. *Astrobiology* **22**, 387-398 (2022).

39. Wagner, A. F. & Wardlaw, D. M. Study of the recombination reaction CH₃+CH₃→C₂H₆. II: theory. *J. Phys. Chem.* **92**, 2462-2471 (1988).

40. Kado, S. et al. Reaction mechanism of methane activation using non-equilibrium pulsed discharge at room temperature. *Fuel* **82**, 2291-2297 (2003).

41. Sherwood Lollar, B. et al. A window into the abiotic carbon cycle - acetate and formate in fracture waters in 2.7 billion year-old host rocks of the Canadian Shield. *Geochim. Cosmochim. Acta* **294**, 295-314 (2021).
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49. Lee, C., Yang, W. & Parr, R. G. Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. *Phys. Rev. B* **37**, 785–789 (1988).

50. Frisch, M. J. et al. Self-consistent molecular orbital methods 25. Supplementary functions for Gaussian basis sets. *J. Chem. Phys.* **80**, 3265–3269 (1984).

51. Frisch, M. J. et al. *Gaussian 09, Revision D.01* (2013).

52. Piasecki, A., Sessions, A., Peterson, B. & Eiler, J. Prediction of equilibrium distributions of isotopologues for methane, ethane and propane using density functional theory. *Geochim. Cosmochim. Acta* **190**, 1–12 (2016).

53. Gilbert, A., Yamada, K. & Yoshida, N. Accurate method for the determination of intramolecular ¹³C isotope composition of ethanol from aqueous solutions. *Anal. Chem.* **85**, 6566–6570 (2013).

54. Gilbert, A., Yamada, K., Suda, K., Ueno, Y. & Yoshida, N. Measurement of position-specific ¹³C isotopic composition of propane at the nanomole level. *Geochim. Cosmochim. Acta* **177**, 205–216 (2016).

55. Boreham, C. J. et al. Canberra, Australia. Methane radiolysis: a model for abiogenic gas generation in the Amadeus Basin. *Record. In 20th August Org. Geochem. Conference: Origins of Oil, Old Organics and Organisms: Program and Abstracts 2018/44* (compd Edwards, D. S. et al.) 33–34. https://doi.org/10.11636/Record.2018.044 (Geoscience, Australia, Canberra, 2018).

56. Etiope, G., Drobniak, A. & Schimmelmann, A. Natural seepage of shale gas and the origin of “eternal flames” in the Northern Appalachian Basin, USA. *Mar. Pet. Geol.* **43**, 178–186 (2013).

57. Schimmelmann, A. et al. Natural geological seepage of hydrocarbon gas in the Appalachian Basin and Midwest USA in relation to shale tectonic fracturing and past industrial hydrocarbon production. *Sci. Total Environ.* **644**, 982–993 (2018).

58. Liu, C., Liu, P., McGovern, G. P. & Horita, J. Molecular and intramolecular isotope geochemistry of natural gases from the Woodford Shale, Arkoma Basin, Oklahoma. *Geochim. Cosmochim. Acta* **255**, 188–204 (2019).

59. Etiope, G., Nakada, R., Tanaka, K. & Yoshida, N. Gas seepage from Tokamachi mud volcanoes, onshore Niigata Basin (Japan): origin, post-genetic alterations and CH₄–CO₂ fluxes. *Appl. Geochem.* **26**, 348–359 (2011).

60. Sherwood, B. et al. Methane occurrences in the Canadian Shield. *Chem. Geol.* **71**, 223–236 (1988).

61. Sherwood Lollar, B. et al. Hydrogeologic controls on episodic H₂ release from Precambrian fractured rocks - energy for deep subsurface life on Earth and Mars. *Astrobiology* **7**, 971–986 (2007).

62. Sherwood Lollar, B. et al. Isotopic signatures of CH₄ and higher hydrocarbon gases from Precambrian Shield sites: a model for abiogenic polymerization of hydrocarbons. *Geochim. Cosmochim. Acta* **72**, 4778–4795 (2008).

63. Holland, G. et al. Deep fracture fluids isolated in the crust since the Precambrian era. *Nature* **497**, 357–360 (2013).

64. Warr, O. et al. High-resolution, long-term isotopic and isotopologue variation identifies the sources and sinks of methane in a deep subsurface carbon cycle. *Geochim. Cosmochim. Acta* **294**, 315–334 (2021).

65. Boreham, C. J. & Edwards, D. S. Abundance and carbon isotopic composition of neo-pentane in Australian natural gases. *Org. Geo-chem.* **39**, 550–566 (2008).

66. Gilbert, A., Yamada, K. & Yoshida, N. Exploration of intramolecular ¹³C isotope distribution in long chain n-alkanes (C₁₁–C₃₁) using isotopic ¹³C NMR. *Org. Geochem.* **62**, 56–61 (2013).

67. Polyakov, V. B. & Horita, J. Equilibrium carbon isotope fractionation factors of hydrocarbons: semi-empirical force-field method. *Chem. Geol.* **559**, 119948 (2021).

68. Thiagarajan, N. et al. Isotopic evidence for quasi-equilibrium chemistry in thermally mature natural gases. *Proc. Natl Acad. Sci. USA.* **117**, 3989–3995 (2020).

## Acknowledgements

This work was supported by the Japan Society for the Promotion of Science KAKENHI grant JP21J22057 (K.T.), JP17H01165 (Y.U.), JP21H01198 (A.G.), and JP21K18646 (A.G.). Kidd Creek samples were collected via funding from the Natural Sciences and Engineering Research Council of Canada Discovery Grant (B.S.L.). Further support from Canadian Institute for Advanced Research (CIFAR) is acknowledged for B.S.L as a Fellow of the Earth 4D Subsurface Science and Exploration Program. Justin Davies, Australia's Nuclear Science and Technology Organisation (ANSTO), Australia, is thanked for managing the gamma-ray irradiation of the methane experiment. Dianne Edwards and Emmanuelle Grosjean reviewed a pre-submission version of the paper, leading to many improvements. C.J.B publishes with the permission of the CEO, Geoscience Australia. Financial support was provided by the U.S. Department of Energy Geosciences program DE-SC0016271 (J.H.).

## Author contributions

K.T., Y.U., and A.G. conceived the project. K.T. and A.G. performed the gas isotopic analysis. K.T., Y.U., and A.G. managed the project and prepared the first draft of the manuscript. K.T. and A.G. collected the samples from Tokamachi mud volcano. B.S.L., T.G., C.J.B., and J.H. provided natural and experimental gas samples. Q.L. conducted theoretical calculations to estimate equilibrium distributions of ¹³C isotopologues. All authors contributed to the final version of the manuscript.

## Competing interests

The authors declare no competing interests.

## Additional information

**Supplementary information** The online version contains supplementary material available at https://doi.org/10.1038/s41467-022-33538-9.

**Correspondence** and requests for materials should be addressed to Koudai Taguchi, Alexis Gilbert or Yuichiro Ueno.

**Peer review information** *Nature Communications* thanks Peter Douglas, John Eiler and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

**Reprints and permission information** is available at http://www.nature.com/reprints

**Publisher’s note** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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47. Urey, H. C. The thermodynamic properties of isotopic substances. *J. Chem. Soc. 562–581 (1947).*

48. Becke, A. D. Density-functional thermochemistry. III. The role of exact exchange. *J. Chem. Phys.* **98**, 5648–5652 (1993).

49. Lee, C., Yang, W. & Parr, R. G. Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. *Phys. Rev. B* **37**, 785–789 (1988).

50. Frisch, M. J. et al. Self-consistent molecular orbital methods 25. Supplementary functions for Gaussian basis sets. *J. Chem. Phys.* **80**, 3265–3269 (1984).

51. Frisch, M. J. et al. *Gaussian 09, Revision D.01* (2013).

52. Piasecki, A., Sessions, A., Peterson, B. & Eiler, J. Prediction of equilibrium distributions of isotopologues for methane, ethane and propane using density functional theory. *Geochim. Cosmochim. Acta* **190**, 1–12 (2016).

53. Gilbert, A., Yamada, K. & Yoshida, N. Accurate method for the determination of intramolecular ¹³C isotope composition of ethanol from aqueous solutions. *Anal. Chem.* **85**, 6566–6570 (2013).

54. Gilbert, A., Yamada, K., Suda, K., Ueno, Y. & Yoshida, N. Measurement of position-specific ¹³C isotopic composition of propane at the nanomole level. *Geochim. Cosmochim. Acta* **177**, 205–216 (2016).

55. Boreham, C. J. et al. Canberra, Australia. Methane radiolysis: a model for abiogenic gas generation in the Amadeus Basin. *Record. In 20th August Org. Geochem. Conference: Origins of Oil, Old Organics and Organisms: Program and Abstracts 2018/44* (compd Edwards, D. S. et al.) 33–34. https://doi.org/10.11636/Record.2018.044 (Geoscience, Australia, Canberra, 2018).

56. Etiope, G., Drobniak, A. & Schimmelmann, A. Natural seepage of shale gas and the origin of “eternal flames” in the Northern Appalachian Basin, USA. *Mar. Pet. Geol.* **43**, 178–186 (2013).

57. Schimmelmann, A. et al. Natural geological seepage of hydrocarbon gas in the Appalachian Basin and Midwest USA in relation to shale tectonic fracturing and past industrial hydrocarbon production. *Sci. Total Environ.* **644**, 982–993 (2018).

58. Liu, C., Liu, P., McGovern, G. P. & Horita, J. Molecular and intramolecular isotope geochemistry of natural gases from the Woodford Shale, Arkoma Basin, Oklahoma. *Geochim. Cosmochim. Acta* **255**, 188–204 (2019).

59. Etiope, G., Nakada, R., Tanaka, K. & Yoshida, N. Gas seepage from Tokamachi mud volcanoes, onshore Niigata Basin (Japan): origin, post-genetic alterations and CH₄–CO₂ fluxes. *Appl. Geochem.* **26**, 348–359 (2011).

60. Sherwood, B. et al. Methane occurrences in the Canadian Shield. *Chem. Geol.* **71**, 223–236 (1988).

61. Sherwood Lollar, B. et al. Hydrogeologic controls on episodic H₂ release from Precambrian fractured rocks - energy for deep subsurface life on Earth and Mars. *Astrobiology* **7**, 971–986 (2007).

62. Sherwood Lollar, B. et al. Isotopic signatures of CH₄ and higher hydrocarbon gases from Precambrian Shield sites: a model for abiogenic polymerization of hydrocarbons. *Geochim. Cosmochim. Acta* **72**, 4778–4795 (2008).

63. Holland, G. et al. Deep fracture fluids isolated in the crust since the Precambrian era. *Nature* **497**, 357–360 (2013).

64. Warr, O. et al. High-resolution, long-term isotopic and isotopologue variation identifies the sources and sinks of methane in a deep subsurface carbon cycle. *Geochim. Cosmochim. Acta* **294**, 315–334 (2021).

65. Boreham, C. J. & Edwards, D. S. Abundance and carbon isotopic composition of neo-pentane in Australian natural gases. *Org. Geo-chem.* **39**, 550–566 (2008).

66. Gilbert, A., Yamada, K. & Yoshida, N. Exploration of intramolecular ¹³C isotope distribution in long chain n-alkanes (C₁₁–C₃₁) using isotopic ¹³C NMR. *Org. Geochem.* **62**, 56–61 (2013).

67. Polyakov, V. B. & Horita, J. Equilibrium carbon isotope fractionation factors of hydrocarbons: semi-empirical force-field method. *Chem. Geol.* **559**, 119948 (2021).

68. Thiagarajan, N. et al. Isotopic evidence for quasi-equilibrium chemistry in thermally mature natural gases. *Proc. Natl Acad. Sci. USA.* **117**, 3989–3995 (2020).

## Acknowledgements

This work was supported by the Japan Society for the Promotion of Science KAKENHI grant JP21J22057 (K.T.), JP17H01165 (Y.U.), JP21H01198 (A.G.), and JP21K18646 (A.G.). Kidd Creek samples were collected via funding from the Natural Sciences and Engineering Research Council of Canada Discovery Grant (B.S.L.). Further support from Canadian Institute for Advanced Research (CIFAR) is acknowledged for B.S.L as a Fellow of the Earth 4D Subsurface Science and Exploration Program. Justin Davies, Australia's Nuclear Science and Technology Organisation (ANSTO), Australia, is thanked for managing the gamma-ray irradiation of the methane experiment. Dianne Edwards and Emmanuelle Grosjean reviewed a pre-submission version of the paper, leading to many improvements. C.J.B publishes with the permission of the CEO, Geoscience Australia. Financial support was provided by the U.S. Department of Energy Geosciences program DE-SC0016271 (J.H.).

## Author contributions

K.T., Y.U., and A.G. conceived the project. K.T. and A.G. performed the gas isotopic analysis. K.T., Y.U., and A.G. managed the project and prepared the first draft of the manuscript. K.T. and A.G. collected the samples from Tokamachi mud volcano. B.S.L., T.G., C.J.B., and J.H. provided natural and experimental gas samples. Q.L. conducted theoretical calculations to estimate equilibrium distributions of ¹³C isotopologues. All authors contributed to the final version of the manuscript.

## Competing interests

The authors declare no competing interests.

## Additional information

**Supplementary information** The online version contains supplementary material available at https://doi.org/10.1038/s41467-022-33538-9.

**Correspondence** and requests for materials should be addressed to Koudai Taguchi, Alexis Gilbert or Yuichiro Ueno.

**Peer review information** *Nature Communications* thanks Peter Douglas, John Eiler and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

**Reprints and permission information** is available at http://www.nature.com/reprints

**Publisher's note** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

© The Author(s) 2022
**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

© The Author(s) 2022