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\begin{document}

\title{Clumped isotope signatures of abiotic methane: the role of combinatorial
isotope effect}



\author[1]{Jabrane Labidi}%
\author[1]{Thomas M McCollom}%
\author[1]{Thomas Giunta}%
\author[1]{Barbara Sherwood Lollar}%
\author[1]{EDWARD DONALD YOUNG}%
\author[1]{will leavitt}%
\affil[1]{Affiliation not available}%


\vspace{-1em}



  
  \date{June 17, 2024}


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Labidi, J.,\textsuperscript{1,*}, McCollom, T.
M.,\textsuperscript{2}Giunta, T.,\textsuperscript{3} Sherwood Lollar,
B.,\textsuperscript{1,4}Leavitt, W.,\textsuperscript{5,6} Young
E.D.,\textsuperscript{7}

\textsuperscript{1} Universit\selectlanguage{ngerman}é de Paris Cité, Institut de physique du
globe de Paris, CNRS, Paris, France.

\textsuperscript{2} Laboratory for Atmospheric and Space Physics,
University of Colorado, Boulder, CO, USA

\textsuperscript{3} University of Brest, CNRS, Ifremer, Geo-Ocean,
F-29280 Plouzané, France

\textsuperscript{4} Department of Earth Sciences, University of Toronto,
Toronto, ON, CANADA

\textsuperscript{5} Department of Earth Sciences, Dartmouth College,
Hanover, NH, USA

\textsuperscript{6} Department of Chemistry, Dartmouth College, Hanover,
NH, USA

\textsuperscript{7} Department of Earth, Planetary, and Space Sciences,
UCLA, Los Angeles, CA, USA.

\textsuperscript{*}Correspondence: labidi@ipgp.fr

\emph{Keywords:} Fischer--Tropsch process; abiogenic methane; multiply
substituted isotopologues

\textbf{Abstract}

Methane clumped isotope signatures of abiogenesis may be diagnostic of
the origin of methane on Earth and other planetary bodies. On Earth,
identifying methane synthesis pathways has proven challenging because of
the lack of unambiguous signatures of the provenance of methane
molecules. The abundances of mass-18 isotopologues of
methane,\textsuperscript{13}CH\textsubscript{3}D
and\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2}, could help
identify occurrences of abiotic methane in nature. We performed
synthesis of abiogenic methane in the laboratory and determined
\selectlanguage{greek}δ\selectlanguage{english}\textsuperscript{13}C, \selectlanguage{greek}δ\selectlanguage{english}D, \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D
and \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2}. We
carried out a set of experiments in hydrothermal conditions between 130
and 300 \selectlanguage{ngerman}°C. The experiments were performed by heating water in the
presence of Fe\textsuperscript{0} powder and CO. The reduction of water
on metallic iron led to the formation of H\textsubscript{2}. CO was
reacted with both H\textsubscript{2} and H\textsubscript{2}O, generating
both CH\textsubscript{4} and CO\textsubscript{2}. We systematically
extracted and quantified methane, and determined \selectlanguage{greek}δ\selectlanguage{english}\textsuperscript{13}C,
\selectlanguage{greek}δ\selectlanguage{english}D \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2}, and
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D of the methane using a
Panorama gas-source mass spectrometer (Nu, Ametek). We observe
\selectlanguage{greek}δ\selectlanguage{english}\textsuperscript{13}C and \selectlanguage{greek}δ\selectlanguage{english}D of methane to be isotopically depleted
relative to the starting materials. The \selectlanguage{greek}δ\selectlanguage{english}\textsuperscript{13}C data
indirectly suggest isotopic equilibrium may have been reached for carbon
isotopes between methane, carbon monoxide and carbon dioxide in our
experiments. In contrast, D/H ratios are inconsistent with equilibrium
isotopic fractionation. This suggests that under our experimental
conditions, hydrogen additions to carbon are governed by kinetics, and
that subsequent D/H re-equilibration was limited. While
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D values approximately track
experimental temperature,
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} values are
displaced far from equilibrium. We find exclusively negative
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} values,
showing deficits down to 40%0 relative to thermodynamic equilibrium. We
interpret the data as evidence for distinct, kinetically induced D/H
pools contributing to methane assembly (a combinatorial effect). The
cumulative D/H fractionations associated with CO (or
CO\textsubscript{2}in nature) hydrogenation likely explain the direction
and magnitude of
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} values during
abiotic methane formation. We suggest that near equilibrium
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D with negative
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} signatures
will help identify methane formed abiotically in nature.

\textbf{Plain language summary}

Methane is observed in various environmental settings on Earth,
including but not limited to hydrothermal fluids and sedimentary
systems. It is also found on other worlds, as it was reported to occur
in Mars's rocks, in Enceladus's geysers, and as of recently, in an
exoplanet's atmosphere. Methane may be formed by life, by the
degradation of organic matter, or by abiotic chemistry. However,
identifying synthesis pathways has proven challenging because of the
lack of unambiguous signatures of the provenance of methane molecules.
We performed synthesis of abiotic methane in the laboratory and
determined its composition. We observe isotopic compositions that are
consistent with carbon isotope equilibrium associated with hydrogen
isotope disequilibrium. These isotope signatures are non-unique, in
contrast with mass-18 isotopologues
data,\textsuperscript{13}CH\textsubscript{3}D
and\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2}. While
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D values approximately track
experimental temperature,
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} values are
extremely depleted, far below equilibrium. The data are explained by
kinetically fractionated D/H pools contributing to methane molecules,
i.e., a combinatorial effect. We suggest that near equilibrium
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D with negative
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} signatures
will help identify methane formed abiotically on Earth and on other
worlds.

\textbf{1. Introduction}

Methane (CH\textsubscript{4}) is observed in a variety of environments
on Earth. Abiotic methane (i.e., synthesized by mechanisms that do not
involve life or thermal decomposition of organic matter), is suspected
to occur in various fluids also containing molecular hydrogen
(H\textsubscript{2}). These fluids are especially prominent in
hydrothermal systems where olivine-rich ultramafic rocks are altered via
serpentinization reactions. For instance, elevated concentrations of
CH\textsubscript{4} and H\textsubscript{2} are observed in
ultramafic-hosted seafloor hydrothermal vent fluids at temperatures
typically exceeding 250 \selectlanguage{ngerman}°C (Charlou et al., 2010; Welhan and Craig,
1979). In these environments, abiotic methane synthesis is thought to
originate from abiotic reactions between CO\textsubscript{2} and
H\textsubscript{2}:

CO\textsubscript{2}+4H\textsubscript{2}=CH\textsubscript{4}+2H\textsubscript{2}O
(1)

At temperatures under 340 °C, thermodynamic equilibrium favors carbon
reduction in fluids that contain H\textsubscript{2} (Klein et al.,
2019). This is the Sabatier reaction, often lumped with a class of
reactions referred to as Fischer-Tropsch-Type synthesis, or FTT, as
reviewed elsewhere (McCollom, 2013; McCollom and Seewald, 2007). Methane
is also observed with H\textsubscript{2} in seeps and fracture waters
from the continental subsurface, at environmental temperatures typically
\selectlanguage{english}[?] 100 degC. In fluids from the Kidd Creek mine (Canada), or in
hyperalkaline fluids issuing from the underlying ophiolite from the Oman
ophiolite, CH\textsubscript{4} and H\textsubscript{2} are present as
major species. The environmental temperatures there allow microbial
methane synthesis, but CH\textsubscript{4} has been suggested as being
dominantly abiotic with only minor mixing with microbial methane on the
basis of C and H stable isotopes of methane and associated gases in
these sites (Etiope et al., 2015; Fritz et al., 1992; Sherwood Lollar et
al., 2008, 2002, 1993). Contributions from microbial methane have also
been identified in gas seeps from the sultanate of Oman on the basis of
microbiological data (Miller et al., 2016).

The abundances of mass-18 methane
isotopologues,\textsuperscript{13}CH\textsubscript{3}D
and\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2,} may be
tracers of methane origin in nature (Stolper et al., 2014; Wang et al.,
2015; Young et al., 2017). The potential for
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} and
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D to demonstrate abiogeneicity
of natural methane on Earth and other worlds is illustrated by
preliminary experimental work. So far, two Sabatier-type experimental
runs have been investigated for
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D versus
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} (Young et
al., 2017). Those methane aliquots were generated by
CO\textsubscript{2}reduction (reaction 1) catalyzed by ruthenium at 70
and 90 \selectlanguage{ngerman}°C following the method of Etiope and Ionescu (2015). The
experiments yielded near-equilibrium
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D values of
\textasciitilde{}+4%0, but
large\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2}
depletions of approximately 70%0 relative to equilibrium. These empirical
values have been thought to reflect a signature acquired by methane
during the abiotic reduction of oxidized carbon in the laboratory and
potentially in nature (Young et al., 2017).

At Kidd Creek, methane samples have compositions that appear consistent
with the Sabatier experimental runs. In the deepest levels of the mine,
CH\textsubscript{4} shows \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D
values of 5.2 \selectlanguage{ngerman}± 0.5\selectlanguage{english}%0 and
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} values are 10
to 30%0 below equilibrium, which has been interpreted as potentially
direct evidence for an abiotic origin of methane (data in Young et al.,
2017; Warr et al. 2021). In contrast, methane from Oman shows negative
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} value and
near-zero \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D. The near-zero was
considered inconsistent with abiotic signatures (Nothaft et al., 2021).
In deep-sea high temperature vents, concordant
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} and
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D data suggested apparent
equilibrium temperatures of \textasciitilde{} 340 \selectlanguage{ngerman}°C (Labidi et al.,
2020). Equilibrium signatures could be acquired during methane synthesis
at high temperature, or may reflect bond reordering after synthesis,
before or during transport.

The Sabatier experiments reported in Young et al., (2017) were performed
at temperatures below 100 °C and in the absence of water. These
experimental conditions are challenging to extrapolate to deep-sea
hydrothermal systems or continental environments for which
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} and
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D data are available.
Consequently, it is not clear how the experimental data should be
applied to the interpretation of natural systems. To constrain the
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} and
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D signatures produced by abiotic
FTT reactions at variable temperatures, we performed controlled FTT
synthesis at hydrothermal conditions in the laboratory at temperatures
varying between 130 \selectlanguage{ngerman}°C and 300 °C.

\textbf{2. Experiments}

The experimental methods employed follow those used previously to study
abiotic methane and other organic syntheses by Fischer-Tropsch-type
(FTT) reactions under hydrothermal conditions (McCollom et al., 2010;
McCollom and Seewald, 2006). Briefly, a flexible-cell reaction cell
composed of a gold bag with titanium fittings is contained inside a
stainless-steel pressure housing (Seyfried Jr et al., 1987). An external
valve connected to the reaction cell with titanium tubing allowed for
collection of volatile gases at the end of the experiments. Prior to the
experiments, the reaction cell components were heated for two hours at
450°C in air to remove organic contaminants and to create an inert
TiO\textsubscript{2} layer on the surface of the titanium fittings.
Previous experiments with the same reaction system have demonstrated
that the reactor itself is not catalytic for organic synthesis under the
experimental conditions (McCollom and Seewald, 2006; McCollom et al.,
2010).

The experiments were performed by heating carbon monoxide (CO) and
native iron (Fe\textsuperscript{0}) in the presence of water at a range
of temperatures. The experiments were initiated by placing 1.5 g of
Fe\textsuperscript{0} (Fe powder, -22 mesh; Alfa Aesar) and
\textasciitilde{}25 g of DI water in a 55 mL gold reaction cell, so the
H\textsubscript{2}O/Fe molar ratio was \textasciitilde{} 50. The native
Fe is included both as catalyst and as a source of molecular hydrogen
(H\textsubscript{2}) since it rapidly reacts with water at elevated
temperatures to form magnetite (Fe\textsubscript{3}O\textsubscript{4})
and H\textsubscript{2} according to the reaction:

3Fe\textsuperscript{0} + 4H\textsubscript{2}O \selectlanguage{english}-
Fe\textsubscript{3}O\textsubscript{4} + 4H\textsubscript{2}. (2)

To promote reaction of Fe\textsuperscript{0}, the water was acidified to
a pH of about 2 with HCl. Individual experiments were conducted over a
range of temperatures from 130degC to 300degC. The target pressure for the
experiments was 15 MPa, but in practice the pressures varied up to 3 MPa
from this value during some experiments.

After sealing the reaction cell with the Ti closure piece, it was placed
in the pressure containment vessel, pressurized at room temperature, and
the entire apparatus was placed horizontally and heated in a furnace.
Once the target temperature was attained, CO was injected through the
sampling valve using the following procedure. First, a loop of
stainless-steel tubing (12.6 mL) was filled with CO at a pressure of
0.93 MPa (the maximum of the pressure regulator on the CO tank) and
attached to the sampling valve. A solution with the same composition as
that in the reaction cell was then pumped into the tubing until pressure
was equal to that within the heated pressure vessel. The valve was then
opened and the CO along with 13 mL of solution was pumped into the
reaction cell, with an equal amount of water bled from the pressure
containment vessel to maintain constant pressure during injection. In
practice, some amount of CO remained in the injection loop after this
procedure, so it is not possible to determine exactly how much CO was
contained in the reaction cell following injection. However, assuming
90\% injection efficiency, the amount of CO injected would have been
about 4.3 millimoles. Considering that 38 mL of water (25 mL initial +
13 mL added during injection) corresponds to \textasciitilde{}2.1 moles
H\textsubscript{2}O, the starting CO/H\textsubscript{2}O molar ratios of
the experiments are estimated to be approximately 2 \selectlanguage{english}?
10\textsuperscript{-3}.

The tank of CO used in the experiments was the same as that used in
McCollom et al. (2010) and has a \selectlanguage{greek}δ\selectlanguage{english}\textsuperscript{13}C of -28%0 V-PDB.
The water used in this study is deionized water from Univ. Colorado tap
water. The \selectlanguage{greek}δ\selectlanguage{english}D of DI water sampled in April 2022 and April 2023 was
--119%0 V-SMOW, similar to measurements of Boulder tap water measured in
older studies (--116\selectlanguage{ngerman}±1\selectlanguage{english}%0; (Landwehr et al., 2014)). The molecular
hydrogen (H\textsubscript{2}) generated from reaction 2 subsequently
reacts with CO to form methane, according to the net reaction:

CO+3H\textsubscript{2}=CH\textsubscript{4}+H\textsubscript{2}O (3)

Reaction 3 is a Fischer-Tropsch reaction that utilizes CO for oxidized
carbon rather than CO\textsubscript{2}. It is distinct from the Sabatier
reaction (reaction 1) that may be similar to reactions in natural
hydrothermal systems where CO\textsubscript{2} is available in large
amounts. Carbon monoxide was used here because it is more active in
hydrothermal FTT reactions than CO\textsubscript{2} (McCollom et al.,
2010), which allowed for obtaining measurable
CH\textsubscript{4}quantities under our experimental timescales. In
addition to CO reduction, the so-called ``water-gas shift reaction''
(McCollom and Seewald, 2006) was prominent under the experimental
conditions; reaction between CO and water led to the oxidation of a
fraction of the CO to form CO\textsubscript{2}:

CO + H\textsubscript{2}O = CO\textsubscript{2} + H\textsubscript{2} (4)

Through a competition between reaction 3 and 4, CO is rapidly and
predominantly converted to CH\textsubscript{4} and CO\textsubscript{2},
with much of the CO\textsubscript{2} precipitating as siderite
(FeCO\textsubscript{3}). Reaction 4 is anticipated to be dominant over
reaction 3 (McCollom et al., 2010).

The experiments were heated for 18 to 138 hours, with longer incubation
times used at lower temperatures to allow for the synthesis reactions to
proceed. Following reaction, the reactors were gradually cooled to room
temperature while pumping water into the containment vessel to maintain
elevated pressure. The pressure was then reduced to \textasciitilde{}5
MPa by releasing water from the pressure containment vessel in order to
allow volatile gases to exsolve and accumulate at the top of the
reaction cell. Within 24 h of experiment termination, the gas phase was
vented through the sampling valve and transferred to 300 mL stainless
steel cylinders (Swagelok) for isotope analysis.

\textbf{3. Isotope Measurements}

Methane was extracted from the stainless-steel storage cylinders on a
vacuum system at UCLA and purified by gas chromatography using
procedures described previously by Young et al. (2017). Gas aliquots of
a given sample were opened to a silica gel-filled stainless-steel U-trap
cooled with liquid nitrogen. Helium carrier gas was used to flush the
sample to the gas chromatograph. Separation was accomplished with a 3m
long, 1/8 in. OD stainless steel column packed with 5 A molecular sieve,
followed in series by a 2m long \selectlanguage{english}[?] 1/8 in. OD stainless steel column
packed with HayeSep D porous polymer. Peaks from the gas chromatograph
were detected with an in-line thermal conductivity detector (TCD).
Because total pressures were \textgreater{} 1 bar in the reaction
vessels, and because samples were dominated by H\textsubscript{2},
purification on the vacuum line had to be performed in sequence of
multiple aliquots. Once collection of a purified aliquot of methane was
complete, this methane was transferred to a glass vial filled with
silica-gel at liquid nitrogen temperature. Another sample aliquot would
then be purified and added to the same glass vial, until a given
reaction vessel had been entirely processed. For the cylinders to be
emptied, 6 to 8 aliquots were processed in this way on the vacuum line.
Samples were then transferred to the inlet of the mass spectrometer
where they were warmed and expanded into the mass spectrometer using a
silica-gel filled cold finger.

Mass spectrometry measurements were done on a Nu Instruments Panorama
with a mass resolving power that allows the simultaneous measurement of
ion currents for
resolved\textsuperscript{12}CH\textsubscript{4}\textsuperscript{+},\textsuperscript{13}CH\textsubscript{4}\textsuperscript{+},\textsuperscript{12}CH\textsubscript{3}D\textsuperscript{+},\textsuperscript{13}CH\textsubscript{3}D\textsuperscript{+}
and\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2}\textsuperscript{+}ion
beams, as described in Young et al., (2016). The analyte peaks are in
most cases baseline resolved from surrounding interferences, and where
there is the potential for a tail effect (in the case
of\textsuperscript{13}CH\textsubscript{5}\textsuperscript{+} tailing
into\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2}\textsuperscript{+}),
previous work shows that the effects on our final results are less than
quoted uncertainties (Young et al., 2016). The measured ratios of these
ion currents yield values for both
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D and
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} as well as
for bulk \textsuperscript{13}C/\textsuperscript{12}C and D/H. The
methane quantities available for isotopic measurements were relatively
low. For standards and samples available in quantities \textgreater{} 80
\selectlanguage{greek}μ\selectlanguage{english}mol, relative abundances
of\textsuperscript{13}CH\textsubscript{3}D\textsuperscript{+}
and\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2}\textsuperscript{+}are
measured for up to 10 and 20 hours, respectively. In these cases,
typical propagated internal uncertainties for the calculation of
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D and
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} are generally
\selectlanguage{ngerman}±0.1\selectlanguage{english}%0 and +-0.5\selectlanguage{english}%0, respectively. In our study, often only 35 to 60 \selectlanguage{greek}μ\selectlanguage{english}mol of
CH\textsubscript{4} were available (Table 1), and uncertainties are
larger, reflecting less favorable counting statistics. The internal
uncertainties range from 0.2%0 and 0.5%0 for
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D, and between 1.1%0 and 3.2%0 for
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2}.

\textbf{4. Results}

For nine experiments conducted at temperatures between 130 and 300\selectlanguage{ngerman}°C,
between \textasciitilde{}5 and \textasciitilde{}80 micromoles of abiotic
CH\textsubscript{4} were obtained. All results may be found in Table 1.
Previous experimental studies of abiotic synthesis have shown that
background sources can contribute significant amounts of methane, which
can be misleading if not carefully quantified (McCollom et al., 2016 and
references therein). In previous experiments employing the same
methodology as used here but without added CO indicated that background
sources contribute \textless{}3 \selectlanguage{greek}μ\selectlanguage{english}moles CH\textsubscript{4} (McCollom et
al., 2010). Our experiments for which we have isotope results have
resulted in the accumulation of 35 to 80 micromoles of methane,
suggesting our sample are dominated by genuinely abiotic
CH\textsubscript{4}, synthesized by CO reduction. A simple mass balance
suggests that the near-complete conversion of the
Fe\textsuperscript{0}powder to magnetite (reaction 2) produced the
H\textsubscript{2}-rich vapor phase, as shown in McCollom et al.,
(2010). This is an essential condition to ensure reaction 3 proceeds
efficiently, allowing the abiotic synthesis of methane (McCollom, 2016).

An inverse correlation between CH\textsubscript{4} quantities and
temperatures is observed (Fig. 1), indicating that experiments at lowest
temperatures allowed the highest accumulations of methane from CO
reduction. The experiments at 275 \selectlanguage{ngerman}°C and 300°C yielded 5 and 10
micromoles of CH\textsubscript{4} respectively, which was insufficient
to characterize isotopologue abundances; the low yields at these
temperatures likely reflect rapid conversion of CO to
CO\textsubscript{2} by the water-gas shift reaction (reaction 4),
short-circuiting reduction to methane. Seven experiments conducted at
temperatures between 130 and 250 °C yielded at least 35 micromoles of
CH\textsubscript{4}, allowing measurements of rare isotopologue ratios.
We interpret the relationship between methane amount and experimental
temperatures (Fig. 1) as a reflection of the lifetime of the injected
CO: upon CO injection, conversion to CO\textsubscript{2} through
reaction 4 begins, precluding methane synthesis. Equilibrium
thermodynamics suggest that reaction 4 goes to completion, hence
CH\textsubscript{4} synthesis occurs in a short time window between
injection of CO and conversion to CO\textsubscript{2}.~ Reaction 4 is
faster at higher temperatures. Evidently, in our experiments conducted
at T \textgreater{} 250°C, CO conversion to CO\textsubscript{2} happens
so rapidly that very little FTT-derived methane is generated.

The \selectlanguage{greek}δ\selectlanguage{english}D of product methane in our experiments vary between --583%0 and
--608%0 versus V-SMOW (Fig. 2), much lower than the starting \selectlanguage{greek}δ\selectlanguage{english}D of water
(--119%0). Hydrogen isotope ratios of methane are positively correlated
with temperatures (Fig. 2). At the highest temperatures, the hydrogen
isotope composition of methane tends to be the most elevated. A clear
outlier to this trend is FT18-4, the experiment conducted at the lowest
temperature (130\selectlanguage{ngerman}°C), which shows the highest \selectlanguage{greek}δ\selectlanguage{english}D of the series (--582%0).
The \selectlanguage{greek}δ\selectlanguage{english}D values for the methane synthesized in this study are different
from those obtained in McCollom et al., (2010) (with values of
\textasciitilde{} --550%0) although experimental conditions except for
temperature were similar. We attribute the small differences as the
simple reflection of variable starting water for the experiments: in the
study of McCollom et al., (2010), the reactant water was purchased from
Fischer\selectlanguage{ngerman}®. The measured \selectlanguage{greek}δ\selectlanguage{english}\textsuperscript{13}C of methane varies between
--44 and --62%0 versus V-PDB (Fig. 2). The carbon isotope values obtained
here are similar to previous results obtained using the same
experimental procedures and starting CO (McCollom et al., 2010). The
\selectlanguage{greek}δ\selectlanguage{english}\textsuperscript{13}C values are weakly correlated with temperature but
interpretations of the correlation should be regarded with caution,
considering the importance of FT18-4, the one outlier on Fig. 2b.

We find \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D values between
1.7\selectlanguage{ngerman}±0.4\selectlanguage{english}%0 (1\selectlanguage{greek}σ, \selectlanguage{english}at 250 \selectlanguage{ngerman}°C) and 4.8±0.2\selectlanguage{english}%0 (1\selectlanguage{greek}σ, \selectlanguage{english}at 130 \selectlanguage{ngerman}°C) for all
experiments except one outlier, FT18-3, with a
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D of 5.8\selectlanguage{ngerman}±0.2\selectlanguage{english}%0 (1\selectlanguage{greek}σ, \selectlanguage{english}at 170 \selectlanguage{ngerman}°C).
Most of the \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D values track the
temperature of abiotic methane synthesis within 1 %0 (Fig. 3a). In
contrast with \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D,
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} values are
exclusively negative, ranging from -3.0 \selectlanguage{ngerman}± 2.5\selectlanguage{english}%0 (1\selectlanguage{greek}σ\selectlanguage{english}) at 210 \selectlanguage{ngerman}°C to
\selectlanguage{english}-32.0+-1.6\selectlanguage{english}%0 (1\selectlanguage{greek}σ\selectlanguage{english}) at 183 \selectlanguage{ngerman}°C (Fig. 3b). The negative
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} values
indicate that the relative amount
of\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} among
methane isotopologues is lower than predicted for a stochastic gas
equilibrated at T \textgreater{} 1000 \selectlanguage{ngerman}°C. The
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} values are
positively correlated with peak experimental temperatures (Fig. 3b); the
magnitude of the disequilibrium relative to stochastic is greater at the
lowest temperatures. The data are shown in the
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D -
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} space on Fig.
4, with natural samples from deep-sea vents (Labidi et al., 2020),
methane from the Kidd Creek mine (Young et al., 2017) and Oman fluids
(Nothaft et al., 2021), for which methane is potentially abiotic in
origin.

\textbf{5. Discussion}

\textbf{5.1 The carbon isotope balance in our experiments}

The \selectlanguage{greek}δ\selectlanguage{english}\textsuperscript{13}C signatures of methane vary around a value of
--55%0 (V-PDB), with FT18-4 as the one outlier with a
\selectlanguage{greek}δ\selectlanguage{english}\textsuperscript{13}C of --44%0 (Fig. 2). All methane samples have
depleted \selectlanguage{greek}δ\selectlanguage{english}\textsuperscript{13}C compared to the starting CO (--28%0). The
isotopic difference between CH\textsubscript{4} and the starting CO of
\textasciitilde{} 25%0 likely
reflects\textsuperscript{13}C/\textsuperscript{12}C fractionation during
methane synthesis. Kueter et al., (2019) determined the
equilibrium\textsuperscript{13}C/\textsuperscript{12}C fractionations
between CO, CO\textsubscript{2} and CH\textsubscript{4} at various
temperatures between 1200 \selectlanguage{ngerman}°C and 300 °C. At 300 °C, the
equilibrium\textsuperscript{13}C/\textsuperscript{12}C fractionation
between CO and CH\textsubscript{4} is --3.1\selectlanguage{english}%0, meaning that
CH\textsubscript{4} should be enriched in \textsuperscript{13}C relative
to CO by 3.1\selectlanguage{english}%0. This is at odds with our observation of a
\textasciitilde{}25\selectlanguage{english}%0 to \textasciitilde{}40\selectlanguage{english}%0 depletion
in\textsuperscript{13}C/\textsuperscript{12}C in
CH\textsubscript{4}relative to the initial reactant CO. Instead methane
\selectlanguage{greek}δ\selectlanguage{english}\textsuperscript{13}C values are broadly consistent with an approach
to\textsuperscript{13}C/\textsuperscript{12}C equilibrium between
methane and CO\textsubscript{2}. This implies that the synthesis of
CO\textsubscript{2} from CO oxidation via reaction 4 was nearly
quantitative so that CO\textsubscript{2} had nearly the
same\textsuperscript{13}C/\textsuperscript{12}C as the initial CO
reactant (--28%0). Subsequent \textsuperscript{13}C/\textsuperscript{12}C
equilibration between CH\textsubscript{4} and this
CO\textsubscript{2}would yield methane with a 25%0 \textsuperscript{13}C
depletion relative to CO\textsubscript{2} at 250 \selectlanguage{ngerman}\textsuperscript{°}C,
and roughly 40 \selectlanguage{english}%0 at 150 \textsuperscript{deg}C (Kueter et al., 2019),
corresponding to predicted methane \selectlanguage{greek}δ\selectlanguage{english}\textsuperscript{13}C values of
\textasciitilde{} --53%0 and -68%0 at 250 and 150 \selectlanguage{ngerman}\textsuperscript{°}C,
respectively. This data supports near-complete conversion of CO to
CO\textsubscript{2} and complete equilibration between
CH\textsubscript{4} and CO\textsubscript{2}. While imperfect, the
magnitude of the shift in \selectlanguage{greek}δ\selectlanguage{english}\textsuperscript{13}C from the initial CO
reactant values to the product methane are generally consistent with
these CH\textsubscript{4}-CO\textsubscript{2} equilibrium values. In
this context, the scatter in the \selectlanguage{greek}δ\selectlanguage{english}\textsuperscript{13}C values, and
imperfect correlation with T (Fig. 2a) would reflect various degrees of
departure from full isotopic equilibrium and/or conversion of CO to
CO\textsubscript{2}. The FT18-4 outlier would reflect the furthest
removal from CH\textsubscript{4} -CO\textsubscript{2} equilibrium,
and/or incomplete conversion of CO to CO\textsubscript{2}, consistent
with this experiment being conducted at the lowest temperature. The
alternative interpretation is that the
CH\textsubscript{4}\selectlanguage{greek}δ\selectlanguage{english}\textsuperscript{13}C values reflect a kinetic
carbon isotope effect involving the reduction of CO. In this case the
correlation with temperature, albeit imperfect, would perhaps reflect a
reservoir effect with a single kinetic fractionation factor, or a
temperature-dependent kinetic fractionation factor.

\textbf{5.2 The hydrogen isotope} \textbf{balance in our experiments}

The \selectlanguage{greek}δ\selectlanguage{english}D signatures of methane range from --610 to --580%0 V-SMOW. If
methane is in D/H equilibrium with experimental water (--119%0 V-SMOW),
the methane \selectlanguage{greek}δ\selectlanguage{english}D would fall between --250 and --350%0 for experiments
conducted at temperatures between 250 and 130\selectlanguage{ngerman}°C (Horibe and Craig,
1995). Thus, the observed \selectlanguage{greek}δ\selectlanguage{english}D signatures of methane of roughly --600%0
reflect unambiguous D/H disequilibrium with water. During FTT synthesis,
methane molecules form via a catalytic surface reaction of sequential
hydrogen additions, presumably sourced directly from H\textsubscript{2}.
It is expected, therefore, that it is the D/H ratio of
H\textsubscript{2} that is a dominant factor in controlling the D/H of
methane in our experiments, not that of water. No D/H measurements of
H\textsubscript{2} are available in this study, but at equilibrium,
CH\textsubscript{4} should be \textasciitilde{} 540%0 higher in \selectlanguage{greek}δ\selectlanguage{english}D than
coexisting H\textsubscript{2} at \textasciitilde{}
200\selectlanguage{ngerman}\textsuperscript{°}C (Horibe and Craig, 1995). In lieu of D/H for
H\textsubscript{2} in our experiments, we may make use of three FTT
experimental studies that have reported D/H of both of
CH\textsubscript{4} and H\textsubscript{2} to infer the likely extent of
isotopic equilibrium between these species. All of these studies report
D/H ratios in CH\textsubscript{4} and H\textsubscript{2} are within 150
\selectlanguage{english}%0 of each other (Fu et al., 2007; McCollom et al., 2010; Taran et al.,
2010), far from the \textasciitilde{} 500\selectlanguage{english}%0 difference expected at
equilibrium. This suggests that abiotic methane synthesized in the
laboratory is not in isotopic equilibrium with H\textsubscript{2}. We
can explore this further if we assume H\textsubscript{2} formed in D/H
equilibrium with water in our experiments. In this case \selectlanguage{greek}δ\selectlanguage{english}D for
H\textsubscript{2} is predicted to be approximately -560%0 at
200\textsuperscript{o}C, in equilibrium with the liquid water \selectlanguage{greek}δ\selectlanguage{english}D of
-119%0 (Horibe and Craig, 1995; Rolston et al., 1976). Since the product
methane has \selectlanguage{greek}δ\selectlanguage{english}D values of approximately --600%0 at this T, it appears that
the CH\textsubscript{4} has D/H similar to that of the reactant
H\textsubscript{2}, as in previous experiments. Under strict isotopic
equilibrium between methane and an infinite pool of H\textsubscript{2},
CH\textsubscript{4} should be \textasciitilde{} 540%0 higher in \selectlanguage{greek}δ\selectlanguage{english}D than
coexisting H\textsubscript{2} at \textasciitilde{}
200\selectlanguage{ngerman}\textsuperscript{°}C. Similar D/H for CH\textsubscript{4} and
H\textsubscript{2} would suggest that either the methane consumed most
or all of the H\textsubscript{2} available locally, or that there is a
D/H kinetic fractionation that offsets the equilibrium fractionation by
nearly 500\selectlanguage{english}%0. Overall, we conclude that a combination of equilibrium
fractionation between H\textsubscript{2}O and product
H\textsubscript{2}, followed by synthesis of methane with similar D/H
than reactant H\textsubscript{2} explains the methane \selectlanguage{greek}δ\selectlanguage{english}D values of
approximately --600%0 obtained here.

The predicted range in \selectlanguage{greek}δ\selectlanguage{english}D values relative to VSMOW for
H\textsubscript{2} formed in equilibrium with the effectively infinite
reservoir of liquid H\textsubscript{2}O from \textasciitilde{} 170 to
250 \selectlanguage{ngerman}\textsuperscript{°}C is \selectlanguage{english}-588 to \selectlanguage{english}-513\selectlanguage{english}%0, respectively, defining a
slope of 0.44\selectlanguage{english}%0/\textsuperscript{deg}C over this temperature range. This is
similar to the observed positive correlation between methane \selectlanguage{greek}δ\selectlanguage{english}D values
with T in our experiments, with a slope of
\textasciitilde{}0.38%0/\selectlanguage{ngerman}\textsuperscript{°}C, lending further support for
the inheritance of D/H from H\textsubscript{2} equilibrated with water,
with the exclusion of the 130°C outlier FT18-4 (Fig. 2). This
interpretation of inheritance of D/H from H\textsubscript{2} convolved
with a kinetic isotope effect is consistent with previous similar
published experiments in which the D/H values of CH\textsubscript{4} and
H\textsubscript{2} are within a few tens of permil of one another (e.g.,
McCollom et al., 2010).

\textbf{5.3 Methane clumped signatures:
does\textsuperscript{13}CH\textsubscript{3}D reflect peak temperature of
methane synthesis?}

We find \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D values to roughly
track with the expected equilibrium values within 1%0 for all experiments
except FT18-3 (170 \selectlanguage{ngerman}°C - see below). Data from the Sabatier experiments
reported in Young et al., (2017) are also included in Figure 3. Although
run under different conditions, the Sabatier experiments also show
equilibrium signatures for \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D.
Overall, the data suggest that FTT processes generate
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D values that are near
equilibrium at the temperature of synthesis. The equilibrium signatures
suggest that despite D/H kinetic isotope effects associated with
hydrogen addition during the assembly of methane molecules,
the\textsuperscript{13}CH\textsubscript{3}D bond-ordering remains
controlled by synthesis temperature. This is an important perspective
for the interpretation of natural methane where abiogenesis is
suspected: unless substantial mixing is involved, one may use
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D values to constrain the
approximate synthesis temperature of abiotic methane.

For FT18-3, the measured \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D of
5.8\selectlanguage{ngerman}±0.4\selectlanguage{english}%0 (2 s.e.) reflects a 3\selectlanguage{english}%0 deviation from an equilibrium value of
2.8\selectlanguage{english}%0 at 170 degC (Fig. 3a). The equilibrium temperature corresponding to
the measured \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D value for FT18-3
is \(25_{-8}^{+5}\) \selectlanguage{ngerman}°C, far from the peak experimental temperatures,
but similar to room temperature (Fig. 3a). We argue against the
possibility of \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D
re-equilibration towards low temperature in FT18-3 since there is no
physical reason why \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D reordering
to room temperature would occur only in this experiment. Like every
other experiment, gases were extracted from the gold reaction cells
within 24 hours of cooling the reactors, stored in stainless steel
containers, and measured for \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D
and \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} at UCLA
within two weeks upon synthesis. On this basis, we argue that the
elevated \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D value for FT 18-3
must reflect an unidentified mass-dependent fractionation of up to 3%0,
rather than re-equilibration.

\textbf{5.4
Large\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2~}deficits
in abiotic methane}

The \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} values
here exhibit pronounced disequilibrium, in sharp contrast with those of
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D. The
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} values are
exclusively of negative sign, ranging from -3.0 \selectlanguage{ngerman}± 2.5\selectlanguage{english}%0 (1\selectlanguage{greek}σ\selectlanguage{english}) at 210 \selectlanguage{ngerman}°C to
\selectlanguage{english}-32.0 +- 1.6\selectlanguage{english}%0 at 183 degC. These values are substantially lower than the
expected equilibrium values of 9.2\selectlanguage{english}%0 and 4.3\selectlanguage{english}%0 at 130 and 250 degC,
respectively. We argue that the
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} deficits are
apparent disequilibrium signatures resulting from combinatorial effects.
In general, combinatorial effects arise when a molecule contains
indistinguishable atoms of the same element, and that these atoms come
from pools with distinct isotope ratios, as has been predicted and shown
for methane previously both from theory and in the laboratory (R\selectlanguage{ngerman}öckmann
et al., 2016; Taenzer et al., 2020; Yeung, 2016). Among the two mass-18
isotopologues of methane, only
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} can be
affected by combinatorial effects, because it is the isotopologue with
two indistinguishable deuterium substitutions for hydrogen.

The root of the combinatorial effect comes from the notation convention
used with clumped isotopes. As an example, consider a sample of methane
gas with \selectlanguage{greek}δ\selectlanguage{english}D = --100 %0 relative to SMOW and \selectlanguage{greek}δ\selectlanguage{english}\textsuperscript{13}C = -10
%0 relative to PDB, corresponding to a measured bulk D/H ratio of
1.40184[?]10\textsuperscript{-4} and
a\textsuperscript{13}C/\textsuperscript{12}C ratio of 0.01112483. In
this example we will use a
measured\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2}/\textsuperscript{12}CH\textsubscript{4}ratio
of 1.11600\selectlanguage{ngerman}×10\textsuperscript{-7}. To calculate the value for
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2}, we compute
the stochastic ratio from the measured bulk carbon and hydrogen isotope
ratios. Isotope-specific mole fractions for singly-substituted
isotopologues are closely approximated as:\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}

\end{center}
\end{figure}

(5)

where R\textsubscript{D} is the bulk D/H ratio and
R\textsubscript{13C}is the bulk
\textsuperscript{13}C/\textsuperscript{12}C ratio. From the properties
of probabilities, the stochastic isotope-specific mole fraction of
\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2,}where isotopes
are distributed randomly across all isotopologues, is:\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}

\end{center}
\end{figure}

, (6)

where \emph{6} refers to the six isotopic isomers for this isotopologue
(\textsuperscript{12}CHDHD,
\textsuperscript{12}CHHDD,\textsuperscript{12}CDHDH, etc.). In this
example, the stochastic value for
x\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} is
1.16547\selectlanguage{ngerman}×10\textsuperscript{-7} and the stochastic value for\emph{x}
\textsuperscript{12}CH\textsubscript{4} =\emph{x} H\textsuperscript{4}
\emph{x} \textsuperscript{12}C is 0.9884432. With a
measured\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2}/\textsuperscript{12}CH\textsubscript{4}ratio
of 1.16000×10\textsuperscript{-7}, we have:

\(CH_{2}D_{2}=10^{3}\left(\frac{\frac{(^{12}CH_{2}D_{2})}{(12\text{CH}_{4})_{\text{measured}}}}{\frac{(^{12}CH_{2}D_{2})}{12\text{CH}_{4})_{\text{stochastic}}}}-1\right)=-16.2\)\selectlanguage{english}%0 (7)

Now let us consider the possibility that in the same gas, the methane
molecules were in fact constructed from two pools of hydrogen with
distinct D/H ratios, with the average D/H being the measured bulk value
in our methane, i.e., \selectlanguage{greek}δ\selectlanguage{english}D = --100 %0. If \selectlanguage{greek}\emph{α} \selectlanguage{english}is the fractionation
factor relating the D/H ratios for the two pools, then:\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}

\end{center}
\end{figure}

(8)

where R\textsubscript{1} is the D/H for one of the reservoirs,\selectlanguage{greek}\emph{αR}\selectlanguage{english}
\textsubscript{1} = \emph{R} \textsubscript{2} is the D/H for the other
reservoir, and \emph{R} \textsubscript{D} is the measured bulk ratio. We
take \selectlanguage{greek}\emph{α} \selectlanguage{english}= 1.5, meaning two hydrogens have a D/H ratio fractionated
by +500%0 relative to the two other hydrogens. Now the true stochastic
ratio, given \emph{a priori} knowledge of equal contributions from the
two pools, is given by the corresponding mole fractions for the two
pools, \emph{x} \textsubscript{D(1)}, \emph{x} \textsubscript{D(2)}etc.,
so Equation (6) becomes:\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}

\end{center}
\end{figure}

. (9)

Similarly, the stochastic mole fraction
for\textsuperscript{12}CH\textsubscript{4} is\emph{x}
\textsuperscript{12}CH\textsubscript{4} =\emph{x}
H\textsubscript{(1)}\textsuperscript{2}\emph{x}
H\textsubscript{(2)}\textsuperscript{2}\emph{x} \textsuperscript{12}C.
In our example, the true stochastic value for \emph{x}
\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} is 1.11885
\selectlanguage{ngerman}×10-7. This is lower by 42\selectlanguage{english}%0 than 1.16545x10-7, the value obtained from
equation (6). Using the stochastic estimate from equation (9), the
example gas has a true
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} of +25%0.

In practice, one would not have any prior knowledge of the actual
distribution of the D/H ratios and would have to apply equation (6)
rather than equation (9) for the calculation of
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2}. The elevated
x\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} from equation
(6) is the cause of the negative
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} value. The
reason for the discrepancy is a mathematical truism (Yeung et al., 2016,
Rockmann et al., 2016, Taenzer et al., 2020). It is straightforward to
show that the actual stochastic isotopologue
ratio\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2}/\textsuperscript{12}CH\textsubscript{4}is
proportional to the square of the geometric mean for the two D/H ratios
(to see this, divide Equation 9 by\emph{x}
\textsuperscript{12}CH\textsubscript{4} =\emph{x}
H\textsubscript{(1)}\textsuperscript{2}\emph{x}
H\textsubscript{(2)}\textsuperscript{2}\emph{x} \textsuperscript{12}C)
while the measured values, by necessity, yield the square of the
arithmetic mean of the two ratios, i.e., the measured bulk D/H ratio
(divide Equation 6 by\emph{x} \textsuperscript{12}CH\textsubscript{4}
=\emph{x} H\textsubscript{4} \emph{x} \textsuperscript{12}C, mindful of
Equation 8). The geometric mean is always less than the arithmetic mean,
unless R\textsubscript{1} = R\textsubscript{2}. Thus in general, where
molecular positions with distinct isotope ratios are indistinguishable,
and the average isotope ratios rather than the site-specific isotope
ratios are all that is accessible by the data (the usual case), the
calculated stochastic ratio of the multiply-substituted molecule to the
major isotopologue
(x\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2}) is
overestimated. This in turn means \selectlanguage{greek}Δ \selectlanguage{english}values derived from these
overestimated stochastic ratios are underestimated. This is the
combinatorial effect.

The building of CH\textsubscript{4} molecules from CO (or
CO\textsubscript{2}) is thought to involve a sequence of steps of carbon
reduction followed by additions of hydrogen molecules (Wang et al.,
2011). The set of reactions can be represented by these steps:

CO \(\rightarrow\) CO* \(\rightarrow\) HCO* \(\rightarrow\)
HCOH*\(\rightarrow\) CH* \(\rightarrow\) CH\textsubscript{2}*
\(\rightarrow\)CH\textsubscript{3}* \(\rightarrow\)
CH\textsubscript{4}*\(\rightarrow\) CH\textsubscript{4} (10)

where *corresponds to a surface-adsorbed species. Here
H\textsubscript{2} is the source of electrons for reduction. This
reaction network allows for the various hydrogens added to methane
during sequential steps to have different D/H ratios. A 500%0 difference
between two of the hydrogens relative to the other two can account for
our observed
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2}deficits of
approximately -40%0. The particular circumstance of
\selectlanguage{greek}δ\selectlanguage{english}D\textsubscript{(1)}= \selectlanguage{greek}δ\selectlanguage{english}D\textsubscript{(2)}= -850%0 and
\selectlanguage{greek}δ\selectlanguage{english}D\textsubscript{(3)}= \selectlanguage{greek}δ\selectlanguage{english}D\textsubscript{(4)}= -350%0 can explain the bulk
methane \selectlanguage{greek}δ\selectlanguage{english}D (\textasciitilde{} -600%0), and the methane clumped isotope
values. According to the theoretical treatments in (Cao et al., 2019)
and (Young, 2019), D/H fractionations associated with the assembly of
methane would plausibly cause such variations in D/H from step to step.
For example, a mixture of early steps involving D/H equilibrium followed
by later H addition steps with purely kinetic isotope effects could
produce the requisite large differences in D/H of \textgreater{} 400%0
among hydrogens (Young, 2019) that can reproduce the observed low
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} values.

The question arises as to whether our interpretation of the source of
low \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} values is
affected by the of the source of carbon, i.e. CO versus
CO\textsubscript{2}. CO\textsubscript{2} methanation begins by reduction
to CO (i.e., the reverse of reaction 4). That additional step occurs
prior to the onset of CO reduction and does not involve hydrogen
additions to carbon. There is no clear physical reason to anticipate
variable
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2}signatures as
the result of the additional step of CO\textsubscript{2}reduction to CO
prior to further reduction by hydrogen. This likely explains why the
Sabatier experiments (run with CO\textsubscript{2}) versus the FTT
experiments here (with CO) can be interpreted together (Fig. 3b).
Indeed, our data suggests the two pathways, from CO to methane and from
CO\textsubscript{2} to methane, may not be very different at all. Our
carbon isotope data suggesting that CO\textsubscript{2} is the precursor
for the\textsuperscript{13}C/\textsuperscript{12}C ratios for product
CH\textsubscript{4} in our experiments in which CO is the initial
carrier of carbon attest to this. Together, the data indicate that it
may be valid to extrapolate our experiments to natural systems, where
CO\textsubscript{2} is likely the primary carbon source for abiotic
methane synthesis.

\selectlanguage{greek}\textbf{5.5 The temperature relationship with
Δ\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2}}\selectlanguage{english}

Experiments with data at the highest temperatures, 210 and 250 \selectlanguage{ngerman}°C,
produced methane with
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} values of
between -3 and -12%0. This is substantially more positive than in
experiments done at 180, 170 and 130 \selectlanguage{ngerman}°C, which produced methane with
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} values of
between -32 and -27%0 (Fig. 3b). The Sabatier experiments at T [?] 90\selectlanguage{ngerman}°C
(Young et al., 2017) yielded methane with
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} values of
roughly -55%0. Taken together, the
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} data show a
clear positive correlation with temperature (Fig. 3b). This may reflect
the possibility that different D/H ratios among pools of hydrogen at
each reaction step are dependent on temperature, with greater
differences at lower temperatures. At \textgreater{} 200\selectlanguage{ngerman}°C, the observed
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} values
require pools of hydrogen atoms that comprise methane to be at most 200
%0 different from one another. At \textless{} 100\selectlanguage{ngerman}°C, the 70\selectlanguage{english}%0
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} deficits of
the Sabatier experiments require at least 700%0 D/H difference between
pools of hydrogen atoms.

The effect of bond reordering is unclear in this set of experiments. One
might imagine that irrespective of experimental temperatures, all
abiotic methane synthesized during FTT experiments is associated with
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} deficits
\textgreater{} 70%0, as the natural consequence of extreme and variable
D/H fractionation factors among the four hydrogen addition steps. This
would be the case if kinetic isotope effects for each step in the
CH\textsubscript{4}-building reaction sequence were fully independent of
temperature. After methane synthesis,
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} values would
approach equilibrium values \textgreater{} 0%0 due to bond reordering.
Bond reordering -- here, homogenization of the D/H among the hydrogens
comprising CH\textsubscript{4} molecules -- would generate
an\emph{a-posteriori} correlation between temperature and
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} as observed
on Fig. 3b. This scenario may not be excluded. Nonetheless, we favor a
temperature-dependent fractionation in D/H between H-addition steps
during CH\textsubscript{4} construction as the source of negative
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} values since
these effects can be seen in the theoretical treatments in Cao et al.,
(2019) and Young (2019).

To summarize, the
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} relationship
with experimental temperature (Fig. 3b) may be explained by two
different scenarios. First, kinetic isotope effects for different
H-addition steps may vary as a function of temperature. This would cause
the resulting combinatorial
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} values to
show a variability naturally correlated to temperature. Alternatively,
an a-posteriori bond reordering may occur in the hydrothermal cells,
from an initial
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} value that is
always ---70%0. If experiments at higher temperature allow further
re-ordering than those at lower temperatures, temperature and
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} will display
a positively correlation.

\textbf{5.6 Hydrogen isotope homogenization in nature: the link with the
laboratory}

The data collected for this study are plotted on figure 4 with published
methane isotopologue data from a variety of geological settings,
including sedimentary basins, deep crystalline environments and
serpentinization sites (Ash et al., 2019, Giunta et al 2019 2021, Young
2017 2019, warr et al 2021, Lin et al 2023). Extreme signatures
associated with active anaerobic oxidation of methane (Giunta et al
2022, Liu et al 2023) were omitted for clarity. Signatures observed in
abiotic methane synthesized in the laboratory are rare in nature (Fig.
4). This might suggest that abiotic methane has been under-sampled so
far, and/or that bond reordering of once-abiotic methane is at play in
natural environments.

Equilibrium values for
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} (and
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D) are seen in most
high-temperature hydrothermal vents with temperatures [?] 250\selectlanguage{ngerman}°C to date
(Fig. 4). Based on experiments, we would anticipate natural abiotic
methane synthesized at \textgreater{} 250°C to have equilibrated
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D values associated with
\textasciitilde{}10%0
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} deficits
relative to equilibrium (Fig. 3b). Based on the experiments, subsequent
bond reordering would seem to be required in order to erase the low
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} values
associated with constructing CH\textsubscript{4} molecules and replace
them with the equilibrium values exhibited by the natural
high-temperature hydrothermal samples (vents) shown in Fig. 4.
High-temperature hydrothermal systems may provide conditions conducive
to bond reordering. Methane is thought to be formed abiotically in fluid
inclusions of basement rocks at temperatures of 300 \selectlanguage{ngerman}°C or higher
(Früh-Green et al., 2022; Klein et al., 2019). Those fluid inclusions
might be maintained at temperatures of 300 °C or above for durations
that may be thousands of years or even far longer, allowing bond
reordering to take place. Because homogenization evidently occurs, the
deficits associated with initial methane synthesis are ``cryptic'',
i.e., they are not observable in nature, only in the laboratory where
experimental durations are short, and reordering does not take place.

Abiotic methane formed and kept at low temperature might have a chance
to preserve deficits in
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2}, should it be
restricted to environments where the kinetics of D/H homogenization
within methane molecules are unfavorable or residence times are short.
This offers a chance to use the combination of
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} and
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D to identify abiotic methane in
nature, as it has been done with microbial methane (Ash et al., 2018;
Giunta et al., 2022, 2019; Young et al., 2017). Abiogenesis may notably
account for signatures observed in methane from \textasciitilde{}25\selectlanguage{ngerman}°C
fluids from the Kidd Creek mine. There, methane samples have
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D values of 5.2 \selectlanguage{ngerman}± 0.5\selectlanguage{english}%0,
translating to temperatures of \textasciitilde{}45 degC. The methane
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} values from
the deepest level in the mine are 10 to 30%0 below equilibrium (data in
Young et al., 2017, also shown here on Fig. 4). These deficits are not
as low as those seen in experiments, and would be compatible with FTT
signatures affected by later bond reordering resulting from processing
by anaerobic methane oxidation (Warr et al., 2021; Young et al., 2017).

Abiogenesis alone is more challenging to reconcile with the observations
from serpentinizing systems in Oman. There, methane is vented to the
surface in \textasciitilde{}35\selectlanguage{ngerman}°C hyperalkaline fluids issuing from the
underlying ophiolite. It carries some of the highest
\selectlanguage{greek}δ\selectlanguage{english}\textsuperscript{13}C yet measured in natural methane (Miller et al.,
2016). Whether microbial methanogenesis can explain the isotopic
features of Oman methane is debated (Etiope, 2017; Miller et al., 2016).
Recent measurements of
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} and
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D were made available on new gas
aliquots from Oman, in Nothaft et al., (2021). The one sample with the
highest \selectlanguage{greek}δ\selectlanguage{english}\textsuperscript{13}C has a negative
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} value and
near-zero \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D (Fig. 4). This is
inconsistent with abiotic signatures reported here: no abiotic methane
made in the laboratory, at any of the investigated temperatures, has
yielded a near-zero \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D (Fig. 3a).
Conversely, microbial methane does exhibit near-zero or negative
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D values (Young et al., 2017,
Giunta et al., 2019) and the Oman data appear consistent with microbial
signatures (Fig. 4). The simplest explanation for the data in Nothaft et
al., (2021) remains that despite the elevated \selectlanguage{greek}δ\selectlanguage{english}\textsuperscript{13}C,
the \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} -
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D signature reflect substantial
contributions of microbial methane in the Oman ophiolite (details in
Nothaft et al., 2021).

\textbf{5.7 Methane clumped signatures in astrobiology}

Our experiments suggest methane produced by abiogenesis on other worlds
may yield apparent
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} deficits.
This is because the difference between equation 6 and 9 remains
insensitive to the absolute value of the D/H ratio; only
D/H\emph{differences} between various hydrogen pools matter in causing
combinatorial effects (Rockmann et al., 2016, Taenzer et al., 2020), not
the absolute values of the source pools. Since D/H differences
inevitably follow from lower-T hydrogen additions to carbon, we expect
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} deficits to
be common anywhere methane is synthesized from the reduction of CO or
CO\textsubscript{2}. This means that on other worlds, where the D/H
ratio of molecular hydrogen is unknown and could be extremely different
from Earth, abiotic methane should still exhibit
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} deficits
comparable to the values observed here (Fig. 3). As on Earth, if this
methane is entrained in a hydrothermal system with ambient temperatures
approaching 300 \selectlanguage{ngerman}\textsuperscript{°}C, this signature may be replaced by
isotopologue equilibrium, a situation that may be uncommon however given
the low temperatures conditions prevailing on most other solar system
bodies.

While biosignatures have been a long-term focus of investigation, both
field and experimental consideration of potential abiosignatures has
been flagged as a critical gap in the search for life beyond Earth
(NASEM, 2022). As such, bond reordering of abiotic methane may be a
useful tool. The in-situ discovery of methane gas in Mars's atmosphere
by Curiosity (Webster et al., 2015) indicates that methane is produced
on the planet, likely from an on-going source. Methane is also detected
in the plumes of Saturn's moon Enceladus, together with molecular
hydrogen (Waite et al., 2017). There, the origin of
CH\textsubscript{4}is thought to be abiotic, derived from hydrothermal
reactions occurring at depth in the oceans of Enceladus, but at
unconstrained temperatures (Waite et al., 2017). A Bayesian analysis of
the escape rates of Enceladus has recently challenged a hydrothermal
origin of methane, and suggested it might potentially be biological
(Affholder et al., 2021). It is not clear that conventional isotopic
tools can yield unambiguous results as to the origin of methane on Mars
and Enceladus, because sound interpretations of bulk isotope ratios by
conventional means require contextual information: what is the meaning
of methane D/H ratios with no constraints on the D/H of the co-existing
molecular hydrogen? What is the overall
\textsuperscript{13}C/\textsuperscript{12}C of the carbon reservoirs
from which methane might evolve? This is underscored by the complex
interpretations of \textsuperscript{13}C/\textsuperscript{12}C ratios
measured on Mars rocks by Curiosity (House et al., 2022). On Mars and on
Enceladus, if methane is synthesized abiotically at high temperature, we
see no physical reason against the appearance of equilibrium
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D and
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} signatures
resulting from reordering, as on Earth. Should
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D and
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} measurements
ever be done for Enceladus methane, any equilibrium temperatures [?] 250\selectlanguage{ngerman}°C
would rule out a microbial origin of methane.

A complication in developing methane isotopologues as tools for
identifying biogenicity to extraterrestrial methane is that
combinatorial effects are also associated with microbial methane and
under the right conditions, with thermogenic methane (i.e., produced
from thermal decomposition of organic matter). All of these processes
can yield large
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} deficits in
the absence of bond reordering because all of them involve multiple
steps of hydrogen addition and draw upon pools of hydrogen with distinct
D/H. Experiments verify this effect for hydrogens with variable D/H
causing
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2}deficits in
microbial methane (Taenzer et al., 2020). Current evidence suggests that
microbial methane is unique in that it shows non-equilibrium
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D values\textsubscript{,}
together with markedly negative
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} values,
probably as the result of classical kinetics involving carbon. The
release of methane during cracking of organic matter in the laboratory
can result in large
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} deficits with
near equilibrium \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D (Dong et al.,
2021), and could thus be confused with purely abiotic methane. In any
case, mass-18 methane isotopologues significantly aid us in
differentiating microbial, thermogenic, and abiotic methane on other
planets or icy moons independent of the unknown
bulk\textsuperscript{13}C/\textsuperscript{12}C and D/H of sources.

\textbf{6. Conclusion}

We performed synthesis of abiogenic methane in the laboratory and
determined \selectlanguage{greek}δ\selectlanguage{english}\textsuperscript{13}C, \selectlanguage{greek}δ\selectlanguage{english}D,
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D and
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} using a
Panorama gas-source mass spectrometer. We carried out a set of 9
experiments in hydrothermal conditions, at temperatures between 130 and
300 \selectlanguage{ngerman}°C. The experiments were performed by heating an aqueous solution in
the presence of Fe\textsuperscript{0} powder and gaseous CO. We find
\selectlanguage{greek}δ\selectlanguage{english}\textsuperscript{13}C and \selectlanguage{greek}δ\selectlanguage{english}D of methane to be isotopically depleted
relative to the starting materials, consistent with equilibrium carbon
isotope fractionation between CO\textsubscript{2} produced in the
experiment and the product CH\textsubscript{4}, and kinetic isotope D/H
fractionations between H\textsubscript{2} and product
CH\textsubscript{4}. We find most of the product methane
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D values trace the temperature
of abiotic methane synthesis within a permil. At the same time, we
observe
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2}deficits of up
to 40%0 relative to equilibrium attributable to a D/H combinatorial
effect occurring during the methane assembly. The combinatorial effect
results from our inability to determine D/H ratios of distinctive pools
of hydrogen that contribute to individual hydrogens on the methane
molecule. The D/H fractionation associated with the various steps of
hydrogen addition to carbon can explain the direction and the magnitude
of the \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2}
signatures of methane that would otherwise be at bond-ordering
equilibrium. In instances of some potentially abiotic methane found on
Earth,, particularly under high T,
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} deficits may
have been generated during methane synthesis, but were subsequently
erased by D/H reordering. In contrast,
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} deficits
generated in our experiments are seen on Earth at sites where the C-H
bond activation was too slow to re-equilibrate
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} to
environmental temperatures. As such these dramatic deficits, when
combined with near-equilibrium \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D
values, might be an avenue for the identification of abiogenesis on
other worlds.

\textbf{Acknowledgments}

This study was supported by NASA Exobiology program grant 80NSSC21K0477
(EDY, WDL, TMM, JL) and NASA Astrobiology program grant 80NSSC18K0828
(TMM).

\textbf{Data availability statement}

All the methane isotope data collected for this study will be available
online via a DOI, on the EarthChem data repository.

\textbf{Captions}

\textbf{Table 1} : Methane isotope data for synthesized abiotic methane.
Bulk \textsuperscript{13}C/\textsuperscript{12}C and D/H are reported
alongside \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D and
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} . Analytical
uncertainties are below 0.1%0 (95\% confidence intervals) for
\selectlanguage{greek}δ\selectlanguage{english}\textsuperscript{13}C and \selectlanguage{greek}δ\selectlanguage{english}D. Propagated uncertainties are listed for
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{13}CH\textsubscript{3}D and
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{12}CH\textsubscript{2}D\textsubscript{2} in 1\selectlanguage{greek}σ. \selectlanguage{english}They
are highly variable and directly reflect counting statistics, i.e.,
CH\textsubscript{4} quantity.

\textbf{Figure 1:} Quantities of synthesized abiotic methane versus
temperature. Data display a negative trend with experiment temperature.
See text for details. Note: there are nine samples here, but only seven
for isotope measurements. The two highest temperatures did not yield
enough methane for mass spectrometry.

\textbf{Figure 2:} \selectlanguage{greek}δ\selectlanguage{english}\textsuperscript{13}C (panel a) and \selectlanguage{greek}δ\selectlanguage{english}D (panel b) of
synthesized abiotic methane versus temperature and observed methane
concentrations. \selectlanguage{greek}δ\selectlanguage{english}\textsuperscript{13}C data display a weak negative
trend with experiment temperature. See text for details.

\textbf{Figure 3:} Relationships between the two mass-18 isotopologues
of methane in experimental samples and the peak experimental
temperatures. Data of other methane synthesized abiotically from Young
et al., (2017) are also shown. Data and uncertainties are from Table 1.

\textbf{Figure 4:} The two mass-18 isotopologues of methane for relevant
experimental and natural data. The FTT data are from this work. The
Sabatier experiments are from Young et al., (2017). Hydrothermal gases
are from the Rainbow, Von Damm and Lucky Strike sites from Labidi et
al., (2020). The Kidd Creek data are only from the 2.9 km level in the
mine, from Young et al., (2017). The Oman data are from various vents in
the ophioloite, and the data can be found in Nothaft et al., (2021).
Other natural data from various geological settings are from Ash et al.,
(2019), Giunta et al., (2019, 2021), Gonzalez et al., (2019)., Lin et
al., (2023), Warr et al., (2021), Young et al., (2017) and Young (2019).

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