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Model A: abundances and δ13C of short alkanesConsidering the cleavage at position m (between the no. m and no. m + 1 carbon atoms) of an n-alkyl chain with n carbon atoms (1 ≤ m 15), the isotopic compositions of gas products are insensitive to the initial kerogen side chain length distribution. For initial values, a δ13C value of −35‰ is applied. The initial chain length is in a normal distribution with a peak of C17 and a standard deviation of σ = 2 carbon atoms. The initial alkane concentrations are assumed to be 0.For simplicity, we assume that since there is no isotopic fractionation within or between the alkyl chains at the beginning of hydrogenolysis, the probability of 13C substitution at any position of any side chain is identical and determined by the initial carbon isotopic composition δ13C. Multiple 13C substitutions on a C–C chain are omitted because consideration of multiple substitutions would drastically increase the modelling complexity. This approximation is valid when the C–C chain is not too long. For example, the ratio between the probabilities of double and single 13C substitution in a C20 chain is ({left[{left(begin{array}{c}20\ 2end{array}right)}{{left(frac{{,}^{13}{{{{{rm{C}}}}}}}{{,}^{12}{{{{{rm{C}}}}}}}right)}}^{2}right]}/{left[{left(begin{array}{c}20\ 1end{array}right)}{left(frac{{,}^{13}{{{{{rm{C}}}}}}}{{,}^{12}{{{{{rm{C}}}}}}}right)}right]}) ≈ 10% for 13C/12C ~ 0.01. Such a chain is long enough that the δ13C of gas products is insensitive to C–C chain length. Numerical simulation was conducted with Mathworks MATLAB 2020a.Model B: bulk and clumped isotopic fractionations of CH4
Conversion of methylene in a long C–C chain to methane is generalised into two steps:$${{{{{rm{R}}}}}}{mbox{-}}{{{{{{rm{CH}}}}}}}_{2}{mbox{-}}{{{{{rm{R}}}}}}{^prime} mathop{longrightarrow}limits^{{{{{{{rm{r}}}}}}}_{a}}_{+{{{{{rm{H}}}}}}}{{{{{rm{R}}}}}}{mbox{-}}{{{{{{rm{CH}}}}}}}_{3}mathop{longrightarrow}limits^{{{{{{{rm{r}}}}}}}_{b}}_{+{{{{{rm{H}}}}}}}{{{{{{rm{CH}}}}}}}_{4}$$
(8)
The first step (step a) is the conversion of the methylene group R-CH2-R′ to a methyl group (RCH3) by accepting a capping hydrogen atom from the hydrogen donor (activated H2); the second step (step b) is the conversion of the methyl group to methane by accepting another capping hydrogen atom. This scheme is highly generalised, and each step may involve multiple elementary biochemical reaction steps, such as the binding of H2 and long alkyl chains to the enzyme, activation of H–H and C–C bonds, and release of the short alkane products from the enzyme. It is beyond the scope of this work to discuss the detailed biochemical reaction steps. But the cleavage and formation of chemical bonds in these steps should be constrained by the observed isotopic patterns.Due to the computational complexity, we did not use the random scission model (Model A) in the simulation involving clumped isotopic fractionation, as explained in the following. A conventional kinetic model of the decomposition of organic matter without considering the constraints of C–C chain lengths is a zero-dimensional problem. Modelling the random cutting of long C–C chains without considering isotopes is a one-dimensional problem, and modelling bulk carbon isotopic fractionation during random cutting (Model A) is a two-dimensional problem. If 13C–13C coupling is included in random cutting, the modelling is a three-dimensional problem; a complex Monte Carlo method has been applied to deal with this problem19. If the 13C–D or D–D coupling is included in Model B, as we wish, it is a problem above the fourth dimension. The complexity of programming and the difficulty of computation make the model unattainable; even if it is achievable, solving this problem is far beyond the scope of this work.Reaction equation Eq. 8 is expanded to the scheme in Fig. 3a to quantify the five most abundant isotopologues in methane (three or more substitutions such as 13CH2D2 or 12CHD3 are ignored due to their low abundances). For the subscripts in Fig. 3a (m, i, and j in ramij or rbmij), the first digit (m = 0 or 1) is the number of 13C atoms involved in the reaction, the second digit (i = 0, 1, or 2) is the number of deuterium atoms connected in the methylene or methyl group, and the third digit (j = 0 or 1) is the number of deuterium atoms in the hydrogen donor.Clumped isotopic compositions of methylene and methane are defined as the following:$$left{begin{array}{l}{{Delta}} {{{{{rm{R}}}}}}{,}^{13}{{{{{rm{C}}}}}}{{{{{rm{HDR}}}}}}^{prime} =frac{({{{{{rm{R}}}}}}{,}^{13}{{{{{rm{C}}}}}}{{{{{rm{HDR}}}}}}^{prime} )({{{{{rm{R}}}}}}{,}^{12}{{{{{rm{C}}}}}}{{{{{{rm{H}}}}}}}_{2}{{{{{rm{R}}}}}}^{prime} )}{({{{{{rm{R}}}}}}{,}^{13}{{{{{rm{C}}}}}}{{{{{{rm{H}}}}}}}_{2}{{{{{rm{R}}}}}}^{prime} )({{{{{rm{R}}}}}}{,}^{12}{{{{{rm{C}}}}}}{{{{{rm{HDR}}}}}}^{prime} )}-1hfill\ {{Delta}} {{{{{rm{R}}}}}}{,}^{12}{{{{{rm{C}}}}}}{{{{{{rm{D}}}}}}}_{2}{{{{{rm{R}}}}}}^{prime} =4frac{({{{{{rm{R}}}}}}{,}^{12}{{{{{rm{C}}}}}}{{{{{{rm{D}}}}}}}_{2}{{{{{rm{R}}}}}}^{prime} )({{{{{rm{R}}}}}}{,}^{12}{{{{{rm{C}}}}}}{{{{{{rm{H}}}}}}}_{2}{{{{{rm{R}}}}}}^{prime} )}{{({{{{{rm{R}}}}}}{,}^{12}{{{{{rm{C}}}}}}{{{{{rm{HDR}}}}}}^{prime} )}^{2}}-1hfill\ {{Delta}} {,}^{13}{{{{{rm{C}}}}}}{{{{{{rm{H}}}}}}}_{3}{{{{{rm{D}}}}}}=frac{({,}^{13}{{{{{rm{C}}}}}}{{{{{{rm{H}}}}}}}_{3}{{{{{rm{D}}}}}})({,}^{12}{{{{{rm{C}}}}}}{{{{{{rm{H}}}}}}}_{4})}{({,}^{12}{{{{{rm{C}}}}}}{{{{{{rm{H}}}}}}}_{3}{{{{{rm{D}}}}}})({,}^{13}{{{{{rm{C}}}}}}{{{{{{rm{H}}}}}}}_{4})}-1hfill\ {{Delta}} {,}^{12}{{{{{rm{C}}}}}}{{{{{{rm{H}}}}}}}_{2}{{{{{{rm{D}}}}}}}_{2}=frac{8}{3} frac{({,}^{12}{{{{{rm{C}}}}}}{{{{{{rm{H}}}}}}}_{2}{{{{{{rm{D}}}}}}}_{2})({,}^{12}{{{{{rm{C}}}}}}{{{{{{rm{H}}}}}}}_{4})}{{({,}^{12}{{{{{rm{C}}}}}}{{{{{{rm{H}}}}}}}_{3}{{{{{rm{D}}}}}})}^{2}}-1 hfillend{array}right.$$
(9)
Note that the isotopic compositions here are expressed in decimals; they should be multiplied by 1000 to give per mil values.The deuterium isotope ratio between the hydrogen donor (denoted with subscript B) and the methylene group (subscript A) is expressed as:$${alpha }_{{{{{{rm{A}}}}}}}^{{{{{{rm{B}}}}}}}=frac{1+{{{delta }}{{{{{rm{D}}}}}}}_{{{{{{rm{B}}}}}}}}{1+{{{delta }}{{{{{rm{D}}}}}}}_{{{{{{rm{A}}}}}}}}$$
(10)
For each reaction step in Fig. 3a, the corresponding rate constants are denoted as kamij for step a or kbmij for step b. Kinetic fractionation factors αkamij = kamij/ka000 and αkbmij = kbmij/kb000 define KIEs. Note that a DKIE is often expressed as kH/kD, which is the reciprocal of the αk nomenclature here. A DKIE may be primary or secondary; a primary DKIE results in αka001 ≠ 1 and αkb001 ≠ 1, and a secondary one results in αka010 ≠ 1 and αkb010 ≠ 1. Kinetic clumped isotope fractionation factors γamij = αkamij/(αka100mαka010iαka001j) and γbmij = αkbmij/ (αkb100mαkb010iαkb001j) define the excessive KIE due to isotope clumping in steps a and b, respectively30.Conversion of the reactant R-CH2-R′ is defined as 1 − f, where f is the residual fraction of R-CH2-R′:$$f=({{{{{rm{R}}}}}}{mbox{-}}{{{{{{rm{CH}}}}}}}_{2}{mbox{-}}{{{{{rm{R}}}}}}{^prime} )/{({{{{{rm{R}}}}}}{mbox{-}}{{{{{{rm{CH}}}}}}}_{2}{mbox{-}}{{rm{R}}}{^prime} )}_{{{{{{rm{initial}}}}}}}$$
(11)
Considering the isotope abundance of D  1 or γb011  > 1, as shown by the Δ12CH2D2 expression in Eq. (13). With this prerequisite, either an inverse primary DKIE (1° DKIE, αka001  > 1, αkb001  > 1) or an inverse secondary DKIE (2° DKIE, αka010  > 1, αkb010  > 1) is necessary, and through numerical simulation, we found that only the inverse 1° DKIE satisfies the above-mentioned δDA, δDB, and Δ12CH2D2 values.Two scenarios (one is the pure stochastic condition, the other is with an inverse 1° DKIE) are modelled (Fig. 3). The parameters are listed in Table 1. For comparison, analytical solutions at the beginning and end of reactions from Eqs. (12) and (13) are presented. The numerical and analytical solutions are nearly identical at the beginning of conversion. There are small differences between the numerical and analytical solutions at the endpoint because the abundance of the hydrogen donor is not extremely excessive. A weak 13C fractionation between the organic precursor and the methane product is obtained with the KIE parameters (Fig. 3b). With such a weak 13C KIE, Δ13CH3D is nearly constant for reaction extent (Fig. 3c). Note that we applied an inverse 13C KIE, as required by the δ13C distribution of the alkane gases (Method 1, Model A). The δD and Δ12CH2D2 values are independent of 13C KIE. Both the bulk and clumped isotopic compositions of methane within the range of reported values are obtained at the organic precursor conversion of 0.65–0.70 as constrained by Fig. 2. More

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