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To understand the chemical origin of life, we studied the probability of amino acid production via *β*-decay from radiocarbon (^{14}C)-containing carboxylic acid. We developed a numerical simulation code that uses a Monte Carlo method for calculating the initial condition (recoil momentum of ^{14}N in ^{14}C *β*-decay) and subsequent dynamical trajectory. We evaluated the productivity of the simplest amino acid [glycine (glycinium)] via ^{14}C *β*-decay in [3-^{14}C]propionic acid as the probability of daughter nuclide ^{14}N remaining in the compound using the developed simulation. As a result, the probability of glycinium production was determined to be approximately 81%, assuming the molecular structures of [3-^{14}C]propionic acid and glycinium with amino group bonding dissociation energy of 3.9 eV estimated using the density-functional theory (DFT) method. Furthermore, according to the calculations with various bonding dissociation potential parameters, a probability of glycinium production of approximately 32% was expected even with a loose amino group bonding dissociation energy of 2 eV.

The first step in the chemical origin of life, the abiotic synthesis of amino acids, is a fundamental problem that unsolved. Oparin proposed the first theoretical solutions to this issue,^{1}^{)} and Urey and Miller gave experimental results.^{2}^{–}^{4}^{)} These responses comprise the production of amino acids by an electric discharge in the presence of methane (CH_{4}), ammonia (NH_{3}), water (H_{2}O), and hydrogen (H_{2}). Although a recent study demonstrated that the primitive Earth consisted of a weakly reducing atmosphere containing nitrogen (N_{2}), carbon dioxide (CO_{2}), carbon monoxide (CO), and water (H_{2}O), this composition differs from the hypothetical atmosphere used in Urey and Miller's experiments.^{5}^{)} Nevertheless, a highly promising hypothesis is the electric discharge synthesis of amino acids, which has been confirmed in such an atmosphere.^{6}^{)} Recently, this production by electric discharge has been partially verified using computational simulations.^{7}^{,}^{8}^{)}

The discovery of amino acids in meteorites, however, lends credence to the extraterrestrial origin of amino acids.^{9}^{–}^{12}^{)} Since homochirality was confirmed in the amino acid of the Murchison meteorite in 1982,^{13}^{)} the origins of these amino acids were discussed with bio-homochirality. In addition, recently, the hypothesis that the intense impacts of extraterrestrial objects on Earth produce amino acids was proposed, which was supported by experimental results with a high-velocity impactor.^{14}^{,}^{15}^{)} A laser-plasma experiment was also employed to suggest and partially verify amino acid formation by the intense impacts of extraterrestrial objects on Earth and the interstellar medium.^{16}^{,}^{17}^{)} Furthermore, ab initio molecular dynamics calculation supports amino acid production by a few km/s of high-velocity impact, which corresponds to the chemical reaction above 1000 K, from primitive biogenic elements, such as isocyanic acid (HNCO).^{18}^{)}

In this study, we propose another possibility of amino acid origin: because ^{14}C converts into ^{14}N in *β*-minus-decay with a half-life of ^{19}^{)} a carboxylic acid-containing ^{14}C in the methyl group (CH_{3}) converted into an amino acid with an amino group (NH_{2}) by ^{14}C *β*-decay. The simplest example of [[3-^{14}C]propionic acid converted into glycine (glycinium)] is presented in Fig. 1 (upper pathway). In ^{14}C *β*-decay, daughter nuclide ^{14}N has momentum due to the recoil of the emitted *β*-electron and antineutrino. If the daughter nuclide ^{14}N bonds with a compound even though it has a lot of recoil momentum, an amino acid is made from the ^{14}C-containing carboxylic acid.

Figure 1. (Color online) Schematic illustration of the structure of a ^{14}C-propionic acid (left) and glycinium production and nonproduction in the *β*-decay of ^{14}C-propionic acid (right).

On Earth nowadays, cosmic rays mainly produce ^{14}C from ^{14}N via the ^{14}N(*n*,*p*)^{14}C reaction,^{20}^{)} and the natural abundance ratio of carbon is around one part per trillion, which varies depending on solar activity and recent nuclear-weapon testing.^{21}^{)} Since the natural abundance of ^{14}C or the natural number of carboxylic acids on the primitive Earth is unknown, even if an amino acid can be produced from a ^{14}C-carboxylic acid, whether those amounts are sufficient to be the origin of life (or a part of it) is unknown and needs to be discussed. In addition, the long half-life of ^{14}C must be a part of such a discussion. However, the unnecessity of nitrogen for the synthesis of carboxylic acid may be an advantage for the amino acid origin; experimental results from the high-velocity impactor^{14}^{,}^{15}^{)} exhibit higher productivity of carboxylic acid than amino acid. However, this study limits our discussion to the production probability of an amino acid from a ^{14}C-carboxylic acid by ^{14}C *β*-decay.

By orbital analysis of recoiled ^{14}N using the Monte Carlo initial momentum determination and subsequent molecular dynamics calculations, we investigated the production probability for glycinium — a protonated form of glycine — which is the staying probability of recoiled ^{14}N in the ^{14}C *β*-decay of the compound. Despite the previously reported molecular dynamics calculations for electro-discharges or high-velocity impacts,^{7}^{,}^{8}^{,}^{18}^{)} the investigation of amino acid productivity in our scenario required a special calculation that traced the specific atom that suddenly converts into a different atom and has a few km/s of high-velocity in a compound with a calculation of electron configuration changes according to atom conversion and interaction between the high-velocity atom and the stopped atom. Because such a calculation is outside the framework of current molecular dynamics calculations, our study provides a rough estimate of glycinium productivity using simple molecular dynamics calculations in the static attractive and repulsive potentials. Instead of making these educated guesses, we experimented with various potential bonding parameters and studied the results' stability, which is the lower limit of the glycinium productivity.

The production and nonproduction of glycinium in the *β*-decay of [3-^{14}C]propionic acid is presented in Fig. 1. Glycinium was produced from [3-^{14}C]propionic acid for the recoiled ^{14}N that remained in the compound (Fig. 1, top). Meanwhile, [3-^{14}C]propionic acid resolved into a carboxymethyl radical and an aminyl radical for the recoiled ^{14}N that escaped from the compound via the ^{14}C–N bond homolysis (Fig. 1, right bottom). Because the typical binding dissociation energy for N–H (4.5 eV)^{22}^{)} is comparable with the maximum energy of recoiled ^{14}N, N–H bonding remains after *β*-decay in most cases. Therefore, in our simulation, we assumed that recoiled ^{14}N escapes as a radical cation form of an aminyl cation radical (_{3}) or a carboxymethylaminyl radical (R-_{2}) and that recoiled ^{14}N is accompanied by three hydrogen atoms (NH_{3}).

The simulations were conducted in three steps, (i) determination of the initial (propionic acid) and final (glycine) molecular configuration by density-functional theory (DFT) calculation, (ii) Monte Carlo determination of the recoil energy and direction of ^{14}N in the *β*-decay of ^{14}C, and (iii) orbital dynamics calculation of recoiled ^{14}N in the three-dimensional potential, which is formed by the bonding and repulsion potentials (Morse potential) of the surrounding atoms. Step (iii) gives a rough estimation of the direction from which it is easy to escape NH_{3}, and repetition of Monte Carlo trials [initial condition given by step (ii)] gives the production ratio of glycinium. Instead, of a roughly estimated glycinium production ratio, the parameters of bonding potentials were changed in a wide range to study the behavior of the production ratio.

The initial and final molecular configurations of [3-^{14}C]propionic acid and glycinium were calculated using a DFT method in the Gaussian 16 package.^{23}^{,}^{24}^{)} From the calculated configurations, the distances between C–^{14}C in [3-^{14}C]propionic acid and the C–N bond in glycinium were 153.2 and 150.3 pm, respectively, as presented in Fig. 1. Because the recoiled NH_{3} started from outside the potential minimum in the glycinium, this different bond length results in a difference in the C–N bonding potential depth in the starting position of the recoiled NH_{3}.

The Monte Carlo method was employed to calculate the initial momentum of recoiled ^{14}N in a ^{14}C *β*-decay. In *β*-minus-decay with a maximum *β*-ray energy *β*-ray energy (*E*) distribution is represented with emitting electron momentum *p* and total *β*-electron energy *W* using the following equation: *Z*, *p* and total energy *W* are expressed, respectively, as follows: *β*-decay of ^{14}C is a Gamow–Teller transition, and spectrum correction factor *α* is a fine structural constant and *R* is a nuclear radius (*A* is a nuclear mass in an atomic unit).^{25}^{–}^{27}^{)}

The Gamow–Teller *β*-decay nuclide, including ^{14}C, has axial-vector-type *β*-neutrino (antineutrino) angular correlations expressed as follows: *v* and *θ* denote the neutrino (antineutrino) velocity and the polar angle of the emission direction with respect to the *β*-ray emission direction, respectively.^{28}^{,}^{29}^{)}

The initial recoil energy and direction of the daughter nuclide ^{14}N in ^{14}C *β*-decay are determined event-by-event using the randomly generated *β*-electron and antineutrino emission energies and directions using these equations for energy distributions *β*-electron and antineutrino *β*-ray energy and NH_{3} recoil energy after 1,000,000 Monte Carlo trials are presented in Fig. 2.

Figure 2. *β*-ray energy (a) and NH_{3} recoil energy (b) distributions by a developed Monte Carlo simulation code with 1,000,000 event-by-event calculations.

The simulation assumes a Morse-type potential for C–N covalent bonding and calculates the probability of NH_{3} remaining or escaping from the C–N bonding potential by a recoiled NH_{3} orbit calculation. The Morse potential is expressed as follows: *a* is defined as follows:

The Morse potential parameters ^{30}^{)}

According to Newton's mechanics, the flight orbit of the recoiled NH_{3} in the potentials is calculated using a four-order Runge–Kutta stepper with 1-attosecond step. The NH_{3} position (*h* is the time step (1 as), and *m* is the mass of NH_{3}. The force *F* is a partial differential of the potential in position *r* expressed as follows:

The orbital calculation was repeated for each Monte Carlo-determined initial recoil momentum of NH_{3}.

Furthermore, these calculations are for a static single molecule. Accordingly, the thermal motion, molecular rotation, and deformation by interaction with neighboring molecules are not considered, although these motions absorb the NH_{3} recoil energy and enhance the probability if glycinum production. In addition, we performed calculations using varying C–N bonding potential parameters to study the results' stability and production lower limits.

To make a case for the bonding or escaping events of recoiled NH_{3} in the compound, two-dimensional plots of the direct distances of recoiled NH_{3} vs curvature ratios at three elapsed times, namely 50, 100, and 150 fs (Fig. 3), are presented before evaluating the production ratio of glycinium to [3-^{14}C]propionic acid. The curvature ratio is defined as the ratio between the direct and orbital distances of the recoiled NH_{3}. In these plots, the calculated events are divided into two groups at a time development of 150 fs.

Figure 3. (Color online) Two-dimensional plot for direct distance vs curving ratio of recoiled NH_{3} calculated using the developed Monte Carlo simulations with different elapsed times: (a) 50, (b) 100, and (c) 150 fs.

We conducted cluster analysis for the 150-fs plots in Fig. 3. Cluster regions and the two-dimensional projection of the NH_{3} flight orbits for the first 100 events in each cluster group are presented in Fig. 4. This analysis elucidates that only the final group (6) consists of events that escaped from the original positions, and in these events, NH_{3} escaped from the compound in the direction where there is nothing. However, other groups (1–5) consist of events that stayed around the original position, and these were considered to bond to the compound and from glycinium. From these results, we set a threshold of 0.7 on the curving ratio [Fig. 3(c)] to distinguish between glycinium formed and unformed groups. We named these groups “bonding” and “escaping”, respectively. The bonding and escaping ratios were evaluated below and above the 0.7 thresholds in NH_{3} orbital calculations for the 150-fs duration.

Figure 4. (Color online) Cluster regions (left) and results of NH_{3} orbit in *XY*-projection in 150-fs plots for these regions (1–6) in Fig. 3. Orbit plots are performed for the first 100 events in each group.

We conducted 10,000 Monte Carlo trials for each parameter condition. The glycinium production probability was calculated to be 81.3% using the Morse potential parameters for the static structure (

Figure 5. (Color online) The NH_{3} remaining ratio, which is the production ratio of glycinium from [3-^{14}C]propionic acid, the in-depth function of C–N bonding potentials; (a)

The production ratios with variations of parameters

The glycinium production ratios as a function of the NH_{3} recoil energies are presented in Fig. 6. The calculations were performed for three potential depths:

Figure 6. Glycinium production ratio as a function of NH_{3} recoil energy with potential depths of 3.9, 3.5, and 3.0 eV.

According to the simulation results using the bonding potential parameters provided by DFT calculations, the largest part (81.3%) of the recoiled NH_{3} stayed in the compound, which was the highest production ratio from the [3-^{14}C]propionic acid. This ratio was almost unchanged by the parameter variations in the equilibrium C–N bond distance (

Only the glycinium production probability for recoiled NH_{3} with an energy of the upper and lower 0.5 eV of the potential depth depended on the initial direction of the recoil momentum, as presented in Fig. 6. Because the NH_{3} recoil energy distribution remained physically unchanged, the probability of glycinium production strongly depended on the bonding potential depth. At a shallow depth of C–N bonding potential, the *β*-ray energy distribution in which lower energy events appear with high frequency causes a rapid decrease in the probability of glycinium production.

For our simulations, we used static molecular structure parameters derived using the DFT approach in the Gaussian 16 package. Due to the shape change of the complex, thermal motion, and changes in the orbital electron configuration, the potential depth of the actual molecules becomes shallower than that of the static molecules. The mismatch in position between the recoiled ^{14}N in propionic acid and the potential minimum of the C–N bonding in the glycinium caused the recoiled NH_{3} to start from a nonminimum point of potential. As a result, the projected glycinium synthesis rate of 81.3% is the maximum limit. Using a dissociation energy of 3 eV for typical C–N covalent interaction,^{31}^{)} approximately 58.6% of the event remained and produced glycinium [Fig. 5(a)]. As can be seen from Fig. 5(a), more than 32.3% of the [3-^{14}C]propionic acid was transformed into glycinium via ^{14}C *β*-decay in the case of loose bond with a 2-eV potential depth.

Our calculations provide a rough estimate of the probability of glycinium production from [3-^{14}C]propionic acid via ^{14}C *β*-decay. Further studies require simulation with a dynamical molecular structure. Experimental quantification is also required to confirm the amino acid synthesis from ^{14}C-carboxylic acids. We will experimentally confirm the production of glycinium from [3-^{14}C]propionic acid. Assuming a 50% production ratio of glycinium, 1.6-nmol glycinium could be produced in 60 days from 37 MBq (15.9 mmol) of [3-^{14}C]propionic acid. This amount of glycinium produced from the highly purified [3-^{14}C]-propionic acid can be measured using a current analysis method, such as a liquid chromatography-mass spectrometry (LC-MS). Our future work includes a verification experiment.

Amino acids other than glycinium can be created from various ^{14}C-incorporated carboxylic acids if glycinium is synthesized from [3-^{14}C]propionic acid via ^{14}C *β*-decay. If amino acids in life are derived from ^{14}C *β*-decay in carboxylic acid, the parity conservation that breaks in *β*-decay could be linked to the biological homochirality of amino acids.

## Acknowledgment

The authors wish to thank members of RIKEN Center for Biosystems Dynamics Research for helpful discussions. This work was partially supported by JSPS KAKENHI Grant Number JP20K12704.

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