https://orcid.org/0000-0003-2993-0044
Figures
Abstract
Formic acid is consistently produced and detected in prebiotic chemistry experiments, constituting a precursor of many carboxylic acids and amino acids. Its behavior with exposure to gamma radiation varies with the pH and solution concentration. This work aimed to model different environmental conditions for formic acid under ionizing radiation using a system of coupled differential equations based on chemical kinetics. An ensemble of radiolysis reaction mechanisms was generated for formic acid at pH 1.5 and formate ion at pH 9, both with radiation doses from 0 to 2 kGy. This was also done for systems with both species in equilibrium, using high molar concentrations, long irradiation times, and large irradiation doses (from 0 to 70 kGy). The results show that these systems can be modeled with a high statistical relationship between the computed solutions and the experimental data; furthermore, the synthesis and degradation of the radiolysis products can be followed. Another dimension of the issue of prebiotic environments was explored using ionizing radiation and analyzing the reactions at various pH values (acidic to basic media). These models allow one to gain insights into the behavior of molecules that are difficult to detect or analyze in the laboratory. Additionally, they offer the possibility of studying potential prebiotic environments.
Citation: Paredes-Arriaga A, López-Islas A, Frias D, Rivera AL, Cordero-Tercero G, Ramos-Bernal S, et al. (2024) Numerical modeling vs experiment of formic acid and formate ion behavior under gamma radiation at several pH values: Implications on prebiotic chemistry. PLoS ONE 19(12): e0315409. https://doi.org/10.1371/journal.pone.0315409
Editor: Parag A. Deshpande, Indian Institute of Technology Kharagpur, INDIA
Received: August 6, 2024; Accepted: November 26, 2024; Published: December 12, 2024
Copyright: © 2024 Paredes-Arriaga et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: Data included in article/supplementary material/referenced in article. Model developed in this work is available at github: https://github.com/A-Paredes-Arriaga/Chemical-kinetics_EDO, DOI: 10.5281/zenodo.13966974
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
Formic acid (HCOOH) is an organic molecule commonly found in the interstellar medium [1], along with other molecules that can be considered essential building blocks of biomolecules, such as methanol (CH3OH), acetic acid (CH3COOH), acetamide (CH3CONH2), and glycine (C2H5NO2), among others [2]. In addition, formic acid has been detected in various comets, such as 67P/Churyumov–Gerasimenko [1], C/2013 R1-Lovejoy [3], and C/1995 O1-Hale-Bopp [4]; the protoplanetary disk of TW Hydrae [5]; and meteorites, such as Murchinson, Allende, Parnallee, Leedey, and Abee [6].
Formic acid and other simple acids can be formed via the radiolysis of carbonates in hydrothermal systems [7, 8], by electric discharges in volcanic dust clouds, and from asteroid impacts [9], all of which are typical scenarios for chemical evolution. Formic acid and formaldehyde are carbonylic compounds often produced in prebiotic simulation experiments under reducing and oxidizing conditions [9–12].
On the geological timescale, the prebiotic era is limited to the time period from around 4.2 to 3.8 Gya ago [13, 14]. The existence of a variety of environments on primitive Earth that could be linked to chemical evolution processes necessitates conducting experiments under different conditions–by varying the pH, pressure, and temperature, among other variables [15]. Ionizing radiation constitutes one of the energy sources that promote the synthesis and degradation of organic molecules [16–18]. The primary contributors to this form of energy on primitive Earth were the radionuclides 40K, 238U, 235U, and 232Th [19–21]. The experiments presented in this work explore the irradiation of carboxylic acid under various pH conditions, concentrations, and dose intensities. These conditions could be present in environments such as acidic lakes [22], alkaline lakes [23], parts of hot spring systems [24, 25], shallow fresh water bodies [26, 27], and oceans [28, 29]. In particular, pH variations in prebiotic simulations are a common feature when subjecting a system to energy sources such as radiation or heat [10, 12, 30–32]. This represents a priority problem to investigate because the solution pH can affect the reactions of a system and, the resulting products [33].
The radiolysis of formic acid has formed the subject of various studies [34–37]. Different reaction mechanisms have been proposed at different pH values [38–40] and concentrations [41] and under oxygen [38] and oxygen-free [42] conditions. In addition, as a reducing agent, formic acid can convert some hydroxyamino acids to their reduced amino acid forms [11]. Formic acid exist in an acid-base equilibrium with the formate ion, HCOO−[43]; with formic acid dominating at pH ≤ 1.5 and the formate ion dominating at pH ≥ 6 (Fig 1).
This work aimed to study the behavior of formic acid, the formate ion and their products under a high ionizing radiation field and different pH conditions and concentrations to simulate prebiotic environments, using a mathematical model to reproduce the experimental data. Given that formic acid, the formate ion, and their mixture entail different sets of reactions (and reaction rate constants), each system constitutes a separate problem of study, relevant to a complex prebiotic system with a variation of its initial conditions. Experimental data from previous studies have been used to validate our numerical models. The study concludes with an inter- and multidisciplinary perspective, addressing problems related to chemical evolution, chemical kinetics, radiation chemistry, differential equations, and computer modeling.
2 Methods
We develop models with different conditions of pH, radiation dose, and concentrations (Table 1); with the aim of modeling the physico-chemical conditions of different probable prebiotic environments as described in the Introduction section. In addition, some of these specific conditions have been determined by experiments previously published by other authors that we can adapt to a prebiotic environment.
Each model is in oxygen-free conditions and room temperature.
2.1 Numerical modeling
We developed numerical models for four chemical systems, all under oxygen-free and room temperature conditions: (1) formic acid at pH 1.5, concentration = 1*10−3 mol/L, maximum irradiation dose = 2 kGy, and dose intensity = 0.90 Gy/min; (2) formate at a pH 9, concentration = 1*10−3 mol/L, maximum irradiation dose = 2 kGy, and dose intensity = 0.90 Gy/min; (3) formic acid and formate mixture at pH 2, respective concentrations = 1.96*10−2 mol/L and 0.04*10−2 mol/L, maximum irradiation dose = 70 kGy, and dose intensity = 266 Gy/min; (4) formic acid and formate mixture at pH 3.75, concentrations of both = 1*10−2 mol/L respectively, maximum irradiation dose = 90 kGy, and dose intensity = 266 Gy/min. We also varied the initial concentration and maximum irradiation dose from the minimum values of 1*10−3 mol/L and 2 kGy, respectively (see Table 1).
The numerical models for formic acid (pH 1.5) and formate (pH 9) were compared with experimental data from the study by Horne et al. (2020). The numerical model at pH 2 was compared with experimental data obtained with the setup presented in Section 2.2. Further, we proposed a reaction mechanism for each chemical system in Table 1 based on a literature review.
Each chemical reaction system was written as a set of coupled differential equations, and each chemical species in a system was represented by one equation [44, 45]. The Eq 1 includes information about the molecules formed, the molecules that decay, and a source term (fi) simulating the radiation source [46, 47]:
(1)
where Xi(t), Xj(t), Xm(t), and Xn(t), is the molar concentration of chemical species i, j, m and n, at time t;
is the rate constant of the reaction between the species i and j,
is the rate constant for the reaction between chemical species m and n that produced the i species.
is the change in the molar concentration of species i at time t. The positive part of the sums represents the formation of the species i as a result of the reaction between the molecules m and n at a rate
. The negative part of the sums represents the degradation of the species i by the reaction of the molecules i and j at a rate
is an external energy source, which in this case is the gamma radiation source.
The source term (fi) is given by the Eq 2 [45, 47, 48]:
(2)
where NA is Avogadro’s number (6.022*1023 molecules), Mi is the molecular mass of species i,
is the molecular mass of water (18.02 g/mol), Gi is the radiochemical yield of species i when the system absorbs 100 eV, and Id is the dose rate (Gy/min).
Each coupled nonlinear differential equation system was solved building a Python (3.7.9) code, employing the solve.ivp function from the numpy library and implementing BDF, an implicit multistep method of variable order (1 to 5) utilizing a backward differentiation formula for the derivative approximation [49]. The general algorithm and full code are available at https://github.com/A-Paredes-Arriaga/Chemical-kinetics_EDO [50]. Finally, the statistical analysis included calculations of the root mean square error (RMSE), R2 value, and standard deviation and a quantile-quantile (Q-Q) plot of the residuals.
2.2 Experimental setup
2.2.1 Preparation and irradiation of formic acid solution.
Formic acid (0.02 mol/L) was prepared using a formic acid solution (95% pure, Sigma-Aldrich, Saint Louis, Missouri, USA) and triple distilled water. The samples were degassed with argon for 15 minutes and then irradiated with gamma rays (60Co source, Gamma-beam 651 PT at the Instituto de Ciencias Nucleares, UNAM). Formic acid aliquots (5 mL, pH 2) were exposed to doses of gamma radiation from 0 to 70 kGy.
2.2.2 Determination of formic acid degradation by titration.
The decomposition of formic acid was measured by titration after exposure to gamma radiation. Sodium hydroxide (0.02 mol/L) was used as the titrant, 0.5% phenolphthalein (Sigma-Aldrich) as the indicator, and formic acid (1*10−3 mol/L) as the standard.
2.2.3 Quantification of carbon dioxide from formic acid solutions after irradiation.
A combined ion-selective electrode, ISE (Orion ™ 9502BNWP), was used for the detection of carbon dioxide (CO2) in the formic acid solutions. Standard CaCO3 solutions (1*10−2 and 1*10−4 mol/L) were prepared to calibrate the electrode. Subsequently, 5 mL of a CO2 buffer solution was added to 50 mL of each solution.
3 Results
3.1 Formic acid at pH 1.5
The reaction mechanism for formic acid at pH 1.5 is represented by Reactions 1.1 to 1.8, based on different studies detailing various possibles routes (Table 2). At a low pH, the solvated electron and aqueous hydrogen react to generate H• radicals (Reaction 1.2). Formic acid is attacked by •OH and H• radicals to form •COOH, which reacts again with the water radicals to form CO2 and O2. CO2 can react with H• to generate the •COOH radical.
Reactions 1.1 to 1.8 were transformed into a system of coupled ordinary differential equations (Eqs 1 and 2). Fig 2 displays the outcomes of computed solutions for the system of equations and depicts the degradation of formic acid under gamma radiation and the resultant production of CO2 and H2.
The experimental data were extracted from the study by Horne et al. (2020), [42].
3.2 Formate at pH 9
The system involving formate at pH 9 comprises reactions that are different from and independent of the radiolysis of formic acid (Table 3). The primary attack is conducted by three water radicals (•OH, H•, and ), forming the
radical. This last radical serves as the basis for the secondary reactions that lead to the formation of oxalate (
). In addition, oxalate reacts continuously with water radicals (Reactions 2.8 to 2.10).
The numerical model indicates the continuous decay of formate and the formation of the oxalate ion as the main radiolysis product (Fig 3). In this case, we also compared our numerical model with the experimental data of Horne et al. (2020) [42].
The dots represent the experimental results of Horne et al. (2020), [42].
3.3 Formic acid and formate at pH 2 and high irradiation doses
Formic acid and formate maintain an acid-base equilibrium at pH 2 (Fig 1). In this scenario, the concentration ratio of formic acid to formate is 98:2, according to the equilibrium established in the studies by Joo et al. (2013, 2014) [43, 55]. Then, if the total molar concentration is 0.02 mol/L, the initial concentration of formic acid is 0.0196 mol/L, and that formate is 0.0004 mol/L.
In this case, the reaction system includes components from the preceding systems. It begins with Reactions 1.1 to 1.8 from the formic acid system and Reactions 2.1 and 2.2 from the formate system. High concentrations of formic acid under ionizing radiation result in the formation of larger molecules [41], (Table 4).
At pH 2, the available concentration of the solvated electron () is low though not null [33]. This species reacts with formic acid to produce the OC•H radical, serving as an intermediate in the reaction that leads to the formation of acids and aldehydes. Formaldehyde (HCHO) and glyoxylic acid (CHOCOOH) have been reported as radiolysis products of concentrated formic acid solutions in previous studies [35].
At pH 2, radical reacts rapidly to produce CO2, via either an electron transfer or disproportionation reaction [42, 56]. The rate constants of k18, k19, and k20 are unknown; they correspond to secondary reactions between many unstable intermediates, making their determination difficult. The numerical stability window of the model caused by the variation of these unknown rate constants is between 1.0*102 s-1 and 1.0*108 s-1. The calculated solutions for formic acid and CO2 show no significant changes in this interval. This stability window is wide because the reactions 3.2–3.4 are at the end part of the chain reaction, suggesting that the main reactions are controlled by the primary attack of the water radicals. The structure of the numerical model requires a number in each rate constant, so in order not to increase the numerical stiffness of the system, we assumed a rate constant of k = 1.0*104 s-1 for the reactions between the intermediates (Reactions 3.2 to 3.4). It is important to note that with this assumption we are not able to give information about the molecules that are produced in these reactions (formaldehyde or glyoxylic acid). If experimental information on these constants were obtained, and the constants were outside of the interval of stability, it would be necessary to resort to other numerical methods or to modify the one used to obtain a stable solution.
All reactions in the formic acid and formate system are expressed as a system of coupled ordinary differential equations (ODEs), according to Eqs 1 and 2:
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
(3.14)
(3.15)
(3.16)
(3.17)
The solutions of the coupled ODEs system of Eqs 3.1–3.17 are shown in Figs 4 and 5. Fig 4 shows the continuous decrease of the total molar concentration of formic acid and formate and the formation of CO2 from 0 to 70 kGy. The total concentration of formic acid and formate are quantified together. CO2 is the principal radiolysis product and is continuously formed.
The lines represent the numerical model, and the dots correspond to the results obtained from the experimental setup used here.
The left axis represents a linear scale, and the right axis involves the same computed solution on a log10 scale to enhance the visualization of formate decay.
The numerical model allows for the separation of the calculated solutions of the formic acid and formate equilibrium. In this case, it is displayed using both linear and log10 scales on the y-axis, with the latter enhancing the visualization of the molecules’ behavior (Fig 5). Additionally, it is possible to follow the molar concentration of each molecule involved in the system.
3.4 Formic acid and formate at a pH of 3.75
Formic acid has an equivalence point (pKa) at a pH of 3.75 [43]. Here, the concentration ratio of formic acid to the formate ion is 1:1. As a theoretical exercise, we developed a numerical model with a high concentration of formic acid at pH 3.75, to approximate the behavior of formic acid and the formate ion mixture under ionizing radiation. The aim of the exercise is to simulate another possible prebiotic environment. From a numerical point of view, we compute models at the two extremes of the pH gradient, when the systems have pure formic acid (section 3.1) and formate ion (section 3.2); and we have the elements to simulate the system when both molecules interact in the same environment at the same initial concentration. With this information we propose a system on the behavior of both molecules in conditions of an acid lake that their radiolysis generates a cluster of other molecules with biological importance.
In a 0.02 mol/L solution, the concentrations of both formic acid and formate are 0.01 mol/L. The reaction mechanism is the same as that of previously mentioned models, following Reactions 1.1 to 1.8, 2.1 and 2.2, and 3.1 to 3.4. Reaction 3.5 is substituted with Reaction 3.6 [33], as shown in Table 5, because above pH 3 the yield of CO2 decreases, and it is replaced by oxalic acid [33, 57].
The model reveals the rapid degradation of formate followed by the degradation of formic acid. The available formate is depleted at 55 kGy, at which point the degradation rate of formic acid increases. The sum of formic acid and formate decay is a continuous function, and the major radiolysis product is oxalic acid. The model also shows a low formation of CO2 (Fig 6A). For further theoretical insights, the initial conditions of this system at pH 3.75 were changed to an initial concentration of 0.001 mol/L for formic acid and the formate ion (1:1) and a total absorbed dose of 2kGy. This calculated solution was compared with the solutions of pure formic acid at pH 1.5 and pure formate at pH 9 (Fig 6B). The three computed solutions reveal the continuous decay of the molar concentrations under gamma irradiation with different degradation rates.
(a) Degradation of formic acid and the formate ion and the synthesis of oxalic acid and carbon dioxide at pH 3.75 under high irradiation doses from 0 to 80 kGy. (b) Formic acid degradation under various pH conditions and doses from 0 to 2 kGy.
3.5 Statistical analysis
Finally, statistical analyses between the experimental data and the numerical calculations were performed (Table 6 and Fig 7). Each model yields an RMSE < 4%. The residual mean and residual standard deviation are < 3% for all the systems, except for the residual standard deviation of the system at pH 2. The residual analysis reveals a close relationship between the calculated and experimental data from the three systems. The Q-Q plot suggests similar behavior for the laboratory experiments and the computational models (Fig 7).
Quantile-quantile plot of residual data from the three models and experimental results: (a) formic acid at pH 1.5, (b) formate ion at pH 9, and (c) formic acid and formate at pH 2.
4 Discussion
Each investigated system requires a different reaction mechanism because the equilibrium between formic acid and the formate ion depends on the pH [55]. Formic acid dominates at pH of 1.5. The exposure of pure formic acid to gamma radiation involves a relatively simple reaction mechanism (Reactions 1.1 to 1.8) and progressive decomposition. The numerical model based on this reaction system shows a continuous decrease in the molar concentration of formic acid from 0 to 2 kGy upon exposure to a γ-radiation field (Fig 1). CO2 and H2 are the main products of the radiolysis of formic acid [41]. The computed solution reveals the formation of these two molecules, with CO2 constituting the major product, even with Reaction 1.8 regenerating the formic acid radical (). The statistical calculations support the correspondence between the numerical model and the experimental data of Horne et al. (2020), with R2 > 0.99.
The aqueous formate solution involves a more complex reaction mechanism than formic acid. At pH 9 and a concentration of 0.001 mol/L, oxalate () is the major radiolysis product [57]. Oxalate is formed by the interaction of two formate radicals (
); it also interacts with water radicals (Reactions 2.8 to 2.10), leading to the formation of other larger molecules. This is the final stage of the reaction mechanism in our numerical model since this study aimed to model the first steps of radiolysis and the main products. The calculated results closely correspond to the experimental data, with R2 > 0.97.
The system at pH 2, comprising the formic acid and formate equilibrium, a high-concentration solution, and at high radiation dose, behaves differently. These variables generate more reactions in the system (Reactions 3.1 to 3.5) and yield more products, such as formaldehyde, glyoxylic acid, and acetaldehyde. The experimental results indicates that the decomposition of formic acid is constant between 0 and 70 kGy and that CO2 is the major radiolysis product. Computational solutions for the coupled system of Eqs 3.1–3.17 can reproduce the general behavior of the experimental results, and monitoring all the molecules of this system, including the formation of each product, is possible. The solution can be observed separately for formic acid and formate, allowing the detailed tracking of each molecule (Fig 6). The calculated solution and experimental data yield an R2 value> 0.96. Notably, all statistical analyses yield a high correlation between the experimental data and the calculated solutions. R2 > 0.96 in each model, and the RMSE is less than 4% for pure formic acid and pure formate and less than 9% for the system at pH 2. The residual mean is less than 3% in each system, validating the model.
We developed the model of formic acid at pH 3.75 to obtain a theoretical approximation of the behavior of formic acid and the formate ion under ionizing radiation. The numerical model (Fig 6A) agrees with the observed decrease in CO2 yield and increase in oxalic acid concentration, indicating that the main radiolysis product is oxalic acid, and that CO2 is formed at a low concentration. Furthermore, it shows that the concentration of formate decreases faster than that of formic acid; this is attributed to the substitution of Reaction 3.5, , with Reaction 3.6,
, restricting the regeneration of formic acid [33]. This model shows faster decay under ionizing radiation than the model at pH 2 (Figs 4 and 6A). It is possible to separate the numerical solutions of formic acid and the formate ion, and Fig 6A indicates that the formate ion decays faster than formic acid. This can be explained by the attack of H• radical on both molecules because the mixture displays competition reactions. The rate constant for the reaction of formic acid with H• is k1 = 4.4*105 s-1 (Reaction 1.3,
, and that for the reaction between the formate ion and H• is k8 = 2.1*108 s-1 (Reaction 2.2,
). The reactions display a difference in the order of magnitude of the rate constant, with the formate ion exhibiting a faster reaction rate than formic acid. Similarly, the reaction of the •OH radical with the formate ion (k7 = 2.6*109 s-1) is one order of magnitude faster than that with formic acid (k2 = 1.4*108 s-1), as shown in Reactions 2.1 and 1.4, respectively. Formic acid and formate at pH 3.75 decay linearly and completely at 89 kGy, whereas at pH 2, 75% of the total molar concentration decays at 70 kGy.
Experimental work for this model, with the radiolysis and identification of all products, can be conducted in future studies. A variation of this model, changing the initial concentration of the total formic acid and formate to 0.001 mol/L, was compared with the calculated solutions of the models involving pure formic acid and pure formate (Fig 6B). The solution of the pure formate ion is the most stable of these systems, probably because the attack of on the formate ion (Reaction 2.3, k9 = 8.0*103 s-1) is slower than the attack of the H• radical on formic acid (Reaction 1.3, k1 = 4.4*105 s-1). The H• radical and the
are in an acid-base equilibrium, where H• dominates in an acidic medium and
dominates in a basic medium. The mixture of formic acid and formate at pH 3.75 decays faster than pure formic acid and pure formate. This can be attributed to the competition kinetics of the reactions because this system involves more molecules reacting to gamma radiation, and there are fewer reactions allowing the regeneration of initial molecules or reaction intermediates.
The dose constant () for each system was also calculated, as show in Fig 6B. The dose constant is a function of the dose and concentration, expresses the reaction kinetics as a pseudo-first-order equation as a first approximation, and is descriptive only. It is an empirical indicator of the resistance of molecules to radiation [58, 59], as given by Eq 4:
(4)
where C is the concentration, C0 is the initial concentration, D is the applied dose (Gy), and
is the dose constant (Gy-1). The obtained constants with Eq 4 have the same order of magnitude and reinforce the abovementioned point, that the 1:1 mixture of formic acid and formate degrades faster than the pure formic acid and pure formate ion systems (Fig 8).
Monitoring every molecule and free radical in a chemical system is essential to understanding chemical evolution processes. One of the study goals was to elucidate the development and implications of each molecule with a possible role in prebiotic chemistry processes. This was based on the importance of formic acid and formate and their radiolysis products, such as other carboxylic acids and amino acids, in biological systems. The agreement between the numerical models and the experimental results supports the proposed reaction mechanisms for the behavior of each system. The numerical models allow one to generate hypotheses regarding the behavior of formic acid and its radiolysis products in various systems with high ionizing radiation fields and pH variations (from acidic to basic media), such as shallow lakes, certain zones of hydrothermal springs, comets, exoplanets, etc.
One should note that the method given herein to model the behavior of organic molecules in aqueous solution under ionizing radiation fields is a simplified approximation of the phenomenon, based only on chemical kinetics and not considering the diffusion coefficients of the molecules in aqueous media.
5 Conclusions
The behavior of formic acid and formate under gamma radiation changes with the pH conditions and molar concentrations used. Our numerical model can reproduce this behavior under such different conditions, where the molecules in each system can undergo numerous reactions. Since these models are based on reaction kinetics, identifying the dominant reaction mechanism is necessary to provide robust solutions. The equilibrium model between formic acid and formate involves more variables than the models for pure formic acid and pure formate. The computed solutions show a strong relationship with the experimental data, and the high statistical significance aids in validating each system and reaction mechanism. The models presented in this work can help elucidate the role played by formic acid and its products in a variety of possible prebiotic environments. Our proposed model thus extends the potential molecules and environments that can be simulated relevant to prebiotic chemistry.
Acknowledgments
A.P.A. acknowledge to CONAHCyT (CVU 929149) for the PhD fellowship and to Posgrado en Ciencias de la Tierra—UNAM for the support of the PhD studies. The authors acknowledge the PAPIIT IN114122 and CONAHCyT 319118 projects. The authors express their gratitude to C. Camargo-Raya for her technical help in the Laboratorio de Evolución Química, ICN-UNAM; and to Dr. B. Leal-Acevedo, Phys. J. Gutiérrez-Romero and M. Sc. M. J. Rodriguez Albarrán for the irradiation of the samples, and E. Palacios-Boneta, M. Cruz-Villafañe and J. Rangel-Gutiérrez for technical support. Finally, the authors would like to thank the anonymous reviewers for their comments and suggestions, which helped to improve the paper. This work was conducted at and supported by the Instituto de Ciencias Nucleares, UNAM.
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