Importance of thermodynamics dependent kinetic parameters in nitrate-based souring mitigation studies.

Souring is the unwanted formation of hydrogen sulfide (H2S) by sulfate-reducing microorganisms (SRM) in sewer systems and seawater flooded oil reservoirs. Nitrate treatment (NT) is one of the major methods to alleviate souring: The mechanism of souring remediation by NT is stimulation of nitrate reducing microorganisms (NRM) that depending on the nitrate reduction pathway can outcompete SRM for common electron donors, or oxidize sulfide to sulfate. However, some nitrate reduction pathways may challenge the efficacy of NT. Therefore, a precise understanding of souring rate, nitrate reduction rate and pathways is crucial for efficient souring management. Here, we investigate the necessity of incorporating two thermodynamic dependent kinetic parameters, namely, the growth yield (Y), and FT, a parameter related to the minimum catabolic energy production required by cells to utilize a given catabolic reaction. We first show that depending on physiochemical conditions, Y and FT for SRM change significantly in the range of [0-0.4] mole biomass per mole electron donor and [0.0006-0.5], respectively, suggesting that these parameters should not be considered constant and that it is important to couple souring models with thermodynamic models. Then, we highlight this further by showing an experimental dataset that can be modeled very well by considering variable FT. Next, we show that nitrate based lithotrophic sulfide oxidation to sulfate (lNRM3) is the dominant nitrate reduction pathway. Then, arguing that thermodynamics would suggest that S° consumption should proceed faster than S0 production, we infer that the reason for frequently observed S0 accumulation is its low solubility. Last, we suggest that nitrate based souring treatment will suffer less from S0 accumulation if we (i) act early, (ii) increase temperature and (iii) supplement stoichiometrically sufficient nitrate.

Souring is the unwanted formation of hydrogen sulfide (H 2 S) by sulfate-reducing microorganisms (SRM) in sewer systems and seawater flooded oil reservoirs. Nitrate treatment (NT) is one of the major methods to alleviate souring: The mechanism of souring remediation by NT is stimulation of nitrate reducing microorganisms (NRM) that depending on the nitrate reduction pathway can outcompete SRM for common electron donors, or oxidize sulfide to sulfate. However, some nitrate reduction pathways may challenge the efficacy of NT. Therefore, a precise understanding of souring rate, nitrate reduction rate and pathways is crucial for efficient souring management. Here, we investigate the necessity of incorporating two thermodynamic dependent kinetic parameters, namely, the growth yield (Y), and F T , a parameter related to the minimum catabolic energy production required by cells to utilize a given catabolic reaction. We first show that depending on physiochemical conditions, Y and F T for SRM change significantly in the range of [0-0.4] mole biomass per mole electron donor and [0.0006-0.5], respectively, suggesting that these parameters should not be considered constant and that it is important to couple souring models with thermodynamic models. Then, we highlight this further by showing an experimental dataset that can be modeled very well by considering variable F T . Next, we show that nitrate based lithotrophic sulfide oxidation to sulfate (lNRM 3 ) is the dominant nitrate reduction pathway. Then, arguing that thermodynamics would suggest that S • consumption should proceed faster than S 0 production, we infer that the reason for frequently observed S 0 accumulation is its low solubility. Last, we suggest that nitrate based souring treatment will suffer less from S 0 accumulation if we (i) act early, (ii) increase temperature and (iii) supplement stoichiometrically sufficient nitrate.

Introduction
Biologic hydrogen sulfide (H 2 S) production due to the activity of sulfate reducing microorganisms (SRM), or the so-called souring process, is a common problem in sewer systems (Jiang et al., 2014) and secondary oil recovery by seawater flooding (Veshareh and Ayatollahi, 2019) due to the odorant, corrosive and toxic nature of H 2 S. Nitrate treatment (NT) is one of the intervention methods to control souring by stimulating nitrate reducing microorganisms (An et al., 2010). Nitrate can suppress souring by various mechanisms such as activating organotrophic nitrate-reducing microorganisms (oNRM) that may outcompete SRM for the available organic matter (Agrawal et al., 2012), and reducing sulfide concentration by stimulating lithotrophic nitrate-reducing microorganisms (lNRM) . Lithotrophic nitrate-reduction based sulfide-oxidation can be associated with biogenic elemental sulfur (S 0 ) (Huang et al., 2015). Due to the corrosive character of S 0 (Lahme et al., 2019;Schmitt, 1991), its accumulation can reduce the efficiency of NT (Dolfing and Hubert, 2017). Therefore, an understanding of likely nitrate reduction pathways as well as kinetics of sulfate and nitrate reduction is essential for designing promising NT plans.
Respiring prokaryotes catalyze redox reactions (called catabolic reactions) to derive energy for growth and maintenance (Jin, 2012). The amount of free energy available from various redox reactions, or Gibbs free energy of catabolic reaction (ΔG cat ) has been used by scientists as a method to compare the likelihood of different metabolisms/pathways. For example, Dolfing and Hubert (2017) used this method to predict nitrate reduction pathways in nitrate-based oil reservoir souring mitigation. Since under typical oil reservoir conditions -ΔG cat of nitrate reduction coupled to acetate oxidation was higher than nitrate reduction coupled to sulfide oxidation, they proposed that under realistic oil field conditions nitrate reduction is more likely to be organotrophic rather than lithotrophic. Dolfing and Hubert (2017) claimed that lithotrophic nitrate reduction coupled to partial oxidation of sulfide to sulfur is an exception and can be more favorable than organotrophic nitrate reduction as far as acetate to sulfide molar ratio is less than 0.001, or the temperature is sufficiently low. Additionally, showing that per mole of nitrate sulfide oxidation to S 0 releases slightly more energy than sulfide oxidation to sulfate, they suggested that S 0 accumulation is likely to occur under nitrate limiting conditions. The assessment of Dolfing and Hubert labels one metabolism/pathway as favorable and the other as unfavorable, and does not allow an energy based quantitative comparison between the occurrence likelihood of each metabolism. We are not aware of any previous research that has used thermodynamics to make a quantitative comparison between the occurrence likelihood of various nitrate reduction pathways in presence of sulfide and S 0 . However, for some other metabolisms such as syntrophic oxidation, iron reduction, sulfate reduction and methanogenesis, Jin and Kirk (2016) and Jin and Kirk (2018) used a thermodynamic limiting factor (F T ) that relates the rate of microbial metabolisms to ΔG cat . The F T coefficient is not the only term that links thermodynamics to metabolism kinetics. Growth yield (Y) is another parameter that controls the kinetics of microbial metabolisms and is a function of ΔG cat and of the Gibbs free energy of anabolic reaction (ΔG an ) (Jin and Roden, 2011;Smeaton and Van Cappellen, 2018). To the best of our knowledge, no previous research work has used growth yield to relate thermodynamics of microbial reactions to their kinetics. Note that Y and F T have been assumed to be constant in biomass explicit microbial kinetic models used over the last decades to simulate souring and its mitigation with nitrate (Hvitved-Jacobsen et al., 2013;Sharma et al., 2008;Veshareh andNick, 2019, 2021a;2021b). As such, the error associated with assuming Y and F T parameters to be constant in simulation of souring process and nitrate-based souring mitigation measures, is unknown.
To address the abovementioned gaps, in this article, we calculate Y and F T to evaluate the range in which they vary under various physiochemical conditions such as electron donor (ED) and electron acceptor (EA) availability, pH and temperature, relevant to sewer systems and petroleum reservoirs. We then use Y to link thermodynamics of sulfur and nitrogen cycle to their kinetics. Using this link, we first revisit the questions raised by Dolfing and Hubert (2017) by illuminating whether nitrate reduction is more likely to be organotrophic or lithotrophic, and whether or not S 0 accumulation during NT is due to a thermodynamic drive. Lastly, we suggest some measures to minimize S 0 accumulation in NT of souring.

Theory
Lithotrophic nitrate reducers can obtain their energy by (i) oxidation of sulfide to sulfur (lNRM 1 ), (ii) oxidation of sulfur to sulfate (lNRM 2 ) and (iii) direct oxidation of sulfide to sulfate (lNRM 3 ). Regardless of whether nitrate is reduced through lNRM or oNRM, it is reduced either to nitrogen gases through denitrification or to ammonium through dissimilatory nitrate reduction (DNRA) (Callbeck et al., 2013). However, various research works (e.g. (Veshareh and Nick, 2019) and (Marietou et al., 2020)) have shown that DNRA is the responsible nitrate reduction pathway in nitrate mitigation of reservoir souring. Therefore, here we do not consider redox reactions related to the denitrification pathway. Lithotrophic nitrate-reducing microorganisms can be autotrophic (laNRM) (Hamilton et al., 2015) or heterotrophic (lhNRM) (Miroshnichenko et al., 2003). Various organic compounds can serve as the ED of SRM and oNRM; however, Dolfing and Hubert (2017) showed that the energy yield of SRM and oNRM metabolisms is independent of the type of the organic ED used. Therefore, here we only consider acetate as the ED, and as the organic carbon source for cell synthesis of organotrophic and lithoheterotrophic metabolisms. Catabolic and anabolic reactions are from, or are based on the methodology introduced by Smeaton and Van Cappellen (2018). Table 1 and 2 list the catabolic and anabolic reactions that represent various metabolisms studied in this work.
Chemical compounds in the aqueous phase can lose or obtain protons or hydroxide due to reaction with water molecules, or combine with other ions or molecules in a process called speciation (Jin and Kirk, 2018). Due to speciation, chemical compounds dissolved in water can exist in various forms or chemical species. As a result, environmental pH can affect the energetics of redox reactions directly by changing the chemical activity of protons, for redox reactions that consume or produce protons, or indirectly, by controlling the speciation of reactants and products. In this work, all reactions are written using dominant chemical species at pH 7. At a neutral pH, hydrogen sulfide (HS − ) occurs in relatively equal proportions as dihydrogen sulfide (H 2 S). Following Smeaton and Van Cappellen (2018), we choose HS − . Catabolic and anabolic reactions are written per mole ED and per mole biomass, Table 1 The catabolic and anabolic reaction, Gibbs free energy and enthalpy for various metabolisms studied in this work. M.J. Veshareh et al. Water Research 206 (2021) 117673 respectively.

Methodology
According to the thermodynamically consistent rate law (Jin and Bethke, 2005;, respiration rate (r) [mol ED⋅s − 1 ] can be written as follows: where v max [mol ED ⋅ (mol biomass) − 1 ⋅ s − 1 ] is the maximum rate of a metabolism, X is the biomass concentration [mol ⋅ (kg water) − 1 ], F K is a kinetic limiting term and F T is a thermodynamic limiting term. According to Monod (1949) and LaRowe et al. (2014) F K and F T can be defined as follows: Where C [mol ⋅ (kg water) − 1 ] is concentration, K is half saturation constant, ΔG cat [J⋅ (mol e − ) − 1 ] is the Gibbs free energy of a metabolism's catabolic reaction under non-standard conditions, F [C ⋅ mol − 1 ] is the Faraday constant, R [J ⋅ mol − 1 ⋅ K − 1 ] is the gas constant, T [K − 1 ] is temperature, and Δψ [V] is the electric potential across the membrane. The subscripts ED and EA denote electron donor and electron acceptor, respectively. Even though the value of Δψ can be different for various low energy environments and for various metabolisms, an evaluation of investigations on several distinct organisms led to the proposition that 120 mV can be considered a representative value for Δψ (Dimroth et al., 2003;Kadenbach, 2003). Toei et al. (2007) and Daniels et al. (1984) reported a value of 118 mV for Δψ. Therefore, here we assume that Δψ is equal to 120 mV for all the considered metabolisms.
Gibbs free energy of a reaction under non-standard condition can be calculated as follow: Where ΔG ∘ is the Gibbs free energy of a reaction under biochemical standard conditions (25 ∘ C, 1 atm, pH 7 and chemical activity of unity (Jin and Kirk, 2018)) and can be calculated by subtracting the sum of Gibbs free energy of formation (ΔG ∘ f ) of substrates from that of products. Q is the reaction quotient. For a hypothetical reaction aA + bB → cC + dD, reaction quotient is equal to: Where [S] is the activity of a reactant/product (S = {A, B, C, D}). Chemical speciation and activity of species are calculated using LLNL Thermodynamic Database (Delany and Lundeen, 1990) and PHREEQC v.3 (pH-REdox-EQuilibrium written in the C programming language (Parkhurst and Appelo, 2013). Activity of S 0 is assumed to be equal to its concentration. Gibbs-Helmholtz equation is used to correct ΔG ∘ for non-standard temperatures: Where ΔH ∘ 298.15 is the enthalpy of a reaction in standard conditions and can be calculated by subtracting the sum of enthalpies of formation of substrates (ΔH ∘ f ) from that of products. The value of ΔG ∘ f and ΔH ∘ f for acetate is obtained from Shock (1995), for inorganic species from Shock et al. (1997), and for biomass from Roels (1980).
The rate of biomass (X) formation is given by: Where µ[s − 1 ] is the specific growth rate and b[s − 1 ] is the specific maintenance rate. The specific growth rate is linked to respiration rate using growth yield (Y) [mol biomass ⋅ (mol ED) − 1 ] through the following relationship: Growth yield Y is dependent on ΔG cat , ΔG an , and energy utilization efficiency of organisms (VanBriesen, 2002). In order to study the effect of changes in chemical (variation in species concentrations and pH) and physical (e.g. temperature) conditions, the Gibbs Energy Dynamic Yield Method (GEDYM) of Smeaton and Van Cappellen (2018) is employed: Where α and β are model parameters and equal to -0.0004 and -0.0694 for a broad range of metabolisms including all major EAs, fermentation, methanogenesis and acetogenesis. Smeaton and Van Cappellen only considered hydrogen as the non-organic ED. Note that GEDYM has not been validated for nitrate-reduction, and sulfideoxidation pathways. However, as the model is valid for all the other metabolisms mentioned above, here we assume that Y for nitratereduction and for sulfide-oxidation pathways follows equation 9 as well.
According to equation 7, the biomass growth depends on the thermodynamic dependent terms of Y and F T . Assuming that microorganisms that derive their energy from the various metabolisms (i) are all present, (ii) have the same kinetic parameters such as v max , K A and K D , and (iii) have the same initial biomass concentration (X), terms Y and F T determine which metabolism proceeds faster. For each metabolism, we consider a set of pH, temperature and concentration of reactants and products in the range observed in petroleum reservoirs and sewer systems, referred to as the base condition.
In order to analyze the temperature effect, the temperature range of  1 to 110 • C is evaluated. This is because contrary to sewer systems, where souring occurs in a relatively narrow temperature range, in petroleum reservoirs souring can occur in a relatively broad temperature range, anywhere between the injection temperature (e.g. between 4 to 25 ∘ C if North Sea water is injected (CLIMATE-DATA.ORG, 2021) to the reservoir temperature. The temperature of subsurface reservoirs depends on their depth (e.g Willems & Nick 2019). Souring in temperatures higher than 110 ∘ C can be ignored as these temperatures preclude microbial activity (Thaysen et al., 2021). Since chemical compounds depending on temperature and pH can appear in water in various forms, we define the base condition based on the sum of various forms. In the base condition, the water phase is saturated with S 0 . Elemental sulfur solubility is calculated by exponential regression of data reported by Kamyshny Jr (2009). Table 3 lists the base condition. C(+4) stands for the sum of carbonate species including carbonic acid (H 2 CO 3 ), bicarbonate (HCO − 3 ), carbonate (CO 2− 3 ) and dissolved carbon dioxide (CO 2(aq) ). S(-2) stands for the sum of sulfide species including dihydrogen sulfide (H 2 S), hydrogen sulfide (HS − ) and sulfide (S 2− ). N(-3) stands for the sum of ammonium (NH + 4 ) and ammonia (NH 3 ), and C(0) stands for sum of acetate and acetic acid. We investigate the variations in Y and F T due to deviations from the base case by changing physical (e.g. temperature) and chemical conditions (e.g. pH and EA concentration). Fig. 1A and B show Y and F T of SRM for various ED (acetate) and EA (sulfate) concentrations for the base case. The value of Y changes significantly from 0.4 for EA = ED concentration of 0.028 M, to zero if EA and ED concentration are both less than 10 − 5 M (Fig. 1A). This is because according to equation 4, by reducing the concentration of EA and ED (reactants of SRM catabolic reaction), and ED (reactant of SRM anabolic reaction), respectively, the energy yield of the catabolic reaction (-ΔG cat ) decreases and the energy demand of the anabolic reaction (ΔG an ) increases. The value of F T follows a similar trend as Y and decreases from a maximum of around 0.5 when ED and EA are high (0.028 M) to around zero (6 × 10 − 4 M) when EA and ED are minimal (10 − 10 M, Fig. 1B). Fig. 1C demonstrates the effect of pH on Y and F T of SRM for the base case. In the pH range of 6.6 to 9.8, F T variations are relatively low and Y shows only a small reduction. Considering the catabolic reaction of SRM and equation 3, F T depends only on ΔG cat , and ΔG cat depends on the activity of HCO − 3 , HS − and acetate. Since the activity of these species is relatively pH independent in the pH range of 6.6 to 9.8, ΔG cat and consequently F T stay relatively constant. The value of Y depends on both ΔG cat and ΔG an . While ΔG cat is relatively constant, in the range of 6.6 to 9.8 due to decrease in NH + 4 (Fig. 2D) ΔG an increases slightly and this leads to a small reduction in Y (from 0.18 to 0.14). For pH values less than 6.6 and above 9.8 decreases in the activity of HCO − 3 and HS − result in increases in Y and F T . For pH values less than 6.6, the decrease in acetate activity cancels out the effect of reduction in HCO − 3 and HS − activity. Therefore, the slope of F T (that is, the absolute value of the derivative of F T ) is slightly sharper for pH values higher than 9.8 (0.19 per unit pH) than pH < 6.6 (0.16 per unit pH). For pH values above 9.8, a significant decrease in NH + 4 activity causes a significant increase in ΔG an . The increase in ΔG an cancels out the decrease in ΔG cat and as a result, the increase in Y values due to an increase in pH above 9.8 (0.04 mol biomass per mol ED) is less than the increase in Y values due to a decrease in pH below 6.8 (0.13 mol biomass per mol ED). Fig. 1D illustrates the impact of temperature on Y and F T of SRM for the base case. Increase in temperature increases the catabolic energy yield (-ΔG cat ) as well as ΔG an . While Y decreases slightly (from 0.19 to 0.17 mol biomass per mol ED) as it depends on both ΔG cat and ΔG an , F T increases by a factor of 2 (from 0.13 to 0.26) since it is only dependent on ΔG cat . Fig. 1E demonstrates how HS − concentration affects Y and F T . As HS − is the product of the catabolic reaction of SRM, the reduction in HS − increases -ΔG cat , and as a result, both Y and F T for SRM increase.

Variations in Y and F T for nitrate reduction
Various nitrate reduction pathways considered in this study have a sufficiently high − ΔG cat such that F T for all of them is equal to one under the conditions for which Y is plotted in Fig. 3. Fig. 3 shows Y for the various nitrate reduction pathways listed in Table 1. For the base case, when the ED concentration increases, growth yield increases for all nitrate reduction pathways. The smallest Y values are associated with laNRM 1 and lhNRM 1 (maximum 0.12 and 0.34, respectively, Fig. 3A), since these two metabolisms have the lowest -ΔG  Table 1. However, the value of ΔG an for laNRM 1 as function of ED concentration (HS − ) varies from -1.1 to 145 kJ per mol biomass. Therefore, while laNRM 1 and lhNRM 1 have the same ΔG cat , laNRM 1 has a lower ΔG an compared to lhNRM 1 (41.84 kJ per mol biomass) for ED concentrations greater than 2.4 × 10 − 5 . A smaller ΔG an for laNRM 1 does not cause a higher Y value compared to lhNRM 1 , as Y depends also on the number of moles of ED (υ) that is utilized in order to synthesize 1 mol of biomass (Smeaton and Van Cappellen, 2018). In laNRM 1 metabolism (υ = 2.1) only [1-2.1Y] fraction of the ED (black dashed line in Fig. 3B) is oxidized for energy production, while in lhNRM 1 metabolism (υ = 0) the ED oxidation only serves for energy production. This can be the reason why Y values of lhNRM 1 are higher than those of laNRM 1 despite having an equal ΔG cat and a higher ΔG an . The Y values for a given ED concentration (e.g. 10 − 4 M ED) are higher for oNRM (0.77, Fig. 3B) than laNRM 3 (0.64, Fig. 3B) since oNRM metabolism has a higher − ΔG ∘ cat (536.0 kJ per mol ED) than laNRM 3 (487.9 kJ per mol ED). The metabolism of oNRM has also a higher − ΔG ∘ cat compared to lhNRM 3 . However, the value of υ is equal to 0.525 for oNRM and 0 for lhNRM 3 . That is, while -ΔG ∘ cat of oNRM is equal to 536.0 kJ per mol ED, only between 50 to 74% of it (solid line, Fig. 3B) is used for energy production. As a result, the energy produced by oxidation of 1 mol ED in oNRM metabolism is in the range of 57 to 82% of that of lhNRM 3 (dotted line in Fig. 3B). Therefore, for a given ED concentration (e.g. 10 − 4 M ED), Y values for lhNRM 3 (1.1 mol biomass per mol ED) are higher than for oNRM (0.77). Similar to the plot of Y versus ED concentration (Fig. 3A), the plot of Y versus EA concentration (Fig. 3C) has a positive slope for all nitrate reduction metabolisms. However, the slope of Y versus log of EA concentration is smaller than the slope of Y versus log of ED concentration. For metabolisms with υ > 0, this is because ED is present in both catabolic and anabolic reaction, whereas EA is only present in the anabolic reaction. For metabolisms that partially oxidize HS − , the stoichiometric coefficient of EA is a quarter of the stoichiometric coefficient of ED. Consequently, ΔG cat is a stronger function of ED concentration than of EA concentration. Fig. 3D illustrates that in general an increase in pH decreases Y for all the nitrate-reducing metabolisms as they are all proton consuming. However, for low pH values (from 3 to 5), the influence of increasing pH on ΔG cat is canceled out by an increase in HS − activity (Fig. 2B) for laNRM 1 and lhNRM 1 . The stoichiometric ratio of HS − to H + is bigger for complete oxidation of sulfide compared to the partial oxidation. In consequence, in the pH range of 3 to 5 the impact of an increase in HS − concentration on ΔG cat of lhNRM 3 and laNRM 3 is higher than the impact of an increase in pH, leading to an increase in Y. The pH increase effect in oNRM metabolism is canceled out by the increase in acetate concentration (Fig. 2C) in the pH range of 3 to 4. The reduction in Y decreases in pH values higher than 10 for all metabolisms due to reduction of NH + 4 concentration. For oNRM metabolism, pH values higher than 10 also reduce HCO − 3 concentration. Consequently, the Y value for oNRM levels off relatively at pH 10. Fig. 1. A) Growth yield (Y) and B) thermodynamic limiting factor (F T ) of SRM calculated using equations 9 and 3, respectively, for various electron acceptor (sulfate) and electron donor (acetate) concentrations, while other influencing parameters such as temperature are equal to the base case. C), D) and E) show the impact of changing pH, temperature and HS − concentration (from that of the base case) on Y (solid lines) and F T (dashed lines), respectively.

Effect of temperature and sulfate concentration
Similar to SRM metabolism, temperature influence on Y value of oNRM, lhNRM 3 , laNRM 3 is relatively insignificant (maximum 3.6, 6.0 and 4.0 % change, respectively). The Y values of lhNRM 1 and laNRM 1 increase significantly from 0.27 to 0.41, and from 0.07 to 0.14, by a reduction in temperature from 110 to 4 ∘ C since the decrease in temperature reduces S 0 solubility (Kamyshny Jr, 2009). Among all nitrate reducing metabolisms, Y of laNRM 3 and lhNRM 3 depend on sulfate concentration. Fig. 3F shows that Y of laNRM 3 is greater than Y of oNRM for sulfate concentrations lower than 10 − 5 M.

Importance of taking into account variations in Y and F T
SRM activity has caused detrimental consequences such as corrosion and reservoir plugging for seawater injection into oil reservoirs (Youssef et al., 2009). In sewer systems, the H 2 S produced by SRM is a major source of odor nuisance (Jiang et al., 2015) and corrosion of concrete sewer pipes (Pikaar et al., 2019). Modeling SRM activity is also essential for mainstream anaerobic digestion technology development (Durán et al., 2020). Microbial sulfate-reduction models have been used to predict the extent of reservoir souring, aiming to minimize the impacts and costs of SRM activity. To the best of our knowledge, in these models Y has been always assumed to be a constant, and the energy that SRM require to maintain transmembrane electric potentials has been ignored (i.e. F T =1) (e.g. (Cheng et al., 2016;Haghshenas et al., 2012;Veshareh et al., 2021)). Underestimation or overestimation of souring in seawater injection can cause erroneous material design for injection/production wells and surface facilities (Johnson et al., 2017). Additionally, inaccurate estimate of souring does not allow efficient treatment design. For example, in nitrate treatment, over usage of nitrate can lead to accumulation of nitrite, which increases corrosion in production wells (Huang and Zhang, 2006). The growth yield of SRM varies significantly from around 0.33 when the concentration of SRM ED and EA is around the maximum concentration observed in typical seawater flooding processes (0.028 M) (Vigneron et al., 2017), to zero for ED and EA concentrations smaller than 1µM (Fig. 1A). In sewer systems, petroleum reservoirs as well as the biofilm of bioreactors, there is invariably a gradient of substrates. As our results show, the value of Y that controls souring in time and space domains is significantly dependent on the substrate concentrations. Assuming a constant Y value -the value of which depends on the laboratory conditions under which Y has been determinedwill thus cause over or underestimation of souring rates in that given time and location. Due to the low exergonicity of SRM catabolic reaction, and depending on the concentration of ED and EA, SRM kinetics will be thermodynamically limited by 60 to around 100% (F T = 0 to 0.4, Fig. 1B). That is, while assuming a constant Y may over or underestimate souring rate, ignoring F T will always lead to overestimation of souring. Therefore, accurate souring simulation requires   3. Growth yield (Y) calculated using equation 9 for organotrophic nitrate reduction (oNRM, circle markers), litho-heterotrophic and autotrophic nitrate reduction coupled to sulfide oxidation to elemental sulfur (lhNRM 1 , * markers and laNRM 1 , triangle markers), litho-heterotrophic and autotrophic nitrate reduction coupled to sulfur oxidation to sulfate (lhNRM 2 , diamond markers and laNRM 2 , x markers), litho-heterotrophic and autotrophic nitrate reduction coupled to sulfide oxidation to sulfate (lhNRM 3 , solid line, laNRM 3 , square markers), for various A) electron donor concentrations (acetate for oNRM, sulfide for laNRM 1 , laNRM 3 , lhNRM 1 and lhNRM 3 , and sulfur for laNRM 2 and lhNRM 2 ), C) electron acceptor (nitrate) concentrations, D) pH, E) temperature and F) sulfate concentrations. B) Left y-axis, the fraction of electron donor that is used for energy production, right y-axis, the ratio of catabolic energy yield of lithoheterotrophic metabolism to that of lithoautotrophic metabolism, for various electron donor concentrations.
considering both of these thermodynamic dependent parameters.
Among various environmental conditions, pH is a classical physiological parameter. In order to tolerate acidic or alkaline conditions, microorganisms require special surface properties that protect cells from proton or hydroxide ions (Golyshina and Timmis, 2005;Horikoshi, 1999). Very low or high pH values restrict microbial reactions due to various reasons. For example, at low pH conjugate acids become abundant and diffuse into cell membrane, destabilizing the membrane and dissipating proton motive force (Russell and Dombrowski, 1980). Our results show that in near neutral pH, with an increase in pH, the SRM rate decreases due to reduction in both F T and Y. The effect of pH on SRM activity has been well studied in the context of wastewater treatment since the pH of waste water can be subject to significant changes due to processes such as fermentation of organics or treatment with alkali (Sharma et al., 2013). Gutierrez et al. (2009) reported that a pH increase from 7.6 to 8.6 and 9.0 reduces the biological sulfate reduction by 30 to 50%.
To further highlight the importance of considering the effect of thermodynamics on souring kinetics, we compare our data with experimental and modelling data published by (Sharma et al., 2014). These authors conducted several batch experiments using a sewer biofilm reactor at various pH values in the range of 4.0 to 9.0 and observed that sulfate reduction rate is maximal at pH 6.3, and decreases when pH deviates from this value. In order to model the pH effect Sharma et al. (2014) used the pH inhibition expression proposed by Angelidaki et al. (1993) for pH values lower than 6.75, and for higher pH values they employed the non-competitive inhibition model of (Siegrist et al., 2002) that is focused on free ammonia. The model developed by Sharma et al. (2014) predicts that souring rate in the range of 6.4 to 8.3 is independent from pH (i.e. is constant), whereas experimental data shows that souring rate decreases with increase in pH in this range (Fig. 4A). The model of Sharma et al. (2014) ignores variations in Y due to pH change and does not take into account any thermodynamic limiting factor. Since the rate of sulfate reduction in the course of Sharma et al. (2014) experiments ( Fig. 2 in their work) is constant, the effect of Y variations can be neglected. To evaluate whether the variations in sulfate reduction rate in the pH range of 6.4 to 8.3 is related to changes in F T , we consider the value of sulfate reduction rate at a reference value and predict other sulfate reduction rates using the following equation: Fig. 4A shows a good agreement between the values predicted using equation 10 and the experimental data. Therefore, considering F T with Δψ = 100 mV can explain changes in sulfate reduction rate in the pH range of 6.4 to 8.3. The strong correlation between F T and rates is probably due to the minimal impact of physiological parameters as pH values are relatively close to neutral. Note that in pH higher than 7, -ΔG cat is less than 45 kJ (Fig. 4B), i.e. the model introduced by (Jin and Bethke, 2003) would predict F T to be equal to zero. Therefore, using the thermodynamic limitation factor proposed by LaRowe et al. (2012) seems to be more appropriate for modeling souring. High pH values also decrease H 2 S liquid to gas mass transfer, increasing the total dissolved sulfide (Ganigue et al., 2011). Higher dissolved sulfide concentrations reduce SRM activity not only due to its toxicity (Kushkevych et al., 2019;McCartney and Oleszkiewicz, 1991), but also by reducing the -ΔG cat of sulfate reduction (Fig. 1E).

Is it valid to use the GEDYM for lNRM pathways?
As mentioned earlier in the methodology section, we assume that GEDYM is valid for estimating the growth yield of lithotrophic metabolisms listed in Table 1. Zeng and Zhang (2005) conducted two batch experiments and measured the growth yield for laNRM 2 to be 0.75 and 0.85. The growth yield values calculated in this work using GEDYM (Fig. 3E), matches these experimental values, suggesting that GEDYM holds true also for lithotrophic nitrate reduction based sulfide oxidation.

What nitrate-reduction pathway is expected to be faster thermodynamically?
Assuming (i) a diverse microbial community that can sustain the various nitrate reduction pathways considered in this study, (ii) all community members have the same kinetic parameters and initial biomass concentration, and (iii) the concentration of EA and ED is equal for all metabolisms, lhNRM 3 is around two times faster than oNRM pathway, for various physiochemical conditions found in typical petroleum reservoirs. This is in disagreement with the work of Dolfing and Hubert (2017) where except for special conditions such as low temperature, or low acetate to H 2 S ratios, oNRM pathway is predicted to prevail. This is because Dolfing and Hubert (2017) consider the pathway with a higher -ΔG cat as the dominant pathway, while not considering the anabolic reaction. Our results highlights that to find the dominant metabolism among metabolic pathways with high catabolic energy yields (i.e. F T = 1), ΔG cat cannot be used directly to reveal the dominant pathway as it has been used for other metabolisms such as methanogenesis or acetogenesis Jin and Kirk, 2018). Note that while the value of Y allows a quantitative comparison between the rate of two metabolisms in a system at its initial condition, the relationship of respiration rate and Y is not linear and rather exponential (equations 1, 7 and (8). Therefore, a two time higher Y of lhNRM 3 compared to that of oNRM can lead to lhNRM 3 domination. This is in agreement with various studies available in the literature. Hubert et al. (2003) injected nitrate into a soured bioreactor (sulfide concentration of 12 mM) and observed that despite injection of 25 mM lactate, nitrate reduction was entirely coupled to sulfide oxidation to sulfate. Lambo et al. (2008) showed that lithotrophic reduction of nitrate always preceded organotrophic reduction of nitrate.  Sharma et al. (2014), modeled by Sharma et al. (2014) and calculated by using equation 10, respectively. B) The Gibbs free energy of catabolic reaction of sulfate reducing microorganisms. C) Left y-axis, circle markers, elemental sulfur (S 0 ) solubility and right y-axis, dashed line, the kinetic limiting factor (F K ) that takes into account the effect of limitation of electron donor (S 0 ) on the kinetics of lithotrophic nitrate-reduction, sulfur-oxidation.
M.J. Veshareh et al. Water Research 206 (2021) 117673 Mathioudakis et al. (2006) by conducting experiments on sour waste water samples (sulfide concentration of around 1mM) reported that nitrate reduction is preferentially autotrophic and that heterotrophic nitrate reduction only commenced after sulfide had been completely oxidized. In another study, Okabe et al. (2003) observed that in sewer biofilm experiments, 65% of the H 2 S produced by SRM residing in deeper layers of the biofilm was oxidized via lNRM in shallower layers. A similar observation has been reported by Garcia-de-Lomas et al. (2007). Dolfing and Hubert (2017) discussed whether lNRM 1 or lNRM 3 is expected to be the dominant lNRM pathways by comparing the -ΔG of acetate driven reduction of sulfate and S 0 . They proposed that because -ΔG of acetate driven reduction of S 0 is less than acetate driven reduction of sulfate, lNRM 1 is the dominant lNRM pathway. They suggested that the dominance of lNRM 1 is more likely at lower temperatures, as the differences between -ΔG of acetate driven reduction of S 0 and that of acetate driven reduction of sulfate, increases with temperature. According to our results (Fig. 3), under all studied conditions, lithotrophic sulfide based nitrate reduction is envisaged to be through lNRM 3 pathways rather than lNRM 1 due to significantly higher Y values (1.6 to 17 times).

Why does elemental sulfur accumulation occur?
The low growth yield of lithotrophic nitrate reduction coupled to partial sulfide oxidation to sulfur (laNRM 1 and lhNRM 1 ) does not imply that nitrate treatment does not cause S 0 production. It merely means that if all the parameters that affect various nitrate reduction pathways are equal, and if oxidation of sulfide to sulfate in a single step is possible, thermodynamically a lower fraction of nitrate reduction is coupled to partial sulfide oxidation and a higher fraction of nitrate is coupled to either complete sulfide oxidation or to oxidation of organic compounds. Veshareh et al. (2021) showed that nitrate treatment by Desulfobacterium autotrophicum and a microbial community of a production water enrichment that contained organotrophic nitrate reducing members from Delta-and Gammaproteobacteria caused S • concentrations of 17 to 22 µM. Jiang et al. (2009) observed that lithotrophic nitrate reduction based sulfide oxidation occurs first by oxidation of sulfide to sulfur (lNRM 1 ), followed by oxidation of sulfur to sulfate (lNRM 2 ). These authors reported that the rate of lNRM 2 is 15% of that of lNRM 1 . Yang et al. (2005) observed that elemental sulfur was the end product of nitrate-based sulfide oxidation. The accumulation of S 0 due to NT has also been reported by Liang et al. (2016). Fig. 3 shows that lNRM 2 metabolisms have higher Y values than lNRM 1 metabolisms due to significantly higher -ΔG cat values, i.e. based on thermodynamics the rate of lNRM 2 should be higher than that of lNRM 1 . Fig. 4C demonstrates why lNRM 2 can be slower than lNRM 1 even though it is thermodynamically more favorable. Sulfur solubility in water is extremely low and increases exponentially with temperature from 6.3 × 10 − 9 at 4 ∘ C to 6.3 × 10 − 7 M at 80 ∘ C (Kamyshny Jr, 2009). By considering 1.7 × 10 − 5 M (Mora et al., 2015) as the half saturation constant for sulfur, and assuming that the environment is saturated with sulfur, the limiting kinetic term (F K ) will be between 2 × 10 − 4 and 4 × 10 − 3 for 4 ∘ C and 80 ∘ C, respectively. That is, if we ignore the dependency of lNRM 1 on any other factor, reduction in temperature from 80 to 4 ∘ C, makes lNRM 1 20 times slower merely because of the reduction in S 0 solubility. Therefore, decrease in temperature does not promote S 0 accumulation only by increasing the growth yield of lNRM 1 due to an increase in Y, but mainly by reducing S 0 solubility and thereby reducing the rate of S 0 oxidation to sulfate. This explains why in sewer systems where the temperature is relatively low (around ambient temperature) NT is associated with elemental sulfur production as shown by Liang et al. (2016) and Jiang et al. (2009). In reservoir souring processes, S 0 accumulation can be expected to occur mainly in injection tubing and near the well bore area. However, the extent of S 0 accumulation can spread to the production well, provided presence of fractures reduce the travel time between the injection and production wells and thereby reducing the water temperature in the production well. Given the abovementioned temperature dependence of S 0 accumulation, one can conclude that heating injection water can remediate the S 0 related corrosion associated with NT of reservoir souring. Note that in Fig. 4 only the range of 4 to 80 ∘ C has been considered because we did not have S 0 solubility data outside this range.

Single-step sulfide oxidation vs two-step sulfide oxidation
Single step nitrate-based oxidation of sulfide (lNRM 3 ) seems to have both the thermodynamic advantage as well as the kinetic advantage as no elemental sulfur is produced as an intermediate. However, Cui et al. (2019) reported that nitrate reducers that catalyze single step sulfide oxidation to sulfate can only harvest two electrons per nitrate reduced as the nitrate reduction pathways stops at nitrite due to a lack of nitrite reductase. The nitrite produced in the single-step oxidation of sulfide to sulfate is then reduced further by nitrate reducers that oxidize sulfide in two-steps. Therefore, two-step sulfide oxidation has a physiological advantage and is expected to always co-occur with the single-step sulfide oxidation as the organisms involved can reduce nitrite. Cui et al. (2019) showed that compared to the two-step sulfide oxidation, single-step oxidation of sulfide is more sensitive to sulfide, suggesting that the physiological advantage of two-step sulfide oxidation when nitrate treatment is applied increases with the souring level. These observations have implications for souring management where (i) low temperatures, (ii) limited nitrate availability compared to sulfide, or (iii) limited travel time between the injection and production well can cause elemental sulfur accumulation. Therefore, we speculate that souring treatment efficiency by nitrate injection can be improved by (i) early treatment, (ii) increasing temperature, (iii) making sure that stoichiometrically enough nitrate is available for sulfide oxidation, and (iv) reducing injection flow rate (in the case of souring in petroleum reservoirs) such that sufficient time is given to lNRM 2 to complete sulfide oxidation (from sulfur to sulfate). Note that half-hearted measures, that include only some of these conditions, may cause even more severe souring problems as for example increase in temperature can increase H 2 S emission from the liquid phase to the gas phase. Zhang et al. (2018) have shown that polysulfide formation substantially increases the rate of organotrophic sulfur reduction to sulfide. Qiu et al. (2020) showed that sulfur-based autotrophic denitrification is substantially enhanced due to polysulfide formation. This is why Zhang et al. (2021) introduced polysulfide as an "electron shuttle" that accelerates sulfur oxidation or reduction. Indeed, the polysulfide formation step should not be ignored in the simulation of sulfide/sulfur/sulfate transformations when the exact value of elemental sulfur accumulation or the rate of accumulation is to be evaluated. However, in this study, polysulfide formation from sulfide and elemental sulfur, and polysulfide oxidation to sulfide has been ignored in the sulfur cycle because rather than estimating the pool sizes of the S 0 species, the aim was to predict the likelihood of elemental sulfur accumulation. Future work can validate the proposed methodology in this paper for polysulfide oxidation and study whether the faster oxidation/reduction rates of polysulfide is due to the thermodynamic advantage of this species under varied physiochemical conditions.

Conclusions
The key findings of this study can be summarized as follows: • Growth yield can be used to link the differences in energetics of microbial reactions with their kinetics for metabolisms that have a sufficiently high -ΔG cat such that thermodynamic limiting factor is equal to one.
• Since the kinetic parameters of souring processes can be substantially affected by thermodynamics, it is essential to couple souring models to a thermodynamic model that takes into account the effect of variations in catabolic energy yield and anabolic energy demand on the metabolism rate. • Compared to other nitrate reduction pathways such as organotrophic nitrate-reduction, based on the magnitude of growth yield, singlestep lithoheterotrophic nitrate reduction driven sulfide-oxidation to sulfate is expected to be the dominant nitrate reduction pathway in sour systems. • Even though nitrate-reduction, sulfur oxidation to sulfate is thermodynamically expected to be faster than nitrate-reduction, sulfide oxidation to sulfur, it is slower due to the low solubility of elemental sulfur. • Elemental sulfur accumulation associated with nitrate-reduction driven sulfide-oxidation is prevalent at lower temperatures due to the exponential decrease of sulfur solubility with temperature.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.