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Hydrogen production by the hyperthermophilic bacterium Thermotoga maritima Part II: modeling and experimental approaches for hydrogen production
 Richard Auria^{1}Email authorView ORCID ID profile,
 Céline Boileau^{1},
 Sylvain Davidson^{1},
 Laurence Casalot^{1},
 Pierre Christen^{1},
 Pierre Pol Liebgott^{1} and
 Yannick CombetBlanc^{1}
 Received: 8 September 2016
 Accepted: 2 December 2016
 Published: 19 December 2016
Abstract
Background
Thermotoga maritima is a hyperthermophilic bacterium known to produce hydrogen from a large variety of substrates. The aim of the present study is to propose a mathematical model incorporating kinetics of growth, consumption of substrates, product formations, and inhibition by hydrogen in order to predict hydrogen production depending on defined culture conditions.
Results
Our mathematical model, incorporating data concerning growth, substrates, and products, was developed to predict hydrogen production from batch fermentations of the hyperthermophilic bacterium, T. maritima. It includes the inhibition by hydrogen and the liquidtogas mass transfer of H_{2}, CO_{2}, and H_{2}S. Most kinetic parameters of the model were obtained from batch experiments without any fitting. The mathematical model is adequate for glucose, yeast extract, and thiosulfate concentrations ranging from 2.5 to 20 mmol/L, 0.2–0.5 g/L, or 0.01–0.06 mmol/L, respectively, corresponding to one of these compounds being the growthlimiting factor of T. maritima. When glucose, yeast extract, and thiosulfate concentrations are all higher than these ranges, the model overestimates all the variables. In the window of the model validity, predictions of the model show that the combination of both variables (increase in limiting factor concentration and in inlet gas stream) leads up to a twofold increase of the maximum H_{2}specific productivity with the lowest inhibition.
Conclusions
A mathematical model predicting H_{2} production in T. maritima was successfully designed and confirmed in this study. However, it shows the limit of validity of such mathematical models. Their limit of applicability must take into account the range of validity in which the parameters were established.
Keywords
 Thermotoga maritima
 Hyperthermophile
 Hydrogen
 Modeling
 Inhibition
Background
Because of the increasing demand for energy, due to economic and population rapid growths and because of the damaging effect of the fossil energies on the environment (global warming), governments are focusing on alternative energy sources for fuels. In response to this dual problem (depletion and pollution), the development of a new, more environmentally friendly, and healthy energy is necessary.
Presently, the development of biofuels from renewable plant biomass is a way to reduce fossil fuel consumption. Among the potential biofuels, hydrogen appears as one of the energy sources for the future. Indeed, hydrogen is highly reactive, with high energy density (122 MJ/kg, compared to 50.1 MJ/kg for methane, 29.7 MJ/kg for ethanol, and 47.3 MJ/kg for gasoline) and is directly convertible into electricity with high efficiency (>80%). In addition, it is a lowcarbon fuel which combustion produces only water, making it an excellent candidate in terms of environmental impact. Overall, the use of hydrogen shows a 10% growth per year, leading to represent 8–10% of total energy in 2025.
Currently, hydrogen production is closely dependent on fossil fuels (natural gas and hydrocarbons). However, new approaches for hydrogen production, such as biophotolysis, photofermentation, and dark fermentation, offer less costly technological solutions in terms of energy balance and are friendlier to the environment. Among these techniques, dark fermentation is of great interest because it allows the biodegradation of complex residues using a broad spectrum of microorganisms and enzymes. In addition, it is performed with abundant, inexpensive, renewable, and biodegradable agricultural waste [1].
To be economically viable, one of the main challenges of dark fermentation is to achieve both high hydrogen productivity and yield. Hydrogen is produced by both mesophile and (hyper) thermophile anaerobic bacteria. In general, the latter ones show slightly lower hydrogen production rates but higher yields. The elevated temperature (70–110 °C) has several advantages by reducing (1) the hydrogen solubility which is known to be a strong inhibitor of growth [2–4], (2) the variety in fermentation byproducts [5] and (3) the sensitivity to contamination by H_{2}consumer and pathogen bacteria present in the waste. Moreover, the high temperature promotes the enzymatic hydrolysis of a wide range of carbohydrates (starch, cellulose, hemicelluloses…) [6–9]. Among the hyperthermophilic anaerobic hydrogenproducing bacteria (>80 °C), Thermotogales were one of the most studied [6, 10–12]. Several Thermotoga sp. (neapolitana, maritima, elfii, petrophila, naphtophila,…) metabolized glucose with high hydrogen yields of 3–4 mol/mol [6, 12–16]. Maximal hydrogen productivity of some Thermotoga strains was reported between 2.7 and 12.4 mmol/L h [6, 14, 16, 17]. Higher hydrogen productivity requires determining the conditions influencing the growth and the metabolism of the Thermotogales. Among them, optimal concentrations of glucose, yeast, sulfur, dissolved hydrogen, carbon dioxide, etc., have to be established [6, 16, 17]. Most Thermotoga species have been reported to reduce elemental sulfur to H_{2}S [18, 19]. Huber et al. [20] proposed that the addition of elemental sulfur stimulates the growth of T. maritima on glucose by reducing the inhibitory effect of hydrogen. Schroder et al. [6] supported this hypothesis by showing that, on glucose, sulfur reduction to H_{2}S stimulated T. maritima growth. On the contrary, Boileau et al. [part I, 21] showed that the addition of small amounts of thiosulfate, as well as some other sulfur sources, allowed a significant increase of T. maritima growth and its hydrogen production. These authors confirmed that the sulfur compound was not used in a detoxification process but rather was assimilated by T. maritima leading to an increase in biomass, and therefore in the amount of hydrogen produced.
To improve the understanding of the biological and physical mechanisms that govern the hydrogen production by dark fermentation, mathematical modeling can be an appropriate tool. The aim of the present study is to propose a mathematical model incorporating kinetics of growth, consumption of substrates, product formations, and inhibition by hydrogen in order to predict hydrogen production depending on defined culture conditions. To the best of our knowledge, the most comprehensive kinetic model predicting the hydrogen production by a thermophilic bacterium was proposed by Ljunggren et al. [3]. This kinetic model takes into account the microbial growth of the extreme thermophilic Caldicellulosiruptor saccharolyticus, its substrate consumption and product formations and, the liquidtogas mass transfer. This model predicted high oversaturation of hydrogen in the liquid (12–34 times the equilibrium concentration) comparable to the experimentally obtained values. The authors have shown that the dissolved hydrogen concentration was a function of the stripping rate and the hydrogen productivity. In the present study, a mathematical model was developed to predict hydrogen production from batch fermentations of the hyperthermophilic bacterium, T. maritima. This model incorporates the kinetics of growth, consumptions of substrates (glucose, yeast, and thiosulfate), and product formations (H_{2}, CO_{2}, H_{2}S, acetate, and lactate). It includes the inhibition by hydrogen and the liquidtogas mass transfer of H_{2}, CO_{2}, and H_{2}S. Most kinetic parameters of the model were obtained from batch experiments without any fitting. The limits of its applicability were clearly established.
Methods
Strain and culture medium
Thermotoga maritima strain MSB8 (DSMZ 3109) was cultivated as previously described [part I, 21]. Basal medium containing, per liter: NH_{4}Cl 0.5 g, K_{2}HPO_{4} 0.3 g, 0.3 g, CaCl_{2} 0.1 g, KCl 0.1 g, NaCl 20 g, MgCl_{2} 0.2 g, yeast extract 1.0 g, and glucose 20 mmol/L was used. Balch trace mineral element solution (10 mL) was added [part I, 21]. The inoculum was obtained from three bottles of 100 mL each, containing 50 mL of liquid culture.
Experimental system, operating conditions, and analytical methods
Experimental system, operating conditions, and analytical methods were specified [part I, 21]. T. maritima was batch cultivated in a similar 2L doublejacket glass bioreactor (FairMenTec, France) with a 1.5L working volume. The temperature was maintained constant at 80 ± 1 °C and pH was controlled at 7 ± 0.1 by the addition of sodium hydroxide (NaOH 0.5 mmol/L). The inlet gas stream of N_{2} was controlled via a massflow meter (Bronkhorst, range 0–500 SCCM), and the one composed of the mixture of N_{2} and H_{2} was prepared using two massflow meters (Bronkhorst, range 0–500 SCCM and 0–100 SCCM, The Netherlands). The stirring was set to 350 rpm. The online measurements of CO_{2}, H_{2}, and H_{2}S concentrations, bioreactor liquid volume, and NaOH consumption are described [part I, 21].
For each experiment, three successive batches were carried out. The first batch was always considered as an adaptation batch, the following two being in general well reproducible.
OD (Optical Density), hydrogen, glucose, acetate, lactate, and hydrogen sulfide concentrations were determined as previously described [part I, 21]. Cell dry weight was calculated from OD data using the same relation of 1 OD unit = 330 mg/L.
Thiosulfate concentration was quantified by ion chromatography (761 Compact IC Metrohm, Metrohm, VillebonsurYvette, France) equipped with a Metrosep Anion Supp1 column (Metrohm).
Mathematical model
The model developed in this study incorporates the kinetics of growth of T. maritima, glucose, yeast extract, and thiosulfate consumptions and product, such as hydrogen (H_{2}), carbon dioxide (CO_{2}), hydrogen sulfide (H_{2}S), acetate, and lactate formations. It takes into account the transfer of H_{2}, CO_{2}, and H_{2}S, as well as the chemical equilibrium between CO_{2} and bicarbonates (HCO_{3} ^{−}) and between H_{2}S and hydrosulfide ions (HS^{−}).
The value of the stoichiometric parameters a, b, c, and d in the Eq. 1 are unknown. The parameter a was estimated using a methodology described in the Determination of kinetic and mass transfer parameters section. Glucose fermentation in T. maritima, under controlled physicochemical conditions, shows that about 5 and 22% of the consumed glucose are converted into biomass and EPS, respectively [22]. These two values of percentages were used to evaluate c and d parameters. The stoichiometric parameter b relates to the lactate production and was determined by the difference between the amount of carbon from the consumed glucose and the amount of carbon found in all the end products: acetate, carbon dioxide, biomass, and EPS.
At low concentration, thiosulfate is a sulfured nutriment for T. maritima growth used for the synthesis of cellular materials [part I, 21]. Thus, this sulfur is incorporated to the biomass according to the elemental composition of T. maritima \(({\text{CH}}_{1.6} {\text{O}}_{0.6} {\text{N}}_{0.2} {\text{S}}_{0.005} )\) determined by Rinker and Kelly [23].
Growth kinetics, substrate, and nutrient consumptions
Product formation in the liquid phase
\({\text{Act}},\;{\text{Lact}},\; \left[ {{\text{H}}_{2} } \right],\; \left[ {{\text{CO}}_{2} } \right],\; \left[ {{\text{H}}_{2} {\text{S}}} \right], \left[ {{\text{HCO}}_{3}^{  } } \right] \;{\text{and}},\; \left[ {{\text{HS}}^{  } } \right]\) are the concentrations in the liquid phase of acetate, lactate, hydrogen, carbon dioxide, hydrogen sulfide, bicarbonate, and bisulfide ions, respectively. \(\left[ {{\text{H}}_{2} } \right]^{*} , \;\left[ {{\text{CO}}_{2} } \right]^{*} \;{\text{and}},\; \left[ {{\text{H}}_{2} {\text{S}}} \right]^{ *}\) are the dissolved concentrations of these compounds at equilibrium. \(Y_{{{\text{ACT}}/{\text{GLU}}}}\),\(Y_{{{\text{H}}_{2} /{\text{GLU}}}} , Y_{{{\text{CO}}_{2} /{\text{GLU}}}}\) are the yields of acetate, H_{2} and CO_{2}. \(Y_{{{\text{H}}_{2} /{\text{H}}_{2} {\text{S}}}} \;{\text{and}}\;Y_{{{\text{H}}_{2} {\text{S}}/{\text{THIO}}}}\) are the stoichiometric H_{2} on H_{2}S yield and the stoichiometric H_{2}S on thiosulfate yield (Eq. 2), respectively.
\({\text{KlaH}}_{2} , \;{\text{KlaCO}}_{2}, \;{\text{and}}\;{\text{KlaH}}_{2} {\text{S}}\) represent the volumetric mass transfer coefficients for H_{2}, CO_{2}, and H_{2}S. K _{1} and K _{2} are the dissociation constants.
The maximum yield of acetate on glucose \(Y_{{{\text{ACT}}/{\text{GLU}}}}^{ \rm{max} }\) was estimated from experiments (see Determination of kinetic and mass transfer parameters) for H_{2} percentage in the gas phase equal to 0. \(Y_{{{\text{H}}_{2} /{\text{GLU}}}}\) and \(Y_{{{\text{CO}}_{2} /{\text{GLU}}}}\) were deduced from \(Y_{{{\text{ACT}}/{\text{GLU}}}}\) using the Eq. 1.
Mass balance in the gas phase
Equilibrium constants and stoichiometric equations
The constants used in the model
Constant  Values 

\(K_{1}\)  1.37 × 10^{−6} mol/kg 
\(K_{2}\)  2.2 × 10^{−7} mol/kg 
\(K{\text{h}}_{2}\)  7.1 × 10^{−9} mol/L/Pa 
\(K_{{{\text{CO}}_{2} }}\)  1.33 × 10^{−7} mol/L/Pa 
\({\text{Kh}}_{2} {\text{s}}\)  2.1 × 10^{−7} mol/L/Pa 
\(D_{{{\text{O}}_{2} }}\)  0.46 × 10^{−4} cm^{2}/s 
\(D_{{{\text{H}}_{2} }}\)  1.4 × 10^{−5} cm^{2}/s 
\(D_{{{\text{CO}}_{2} }}\)  0.66 × 10^{−4} cm^{2}/s 
\(D_{{{\text{H}}_{2} {\text{S}}}}\)  0.46 × 10^{−4} cm^{2}/s 
\({\text{pH}}\)  7 
\(R\)  8.314 × 10^{3} Pa L/mol°K 
\(T\)  80 °C 
\(V_{\text{g}}\)  0.5 L 
\(V_{\text{l}}\)  1.5 L 
Results and discussion
Determination of kinetic and mass transfer parameters
Yields and kinetic parameters
Parameters of the model
\(\mu_{\rm{max} }\) (h^{−1})  \(\mu_{\text{d }}\) (h^{−1})  \(m_{\text{GLU}}\)** mmol/g/h  \(Y_{{{\text{X}}/{\text{GLU}}^{\text{a}} }}\) (g/mol)  \(Y_{{{\text{X}}/{\text{YEAST}}}}\) (g/g)  \(Y_{\text{X/THIO}}\) * (g/mmol)  \(K_{\text{sglu}}\) mmol/L  \(K_{\text{syeast}}\) g/L  \(K_{\text{sthio}}\) mmol/L  \(\left[ {{\text{H}}_{{ 2 {\text{crit}}}} } \right]\) mmol/L  N 

0.9 ± 0.05  0.05 ± 0.02  2.2  20.9 ± 3.2  0.67 ± 0.11  3.617 ± 0.18  5.7 ± 1.1  0.30 ± 0.1  0.052 ± 0.01  1.44 ± 0.01  1 
\(\mu_{ \rm{max} }\) and μ _{d} were obtained for yeast extract, thiosulfate, and glucose concentrations of 4 g/L, 0.12, and 60 mmol/L, respectively with a stripping rate of 100 mL/min. The maximum growth rate of T. maritima (\(\mu_{max }\) = 0.7 − 0.9 h^{−1}), measured in this study, was comparable to those obtained by Huber et al. [20] for T. maritima (0.6 h^{−1}) and, T. naphthophila (0.7 h^{−1}) and T. petrophila (0.77 h^{−1}), two species very closely related to T. maritima [13]. Since, in these experiments, the conditions were meant to allow obtaining the highest possible \(\mu_{{{ \rm{max} } }}\), and in the present model, \(\mu_{{{ \rm{max} } }}\) was chosen at 0.9 h^{−1}. This value is higher than the ones obtained with C. saccharolyticus (0.2–0.5 h^{−1}), an extreme thermophile H_{2}producing bacterium among the most studied [3, 24, 25]. This fairly high growth rate is an advantage for this H_{2} producer that allows the reduction of the H_{2}production time. For our model, the average value of \(\mu_{\text{d}}\), obtained from experiments, was estimated at 0.05 h^{−1}. Very few values of cell death rate (\(\mu_{\text{d}}\)) of hyperthermophilic microorganisms are available in the literature. This parameter ranges from 0.014 to 0.105 h^{−1} [3, unpublished data].
\(Y_{{{\text{X}}/{\text{THIO}}}}\) of 3.167 g/mmol was obtained by Boileau et al. [part I, 21] from batch culture experiments. \(Y_{{{\text{X}}/{\text{THIO}}}}^{\text{X}}\)(Eq. 8) represents the equivalent sulfur of the thiosulfate incorporated into the biomass and was calculated from the elemental composition of T. maritima [23]. It is equal to 10.47 g biomass/mmol thiosulfate. \(Y_{\text{X/THIO}}^{{{\text{H}}_{2} {\text{S}}}}\) was determined from the difference between the total yield of thiosulfate \(Y_{\text{X/THIO}}\) and \(Y_{\text{X/THIO}}^{\text{X}}\), considering that all sulfur not incorporated into the biomass was reduced into H_{2}S (Eq. 2). \(Y_{{{\text{X}}/{\text{THIO}}}}^{{{\text{H}}_{2} {\text{S}}}}\) was equal to 5.52 g biomass/mmol thiosulfate. \(K_{\text{sthio}}\), obtained in this study from batch experiments, was of 0.052 mmol/L (Additional file 1: Fig. S3).
In this study, two sulfur sources for T. maritima growth are available: thiosulfate added to the culture medium and sulfur (cysteine, etc…) in the yeast extract. In the previous paper [part I, 21], the authors determined an equivalent thiosulfate concentration of 0.03 mmol per g of yeast extract (\(\varepsilon_{2}\), Eq. 7). A similar approach was used to determine the equivalent content of glucose in 1 g of yeast extract (Fig. 1). This value was obtained using the linear regression \(X = 20.4\;{\text{Glu}} + 11.4\) by extrapolating the line to a cell mass concentration (\(X\)) equal to 0. Therefore, the equivalent glucose per g of yeast extract was of 0.56 mmol (\(\varepsilon_{1}\), Eq. 5).
Determination of \(Y_{\text{ACT/GLU}}\) and [H_{2crit}] the critical dissolvedH_{2} concentration
\(Y_{\text{ACT/GLU}}^{ \rm{max} }\) was obtained from the experimental correlations for \({\text{H}}_{2}\) = 0% (Fig. 3). It is equal to 1.38 mol/mol. \(Y_{{{\text{H}}_{2} /{\text{GLU}}}}^{ \rm{max} }\) and \(Y_{{{\text{CO}}_{2} /{\text{GLU}}}}^{ \rm{max} }\) were deduced from \(Y_{{{\text{ACT}}/{\text{GLU}}}}^{ \rm{max} }\) using the Eq. 1. Thus, \(Y_{{{\text{H}}_{2} /{\text{GLU}}}}^{ \rm{max} }\) and \(Y_{{{\text{CO}}_{2} /{\text{GLU}}}}^{ \rm{max} }\) are equal to 2.76 and 1.38 mol/mol, respectively. Mars et al. [16] measured similar values of \(Y_{\text{ACT/GLU}}^{ \rm{max} } \;\left( {1.4\;{\text{mol}}/{\text{mol }}} \right),\; Y_{{{\text{H}}_{2} /{\text{GLU}}}}^{ \rm{max} }\) (2.9 mol/mol), and \(Y_{{{\text{CO}}_{2} /{\text{GLU}}}}^{ \rm{max} }\) (1.6 mol/mol) for T. neapolitina growing on glucose in batch culture. Values of \(Y_{{{\text{H}}_{2} {\text{S}}/{\text{THIO}}}}\) and \(Y_{{{\text{H}}_{2} /{\text{H}}_{2} {\text{S}}}}\) were deduced using the Eq. 2. Both values are equal to 2 mol/mol.
Because dissolved H_{2} (\(\left[ {{\text{H}}_{2} } \right]\)) cannot be measured, the parameter N (Eq. 4) cannot be directly evaluated. Datasets from five experiments were used to evaluate the maximum of H_{2} productivity, which was compared to the maximum H_{2} productivity obtained from the mathematical model for N values of 0.5, 1, and 1.5. For these experiments, at least one substrate (thiosulfate, glucose, or yeast extract) is the growthlimiting factor [part I, 21]. For each experimental condition, the average of the difference between the maximum of H_{2} productivity obtained from the model and from the experiment was 2.2, 0.2, and 1.5 mmol/L/h for N equal to 0.5, 1, and 1.5, respectively. Ljunggren et al. [3] estimated, using another approach with C. saccharolyticus, the parameter N, from experiments with 5 g/L of glucose and different stripping rates, ranged from 20 to 100 mL/min. These authors obtained a mean value of N (4.5) with a large standard deviation (0.84–16.24). For the rest of our study, a value of N equal to 1 was chosen.
Determination of the volumetric mass transfer coefficient
\({\text{KlaO}}_{2}\) was determined by measuring the dissolved O_{2} concentration in the sterile culture medium [27]. It was obtained in the 2L bioreactor by either physical absorption or desorption of oxygen. Airflow rate was tested in a range from 20 to 500 mL/min at a constantspeed agitation (350 rpm).
From the Eqs. 28 and 29, the \({\text{KlaH}}_{2}\) was of 114 h^{−1} for a \(Q_{{{\text{N}}_{2} }}\) of 100 mL/min and a stirring rate of 350 rpm. This value is significantly higher than those usually reported in the literature for similar systems [3, 28]. Ljunggren et al. [3] measured a \({\text{KlaH}}_{2}\) value of 9 h^{−1} at 70 °C for the same \(Q_{{{\text{N}}_{ 2} }}\) and stirring rate. A \({\text{KlaH}}_{2}\) of 17 h^{−1} was obtained for \(Q_{{{\text{N}}_{ 2} }}\) of 100 mL/min, but for lower temperature (35 °C) and with an unknown value of stirring rate (use of a stir plate) [28]. In our study, the higher \({\text{KlaH}}_{2}\) could be explained by the following: (1) the hightemperature experiments (80 °C) increasing the diffusion coefficient of H_{2} in water, (2) the use of a 3cmlong fritted cylinder gasdispersion stone with 4–60 μM pore size allowing the formation of numerous very small bubbles of gas and, (3) the use of two axial impellers promoting an effective mixing. \({\text{KlaCO}}_{2}\) and \({\text{KlaH}}_{2} {\text{S}}\) were determined in the same way as \({\text{KlaH}}_{2}\).
Model validation
The mathematical model developed in this study must be able to both provide new knowledge and predict the optimal operating conditions of the H_{2} production by T. maritima. For this, the mathematical model was validated for various glucose, yeast extract, and thiosulfate concentrations and inlet N_{2} flow rates. Forty experiments were carried out with different concentration ranges of glucose (2.5–63 mmol/L), yeast extract (0.2–8 g/L), thiosulfate (0.01–2 mM), and inlet N_{2} flow rates (17–190 mL/min). These various operating conditions correspond to situations where one or none of these compounds (glucose, yeast extract, and thiosulfate) is the limitinggrowth factor.
In the model, equations related to sulfur have been taken into account. To experimentally confirm the model, the thiosulfate disappearance needs to be monitored. In order to do so, experiments with increased thiosulfate concentration to 20 mmol/L were performed (Glucose 60 mmol/L, Yeast extract 4 g/L, and \(Q_{{{\text{N}}_{ 2} }}\) 100 mL/min). In this experiment, none of the compounds of interest was the limiting factor for growth. In this case, 4 mmol/L of thiosulfate were consumed and the total H_{2}S production measured at the end of the fermentation was of 3.5 mmol. Similar maximum of biomass and H_{2} productivity, acetate and H_{2} productions were obtained in the same conditions with 0.12 mmol/L thiosulfate (data not shown), corroborating that thiosulfate was not the growthlimiting and or inhibiting factor. The addition of 4 g/L of yeast extract increases the equivalent initial total thiosulfate concentration (\({\text{Thio}} + \varepsilon_{2}\)) from 20 to 20.24 mmol/L. A negligible part of this thiosulfate (0.07 mmol/L) is incorporated as sulfur into the biomass \(\left( {{\text{CH}}_{1.6} {\text{O}}_{0.6} {\text{N}}_{0.2} {\text{S}}_{0.005} } \right)\) [23], the remaining consumed thiosulfate (3.93 mmol/L) should be converted into H_{2}S (7.86 mmol/L, Eq. 2). However, the total H_{2}S production measured experimentally (3.5 mmol) is about 2 times lower, showing that not all the thiosulfate was converted into H_{2}S and that probably some other unknown sulfur compounds are produced during the fermentation.
Prediction of the mathematical model
In this study, a mathematical model of the fermentation of T. maritima has been written. This model has been proved to be in agreement with the experiment in a certain range of concentrations of glucose, yeast extract, and thiosulfate, and of the inlet gas (N_{2}) flow rate. The final purpose of this model is to provide, among other things, a mean of predicting specific H_{2} productivity linked to H_{2} inhibition for T. maritima in various situations.
Studies have showed that a decrease of N_{2} stripping rate (i.e., mass transfer coefficient, KlaH_{2}) resulted in a lower productivity and yield of H_{2} [3]. Our results showed that between 100 and 180 mL/min, no change of experimental maximum H_{2} productivity and glucose consumption rate was observed, while a strong decrease (about 50%) was recorded when \(Q_{{{\text{N}}_{2} }}\) was reduced to 20 mL/min (Additional file 1: Fig. S4). The model was used to simulate the specific H_{2} productivity and the inhibition by H_{2} versus the N_{2} inlet flow rate (\(Q_{{{\text{N}}_{2} }}\), 5–100 mL/min) for different operating conditions (0 < Gluc < 20 mmol/L or 0.1 < Yeast < 0.5 g/L or 0 < Thio < 0.06 mmol/L). These ranges correspond to the concentrations where the model was accurately validated (Fig. 4a–e). Initial biomass concentration, volume of liquid of bioreactor, and volume of gas (headspace) were set to 31.6 mg/L, 1.5, and 0.5 L, respectively.
In almost all the cases, the increase in concentration of the limiting factor leads to a notable increase in specific H_{2} productivity. However, this increase, when the produced H_{2} is slightly flushed by the inlet gas stream (5 mL/min), leads to a strong relative inhibition. Whatever the operating conditions, when N_{2} stripping rate increases from 5 to 100 mL/min, the inhibition by H_{2} strongly decreases from 33 to 5% (Fig. 6a–c). The combination of both variables (increase in limiting factor concentration and in inlet gas stream) leads up to a twofold increase of the maximum H_{2} specific productivity with the lowest inhibition.
This model can be extended beyond the upper limits for nonlimiting concentrations of glucose (>20 mmol/L), thiosulfate (>0.06 mmol/L), and yeast extract (0.5 g/L). However, in these cases, the mathematical model will overestimate the production of hydrogen of about 30% (Fig. 4c). Moreover, this model cannot be validated without sparging or for low stripping rates (<5 mL/min). Indeed, for these conditions, gastransfer diffusion (H_{2}, CO_{2}, N_{2}, and H_{2}S) becomes predominant. This parameter is not taken into account in our model. In this case, H_{2} inhibition will be important and will affect strongly the specific H_{2} productivity.
Conclusions
Batch fermentations of T. maritima were successfully simulated using a mathematical model that incorporates the kinetics of growth, consumptions of substrates (glucose, yeast extract, and thiosulfate) and product formations (H_{2}, CO_{2}, H_{2}S, acetate, and lactate). Except for one, all of the model parameters were determined experimentally. However, the limits of the validity of this model were clearly established, and it is within the ranges when glucose or yeast extract or thiosulfate limit the growth. Anyway, the development of structured (mechanistic) models for quantifying microbial growth kinetics are still limited because the mechanism of cell growth is very complex and is not yet completely understood. Moreover, the defined mineral media are formulated so as to allow microorganisms to synthesize their cellular components from single sources of carbon, sulfur, ammonium, phosphorus… Usually, only one of them limits the maximum quantity of biomass that could be produced, with all other nutrient in excess. We focused our interest on four state variables (glucose, yeast extract, thiosulfate, and hydrogen concentrations), but we should not dismiss the potential other limiting factors present in the medium. In the future, it would be interesting to research other potential limiting factors and to take into account their influence on the model. For example, Rinker and Kelly [23] demonstrated that the lower NH_{4}Cl concentration (0.5 g/L our study) would be limiting for the growth of T. maritima.
From now on, this model can be use, in the limits of its validity, to predict, depending on the scenario, many elements, and in particular, specific H_{2} productivity, dissolvedH_{2} concentration, etc.
Abbreviations
 \({\text{Act}}\) :

acetate concentration (mol/L)
 [CO_{2}]:

CO_{2} concentration in the liquid phase (mol/L)
 \(\left[ {{\text{CO}}_{2} } \right]^{*}\) :

concentration of dissolved CO_{2} at thermodynamic equilibrium (mol/L)
 \({\text{D}}_{{{\text{O}}_{2} }}\) :

coefficient of diffusion in water for O_{2} (cm^{2}/s)
 \(D_{{{\text{H}}_{2} }}\) :

coefficient of diffusion in water for H_{2} (cm^{2}/s)
 \(D_{{{\text{CO}}_{2}}}\) :

coefficient of diffusion in water for CO_{2} (cm^{2}/s)
 \(D_{{{\text{H}}_{2}{\text{S}}}}\) :

coefficient of diffusion in water for H_{2}S (cm^{2}/s)
 \({\text{Glu}}\) :

glucose concentration (mol/L)
 H_{2} :

H_{2} concentration in the gas phase (mol/L)
 \(\left[ {{\text{H}}_{{2{\text{crit}}}} } \right]\) :

critical dissolvedH_{2} concentration (mol/L)
 [H_{2}]:

H_{2} concentration in the liquid phase (mol/L)
 [H_{2}S]:

H_{2}S concentration in the liquid phase (mol/L)
 [HCO_{3} ^{−}]:

\({\text{HCO}}_{3}^{  }\) concentration in the liquid phase (mol/L)
 \(\left[ {{\text{HS}}^{  } } \right]\) :

HS^{−}concentration in the liquid phase (mol/L)
 \(\left[ {{\text{H}}_{2} } \right]^{*}\) :

concentration of dissolved H_{2} at thermodynamic equilibrium (mol/L)
 [H_{2}S]^{*} :

concentration of dissolved H_{2}S at thermodynamic equilibrium (mol/L)
 K _{sglu} :

saturation constant of glucose (mol/L)
 K _{syeast} :

saturation constant of yeast extract (g/L)
 K _{sthio} :

saturation constant of thiosulfate (mol/L)
 KlaH_{2} :

volumetric mass transfer coefficient for H_{2} (h^{−1})
 KlaCO_{2} :

volumetric mass transfer coefficient for CO_{2} (h^{−1})
 KlaH_{2}S:

volumetric mass transfer coefficient for H_{2}S (h^{−1})
 \(K_{1 }\) :

dissociation constant for CO_{2} into bicarbonate (mol/kg)
 \(K_{2 }\) :

dissociation constant for H_{2}S into bisulfide (mol/kg)
 Kh_{2} :

Henry’s constant for H_{2} (mol/L/Pa)
 \(K_{{CO_{2} }}\) :

Henry’s constant for CO_{2} (mol/L/Pa)
 Kh_{2}s:

Henry’s constant for H_{2}S (mol/L/Pa)
 Lact:

lactate concentration (mol/L)
 \(m_{\text{GLU}}\) :

maintenance coefficient (mol/g/h)
 N :

exponential parameter
 \(Q_{\text{g}}\) :

outletgas flow rate (L/h)
 \(Q_{{{\text{N}}_{ 2} }}\) :

inlet gas (N_{2}) flow rate (L/h)
 \(Q_{{{\text{H}}_{2} }}\) :

outletgas (H_{2}) flow rate (L/h)
 \({Q_{{\text{CO}}_{2}}}\) :

outletgas (CO_{2}) flow rate (L/h)
 \(Q_{{{\text{H}}_{2} {\text{S}}}}\) :

outletgas (H_{2}S) flow rate (L/h)
 R :

gas constant (Pa L/mol °K)
 T :

temperature (°K)
 Thio:

thiosulfate concentration
 V _{l} :

freeliquidspace volume in the reactor (L)
 V _{g} :

freegaseousspace volume in the reactor (L)
 X :

cell mass concentration (g/L)
 Yeast:

yeast extract concentration (g/L)
 Y _{X/GLU} :

biomass yield (g/mol)
 Y _{X/YEAST} :

yeast extract yield (g/g)
 Y _{X/THIO} :

thiosulfate yield (g/mol)
 \(Y_{{{\text{X}}/{\text{THIO}}}}^{\text{X}}\) :

thiosulfate yield (equivalent sulfur from thiosulfate incorporated into the biomass) (g/mol)
 \(Y_{{{\text{X}}/{\text{THIO}}}}^{{{\text{H}}_{2} {\text{S}}}}\) :

H_{2}S yield (equivalent sulfur from thiosulfate released as H_{2}S) (g/mol)
 \(Y_{{{\text{ACT}}/{\text{GLU}}}}\) :

acetate yield (mol/mol)
 \(Y_{{{\text{H}}_{2} /{\text{GLU}}}}\) :

H_{2} yield (mol/mol)
 \(Y_{{{\text{CO}}_{2} /{\text{GLU}}}}\) :

CO_{2} yield (mol/mol)
 \(Y_{{{\text{H}}_{2} /{\text{H}}_{2} {\text{S}}}}\) :

stoichiometric H_{2} on H_{2}S yield (mol/mol)
 \(Y_{{{\text{H}}_{2} {\text{S}}/{\text{THIO}}}}\) :

stoichiometric H_{2}S on thiosulfate yield (mol/mol)
 \(Y_{{{\text{ACT}}/{\text{GLU}}}}^{ \rm{max} }\) :

maximum yield of acetate (mol/mol)
 \(Y_{{{\text{H}}_{2} /{\text{GLU}}}}^{ \rm{max} }\) :

maximum yield of H_{2} (mol/mol)
 \(Y_{{{\text{CO}}_{2} /{\text{GLU}}}}^{ \rm{max} }\) :

maximum yield of CO_{2} (mol/mol)
 \(Y_{{{\text{H}}_{2} {\text{S}}/{\text{THIO}}}}\) :

stoichiometric H_{2}S on thiosulfate yield (mol/mol)
 \(Y_{{{\text{H}}_{2} /{\text{H}}_{2} {\text{S}}}}\) :

stoichiometric H_{2} on H_{2}S yield (mol/mol)
 \(\varepsilon_{1}\) :

equivalent concentration of glucose in the yeast extract (mol/L)
 \(\varepsilon_{2}\) :

equivalent concentration of thiosulfate in the yeast extract (mol/L)
 μ :

specific growth rate (h^{−1})
 μ _{max} :

maximum specific growth rate (h^{−1})
Declarations
Authors’ contributions
RA: model development and manuscript writing. CB, PC, and PPL: data collection and analysis of the results. SD: conception, design, and mass transfer experiments. YCB: planning of the fermentation experiments and parameter estimations. LC: manuscript writing. All authors contributed to revision of the manuscript and approved the text and diagrams for submission. All authors read and approved the final manuscript.
Acknowledgements
This work received financial support from the French Research National Agency (ANR) (Project: ANR2010BIO00503, HYCOFOL_BV).
Competing interests
The authors declare that they have no competing interests.
Availability of supporting data
All data generated or analyzed during this study are included in this published article and its supplementary information (Additional file 1).
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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