Kinetics, multivariate statistical modelling, and physiology of CO 2 -based biological methane production

(cid:129) Gas-to-gas conversion processes are analyzed with respect to bioenergy production. (cid:129) CO 2 -BMP modeling is performed and model validity is discussed. (cid:129) Multivariate data analysis and biological gas conversion mechanistic is integrated. (cid:129) Gas limitation and liquid limitation in pure culture biological CH 4 production are highlighted. (cid:129) Continuous culture CH 4 bioprocessing from H 2 /CO 2 is discussed. Conversion of surplus electricity to chemical energy is increasingly attracting attention. Thereof, biological energy conversion and storage technologies are one of several viable options. In this work, the inherent challenges faced in analyzing the CO 2 -based biological methane production (CO 2 -BMP) process for energy conver- sion and storage are discussed. A comprehensive assessment of key process parameters on several CO 2 -BMP process variables was conducted. It was found that literature data often misses important information and/or the required accuracy for resolution of the underlying mechanistic e ﬀ ects, especially when modelling reactor dependent variables. Multivariate dependencies inherently attributable to gas-to-gas conversion bioprocesses are particularly illustrated with respect to CO 2 -BMP. It is concluded that CO 2 -BMP process modelling requires the application of process analytical technology. The understanding of the CO 2 -BMP mechanistic process is discussed to assist with the analysis and modelling of other gas-to-gas conversion processes. The ﬁ ndings presented in this work could aid in establishing a biotechnology-based energy to gas conversion and storage landscape.


Introduction
Converting surplus electricity to chemical energy is increasingly attracting attention [1]. In this frame, chemical or biological energy conversion and storage technologies for the power-to-gas concept are one of several viable options [2,3]. Due to decreasing reserves of fossil fuels and growing awareness for global warming, carbon dioxide (CO 2 ) utilization has become a topic of industrial relevance [4]. An effective reduction of CO 2 emissions will be achieved in the long term if renewable energy production can be linked with power conversion and storage technologies. Furthermore, the production of renewable energy is significantly more carbon neutral when compared to fossil fuel-based energy production [5,6]. Therein, a renewable energy production molecular hydrogen (H 2 ) respectively [7]. Recent advances in bioprocess technology [2,3,8,9] and the development of biorefinery concepts favored the development of 5th generation biofuels, which employ microorganisms to convert gaseous substrate(s) to gaseous end products. 5th generation biofuels encompass CO 2 -based biological methane (CH 4 ) production (CO 2 -BMP) and H 2 production from C 1 compounds [9,10]. CO 2 -BMP and H 2 production from C 1 compounds are known to be the only gaseous biofuel production technologies that have no immediate requirement for photosynthesis. Thus, integrating surplus renewable power conversion with CO 2 capture and storage can be performed by applying the CO 2 -BMP process.
The CO 2 -BMP process is characterized by utilizing hydrogenotrophic methanogenic archaea (methanogens) for CH 4 production [9]. Because CO 2 -BMP is a bioprocess, it encompasses distinct and emergent advantages compared to its chemical counterpartthe Sabatier process. One such advantage is the autocatalytic regeneration of methanogens accompanied by CH 4 production [9,[11][12][13][14]. In this process, methanogens exhale CH 4 as a metabolic end product of their energy conserving metabolism while fixing a variable part of CO 2 in the form of biomass [14][15][16]. Therefore, the production of CH 4 is essential for the survival of the organisms. The CO 2 -BMP process can be carried out by an enrichment culture [17][18][19][20][21][22][23] or pure culture of methanogens [9,24] and benefits from its ability to convert CO 2 and H 2 to CH 4 at very high volumetric methane evolution rates (MERs) while in continuous culture [25,26]. An additional advantage is the mild bioprocessing conditions (e.g. temperatures from approx. 0°C to 122°C) that can be applied during CO 2 -BMP [27,28].
High purity H 2 and CO 2 can be employed as substrates for the CO 2 -BMP process [9,12,26]. It has also been shown that the CO 2 by-product of the anaerobic digestion process can be microbiologically transformed to CH 4 at different conversion efficiencies and MERs [13,21,24,29]. However, it has been noticed that the technology readiness level (TRL) of the different microbiological biogas converting technologies can vary tremendously [24]. Although direct microbiological biogas conversion in anaerobic digesters was shown to be possible, the MER and CH 4 concentration in the offgas remained negligible [20,24]. On the contrary, microbiological biogas conversion by using pure [13] or enrichment cultures [21,23] of methanogens was shown to be efficient. Drawbacks of using enrichment cultures for microbiological biogas conversion are the ambiguous adaptation procedures, the time it takes for the culture to adapt to certain conditions, and unintended side reactions that occur within the enrichment [24,30]. Eventually, pure cultures of methanogens were not only applied in microbiological biogas upgrading [13,31], but were also utilized for conversion of CO 2 from industrial flue gases [13]. While pure cultures have been used for the conversion of chemical species, it should be noted that the CO 2 -BMP process results in a different product formation kinetic [32,33] when compared to liquid-based continuous culture bioprocessing [6,34]. Therefore, many challenges in the analysis of production kinetics, physiology, scale-up, and modelling of the CO 2 -BMP process have emerged [8].
The first aim of this study was to comprehensively assess the effects of key process parameters (KPP) on several CO 2 -BMP process variables, which were obtained from literature, on continuous culture bioprocessing. Second, this study discusses the multivariate dependencies inherently attributable to CO 2 -BMP gas-to-gas conversion bioprocesses. Third, it is shown that the presented models possess limits that prevent a simple analysis of the CO 2 -BMP process. Fourth, the application of multivariate data analysis and modelling CO 2 -BMP process is thoroughly discussed. It was of great interest to review and refine the understanding of the kinetic aspects involved in gas converting bioprocess technologies and to better control and avoid undesired or uncontrolled limitations of the CO 2 -BMP kinetics.
The novelty of this contribution goes beyond bioprocess modelling. Here, a critical analysis of literature on CO 2 -BMP in pure culture was performed. It is shown that both liquid and gas limitations need to be carefully considered when attempting CO 2 -BMP bioprocessing. Examples on how to model the CO 2 -BMP processes are given and it is shown that wrong conclusions have often been drawn due to an application of erroneous results. It is discussed that during CO 2 -BMP modelling an in depth understanding of the biology and the process is required and that the physiology of the target organism must be carefully considered to cope with the multivariate nature of this process. Finally, it is shown that biological gas-to-gas conversion and energy storage processes must be scaled by linking kinetics, modelling, and physiology.

Material and methods
First, the existing literature of pure culture CO 2 -BMP, independent of bioreactor conditions and scale, was reviewed with an in depth examination of methanogenic strains, bioprocess setup, and growth conditions. Second, pure culture CO 2 -BMP data was extracted from literature [11,12,25,26,32,[35][36][37][38][39][40][41][42][43][44]. Third, the data was applied for qualitative and quantitative assessment and subsequent modelling. A list of comprehensively extracted results from literature is provided in Supplementary Material 1. From all literature reports on pure culture CO 2 -BMP, only the data on continuous culture experiments were analyzed as the stability of process variables in steady state allowed for a precise quantification. Closed batch and fed-batch CO 2 -BMP experiments were not considered.

Definition of parameters and units
The following variables and KPPs were extracted or calculated based on the information provided in literature: the gassing rate per ). Y (x/CH4) was used to assess the flux of the carbon into biomass and into CH 4 on a C-molar level for all the cultivations performed with Methanothermobacter marburgensis [11,12,26]. Although the analysis of Y (x/CH4) was possible for experiments reported before [11,12,26,35], Y (x/CH4) could not be retrieved or calculated from all of the experiments presented in Supplementary Material 1 because C-molar biomass productivity (r (x) ) [C-mmol L −1 h −1 ] had not been reported. However, Y CH4 that was defined as the quotient of µ to qCH 4 [15] could be retrieved from literature. Most KPPs and variables could be directly extracted from literature without the necessity to convert results [11,12,26,35]. In some cases the conversion of extracted literature data into aforementioned molar units was performed.

Data validation procedure
Data was curated according to the degree of reduction balance (DoR-balance) and carbon balance (C-balance) by applying manual data quality control steps. These mass balance curation steps could only be performed were the relevant information was provided in literature. The relevant bioprocess and physiological parameters were then presented after a data quality assessment based on published methodologies [9,45]. Data curation also involved a thorough qualitative selection procedure where an assessment step analyzing the data by using the MER/MER max concept was implemented. The MER/MER max ratio presented is the dimensionless quotient of MER to the maximum possible volumetric CH 4 production rate (MER max [mmol L −1 h −1 ]) according to the reaction stoichiometry and experimental settings neglecting biomass formation [11,12,26]. The MER/MER max concept for apparent gas conversion to maximum theoretical gas conversion has been previously introduced [9,26]. The MER/MER max concept was used to identify outliers according to the percentage of MER in relation to MER max . The resulting quotient is referred to as MER/MER max and is plotted against the CH 4 offgas content in Vol.-%. This characteristic graph changes with different H 2 to CO 2 gas inflow ratios. This is due to the fact that, based on the presented assumption(s), full gas conversion can only be achieved when a H 2 to CO 2 gassing ratio of 4:1 is applied [9]. Even though the data was extracted from literature for CO 2 -BMP modelling purposes, re-calculation of the data was necessary to be able to equalize the entries for subsequent qualitative and quantitative analyses. This method overestimates MER for all other data that were not calculated based on the r inert correction factor [9,26]. The r inert correction factor accounts for the fact that stoichiometric gas contraction occurs during conversion. It is needed to calculate the MER based on the educt gas inflow and the CH 4 offgas composition [12,26]. However, if Y (X/CH4) is assumed to be 1-5% of the total carbon flux into the biomass, the error on MER quantification is relatively small [11,12,15,26,36,45]. This approach was applied to reject data with highly deviating DoR-or Cbalances.

Multivariate statistical analyses
Principal component analysis (PCA) was used to cluster the KPPs and variables and to visualize the variability of the CO 2 -BMP data. Subsequently, the data will be treated using multiple linear regressions (MLR) to obtain models and describe the MER and qCH 4 for different parameter spaces and reactor configurations. PCA and MLR modelling was conducted by using DataLab (Ipina GmbH, Pressbaum, Austria (www.datalab.com)) and Design Expert 8.0.7.1 (Stat-Ease, Inc., Minneapolis, USA). Data imputation was performed according to the DataLab data imputation routine using mean fill functions only for columns where 15% empty cells or less of the cells were missing data (values shown in Supplementary Material 1). PCA was performed on a qualitatively curated data set from CO 2 -BMP continuous culture (Supplementary Material 1). After removing erroneous data, the CO 2 -BMP data set used for PCA consisted of 172 continuous culture conditions (n = 172). Data extracted from literature did not provide enough information to accurately sort the data sets according to the following conditions: ammonia concentration, titration liquids, sulphur concentration, liquid dilution rates, gas flow rates, and ORP. Data size for multivariate analyzes was comfortably high (10 data points per KPP or variable, [46,47]) to allow data substantiation concerning multifactorial dependencies. The data set used for multivariate analyses of M. marburgensis CO 2 -BMP comprised 159 continuous culture conditions (n = 159, Supplementary Material 1). In general, data was checked for multinormality and skewness as well as for linearity of individual variables to individual process factors. All PCA analyses were performed based on a correlation matrix obtained though standardization of data. From a PCA bi-plot, which is based on a correlation matrix, the cosine of the angle between the loadings represents the correlation between process factors and/or dependent variables. Loadings are the sum of the eigenvector multiplied by the square root of the eigenvalue. For PCA, the data set was differentiated according to pre-defined classes. The differentiation into pre-defined classes was necessary due to the related variables being setup dependent, especially for data on gas transfer-related variables. Mixing data from different setup specific CO 2 -BMP cultivations would generate inaccurate models. The pre-defined classes of the data set were introduced based on the various setup conditions and associated bioreactor volumes. Experiments previously performed for M. marburgensis that vary these conditions were carried out as follows: continuous culture in a 2L bioreactor [12], design of experiments (DoE) in a 2L bioreactor [11], continuous culture in a 10 L bioreactor [26], cell retention in a 10 L bioreactor [26], and DoE in a 10 L bioreactor [35]. From Supplementary Material 1 data subsets for qCH 4 vs D were established by segregating the kinetic limitation faced by Y (x/CH4) . The data subsets were used to identify and subsequently demonstrate novel insights into mechanistically inherent aspects of the kinetic limitations occurring during CO 2 -BMP. Prior to PCA and MLR analyses the data was analyzed for homoscedasticity by visual data inspection of the corresponding graphs that are provided in Supplementary Material 2. Multicolinearity was assessed using the variance inflation factor (DataLab, Ipina GmbH, Pressbaum, Austria).

Results
Since the publication of the simple unstructured mathematical model for a continuous pure culture CO 2 -BMP process [32], new approaches have been reported for the cultivation of methanogenic archaea converting CO 2 . The model in Schill et al. [32] describes growth and productivity of M. thermoautotrophicus in a gas-limited state as function of KPPs such as D or gassing rate. In general, many studies on CO 2 -BMP focused on fed-batch or continuous culture modes [26,32,33,36]. The primary goal of these studies was to induce gaslimited or liquid-limited conditions and derive quantitative physiological variables. In some cases, these studies also examined the underlying thermodynamic and metabolic constraints of biological methanogenesis [13,26,33,35,36,48,45]. However, the dual nature of limitations (gas transfer-based or liquid-based) or inhibitions that can be faced upon biomass growth pose challenges for the development of a robust and scalable technology [8].
Continuous culture CO 2 -BMP data are shown in Fig. 1. These data were plotted according to the quotient of MER/MER max to CH 4 offgas. M. marburgensis continuous culture data fits the MER/MER max to CH 4 offgas relationship. This is a consequence of the method applied for the calculation of MER via r inert gas flow as described previously [9,12,26]. Although the r inert correction factor was shown to be fairly correct for low biomass concentrations between 1 and 5% [15,36], and the DoRbalances were shown to not vary greatly [11,12,26], the calculation could become more erroneous if the Y (x/CH4) is higher, as could be the case for other methanogenic archaea such as Methanosarcina barkeri [15]. In addition, it poses limitations in terms of quantification accuracy and the ability to identify physiologic effects during a process operation. Furthermore, if MER and MER max values are calculated by neglecting r (x) , the proportion should be true. If MER is measured, but MER max is calculated from literature data neglecting biomass formation, then the ratio is underestimated because the calculated MER max is higher than the real MER max . Therefore, it must be noted, whether or not the MER was calculated from literature data or taken directly from the publication.
Physiological effects cannot be quantified if the C-balance variability is too high. In fact, the accuracy required for C-balancing needs to be assessed as a function of the Y (x/CH4) resolution target [8]. Therefore, enhanced C-and DoR-balancing would benefit the modelling of MER and r (x) as a function of KPP. Calculations of MER/MER max from literature data were expected to possess a slight offset from the concept graph line as some systematic differences are inherent to the calculation process. However, even with this in mind, some data could not be closely fitted as can be seen in Fig. 1. An interval around full conversion of reactive gases was set in order to compensate for neglecting r (x) . Subsequently, only data fitting within this interval were presented and retained in the final data set. The final data set (Supplementary Material 1) will be used for the modelling of CO 2 -BMP and the multivariate data analysis of process variables.

Gas transfer-limited versus liquid-limited biomass growth
The appearance of dual (gas transfer-based or liquid-based) limitation mechanisms, which are inherent to CO 2 -BMP processes pose challenges for process analytical technologies (PAT) and quantification towards the development of a robust, controlled, and automated bioprocess [8,9,35]. Therefore, in order to accurately quantify the kinetics of gas converting bioprocesses, it is important to know the actual limitation at either a given time point or as a function of the process parameters applied to allow for control of the biocatalytic activity [35]. This strategy allows scaled feeding of the organism according to physiologic demand and avoids undesired limitations in the process reaction kinetics. Data from different CO 2 -BMP processes [11,12,26,42,43] are shown as individual plots of qCH 4 as a function of D in Fig. 2.
As an example, data extracted from Peillex et al. shows that the data points calculated for qCH 4 have an unusually high variation at the same D [43]. Although data obtained from Peillex et al. fit the MER/MER max concept (Fig. 1, qCH 4 data values were found to be more than 400% above the qCH 4,max reported for M. marburgensis in continuous culture [12,35]. Although qCH 4,max estimation of M. marburgensis was performed by using dynamic process conditions [12,48,49] or via a controlled liquid-limited condition [35], it can be considered that a maximum standard deviation of 10% is expected on the reported values. Therefore, qCH 4 values obtained by Peillex et al. are not likely to reflect physiological characteristics of M. marburgensis and will therefore not be used for modelling. A possible cause for the qCH 4 deviation in Peillex et al. could be an erroneous determination of biomass. In fact, if biomass concentration was under-evaluated by a factor of ten, then the qCH 4 values would fit results reported elsewhere [11,12,26,35] at the given process conditions. However, higher values of qCH 4 of M. marburgensis were reported for fed-batch experiments [48,50]. It is believed that during dynamic fed-batch experiments the quantification challenge (i.e. the measurement or calculation of volume compensation) has an impact on the quantification of process responses. This assumption could be verified by analyzing the error propagation over the multiple quantification steps and assumptions reported elsewhere [48,50].
When analyzing the model proposed by Schill et al. during gaslimited CO 2 -BMP, a linear relationship between qCH 4 and D is expected. This characteristic is associated to the autobiocatalytic reactions at a fixed Y (x/CH4) and gas transfer rate (GTR). Essentially, a culture will reach different values for x at equilibrium with different applied D (i.e. x decreasing with an increasing D as a consequence of wash out) [12,32]. This phenomenon can be observed in Fig. 3, where data from M. marburgensis [11,12,26,35] can be dissected into two broad trends for qCH 4 as a function of D depending on the type of limitation affecting biomass growth.
Although simple linear relationships generally describe liquid substrate based bioprocess development [51], the specific product formation with respect to D in a gas-or liquid-limited bioprocess is different [12,32,36,52]. In Fig. 3, it can be seen that linear relationships cannot be trivially applied to CO 2 -BMP processes and that this relation depends mostly on the limitation faced by r (x) . It is well known that qCH 4 can  vary greatly at a given GTR independent of D when a liquid-limitation or inhibition occurs [11,35]. However, under gas-limited conditions qCH 4 is linearly dependent on D at a slope proportional to Y (x/CH4) [12,32,36]. This is also shown in Fig. 3.
The maintenance energy of a liquid-limited pure culture can be determined by plotting the specific product productivity as a function of D. However, maintenance energy is defined as the energy for metabolic functions not related to growth. Hence, for CO 2 -BMP it would correspond to the Gibbs free energy inherent to the material flow used neither for biomass or product formation. A Y (x/CH4) of zero would be experimentally required to allow quantification of the maintenancerelated metabolism. Additionally, the kind of limitation (gas or liquid) or inhibition in Fig. 3 could be elucidated when analyzing qCH 4 as a function of D. However, it is extremely challenging to assure that the examined cultures are solely gas-or liquid-limited, since not only proper biomass, vvm, CH 4 offgas quantification, and subsequently analytics must be taken into account, but a priori knowledge about the setup of interest is also required.

Bioreactor setupanother limitation towards high MER
The correlation of the MER as a function of the volume dependent gassing rate, vvm, is another relationship that is often presented for analyzing CO 2 -BMP [12,25,26,42,53]. An analysis of MER as a function of vvm is shown for the data obtained from literature in Fig. 4.
In literature, linear relations were often shown for MER as a function of vvm [26,36]. However, this relation is not only strictly setup dependent, but also only holds true within a limited range of vvm increase and is furthermore known to impact the CH 4 offgas. In a specific CO 2 -BMP setup with a specified GTR, a maximum gassing rate can be applied. At higher gassing rates, increases of MER will not occur. In continuously stirred tank reactors (CSTRs), this is caused by flooding of the stirrer [54]. Flooding of the stirrer describes the phenomenon that for a given agitation speed with increasing vvm a set-up specific gas to liquid mass transfer maximum (flooding point) will be reached. Beyond the flooding point additional gas supplied to the bioreactor will not anymore be able to be transferred to the liquid phase. The additionally supplied gas will cause the aggregation of bigger-sized gas bubbles that ascend around the stirrer axis and escape the bioreactor. Fig. 4 shows the comparison of the relation between vvm and MER in two different CSTR setups with deviating mixing systems and therefore different mixing efficiencies. While in one system (Fig. 4a), due to flooding of the stirrer, additional gassing did not contribute to an increase of the MER, the other setup showed a steady increase of MER over vvm as additionally provided gas could also be effectively transferred into the liquid phase (Fig. 4b). However, a trend towards a maximum reachable MER can also be seen here as the curve is flattening with an increasing gassing rate [26].
The gas residence time should also be taken into consideration since at higher vvm the contact time between gas bubbles and liquid is consequently reduced. The mix of these physicochemical limitations implies that for a given reactor setup (with a defined maximum GTR) a maximum MER exists at which a targeted CH 4 offgas quality can be consequently reached. This is because offgas quality is always affected by the interplay of GTR and the average residence time of the reacting gases if sufficient biocatalyst is available [8]. These mechanistic constraints need to be considered and reflected in the models of gas converting bioprocesses particularly in the case of CO 2 -BMP. Unfortunately, such trends were not described or translated within the existing models available [11,26,32,36] which are often restricted to a limited operational space and are unsuited for extrapolating knowledge for the purpose of process operation or scale up activities in different reactor setups.

PCA of CO 2 -BMP continuous culture data
PCA is a statistic tool used for multivariate data analysis and that can be utilized to identify correlations and loadings among process parameters and dependent variables [46,47]. In this section PCA was applied to three different data sets. The data sets used for the PCAs are available in Supplementary Material 1. An overview of all process parameters and dependent variables that could be applied in the PCA is shown in Table 1.
The PCA for Fig. 5a comprised 172 independent continuous culture steady state conditions performed with different methanogenic strains. A total of six principal components (PCs) are necessary to explain 77.70% of the total variability (Supplementary Material 3), which renders the cluster challenging to interpret due to the numerous dimensions involved. When analyzing PC 1 and PC 2, only 44.77% of the total variability of the dataset can be explained. This denotes a strong multivariate nature of variables and parameters in the CO 2 -BMP processes, which can generally be extended to all gas converting bioprocesses. This is because gas converting bioprocesses are not only dependent on the kinetics of gas to liquid mass transfer but also on the physiology of the biocatalyst. In Fig. 5a, it is difficult to recognize a clear clustering of KPPs with respect to MER or qCH 4 . For a CO 2 -BMP process performed at the same temperature for a given reactor setup, MER should tendentiously cluster with vvm, pressure, and agitation. Alternatively, qCH 4 should share a correlation with D or Y (x/CH4) , as it was shown in Fig. 3.
Such constraints could not be observed in Fig. 5a. The only conclusion that can be drawn from the CO 2 -BMP process data reported in literature is that great data variability might occur from the different experimental approaches and setups used by the different authors. This led to the intention of retrieving a more compact PCA analysis. To obtain a clustering between the KPP of CO 2 -BMP and MER and qCH 4 (please also refer to Fig. 1, a dataset using only data from M. marburgensis was applied (Supplementary Material 1). The results of this analysis showing PC1 and PC2 in a bi-plot are presented in Fig. 5b. Therein, three correlating clusters can be identified. Cluster 1 is composed of factors 1, 10, and 11, and therefore represents a combination of MER, H 2 /CO 2 gassing rate, and pressure. Cluster 2 is composed of factors 8 and 12, wherein r (x) and TE are correlating. Cluster 3 is composed of factors 4, 7, 9, 16, and 18, where all liquid relevant process factors such as ORP, DS, and D are clustering with the dependent physiological variables qCH 4 and Y (x/CH4) . Nevertheless, this case also needs six PCs to explain 78.44% of the total variability. Hence, a detailed analysis would imply examining all of the combinations and permutations of PC1 to PC6. However, this exercise poses a certain challenge for the interpretation of results in such a multi-dimensional space. When considering the contribution of communalities to the individual PCs, it turned out that gas related process parameters (vvm, agitation, pressure, MER, CH 4 offgas) as along with D and TE contribute to the first two PCs.
Finally, a third PCA was performed. This dataset consisted of data from gas-limited M. marburgensis cultures only. The relationship of GTR related process factors, aforementioned KPPs, and variables are shown in Fig. 6. In this PCA only two PCs were necessary in order to explain 68.49% of the total variability (Supplementary Material 3). In Fig. 6, a clustering of factors 1, 10, and 11 can be identified. The results presented in Fig. 6 are supported by findings reported in literature for gaslimited conditions during CO 2 -BMP in continuous culture operations [26,36,52]. The results of the PCA presented in Fig. 6 suggest that a proper multivariate model of continuous culture CO 2 -BMP could be obtained by applying MLR fitting methods. However, the presented data clearly showed that best results are obtained when using data restricted to a defined physiological state. The more variable the underlying dataset was in terms of physiological states, different reactor setups, or various strains used, the more difficult it was to obtain PCA results that could be set in a logical context to what is found in literature.

Modelling of MER
To highlight the complex interdependencies between process variables and responses in CO 2 -BMP, multivariate models are presented for MER and qCH 4 . Based on the results of the PCA analyses and the high number of data available for M. marburgensis continuous culture experiments, the latter dataset was subsequently used for MLR. The results are shown in Supplementary Material 2.
MLR analysis lead to the following results: vvm, pH, temperature, agitation, DS, TE, and pressure were found to significantly influence MER (89.3%, r 2 = 0.8927). However, it must be noted, that this model  was established from continuous culture CO 2 -BMP results that were based on different bioreactor setups and geometries [11,12,26,35]. Furthermore, the data density towards higher MER values decreases. In Fig. 7a, the MER of an M. marburgensis continuous culture utilized for CO 2 -BMP is shown as a function of vvm and pressure for gas-limited conditions. The significant ANOVA model of MER of M. marburgensis is shown in Supplementary Material 2. While the overall positive influence of vvm and reactor pressure on the MER is correctly reflected by the model plot presented in Fig. 7a, the exact correlation between vvm and MER is obviously wrong. The presented model predicts an exponential increase of the MER with vvm, while the data collected during experiments with a certain setup (Fig. 4) showed the opposite trend, a flattening of the curve towards higher vvm. As previously explained, this is an inevitable consequence of the flooding phenomena occurring in CSTR reactors at a certain vvm. However, since data from several different CSTR setups with different individual flooding points were used as input for the MLR analysis, the correlation between vvm and MER is erroneously predicted. To overcome this problem, a sub-dataset, consisting of data collected with a single bioreactor setup, was used to perform a new MLR analysis. The outcome is shown in Fig. 7b. In this case, the experimentally determined correlation between vvm and MER is now properly reflected but therefore only valid in the design space. This shows that modelling of gas transfer, and therefore setup dependent variables like MER, should be performed for specific bioreactor setups while taking into account the underlying mechanisms of gas-liquid mass transfer as well as the residence time of reacting species. GTR mechanisms can, among others, be affected by reactor geometry, operation mode, working volume, broth rheology, agitation system, and sparging.

Modelling of qCH 4
Another MLR model was established for qCH 4   and D is shown in Fig. 8 and the ANOVA models are presented in Supplementary Material 2. Temperature, ORP, gassing ratio, and DS were not found to be significant.
The model equation for qCH 4 (Supplementary Material 2) shows that agitation also affected qCH 4 in several ways. Agitation increases the k L a, which influences r (x) and MER. It has been observed that increasing agitation had negative effects on r (x) [11]. These significant qCH 4 model terms in the equation can be explained by the multivariate nature of external influences affecting the physiology of methanogens, e.g. pH, ORP, temperature or pressure [14][15][16]. Due to the multivariate analysis of existing CO 2 -BMP data, it becomes obvious that such influences would require the employment of sensitive analytical methods (e.g. TE analytics) for the liquid phase [55] and fine quantification of gas flow and composition [8] to be able to enhance the overall accuracy of process elemental balancing. This would enable the resolution of small variations of Y (x/CH4) as a function of input parameters and/or to compensate for the eventual lysis of biomass which would significantly affect r (x) determination and subsequent Y (x/CH4) calculation [8].

Discussion
The above-mentioned constraints clearly show, that for a gas converting bioprocess, such as CO 2 -BMP, the two main kinetic determining limitations, gas-and liquid-limitation, need to be considered for modelling the overall process kinetics. A summary of possible issues, their interpretation and tasks that could occur during analysis and modelling of the CO 2 -BMP process is given in Table 2. However, it has to be noted that it is the gas to liquid mass transfer that is limiting MER and not the physiological capacity of the methanogens [11,12,26,32,33,35,56].
In a CO 2 -BMP bioprocess, the biomass acts as an autobiocatalyst and needs to be properly handled to exploit the full biocatalytic activity of the organism. Therefore, inhibitory or limiting liquid-based compounds would need to be quantified with sophisticated PAT and methods [12,49,55]. After biomass is grown in CO 2 -BMP fed-batch cultures [48,50], the continuous culture CO 2 -BMP process will enter a H 2 -based gas limitation phase [26]. Therefore, the growth medium for methanogens is generally aimed to be non-liquid limiting, and eventually noninhibitory, as one of the main goals of CO 2 -BMP is to achieve maximum MER for subsequent bioprocess scale-up. Therefore, overfeeding of minerals is often applied to avoid such liquid-based limitations. An example of such a constraint is presented in Fig. 9.
MER of M. marburgensis from continuous culture experiments was plotted as a function of feeding ratio with a sulphide flow rate (S in ) to r (x) . It clearly shows, that the highest MER values were obtained at a S in /r (x) feeding ratio of either > 0.001 and < 0.017 which is close to the elementary composition found in the biomass of methanogens [57]. Higher feeding rates are thus not necessary. This could also be an indication that sulphide overfeeding was affecting the quantification of physiological responses shown for CO 2 -BMP in literature. Such findings could also be because of a variation observed in physiologic responses that could explain why none of the models shown above are valid. The equilibrium of sulphide species in aqueous phase and their interaction with TE needs to be dissected as a function of the pH, temperature, vvm, and ORP [58]. However, only one attempt has been made to account for H 2 S/HS -/S 2equilibrium when performing elemental balancing in CO 2 -BMP [56]. The negative effect on either MER or r (x) could not be precisely determined for DS even though modelling indicates the possibility that the latter parameter was negatively influencing MER (Supplementary Material 2). Recently, sulphide and TE interactions were determined during CO 2 -BMP fed-batch bioprocessing. This was done to avoid physiologically unfavorable KPP settings [48]. Even so, this attempt did not fully dissect the complex sulphide and TE interactions in CO 2 -BMP processes. During CO 2 -BMP modelling, the gas transfer limitations are also of concern. As it was shown before, gas transfer related variables, such as MER or CH 4 offgas, were found to be strongly setup dependent [59]. Modelling across different reactor setups can consequently lead to erroneous results if the influences on the system are not properly characterized.
Without a combination of PAT and experimental approaches it is difficult to unscramble liquid and gas transfer related influences, which could easily lead to misinterpretation of process factor correlations [11,12,48,49]. Proper modelling for CO 2 -BMP therefore requires prior detailed knowledge about both the bioreactor setup and the physiology of the applied strain. This, however, creates the need for analytical tools that allow for the balancing of individual compounds, particularly for carbon and hydrogen molar fluxes, to a very accurate level. The presented results show that models based on literature data often lead to erroneous predictions and conclusions.
Previous approaches for modelling CO 2 -BMP neglected parameters such as the influence of liquid limitations on the performance of the system, assuming the culture to be solely gas-limited [32]. While this approach can deliver valuable results, it is very limited in applicability, since the constraints that need to be made to keep the assumptions valid are narrow and are difficult to achieve. This is especially true as several studies have shown a strict separation of gas transfer limitations and liquid limitation. This interdependency adds a great deal of complexity to the modelling of any gas converting bioprocess and is particularly true for CO 2 -BMP.

Conclusions
This work shows the inherent challenges faced in modelling CO 2 -BMP. The most important aspect is the dependency of the performance on both, gas transfer limitations and liquid-based limitations. Utilizing PAT is inevitable in order to discriminate between these two factors. Implementing a real-time biomass sensor to correct the r inert calculation method for the MER, where biomass formation is currently neglected, could result in an improved C-and DoR-balancing and would allow performing accurate and timely k L a determinations. Literature data often misses important information and/or the required accuracy for resolution of the underlying mechanistic effects, especially when modelling reactor dependent variables. Modelling can only be based on a mechanistic understanding of a particular process. Otherwise, modelling misinterpretation might occur. Understanding the mechanistic effects of CO 2 -BMP could therefore assist the analysis and modelling of other gas-to-gas conversion bioprocesses. Table 2 Overview of issues, their interpretation and the required action that could be occurring during analysis and/or modelling of CO 2 -BMP.