Correspondence: Marion Leclerc, Institut National de la Recherche Agronomique (INRA), UMR1319, Micalis, 78350 Jouy-en-Josas, France. Tel.: +33 1 697 9706; fax: +33 1 486 4545; e-mail: firstname.lastname@example.org
Butyrate is the preferred energy source for colonocytes and has an important role in gut health; in contrast, accumulation of high concentrations of lactate is detrimental to gut health. The major butyrate-producing bacterial species in the human colon belong to the Firmicutes. Eubacterium hallii and a new species, Anaerostipes coli SS2/1, members of clostridial cluster XIVa, are able to utilize lactate and acetate via the butyryl CoA : acetate CoA transferase route, the main metabolic pathway for butyrate synthesis in the human colon. Here we provide a mathematical model to analyse the production of butyrate by lactate-utilizing bacteria from the human colon. The model is an aggregated representation of the fermentation pathway. The parameters of the model were estimated using total least squares and maximum likelihood, based on in vitro experimental data with E. hallii L2-7 and A. coli SS2/1. The findings of the mathematical model adequately match those from the bacterial batch culture experiments. Such an in silico approach should provide insight into carbohydrate fermentation and short-chain fatty acid cross-feeding by dominant species of the human colonic microbiota.
The human colonic microbiota has a number of important roles, including the anaerobic degradation of carbohydrates not absorbed in the upper digestive tract. This results in the production of short-chain fatty acids (SCFA), mainly acetate, propionate and butyrate, which have been recognized for their health-promoting effects (Topping & Clifton, 2001). In the fermentation process, lactate is a key intermediate produced via the homofermentative or heterofermentative pathways. Lactate participates in a reversible reaction with another key component of the fermentation process, pyruvate. It is thus a precursor of the formation of acetate, propionate and butyrate in species such as Megasphaera and Veillonella. Lactate can also be a cosubstrate for the sulphate-reducing species Desulfovibrio piger (Marquet et al., 2009).
Within the last decade, butyrate-producing bacteria, mostly belonging to the Firmicutes phylum, have been isolated from the human gastrointestinal tract (Barcenilla et al., 2000). Among them, Faecalibacterium prausnitzii and Roseburia spp. are unable to use lactate (Duncan et al., 2004a, b). Conversely, Anaerostipes caccae, Eubacterium hallii L2-7 and the proposed new species Anaerostipes coli SS2/1 (Walker et al., 2011), are lactate-fermenting, butyrate-producing bacteria. Eubacterium hallii and A. coli belong to clostridial cluster XIVa (Lachnospiraceae). Anaerostipes coli is a dominant member of the colonic microbiota recognized for its importance in butyrate production (Walker et al., 2011). The abundance of E. hallii in human faeces ranges from 1% to 3% of total bacteria (Harmsen et al., 2002; Louis & Flint, 2009).
Butyrate can be synthesized from two metabolic pathways: butyrate kinase and butyryl CoA : acetate CoA transferase (Miller & Wolin, 1996; Diez-Gonzalez et al., 1999; Duncan et al., 2002). In vitro studies have identified the latter mechanism as the dominant one in the human colonic ecosystem (Louis et al., 2004). Pyruvate is the central pivot of butyrate synthesis. It is oxidized to acetyl coenzyme A (acetyl-CoA), which is further routed to acetate or butyrate. Acetate is produced via acetate kinase. This pathway generates energy in the form of ATP. For butyrate formation, two molecules of acetyl-CoA are condensed to one molecule of acetoacetyl-CoA, and subsequently reduced to butyryl-CoA. Butyryl CoA : acetate CoA transferase transfers the CoA moiety to external acetate, leading to the formation of acetyl-CoA and butyrate (Duncan et al., 2002). In addition to restoring NAD+ in the cell, more energy may be obtained from butyrate formation in bacteria with a membrane-associated NADH : ferredoxin oxidoreductase-linked proton motive force (Seederof et al., 2008; Louis & Flint, 2009).
In vivo and in vitro studies of the human gastrointestinal tract are hampered by both ethical and technical constraints, including the low percentage of cultured bacteria (Suau et al., 1999; Eckburg et al., 2005). Mathematical in silico models based on biological knowledge will help in understanding and predicting the patterns of carbohydrate fermentation and metabolite cross-feeding occurring in the human gastrointestinal tract. Mathematical models have been used previously to study anaerobic reactors (see, e.g. Batstone et al., 2002). Given the processing similarities between anaerobic bioreactors and the human large bowel, the anaerobic digestion modeling represents a framework for modelling the human colonic ecosystem. Up to now, there have been very few dynamic models for the metabolic reactions driven by colonic bacterial species. Amaretti et al. (2007) developed a mathematical model to describe the kinetics in Bifidobacterium adolescentis MB 239 grown on different substrates. Other mathematical approaches have analysed the butyrate production from labelled substrates (Duncan et al., 2004a, b; Morrison et al., 2006). Recently, 13C-labelled glucose was used to map the fermentation metabolic fluxes in human colonic microbiota (de Graaf et al., 2010).
Complementary to animal studies (e.g. Le-Blay et al., 1999), in silico models that predict butyrate formation should provide valuable insights into the structure and dynamics of human colonic microbiota, suggest interesting additional in vitro or in vivo experiments, and help in developing strategies to promote gut health.
The mathematical model construction process should obey the principle of parsimony (also known as Occam's razor), striving to build the least complex model that can still adequately represent the system under study. The aim of the work reported here was to develop a parsimonious mathematical model to describe the conversion of lactate and acetate into butyrate by key human colonic bacterial species. The model was constructed based on the butyryl CoA : acetate CoA transferase pathway for butyrate formation and was assessed by computing in vitro data obtained for E. hallii L2-7 and A. coli SS2/1 (Duncan et al., 2004b).
Materials and methods
Figure 1 illustrates a simplified scheme of the butyryl CoA : acetate CoA transferase pathway. The kinetic model developed is an aggregated representation of this pathway. A simplified schematic representation is displayed in Fig. 1 (inset box). The dynamics of all variables depends only on the concentrations of lactate and the number of bacteria (OD).
The mathematical model was derived by formulating mass-balance differential equations for a batch system. The model equations are then:
The model variables (also called state variables) are the bacterial concentration xla in OD650 nm, assumed to be proportional to molar concentration, lactate concentration sla, acetate concentration sac and butyrate concentration sbu in mM.
The function ρla is the consumption rate of lactate. This is expressed by the standard Monod equation:
The parameter km is the lactate consumption rate constant. It is expressed in mM lactate per OD h−1. The Monod constant K is in mM lactate. Y (OD mM−1 lactate) is the biomass yield according to the consumed lactate. The yield factors Yac (mM acetate mM−1 lactate) and Ybu (mM butyrate mM−1 lactate) represent the molar change of acetate and butyrate to the consumed lactate. The yield factors are related to the stoichiometry of the metabolic conversions. Based on the reactions for butyrate production (Papoutsakis, 1984) and lactate consumption (Costello et al., 1991), the following set of reactions is derived to group the intermediate steps of butyrate:
It should be noticed that reactions(a) and (b) are used as a simplification of the complete mechanism in order to provide stoichiometric relationships between metabolites. In addition to the above reactions, carbon from lactate contributes to bacterial growth.
Let (1−fla) be the fraction of lactate used for bacterial growth. Denote by ηa the part of the remaining fraction fla of lactate that is converted in reaction (a). Then, the part involved in reaction (b) is ηb=1−ηa.
Exploiting the stoichiometry, we get:
The biomass yield factor Y is decomposed as:
where α is a conversion factor between OD and bacterial concentration and β is the number of bacteria produced per mole of lactate consumed for the bacterial growth.
Note that Yac is a net factor representing the difference between production and consumption of acetate. Thus, if Yac is positive, it means that lactate is routed to reaction (a) in a higher proportion than to reaction (b).
Stoichiometry imposes the following constraints on Yac and Ybu:
Moreover, assuming that the fraction of lactate devoted to bacterial growth is low (fa is close to 1) we get:
which gives an approximate relation between the number of butyrate moles produced per time unit (dnbu) and the number of moles of lactate and acetate utilized per time unit (−dnla and −dnac):
which means that bacterial concentration xla is related to lactate concentration sla by a linear relationship with slope −Y. In the same way, acetate concentration sac and butyrate concentration sbu are related to sla by linear relationships with slope −Yac and −Ybu, respectively.
The mathematical model does not take into account biomass decay, and no distinction between d-lactate and l-lactate is attempted. For E. hallii L2-7, the sum of the two isomers is considered a sole substrate. In the case of A. coli SS2/1, d-lactate is the sole substrate, because this strain was not able to utilize l-lactate.
Use of experimental data to estimate the model parameters
The parameters to be estimated are km, Y, Yac, Ybu and K. Experimental data formerly obtained (Duncan et al., 2004b) were used to estimate these parameters. The published data are means of triplicates with pure cultures of either E. hallii L2-7 (DSM 17630; accession number AJ270490) or A. coli SS2/1 (DSM 23942) providing the inoculum. Eubacterium hallii L2-7 was isolated from a 2-year-old healthy infant (Barcenilla et al., 2000) and A. coli SS2/1 from a healthy adult female (Louis et al., 2004) consuming a westernized diet. Bacterial strains were grown in YCFA medium containing (per 100 mL) 0.1 g casitone, 0.25 g yeast extract, 0.4 g NaHCO3, 0.1 g cysteine, 0.045 g K2HPO4, 0.045 g KH2PO4, 0.09 g NaCl, 0.009 g MgSO4·7H2O, 0.009 g CaCl2, 0.1 mg resazurin, 1 mg haemin, 1 μg biotin, 1 μg cobalamin, 3 μg p-aminobenzoic acid, 5 μg folic acid and 15 μg pyridoxamine. Final concentrations of the SCFA in the medium were 46 mM acetate, 9 mM propionate and 1 mM each of iso-butyrate, iso-valerate and valerate. The medium was adjusted to pH 6.7 and placed in Hungate tubes that were flushed with CO2 and heat-sterilized. Heat-labile vitamins were added after the medium was autoclaved to give a final concentration of 0.05 μg mL−1 thiamine and 0.05 μg mL−1 riboflavin. The glucose-supplemented medium as a carbon source contained a final concentration of 10 mM. Each test bacterial strain was inoculated into triplicate tubes and growth measured spectrophotometrically as A650 nm. Growth rates were calculated in exponential phase. The medium was supplemented with 35 mM dl-lactate (pH 6.7±0.1), of which approximately 16 mM were in the l-lactate form. The in vitro experiments were carried out in Hungate tubes under anaerobic conditions. Fermentation was monitored by synchronous measurements of the OD650 nm and concentrations of glucose, lactate (dl and l), acetate and butyrate. Concentrations of l-lactate and glucose were determined by enzymatic methods. Concentrations of SCFA and dl-lactate were determined by capillary GC. d-Lactate was estimated as the difference between total lactate and l-lactate concentrations.
Parameter estimation focused on E. hallii L2-7 and A. coli SS21/1 experiments in dl-lactate-containing medium (fig. 1b and e in Duncan et al., 2004b). Data in the lag phase were not considered.
The linear relationships between concentrations were assessed using total least squares (Vanhuffel & Vandewalle, 1989; Björck, 1996). The usual least squares method was not used because in the linear relationships that we analyse, the independent variable (lactate concentration) is not error-free. The total least squares provided estimates of the initial concentrations and of the yields. The estimated mean square measurement errors were then calculated. A parameter estimation of the complete model was then carried out by the maximum likelihood approach. The previously estimated Y, Yac and Ybu values were used to compute plausible starting values in the optimization step. Similarly, for the initial bacterial, lactate, acetate and for butyrate, the estimated values together with estimated SDs generated plausible values of the initial concentrations.
In many situations, the concentration of lactate will be very small compared with the Monod constant K. An estimation was therefore performed for the complete model and also for a simplified version where the following approximation:
valid when sla<K, was made. Note that in this case, only the ratio km/K can be estimated.
The cost function to be optimized was derived, assuming that the measurement errors followed a normal distribution with unknown and diagonal covariance matrix (e.g. Walter & Pronzato, 1997). The Matlab® toolbox ideas (Muñoz-Tamayo et al., 2009) was used to estimate the model parameters. ideas uses a quasi-Newton unconstrained optimization method for its optimization step. Once the estimation is completed and the constraints imposed by stoichiometry are satisfied, the approximate 95% confidence intervals for the parameters are calculated based on the computation of the Fisher information matrix (e.g. Walter & Pronzato, 1997). ideas can be freely downloaded from the web page http://www.inra.fr/miaj/public/logiciels/ideas/index.html.
Results and discussion
The construction of the mathematical model describing the kinetics of butyrate production by E. hallii L2-7 and A. coli SS2/1 was based on standard models of anaerobic reactors. We adapted such models to the specificities of human colonic bacteria. For instance, lactate contribution to butyrate is not considered in mathematical models for anaerobic reactors (Costello et al., 1991; Skiadas et al., 2000), whereas lactate is an important precursor of butyrate production in the human colon.
Utilization of lactate and acetate to produce butyrate in the metabolic pathway catalysed by E. hallii L2-7 and A. coli SS2/1 is shown in Fig. 2. Both strains were provided with 35 mM of dl-lactate, from which approximately 16 mM was in the l-lactate form. Eubacterium hallii L2-7 utilized all the available lactate, whereas A. coli SS2/1 could only utilize d-lactate. The data of Duncan et al. (2004b) also showed that in spite of the difference in lactate utilization by the bacterial strains, the butyrate produced is about half the number of reacting moles of lactate plus acetate in both cases. The addition of glucose to lactate-containing medium repressed lactate utilization by E. hallii L2-7 but had little effect on lactate metabolism by A. coli SS2/1.
The linear relationships between concentrations are corroborated by the experimental data for both strains (Fig. 3). Estimated yield factors, initial concentrations and measurement error SDs are given in Table 1. These results agree with the structure of our model based on the butyryl CoA : acetate CoA transferase pathway.
Table 1. Total least squares estimates of the yield factors: measured initial concentrations, estimated initial concentrations and estimated mean quadratic errors for biomass and substrates
Y (OD mM−1 lactate)
Yac (mM acetate mM−1 lactate)
Ybu (mM acetate mM−1 lactate)
Eubacterium hallii L2-7
Anaerostipes coli SS2/1
sla (mM) Measured Estimated (error)
xla (OD) Measured Estimated (error)
sac (mM) Measured Estimated (error)
sbu (mM) Measured Estimated (error)
E. hallii L2-7
Anaerostipes coli SS2/1
Maximum likelihood estimation was performed for several starting values of the parameters, and for several values of the initial concentrations (Table 1). For each parameter, the mean and SD of the estimates were computed. For both strains, the km parameter was the most sensitive to the initial concentration, with an SD of 15%. For the other parameters, the SDs ranged from 3% to 14% of the mean. A multistart routine on the initial guess of the parameters was implemented to reduce the risk of finding local minima. The Monod model, in its simplified version, adequately represents the in vitro data from both strains (Fig. 4). Data corresponding to A. coli SS2/1 are more scattered, thus the fit is less satisfactory than for E. hallii L2-7. A discrepancy for acetate concentration was observed for both strains. In the case of E. hallii L2-7, this mismatch occurs for the first three sampling points. According to the model, acetate concentration will either always increase or always decrease. This transient mismatch may be due to measurement uncertainty, but most likely to the presence of carbon sources in YCFA medium from the yeast extract, not accounted for by the model. For E. hallii L2-7, the mathematical model slightly overestimated the biomass concentration at the final time point. Apart from measurement uncertainty, this could be explained by the fact that the kinetic model considers that biological activity is growth-associated. As bacteria can remain metabolically active without cell division, the OD in batch cultures is not always a very good indicator of bacterial activity.
Table 2 shows model parameters estimated with the quadratic kinetic function in Eqn(14) with their approximate 95% confidence intervals. As expected, the yield factor estimates are very close to the values obtained with total least squares computation only. The confidence intervals are narrow for both strains.
Table 2. Estimates of the model parameters for the simplified kinetic model with their approximate 95% confidence intervals
km/K (1 per OD h−1)
Y (OD mM−1 lactate)
Yac (mM acetate mM−1 lactate)
Ybu (mM butyrate mM−1 lactate)
Eubacterium hallii L2-7
1.205 ± 0.082
0.013 ± 0.001
−0.556 ± 0.061
0.666 ± 0.032
Anaerostipes coli SS2/1
1.412 ± 0.148
0.015 ± 0.001
−0.404 ± 0.194
0.678 ± 0.058
We also assessed the parameter accuracy for the complete Monod model (results not shown here). The SDs of the yield factors compare with those of the simplified model. However, km and the affinity constant K show high SDs, as was expected. Experimental data with higher lactate concentrations would be necessary to provide correct estimates of these parameters.
The equations from the simplified set of reactions for butyrate production have enabled us to present stoichiometric relationships between metabolites. For instance, the number of moles of butyrate formed is approximately half the number of reacting moles of lactate plus acetate. This relationship was indeed verified by the experimental results. In addition, the stoichiometry is in agreement with observations on other butyrate-producing bacteria grown in glucose or fructose, where the conversion of acetate and bacterial growth ceased with the disappearance of glucose or fructose (Duncan et al., 2004a; Falony et al., 2006) or, in this case, lactate. It can be deduced that, in a given bacterial strain, lactate may be the sole carbon source to produce butyrate; only if the acetate kinase activity is high enough to produce acetate for the conversion of butyryl CoA to butyrate (by butyryl CoA : acetate CoA transferase).
The model adequately describes the kinetics of butyrate production by E. hallii L2-7 and A. coli SS2/1. However, more experimental data are necessary to perform a formal validation of the model.
The parameters were estimated from in vitro data obtained under a specific set of conditions and these could change for experiments carried out under other conditions. For instance, lactate consumption rate and SCFA production are known to depend on hydrogen partial pressure (Costello et al., 1991; Macfarlane & Macfarlane, 2003). Unfortunately, for the experiments studied here, hydrogen was not measured and its effect not evaluated. It has been shown that lactate utilization and fermentation processes by human faecal microbiota depended on pH (Belenguer et al., 2007), and it has been reported that the estimates of kinetic parameters can also change with pH conditions (Amaretti et al., 2007). The model presented here does not take into account the influence of pH or of hydrogen partial pressure on the kinetics. Empirical mathematical expressions show that the effects of pH and H2 on the kinetic rates have been developed for anaerobic reactors (Costello et al., 1991; Batstone et al., 2002) and can be used for our model. Additional experimental work (Fig. 5) has revealed that E. hallii L2-7 growth rate decreased from 0.18 h−1 at pH 6.5 to 0.13 h−1 at pH 5.5 at an initial concentration of 35 mM dl-lactate. Surprisingly, A. coli SS2/1 was stimulated at pH 5.5 (0.31 h−1) compared with pH 6.5 (0.13 h−1). Very interestingly, this suggests that the relative contribution of these two species to lactate utilization may vary with pH and, consequently, in proximal (pH 5.5) and distal colon (pH up to 6.8).
Environmental conditions impose thermodynamic limitations on the spectrum of fermentation products. The Gibbs free energy change (ΔG′) of the reactions is the driving force that determines the metabolite concentration profile. Thus, the initial concentration and accumulation of metabolites will strongly affect the reaction progress. In vitro experiments with the ruminal butyrate-producing bacterium Megasphaeraelsdenii showed the impact of the initial concentration of acetate on butyrate metabolism. Under low initial concentrations of extracellular acetate (<4 mM), the production rate of acetate was higher than its consumption rate. When the initial concentration increased, the production decreased, whereas consumption increased concomitantly with an increased butyrate production (Hino et al., 1991). For the experimental data analysed here, the acetate consumption rate was higher than its production rate, indicating that exogenous acetate is required to achieve maximal growth. In the kinetic model, this implies a negative value for Yac yield factor, so ηa is <0.5. Under conditions where acetate production is higher than consumption, the model structure advocated here will remain identical, but ηa will be >0.5 and Yac will become positive. This shift in the reaction process can be handled mathematically using thermodynamic-based models as proposed by Rodríguez et al. (2008).
Application of the model and perspectives
Due to a number of ethical and technical limitations, including the limited number of cultured species available for metabolic studies, the human colon has remained largely unexplored for years. However, there is recent evidence that many representatives of the most abundant and metabolically active species have been isolated (Flint et al., 2007; Tap et al., 2009; Walker et al., 2011) and their genomes sequenced within the Human Microbiome Project (Turnbaugh et al., 2007). In vitro models based on experimental data from these isolates have proved very useful to study metabolism and interspecies interactions (Belenguer et al., 2006; Falony et al., 2006) and to provide better representations of the complexity of the gut ecosystem (Macfarlane & Macfarlane, 2007). Furthermore, simplified models in gnotobiotic rodents provide central information on in vivo bacterial metabolism and its consequences for host epithelial genetic responses (Mahowald et al., 2009). However, extrapolation of these studies to the human colonic ecosystem is not trivial. In silico models, such as the one proposed here, offer complementary approaches that can be used to address the complexity of the large intestine.
Metagenomic studies of the microbiota from the human gastrointestinal tract confirmed the importance of the metabolic pathway studied here: the genes involved in butyrate production demonstrated high odds ratios compared with all available bacterial genomes or human genome (Gill et al., 2006), and European MetaHit project (data not shown).
The mathematical model presented here focuses on E. hallii L2-7 and A. coli SS2/1; both are part of the dominant phylogenetic core of the human intestinal microbiota (Tap et al., 2009; Walker et al., 2011). These two strains differ in their affinity for lactate enantiomers. Whereas E. hallii was able to use both d- and l-lactate, A. coli SS2/1 only utilized d-lactate. Nevertheless, the model takes these differences into account. Even if the amount of data studied here is limited, the mathematical model structure should not be restricted to E. hallii L2-7 or A. coli SS2/1 as it could be applied to other phylogenetically distinct bacterial groups that share enzymatic reactions. Kinetic rates may differ between species, as demonstrated for the two strains analysed here; however, the model structures should be identical. Other observed differences between E. hallii and A. coli are their response to pH, in addition to A. coli showing greater resistance to glucose repression. Further refinement of the model will be required to take this behaviour into account.
Additional data are required to extrapolate our results to the human colonic ecosystem. Interesting data to be integrated in our approach include growth experiments with other key butyrate-producing bacteria that are unable to utilize lactate, such as Roseburia intestinalis and F. prausnitzii. In vitro experiments on microbial interactions between hydrogen utilizers and butyrate producers as presented by Chassard & Bernalier-Donadille (2006) provide useful data to build more representative models. Mathematical model approaches such as the one we developed to study homoacetogenesis reaction by Blautia hydrogenotrophica (Muñoz-Tamayo et al., 2008) could also be coupled with the model presented here to allow valuable cross-feeding interactions to be studied.
An important application of the present work is the integration of butyrate metabolism into a model that includes the complete trophic chain of carbohydrate fermentation as well as transport phenomena such as SCFA absorption (Muñoz-Tamayo et al., 2010). The estimation of the parameters of the resulting complex model is particularly challenging because of the scarcity of data. The analysis of in vitro bacterial growth experiments, such as the present study, is essential in the definition of prior knowledge to tackle the estimation problem for the complete model. This model could then address the impact of different carbon sources with butyrogenic effects. Such an approach would contribute to the development of strategies for enhancing butyrate production in relation to its health-promoting effects.
R.M.-T. was supported by a PhD fellowship of INRA Departments MICA and MIA.