The ability of microbial species to consume compounds found in the environment to generate commercially-valuable products has long been exploited by humanity. The untapped, staggering diversity of microbial organisms offers a wealth of potential resources for tackling medical, environmental, and energy challenges. Understanding microbial metabolism will be crucial to many of these potential applications. Thermodynamically-feasible metabolic reconstructions can be used, under some conditions, to predict the growth rate of certain microbes using constraint-based methods. While these reconstructions are powerful, they are still cumbersome to build and, because of the complexity of metabolic networks, it is hard for researchers to gain from these reconstructions an understanding of why a certain nutrient yields a given growth rate for a given microbe. Here, we present a simple model of biomass production that accurately reproduces the predictions of thermodynamically-feasible metabolic reconstructions. Our model makes use of only: i) a nutrient' s structure and function, ii) the presence of a small number of enzymes in the organism, and iii) the carbon flow in pathways that catabolize nutrients. When applied to test organisms, our model allows us to predict whether a nutrient can be a carbon source with an accuracy of about 90% with respect to in silico experiments. In addition, our model provides excellent predictions of whether a medium will produce more or less growth than another () and good predictions of the actual value of the in silico biomass production.
Our phenomenological model expresses the impact that different carbon sources (or nutrients) have on a microbe' s ability to grow using only information on the chemical structure of these carbon sources and on the ability of a microbe to catabolize these nutrients. Our model is built to reproduce the predictions of flux balance analysis calculations (FBA) on metabolic reconstructions, thus it will suffer from the same limitations. Indeed, while FBA is a powerful tool to investigate microbial metabolism and microbial growth in particular, it has a number of limitations when estimating growth rates and the effect of media and environmental conditions on growth. It is well-known that microbes need a minimal medium and a carbon source in order to grow. Minimal media have been described for a number of species and contain essential chemical species without which the species would not be able to grow [20], [29]. The growth rate of a microbe, however, depends on many other factors including the uptake rate of nutrients, temperature, regulation, the availability of oxygen, etc. FBA is a linear optimization method that predicts the maximal conversion of a set of carbon sources into biomass with a fixed minimal medium. In order for FBA conversion rates to reproduce empirical growth rates, one needs to consider an additional ATP maintenance flux which is obtained by fitting FBA results to empirical growth rates obtained for a certain temperature, minimal medium, and carbon source uptake. While ATP maintenance rates obtained for a specific minimal medium have been shown to give accurate predictions of growth rates in different minimal media for some organisms [8], in principle one cannot assume that they are valid for predicting growth rates for arbitrary minimal media. Additionally, because metabolic reconstructions do not consider regulatory constraints, FBA will predict that a microbe is capable of uptaking two different sugars simultaneously, while it is well-known that if there is more than one sugar carbon source at high enough concentrations, the organism will exhaust the preferred one before consuming the others [30]. A microbe will, however, consume multiple sources of carbon other than sugars simultaneously and as a consequence grow faster [31]. It has been shown that steric constraints can already reproduce diauxic growth in some organisms in the presence of multiple sugars [32], however, there is no general framework that is able to deal with this issue when using FBA. To develop a phenomenological model that is able to reproduce maximal biomass conversion rates per carbon source under aerobic conditions, we investigate the maximum amount of biomass that can be produced by an organism in the presence of a minimal medium, oxygen and one or more carbon sources including at most one sugar. To this end, we run FBA on metabolic reconstructions in which we remove ATP maintenance [27] (see Methods for specific details). We concentrate on a training set of four species for which there are high-quality metabolic reconstructions available, and which cover a wide range of microbial phylogeny: a gram-negative bacterium (E. coli [8]), a gram-positive bacterium (B. subtilis [9]), an eukaryote (Saccharomyces cerevisiae [21]), and an archaeon (Methanosarcina barkeri [24]). We validate our model on a test set of three species for which we also have high-quality metabolic reconstructions—Helicobacter pylori, a gram-negative bacterium [20], Staphylococcus aureus, a gram-positive bacterium [22], and Mycobacterium tuberculosis, an acid-fast gram-positive bacterium [23]. The rationale for choosing a small number of species for model building and validation is that lower quality reconstructions are likely to have significant gaps that could incorrectly bias the determination of the model. Additionally, the aggregate set of nutrients available for these reconstructions is of 352 nutrients, which cover all nutrient types and 90 out of 97 pathways available in KEGG [33]–[35], thus ensuring that our model is comprehensive. We believe that the development of a systems-level model of microbial metabolism is not only complementary to current approaches but offers some advantages with respect to them. Specifically, while FBA run on metabolic reconstructions already has the capability of predicting maximum biomass yield, our model has the advantage that since it does not consider microscopic details of the metabolism of a species it is directly applicable to any other organism growing under the conditions of our analysis. In fact, we show that solving over a thousand linear equations under constraints can be well approximated by a simpler model whose principles are easier to understand. As such, our model offers the possibility of uncovering universal features of the metabolism of organisms that other computational approaches are not capable of. The mathematical model we develop is thus a valuable tool from both fundamental and applied perspectives, since it can help understand the metabolism of organisms for which a metabolic reconstruction is not available, or guide the process of validating metabolic reconstructions.
The first step is to build a model that predicts whether an individual nutrient can be used as a source of carbon by species (see Fig. 1). If it can, we say that belongs to the group of nutrients that contribute to growth in; otherwise we say that belongs to the group of nutrients that do not contribute to growth in species. We use flux balance analysis on the species in the training set to empirically determine which nutrients belong to and which ones to (see Materials and Methods). We then use these data to build the model. In this section we describe the model (which we summarize in Fig. 2) and validate it with the species in the test set. The next step is to determine for those nutrients that produce growth what is the maximal production of biomass when that nutrient is the only source of carbon. In fact, if a nutrient is the carbon-limiting source in a given medium, the biomass yield of the nutrient must be related to the number of carbons in the nutrient. In Fig. 6, we display the in silico biomass production and the biomass yield of all nutrients as a function of the number of carbon atoms they contain, for the species in the training and test sets (we do not consider M. barkeri in our model for biomass yield because M. barkeri can only grow anaerobically, and this has a significant effect on the energy used to polymerize the proteins and nucleic acids in the biomass). It is visually apparent from Fig. 6 that there is a strong correlation between the number of carbons in a nutrient and the biomass production. We thus model the biomass production induced by nutrient as (3) where is the (nutrient-independent) average biomass yield of the nutrients in The data suggests the existence of an upper bound for the biomass yield () as a function of the number of carbons (blue line in Figs. 6,7, and 8). The nutrients frequently found at or close to this upper bound are sugars, alcohols, and other compounds with hydroxyl groups. Many of the hydroxyl groups on these compounds are typically oxidized in order to reduce NAD to NADH, which is an important source of energy, and will in turn increase the biomass yield. For simplicity, we obtain as the average biomass yield for the sugars uptaken by an organism. The average nutrient has of the efficiency of these high-efficiency sugars, but nutrients vary wildly in their yield (Fig. 7). We consider one main factor that contributes to this variation in biomass yield: the number of carbons in a nutrient that are effectively catabolized. Finally, we want to model the maximal biomass production when there is more than a single source of carbon present in the medium, or in other words when organisms grow in a complex medium. We consider complex media in which nutrients are restricted to five classes: sugars, fatty acids, amino acids, purines, and pyrimidines, partly because they are commonly used in growth rate/biomass yield experiments, and partly because it simplifies the analysis of the results. In addition, complex purines and pyrimidines are a mixture of sugars and nucleobases, and as we are already considering sugars, we only use the simple nucleobases. The simplest plausible model for the contribution of nutrients to the biomass production is one in which each nutrient has an independent contribution to the biomass production. For each species, we calculate the biomass production on a complex medium containing nutrients using (5) We estimate for a nutrient from class using the average yield for that class in the organisms in the training set (6) where is the number of nutrients of class that are taken up by species, and represents the average over species in the training set. For tryptophan and for purines, we use the effective number of carbons when calculating yield, as described previously. To test this model, we randomly generate ensembles of complex media as described in the Data and Methods section. For each medium and for each species, we calculate; we use FBA to find the actual in silico biomass production of the organism on the complex media. In Fig. 9 we show how compares to for each species, and for each of the complex media we generate. In these comparisons we also include predictions for the species in the test set (with the exception of S. aureus, for whose reconstruction it is not clear what units were used for the biomass production). We find a very strong correlation (using the Spearman rank correlation coefficient) between the predictions of our model and in silico experimental results for training set species (E. coli, B. subtilis, and S. cerevisiae) and test set species (H. pylori and M. tuberculosis). This indicates that the model accurately captures which media will result in faster/slower production of biomass. For each species, however, the model systematically under or over-predicts growth. A regression of the predicted biomass production versus experimental in silico production indicates that, with: for E. coli, for B. subtilis, for S. cerevisiae, for H. pylori, and for M. tuberculosis. The parameters used to make the predictions for the individual nutrients in the linear model were trained on three species. The linear model consistently under-predicted the in silico biomass production for these three species, and more so for media containing more nutrients. This is a strong indicator that the nutrients that are uptaken are used synergistically by the in silico organism to produce more biomass than expected, showcasing the effect of catabolic pathways being highly connected in the metabolic network. A more complete model for predicting biomass production in complex media will therefore need to take into account synergistic interactions among catabolic pathways. Our model sheds light on several questions related to the impact of nutrients on the biomass production of microbes. Our approach treats microbial metabolism as a “black box” that uses nutrients to reach optimal biomass production. Because the model does not take species-specific details into consideration, it is useful for generating predictions for any microbe, something that would be impossible with any existing modeling approach. To illustrate how one would proceed to extrapolate to “new” organisms and to show what kind of insights one could obtain, we generate predictions for four organisms that lack a metabolic reconstruction: R. palustris (a gram-negative bacterium), L. monocytogenes (a gram-positive bacterium), D. discoideum (an eukaryote), and T. acidophilum (an archaeon). We find that, overall, these species are predicted to take up fewer nutrients than the species in the test set (Fig. 10A). This is a consequence of the limited annotation that the authors of TransportDB could use for the predicted protein transporters. One example of this limitation is that for all four species, there were many transporters predicted to take up amino acids, but there was no indication of which amino acids the transporters were specific for. Therefore, for the sake of prediction, we consider that all four species uptake all twenty natural amino acids. We then use the model of catabolic potential to predict whether each of these nutrients could be a source of carbon. In Fig. 10B we show the number of nutrients that belong to one of the nutrient classes previously described and whether these nutrients are or. For the fatty acids, none of which were predicted to be uptaken by any of the four species, we examined the enzymes available and found that only D. discoideum contains the enzymes for -oxidation, and could therefore catabolize fatty acids. Finally, we combine our models of biomass yield of nutrients, and of biomass production on complex media (Fig. 11). We choose four complex media which contain a different number of the same nutrients we used for the randomly-generated complex media, namely: sugars, fatty acids, bases, and amino acids. The four media contain: 1) glucose; 2) glucose and hexanoic acid; 3) glucose, hexanoic acid, adenine, guanosine, cytosine, and thymine; 4) glucose, hexanoic acid, adenine, guanosine, cytosine, and thymine and the twenty natural amino acids. We estimate the biomass production in the same manner that we described earlier for the randomly-generated complex media. We find that the biggest influence on the biomass production is the number of carbons available in the nutrients present in the medium; because we are now adding 20 amino acids to medium 4, the biomass production increases almost -fold, on average, in that case. Note that these predictions for biomass production are based on the biomass yield per carbon available in the nutrients present in the medium. Biomass yield is a time independent quantity that cannot be directly associated to growth rate. However, because biomass yield gives an upper limit for growth rate [48], our model can be used as a baseline for researchers to explore and model growth rates.
Microbes use nutrients found in their environments to grow. Understanding and developing quantitative models of this process is of fundamental importance in cell biology, physiology, medicine, evolution, synthetic biology and bioengineering, not to mention of practical importance to those that need to grow microbes in laboratories. Given the diversity of the microbial world and the number of combinations of nutrients available in their environments, this seems too difficult a problem. In this study, we focused on how microbes might catabolize nutrients to obtain carbon for biomass production. Our model comprises three levels, with each level building up on the results from the previous one. The first level concerns whether a nutrient will be catabolized. The second level concerns whether all of the carbon in a catabolized nutrient is available for biomass production. The final level incorporates the biomass yield for selected classes of nutrients and enables us to make a prediction on the biomass production of a microbe in a complex medium. To validate our approach, we compare the predictions of the complete model with in silico predictions of growth in complex media of species that are not part of the training set. Our results on these species are excellent predictions of which media will produce more/less growth. Finally, we looked at the biomass yield of sugars, amino acids, purines, pyrimidines, and fatty acids. We found little variation in the yield of these nutrients amongst different species, and were able to postulate a model for the biomass production of a microbe on a complex medium containing any number of these important nutrients. All of these nutrients have separate catabolic pathways, with the exception of some groups of amino acids which share a catabolic pathway. The fact that the in silico microbial biomass production was more than our model predicts indicates that each of these catabolic pathways can work together synergistically to improve the biomass production of the microbe. The ability of our model to predict biomass production on complex media must be balanced by an understanding of its limitations. First, we model microbial metabolism as a black box, and therefore we do not account for absent pathways that biosynthesize some biomass components. The absence of such pathways would require that the corresponding biomass components be made available in the medium used, but our model cannot be used to predict which of these are indeed required. This means we cannot use the model in its current form to predict the minimal medium that is needed for a microbe to grow. Second, microbial species are sometimes identified by the nutrients they take up and excrete. The ability of a microbe to transport such compounds in and out of the cell is largely dependent on the specific protein transporters available. A separate body of work exists, largely in the form of TransportDB [39], and TCDB [49], [50], which will enable the researchers to predict the proteins that transport specific nutrients. We incorporate such predictions of nutrient transport into our model to make predictions for specific species that lack a metabolic reconstruction. Importantly, while knowledge of transporters can enable us to predict which nutrients in the medium can be taken up, we are currently unable to predict which nutrients are excreted as that would in principle require knowledge of the full metabolic network. Third, the stoichiometric method we describe for generating the biomass data cannot be used to predict the growth rate of a microbe because kinetic information is not included. However, our model provides a baseline to which one can add kinetic information. For example, the rate-limiting step in poor growth media is likely to be the rate at which a microbe takes up nutrients. Therefore one can use the kinetics of nutrient transport with our prediction of biomass yield to predict growth rate. All this notwithstanding, we believe that our approach and models open the door to significant advances in the quantitative modeling of microbial metabolism, and eventually of the metabolism of more complex organisms. In particular, our models could be extended to consider whether a nutrient acts, not only as a source of carbon, but also as a source of nitrogen or energy, or directly as a component of the biomass. Our models could also be extended to include more detailed information about pathways, or to consider functional metabolic modules [51]–[53] instead of pathways.
Stoichiometric methods have been widely used in metabolic engineering for over 20 years [59], the most used of which is Flux Balance Analysis (FBA) [26], [27], [60]. FBA aims at determining the fluxes through each one of the metabolic reactions in an organism. Thus, FBA relies on the determination of a stoichiometric matrix that represents all the reactions and metabolites in an in silico organism, and a vector of uptake fluxes. In the matrix, each row corresponds to a reaction, and each column to a metabolite. is the stoichiometric coefficient of metabolite in reaction. The vector of uptake fluxes can have a non-zero entry for every transport reaction that moves a nutrient into the in silico organism. As a simple example, consider a metabolic network with three reactions: (7) where represents the transport reaction for uptaking Glucose from the environment. The stoichiometric matrix for metabolic network (7) is: (8) Assuming a steady-state concentration of every metabolite and requiring mass-conservation we must impose that (9) where is the set of fluxes through each metabolic reaction. In order to solve Eq. (9), we need to provide the vector. The default flux for each uptake reaction is, meaning that the nutrient is not in the medium or that it cannot be uptaken by the organism. We set if nutrient is present in the medium and could be taken up by the organism. The system in Eq. (9) has many solutions. In our analysis, the biologically relevant metabolic state is the one that maximizes biomass production [59]. The problem of finding the fluxes through the reactions with a cost function that have to satisfy a number of constraints is a standard linear optimization problem that can be numerically solved using the subroutines provided in GLPK [61]. In order for us to model whether a nutrient can be a source of carbon and the biomass yield of the nutrient, we control the data in two ways. First, for some organisms the uptake of specific nutrients can lead to without being considered significant [8]. Secondly, ATP hydrolysis is integrated into the biomass of the in silico organisms, and is typically trained to better match empirical results based on growth rates. We are not exploring growth rates and we thus adjust the ATP hydrolysis in the biomass of the in silico organisms in order to better support the conclusions we reach. These controls are explained in detail in Text S1. We tested our model on a large number of complex media generated using the nutrients available for uptake in the in silico organisms. Because the number of possible combinations of these nutrients was too large for us to test each one computationally, we considered an ensemble of 1000 randomly generated complex media. For each complex medium, every nutrient was made available for uptake with the probability, and was excluded with the probability. We generated ensembles for, , and (for a total of 3000 random media). Sugars present an unusual case because a microbe such as E. coli has been known to exhibit diauxic growth [65]–[67]. This means that microbes regulate sugar uptake so that, despite having various sugars present in the medium, they will only take up one sugar at a time. Our model does not take gene regulation into account, and therefore we manually limited the number of sugars presents in each complex medium to one, specifically glucose.
The ability of microbial species to consume compounds found in the environment to generate commercially-valuable products has long been exploited by humanity. The vast untapped diversity of microbial species offers a wealth of potential resources. However, little is known about most microbial species. While the metabolic network of an organism can be studied to find its nutritional requirements, we lack a reliable metabolic reconstruction for most species. We use in silico organisms to systematically explore whether an arbitrary nutrient can stimulate growth as a single source of carbon, and how effectively it can be used by the organism. We find that we can predict whether a nutrient is a source of carbon and the biomass yield of that nutrient with a simple model that transcends the diversity of species and their environments. Our model for catabolic potential can therefore be used as a baseline model for any microbe for which we lack a metabolic reconstruction.
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Phenomenological Model for Predicting the Catabolic Potential of an Arbitrary Nutrient