PRUNING GENOME-SCALE METABOLIC MODELS TO CONSISTENT AD FUNCTIONEM NETWORKS

Transcription

1 309 PRUNING GENOME-SCALE METABOLIC MODELS TO CONSISTENT AD FUNCTIONEM NETWORKS SABRINA HOFFMANN ANDREAS HOPPE HERMANN-GEORG HOLZHÜTTER Institute of Biochemistry, Medical Faculty of the Humboldt University, Charité, Monbijoustr. 2, Berlin, Germany Metabolic networks represent a set of reactions and associated metabolites that may occur in a given cell or tissue. They are frequently reconstructed from pure genomic data without thorough biochemical validation. Such genome-scale metabolic networks may thus either lack relevant or contain non-existent reactions and metabolites. Filling gaps and removing falsely predicted reactions can be a cumbersome procedure. On the other hand, using the network to build mathematical models addressing a specific problem (e.g. analyzing changes in the level of cellular ATP at substrate depletion) it may turn out that the network comprises more reactions and metabolites than actually needed or, on the contrary, that essential reactions are missing. Therefore, we propose a method to prune the whole network to a smaller sub-network which contains no dead ends and blocked reactions, i.e reactions that may neither proceed in forward nor backward direction. Inspection of this reduced network reveals its actual functional capabilities in terms of producible metabolites. We apply our method to a genome-scale metabolic network of E. coli. Depending on the choice of the exchangeable metabolites, composition of the external medium, and type of thermodynamic constraints we obtain different reduced network variants that may serve as a basis for flux balance models. Keywords: FBA; flux balance analysis; ijr904 Escherichia coli; genome-scale metabolic model; consistency; minimal flux mode; MinMode; flux minimization. 1. Introduction The process of metabolic network reconstruction from genomic information has significantly advanced and the number and size of such published networks increased considerably during the last few years [2, 4, 5, 16, 17, 19]. While traditional metabolic modelling implied a step by step construction of a specific pathway [18, 22], the so-called genome-scale metabolic models are reconstructed by a top-down procedure. Such genomic reconstructions exploit sequence homologies to genes of enzymes and membrane transporters already known in other cell types. Organism specific collections of automatically assigned biochemical reactions on this level can be downloaded from resources such as KEGG [26] or BioCyc [24]. For

2 310 S. Hoffmann et al. a detailed review on the reconstruction of microbial genome-scale metabolic models the reader is referred to Francke et al. [6]. As opposed to the traditional bottomup models the purpose of genome-scale models is not the exploration of a specific question but rather to collect all available biochemical reactions of a certain cell type. Obviously, for some parts of the network the available knowledge is fragmentary. Assignment of genes to enzymes can be wrong, the expression of certain genes can be inhibited due to DNA methylation, enzymes may catalyze different chemical reactions than already reported for other cell types, or important metabolic genes may have escaped identification because of too large dissimilarities of sequences, to name a few reasons for errors in genome-wide reconstructed metabolic networks. Despite of these errors genome-scale metabolic models, especially the manually curated ones, contain a complete and correct description of many cellular functions which may serve as model objectives. Here, we propose an approach to identify these complete parts and to elucidate the functional capabilities of a given genomescale metabolic model. To this end we apply a two-step strategy which consists of pruning the network of all blocked reactions, i.e. reactions that may neither proceed in forward nor backward direction, followed by the determination of its synthesizing capacity given by the set of metabolites for which a net synthesis is possible. We demonstrated this procedure with a genome-scale metabolic network of E. coli. Furthermore, we imposed different constraints on the composition of the external medium, exchangeable metabolites, and flux directionalities and found remarkable differences in the size and synthesizing capacities of the resulting metabolic models. 2. System and Methods Definition of the FBA model. Because of sparsely known enzyme-kinetic details, constraint-based modeling currently represents the method of choice to analyze large-scale metabolic models. Basically, this method requires only knowledge of the network topology, which can be described mathematically formalized and compact by the stoichiometric matrix S, an m n matrix where m corresponds to the number of metabolites (rows), n to the number of reactions (columns) or fluxes. Its positive or negative elements S ij specify the amount of metabolite M i formed or consumed in reaction j, respectively. Neglecting the spatial distribution, the time-dependent change of the metabolite concentrations is determined by the kinetic equation system: d[m i ] dt = j S ij v j b i, (1) where v j refers to the flux rate of the j-th reaction, [M i ] denotes the concentration of metabolite M i, and b i refers to the unspecific metabolic use of metabolite M i not covered by the chemical reactions in the considered network. A common assumption of constraint-based modeling approaches is the so-called flux balance constraint that assumes a steady-state behavior, i.e. the metabolite concentrations remain constant

3 Pruning Genome-Scale Metabolic Models 311 over time: S ij v j b i = 0 i N m (2) j No metabolic use (b i = 0) will be referred to as strict steady state condition for metabolite M i. In biological systems, this strict assumption is valid if the characteristic time constant for changes of the metabolic output is much larger than the time constant of metabolic conversions. As this condition is actually not met even under conditions of cellular growth (because the amount of all metabolites has to be increased) several studies relaxed the flux balance constraint (Eq. 2) by allowing accumulation of all metabolic species, i.e. assuming b i 0 [12, 13, 15]. Although quite unusual, variable b i might also take negative values and represent a metabolic repository from which metabolites may be fed into the system. Both, the availability of a repository and further metabolic use imply an exchange of the metabolite over systems boundary. This is actually true in both directions for metabolites whose concentrations are large enough to remain constant despite consuming and producing fluxes, e.g. compounds of the external media. Therfore, for these compounds variable b i is unconstraint and no flux balance constraint is applied to these so called external metabolites. Furthermore a flux-balance model has to specify the metabolic output of the network, i.e. a set of metabolites delivered by the network either as cellular building blocks for macro-molecules, degradation products (e.g. of toxic compounds), or exported material that represent metabolic functions. As these metabolites are required by the cell, their Fig. 1. Characteristics of a flux balance model metabolic use has to be greater than zero, e.g. b i = 1. Fig. 1 summarizes the necessary definitions of a flux balance model. In addition, we consider thermodynamic information to constrain the reversibility of reaction directions as described in the next subsection. The definition of an appropriate objective function, i.e. the measure which is assumed as optimized in cellular system, is most crucial for constraint-based optimization analysis. Possible functions include the maximization of biomass production and minimizing the sum of fluxes [10].

4 312 S. Hoffmann et al. Determination of blocked reactions and non-producible metabolites. The method used to determine blocked reactions and non-producible metabolites is based on the concept of minimal flux modes [9]. These modes are minimal flux distributions required either for the net synthesis of a single metabolite, so called metabolite minimal modes (MetabMinMode), or to maintain an unit flux through a given single reaction (either in forward or backward direction) as reaction minimal modes (ReactMinMode). The advantage of this approach lies in its simplicity and extensibility. Here, we applied additional constraints and focussed on model inconsistencies instead of predicting physiological flux distributions. If for a given metabolite the MetabMinMode does not exist the metabolite is called non-producible. If for a given reaction the ReactMinMode does neither exist in forward nor backward direction the reaction is called blocked. It has to be noted that non-existence of a minimal flux distribution excludes the existence of any flux distribution, our criteria for the determination of blocked reactions and producible metabolites are thus sufficiently general. Specification of metabolic use, external metabolites and flux directions. All models used in this study consider only extracellularly located metabolites as external. For the remaining set of internal metabolites two different assumptions on metabolic use are investigated: In the first situation, denoted by (U bio ), metabolic use is assumed (b i 0) only for metabolites serving as biomass precursors according to Reed et al. [19]. All other metabolites are strictly balanced with zero metabolic use (b i = 0). In the second situation (U all ), metabolic use is allowed (b i 0) for all metabolites. In our analysis, we included three external media differing in available carbon sources. Environment E gluc stands for a glucose minimal media (composition and concentrations taken from Henry et al. [8]). A slightly richer medium E rich contains the following carbon sources in addition to glucose used in E gluc : ac, akg, lac-d, lac-l, glyc, mal-l, pyr and succ, each at 0.02M. E cplx represents a complex medium that includes the carbon sources of E rich plus a larger number of other exchangeable species as defined by Reed et al. [19]. All three media contain unlimited oxygen, phosphate, sulfate, nitrogen, potassium, iron, and sodium. The concentrations of the external metabolites are given in the supplementary material. Three different variants of restrictions on flux directions are considered. In the first variant, denoted with R all, no restrictions are imposed on the directions, i.e. all fluxes may proceed in both forward and backward direction. In the second variant, denoted with R irr, fixed heuristic reversibility constraints as proposed in the original ijr904 model [19] are used. In addition, as proposed by Reed et al. [20], for 17 reactions a forward and backward directions are constrained to prevent thermodynamically infeasible cycles. In the third variant, denoted with R tr, no a priori a Reactions: VPAMT, ALARi, LCADi, ACCOAL, GALUi, ADK, CYTDt2, ABUTt2, GLUt4, INSt2, ADNt2, PROt4, SERt4, THMDt2, THRt4, URAt2, URIt2

5 Pruning Genome-Scale Metabolic Models 313 assumptions on the reversibility of reactions are made. Instead, MetabMinModes and ReactMinModes are calculated with the constraint of thermodynamic realizability (TR) [11], i.e. metabolite concentrations are restricted to physiologically feasible ranges and must be determined such that the changes of Gibb s free energies ( G r ) j are consistent with flux directions. The Gibb s free energy depends on the standard Gibb s free energy ( ) G 0 r j and active concentrations [M i] of the reactants by the formula [1]: ( G r ) j = ( G 0 r )j M + RT ln [M i ] S ij, (3) where R is the gas constant, T the temperature, and S ij the stoichiometric coefficient of metabolite M i in the j-the reaction. The linear problem to be minimized reads as follows: minimize N v j j=1 subject to v j 0 j N n, i=1 N S ij v j b i = 0 j=1 i N m if the k th MetabMinMode: if the r th ReactMinMode: b k = 1 v r = 1 if U bio : if U all : b i = 0 i / M biomass b i 0 i N m b i 0 i M biomass if R tr : if R irr : 0 v j + αd j α, j N n 0 ( G r ) j + αd j α, j N n v j = 0 j R irr ( G r ) j = ( G 0 r )j + n i=1 cs ij i c max i c i c min i, i N m where n is the number of reactions and m the number of metabolites; for any 1 j n, d j is a binary variable; c i = RT ln([m i ]) is a coefficient calculated from the active concentration of metabolite M i, c max i and c min i are minimal and maximal values related to a realistic concentration range of metabolite M i ; α is set to a positive number which is larger than any possible flux value and larger than any possible Gibb s energy value, and it can easily be shown that the constraints 0 v j +αd j ) α and 0 ( G r ) j +αd j α are equivalent to v j 0 sgn(v j ) = sgn (( G r ) j. Physiological concentration ranges were available for 22 internal metabolites (given in Kümmel et al. [14]) and 10 external metabolites (given in Henry et al. [8]). For the other metabolites generic concentration bounds were used based on typical cellular concentration ranges reported in the literature: 5 µm 2 mm. Standard Gibb s free energies computed by Henry et al. [8] are used.

6 314 S. Hoffmann et al. Definitions: Metabolic network Flux-balance model Metabolic output Non-producible metabolite Blocked reaction Set of metabolites and reactions combined by the stoichiometric matrix A metabolic network combined with definitions of exchangeable metabolites, metabolic output, declaration of irreversible reactions, and further constraints that are optimized towards a flux objective to predict steady-state flux distributions A metabolite whose production is an essential metabolic function. A metabolite for which net production is not possible by any flux distribution with respect to given model assumptions determined by the computation of a MetabMin- Mode Reaction that may neither proceed in forward nor in backward direction with respect to given model assumptions determined by the computation of a ReactMin- Mode 3. Results We based our studies on the genome-scale metabolic network ijr904 of E. coli [19] which already has been analyzed in several studies [7, 20, 23, 25]. The network encompasses 931 reactions and 618 metabolites. The problem is that this network contains a large number of blocked reactions and non-producible metabolites. For example, 408 blocked reactions were reported for a flux model of a previous network (ije660a) when biomass production is maximized under aerobic growth conditions in a glucose minimal medium [3]. The fact that such a large part of the network is comprised of disabled reactions hampers an unbiased statistical analysis of flux distributions, e.g. when analyzing the impact of enzyme mutants. Therefore, we pruned the network to its consistent core by eliminating all blocked reactions based on the calculation of ReactMinModes for all reactions of the network (see Table 1. Numbers of blocked reactions for different environments, exchangeable metabolites, and reversibility constraints. R all R irr R tr U bio U all U bio U all U bio U all E cplx E rich E gluc E zero subsection 2). In total, we constructed 18 sub-networks from the original ijr904 network using different constraints on exchangeable metabolites, environment (medium composition) and flux directions explained in subsection 2 and summarized in Fig. 3. The large impact of the various constraints on the number of blocked reactions is depicted in Table 1. The minimal number of blocked reactions is 20 if metabolic use

7 Pruning Genome-Scale Metabolic Models 315 is assumed for all metabolites (case U all ), the direction of fluxes is not restricted (R all ), and the cells grow in a complex medium (E cplx ). In contrast, 377 reactions are blocked if metabolic use is restricted to precursor metabolites for biomass production (case U bio,r irr,e gluc ). Removal of blocked reactions does not affect reactions involved in internally balanced reactions. Internal cycles that are thermodynamically infeasible are excluded automatically if the calculation of the ReactMinModes is performed under the TR constraint and the principle of micro-reversibility is taken into account [11]. For other types of thermodynamic constraints on flux directions, the internally balanced cycles can be identified by ReactMinModes that remain if the concentration of all external substrates is put to zero (case E zero, see last row in Table 1). Intriguingly, for the fully reversible network there are 367 = reactions belonging to internal cycles. Blocked reactions were removed from the original model by canceling the respective column in the stoichiometric matrix S. The synthesizing capacity of these reduced networks is given by the total number of producible metabolites, i.e. metabolites for which a MetabMinMode can be found (see subsection 2). Fig. 2 illustrates the Fig. 2. Producible and non-producible metabolites for complex metabolic input (Ecplx) synthesizing capacities of the original, complete network (cpl) and the various reduced networks (red) for cells growing in a complex medium (E cplx ). The weaker the applied constraints, the higher the number of producible metabolites. In U all there is no difference of producible metabolites between the reduced and the original

8 316 S. Hoffmann et al. model, whereas for U bio the number of producible metabolites is lower in the reduced model. This decrease results from the strict flux balance constraint for non-biomass producing metabolites assumed for U bio. In contrast to the calculation of the ReactMinModes, for the calculation of a Metab- MinMode, metabolic use is assumed for the corresponding metabolite to drain it out of the network. Therefore, blocked reactions always include non-producible metabolites although reactions including non-producible metabolites are not necessarily blocked. If synthesis of a metabolite is not possible itself, subsequent reactions may only proceed if a reaction exists that regenerates the consumed non-producible metabolite. If this is the case, a so-called regeneration cycle exists and the reactions of which may carry a non-zero flux and thus will not be identified as blocked. For example, the reduction of ATP (atp) to its desoxy form (datp) is driven by an oxidation of the cofactor thioredoxin (trdrd/trdox): RNTR1: TRDR: atp + trdrd --> datp + h2o + trdox h + nadph + trdox --> nadp + trdrd The oxidized as well as the reduced form of thioredoxin are non-producible. However, there exists a NADPH (nadph) dependent thioredoxin reductase that catalyzes the regeneration of thioredoxin (TRDR) and thus enables a flux through RNTR1. Therefore, after elimination of blocked reactions the reduced model only comprises nonproducible metabolites involved in such regenerating cycles that may contribute to the synthesis of other metabolites while lacking an own de novo synthesis. The Fig. 3. Overview over the obtained results with n and m denoting the total number of reactions and metabolites, respectively, m stands for the number of metabolites that have been removed from the original model and # indicates the number of non-producible metabolites.

9 Pruning Genome-Scale Metabolic Models 317 number of metabolites removed from the original network and the remaining number of non-producible metabolites for the 18 considered different combinations of constraints among others are depicted in Fig. 3. It has to be noted that the 49 biomass relevant metabolites are producible by all reduced network variants. However, in none of the reduced networks all metabolites are producible. For the variant with assumed metabolic use of all metabolites (U all ), complex substrate composition (E cplx ) and heuristic a priori reversibility constraints (R irr ) in total 64 nonproducible metabolites have been removed and only 15 of the remaining 554 metabolites are non-producible. As reasoned above, these 15 metabolites must be involved in regenerating cycles, which according to the applied reversibility constraints, are thermodynamically feasible. Determination of moiety conservation relations [21] of the reduced stoichiometric matrix allows to group these 15 non-producible metabolites into three sets containing: 1) the acetyl carrier protein (ACP) and metabolites coupled to this protein b, 2) the reduced and oxidized form of thioredoxin (trdrd and trdox) and 3) the loaded and unloaded form of L-Glutamyl-tRNA (trnaglu and glutrna). Hence, there are actually only three metabolites (for example ACP, trdrd, and trnaglu) which, if producible, would render all metabolites of the model producible. However, without any further modifications the cellular functionality, in terms of metabolic output by this model variant, can be described by a combination of the 554 producible metabolites. The model above lacks 33 metabolites present in the respective model without constraints on reversibility (E cplx, U all, R all ). To determine which metabolites could additionally be produced if one reversibility constraint is relaxed, we investigated the synthesizing capacity of the completely reversible model, modified by dropping the reversibility for reaction r i (E cplx, U all, R all \ {i}). Iteration over all reactions assumed to be irreversible in R irr showed 23 reactions whose irreversibility blocks the synthesis of exactly one metabolite. We modified the model to allow these 23 reactions to proceed in both directions by removing their reversibility constraint. After this modification 9 additional metabolites can be produced by the model. 4. Discussion We propose a method to prune a given network to its consistent core for flux-balance analysis. We defined the consistency of a flux-balance model by requiring that (i) all metabolites forming the studied metabolic output of the model are producible under steady-state conditions and (ii) a non-zero flux through all reactions is possible. An even more rigorous criterion for consistency of a flux-balance models based on genome-scale networks had been proposed by Kumar et al. [15] demanding producibility of all metabolites in the cytosol, as well as producibility and degradability of all metabolites in intracellular compartments other than the cytosol. However, b acacp, actacp, palmacp, myrsacp, hdeacp, malacp, ddcaacp, tdeacp, octeacp and 3hrmsACP

10 318 S. Hoffmann et al. the idea behind this criterion is that during growth the molar amount of every metabolite increases as the cellular volume increases and the concentrations remain constant (dilution effect). This criterion is useful if the knowledge presented by genome-scale metabolic networks is considered to be complete. In the context of a specific question, e.g. biomass production, producibility of a relevant set of metabolites is sufficient. By removing reactions and non-producible metabolites according to the specific constraints formulated in the corresponding flux-balance model we reduced the original metabolic network of E. coli to a core network capable of providing a metabolic output in accordance with the defined exchange processes in the original network. This is demonstrated by the fact that irrespective of the applied constraints the reduced network still allows for the synthesis of the 49 metabolites required for biomass production the functional output whose investigation motivated the compilation of the ijr904 network. However, metabolic functions other than biomass production may be of interest as well. The full spectrum of possible metabolic functions requiring a net synthesis of output metabolites can be directly inferred from the synthesizing capacity of the pruned network. As demonstrated for the E. coli network this synthesizing capacity depends strongly upon the constraints placed to the substrate composition of the extracellular space, directionality of the fluxes, and exchangeability of metabolites. Our calculations have shown that under certain constraints only three metabolite sets of the E. coli network remain for which producibility has to be somehow enabled in order to render all metabolites producible. These are the acetyl carrier protein, thioredoxin (in reduced or oxidized form) and L-Glutamyl trna (loaded or unloaded with the amino acid). Obviously, these are compounds that do not represent metabolites in a strict sense and thus should not be included into the metabolic network. This issue shows a weak point in the definition of metabolic networks, a definition which usually is confined to chemical compounds having a lower molecular weight of several hundreds of Dalton. On the other hand, gene regulatory or signaling networks are composed of interacting proteins and nucleic acids also without explicitly taken into account the synthesis and degradation of these macromolecules. The reconstruction of cellular reactions networks modelling the turnover of high-molecular weight compounds as proteins, nucleic acids and phospholipids is currently no man s land. To overcome this situation is certainly a great challenge for future work in computational systems biology. References [1] Atkins, P. W. and De Paula, J., Atkins Physical Chemistry,Oxford University Press, [2] Becker, S. A. and Palsson, B. O., Genome-scale reconstruction of the metabolic network in Staphylococcus aureus N315: an initial draft to the two-dimensional annotation, BMC Microbiol, 5(1):8, [3] Burgard, A.-P., Nikolaev, E.-V., Schilling, C.-H., and Maranas, C.-D., Flux coupling

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