A metabolism is a complex network of chemical reactions. This network synthesizes multiple small precursor molecules of biomass from chemicals that occur in the environment. The metabolic network of any one organism is encoded by a metabolic genotype, defined as the set of enzyme-coding genes whose products catalyze the network's reactions. Each metabolic genotype has a metabolic phenotype. We define this metabolic phenotype as the spectrum of different sources of a chemical element that a metabolism can use to synthesize biomass. We here focus on the element sulfur. We study properties of the space of all possible metabolic genotypes in sulfur metabolism by analyzing random metabolic genotypes that are viable on different numbers of sulfur sources.We show that metabolic genotypes with the same phenotype form large connected genotype networks - networks of metabolic networks - that extend far through metabolic genotype space. How far they reach through this space depends linearly on the number of super-essential reactions. A super-essential reaction is an essential reaction that occurs in all networks viable in a given environment. Metabolic networks can differ in how robust their phenotype is to the removal of individual reactions. We find that this robustness depends on metabolic network size, and on other variables, such as the size of minimal metabolic networks whose reactions are all essential in a specific environment. We show that different neighborhoods of any genotype network harbor very different novel phenotypes, metabolic innovations that can sustain life on novel sulfur sources. We also analyze the ability of evolving populations of metabolic networks to explore novel metabolic phenotypes. This ability is facilitated by the existence of genotype networks, because different neighborhoods of these networks contain very different novel phenotypes.We show that the space of metabolic genotypes involved in sulfur metabolism is organized similarly to that of carbon metabolism. We demonstrate that the maximum genotype distance and robustness of metabolic networks can be explained by the number of superessential reactions and by the sizes of minimal metabolic networks viable in an environment. In contrast to the genotype space of macromolecules, where phenotypic robustness may facilitate phenotypic innovation, we show that here the ability to access novel phenotypes does not monotonically increase with robustness. In any biological system, genotypes contain the information needed to make phenotypes. The relationship between genotype and phenotype is also known as a genotype-phenotype map . The abipopulations of networks "discover" metabolic innovations in metabolic genotype space. Specifically, we are interested in how the rate of discovery depends on the robustness of a metabolic system. This robustness indicates a metabolic network's ability to preserve its biosynthetic capacity upon random removal of reactions. Previous work on macromolecules showed that the robustness of a molecule's phenotype to mutations can accelerate the rate at which evolving populations discover new phenotypes [In this work we concentrate on the genotype-phenotype maps of metabolic networks involved in the utilization of sulfur. We have two aims. Aim 1 is to examine how general earlier observations about the genotype-phenotype map of carbon metabolism are ,6. We doenotypes ,8. We wi4+ to the reduced state S2-. This versatility in oxidation state may explain the diversity of sulfur metabolism and why it is involved in both anabolism as well as catabolism. In catabolism, depending on the environment, sulfur can be used as an electron acceptor or an electron donor, and in some cases even both as donor and acceptor. In anabolism, sulfur must first be reduced in a sequence of energetically expensive steps before being incorporated into biomass [Carbon metabolism comprises so many reactions that the computational demands of studying population processes in its genotype space are too high for current computational technology. Sulfur metabolism, in contrast, comprises a smaller number of chemical reactions, which renders the computational analysis of population processes more tractable. Despite being involved in fewer reactions, sulfur is no less essential to biological organisms than other elements, such as carbon or nitrogen. Sulfur is a versatile and integral element in the biochemistry of organisms ,10. Its  biomass .Sulfur is present in two major constituents of biomass, the amino-acid cysteine, which confers stability to proteins through disulfide bonds, and the amino acid methionine, which is the first amino acid of many proteins. Sulfur is also a part of S-adenosylmethionine . This compound is a cysteine metabolite that is a major methyl donor to the methyl carrier metabolite tetrahydrofolate, which is indispensable for amino acid synthesis, and for the methylation of biomolecules. Furthermore, sulfur is the active element in coenzyme-A, an acyl carrier metabolite involved in the calvin cycle and in lipid synthesis. Sulfur is also present in the active core of iron-sulfur proteins, which are involved in a number of important reactions. Examples include nitrogenase, which enables the fixation of nitrogen, and hemoglobin, which enables the transport of oxygen. Another prominent molecule involving sulfur is glutathione, a peptide responsible for protection against oxidative stress in cells.We next outline the order of our analyses in the Results section. First we introduce two concepts that allow us to estimate some properties of genotype space organization. The first is that of a minimal metabolic network. This is a metabolic network from which no reactions can be removed without destroying its viability in a given environment, that is, its ability to synthesize all essential biomass molecules. The second concept is that of a superessential reaction. For our purpose, a superessential reaction is an essential chemical reaction that occurs in all minimal networks. After these preliminary analyses, we demonstrate the existence of long phenotype-preserving paths through metabolic genotypes space that allow exploring this space through many single phenotype-preserving mutations (aim 1). The maximum length of these paths and metabolic network size can be estimated and varies linearly with the number of superessential reactions. We show that the robustness of metabolic networks depends both on their size and on the average size of minimal metabolic networks viable on a given number of sulfur sources. Next, we show that the existence of neutral paths allows evolving metabolic networks to encounter an increasing number of novel phenotypes (aim 2). We finally explore the relationship between robustness and a population's ability to access novel phenotypes through changes in a network's reactions. In contrast to macromolecules, where robustness may facilitate phenotypic innovation ,8, we figenotype as the set of biochemical reactions that may take place in an organism, and that are catalyzed by gene-encoded enzymes. The set of all reactions used in this work is a subset of 1221 reactions out of 5871 reactions we curated previously [We follow an approach taken in a previous study of large-scale metabolic networks . We defieviously . These ri in this vector corresponds to a reaction. Its values ('0') or ('1') at position i indicate the inability or ability of the organism to catalyze the corresponding reaction  that allow the network (metabolic genotype) to synthesize all biomass components, if one of the sulfur sources is provided as the sole sulfur source to the organism. We represent this phenotype as a binary vector of length 124 whose entry at position i indicates viability if sulfur source i is the sole sulfur source . Specifically, we generated 100 minimal networks for 10 random sets of sulfur sources with the same number S - but not necessarily identity - of sources. We note in passing that such networks often also happen to be viable on additional sulfur sources that we did not require them to be viable on  where it ranges from avg D= 0.2 to avg D= 0.8, than for metabolic networks subject to many environmental demands (S = 60) where it ranges from avg D= 0.2 to avg D= 0.4.We generated 1000 random minimal metabolic networks viable on a given number min Nas a function of the number of sulfur sources they are viable on. It ranges from min N= 14 reactions for S = 1 to min N= 87 reactions for S = 60. By definition, all reactions in a minimal network are essential, but some of these reactions are special because they occur in all minimal networks viable on a given set of sulfur sources. We call these reactions superessential reactions [S on a network. The number of superessential reactions SE Rincreases with S, but it is generally much lower than the total number of reactions in a minimal metabolic network. For example, at S = 1, 4 out of 14 reactions are superessential. At S = 60, 44 out of 87 reactions are superessential. We will show that the number of superessential reactions plays an important role in one of our analyses below.Figure We next extended our previous work on carbon metabolism to ask about the existence of genotype networks in the space of sulfur-involving reactions, and of neutral paths that traverse such networks while preserving a metabolic phenotype. We define a neutral path as a series of mutations (reaction additions or deletions) that leave a phenotype intact Figure . We emphN of reactions in a network, and on the environmental demands on the network? To answer these questions, we performed 200 random walks of 10 000 mutations each for metabolic networks of various sizes, and for various environmental demands. Specifically, for networks of each size we performed 20 random walks for each of 10 different sets of S of sole sulfur sources that we required the network to be viable on. Each random walk started from a random initial viable metabolic network comprising N reactions (see methods for details). We allowed N to vary by no more than one reaction during the random walk. Moreover, each step in the random walk had to preserve viability. Finally, none of the steps was allowed to decrease the distance to the starting network, in order to maximize the distance from this network (see methods for details).We are especially interested in two questions. How far does a neutral path typically lead through genotype space? And how does this distance depend on the number max Dobtained in such random walks for networks up to 300 reactions, where we required viability on S=1, 5, 10, 20, 40, or 60 different sole sulfur sources. This distance is in general large. For example, max Dis greater than 0.7 for all metabolic networks with more than 200 reactions. For each value of S, the data point at the smallest value of N  corresponds to the minimal metabolic networks we discussed earlier. Perhaps surprisingly, these minimal networks can not only be very diverse, as we saw earlier, but neutral paths starting from any one such network can also reach far through genotype space. For example, the maximal length of neutral paths is max D= 0.65 for minimal metabolic networks viable on S = 1 sulfur source, and still a sizeable max D= 0.38 for metabolic networks viable on S = 60 sulfur sources. To provide a point of reference, the E. coli metabolic network has 142 reactions involving sulfur. Random viable metabolic networks of this size would have maximum genotype distances between max D= 0.96 (for S = 1) and max D= 0.60 (for S = 60).Figure We asked whether the maximal genotype distance of networks of a given size, as well as their robustness to reaction removal can be predicted from properties of the underlying minimal networks. The answer is yes.max Dbetween metabolic networks of the same size increases with metabolic network size N. The solid lines show the relationship between the maximal genotype distance max Dand metabolic network size N as predicted by the equationFigure SE Ris the number of reactions that are super-essential for a given environmental demand S. We had estimated this number in our previous analysis of minimal networks , robustness ranges from ρ = 0.9 under low environmental demands (S = 1) to ρ = 0.2 under high environmental demands (S = 60). The relationship between ρ and network size N can be explained by noting that ρ = 1-Ress/N where ess Ris the number of essential reactions. We find that ess Rdecreases linearly with increasing metabolic network size -Nm, In this equation, min Nis the average size of minimal metabolic networks (estimated above for given S) and m is the rate at which the number of essential reactions decreases with increasing metabolic network size  with increasing N.This relationship means that network robustness is a linear function of the ratio S of sulfur sources, and on the genotype networks they form. Long paths through a genotype network can contribute to evolutionary innovation in metabolic phenotypes, if many novel phenotypes can be encountered near such a path. We next asked whether this is the case, and how this number of novel phenotypes depends on environmental demands on a network. We consider a phenotype to be novel if it confers viability in a set of new sulfur sources, in addition to those required by the environmental demands imposed on the metabolic network.Thus far, we have concentrated on the characteristics of individual sets of genotypes viable on a given number We first introduce the notion of a (1-mutant) neighborhood around a metabolic network genotype, which comprises all networks that differ from the genotype by a single reaction Figure . BecauseG and k Gon the same genotype network, where k Gis a metabolic network derived from G through k random reaction changes. That is, we determined the fraction of novel phenotypes that occurred in the neighborhood of only one but not the other network. Below we refer to it as the fraction of novel phenotypes unique to one neighborhood. If this fraction is very small even for large k, then networks in different regions of a genotype space will have mutational access to similar novel phenotypes. Figure S. Figure S on which viability is required. It also does not depend strongly on metabolic network size (results not shown). In sum, the neighborhood of moderately different metabolic networks contains very different novel phenotypes.We asked how different are the novel phenotypes in the neighborhood of two metabolic networks The question of how robustness relates to phenotypic variability has raised considerable interest in recent years ,20. Macrgenotype as its fraction of non-essential reactions. Analogously, we now consider the robustness of a metabolic phenotype as the average fraction of non-essential reactions of all networks with this phenotype [S of sulfur sources on which they are viable can serve as a proxy for phenotypic robustness. The greater a phenotype's S is, the smaller is its robustness.Above we considered the robustness of a metabolic henotype . We showWhen analyzing how evolving populations explore a genotype network, we need to distinguish between two different kinds of populations. The first are populations where the product of population size and mutation rate is much smaller than one. For brevity, we refer to such populations here as small populations. The second are populations where this product is much greater than one. We refer to these as large populations.N and number of sulfur sources S. Specifically, for each N and S, we carried out 200 random walks of 10 000 mutations each . Figure S, indicating that metabolic networks under few and many environmental demands encounter fewer novel phenotypes than under an intermediate number of environmental demands (S≈20). The cumulative number of novel phenotypes depends strongly on metabolic network size for S < 20, where larger metabolic networks encounter more novel phenotypes throughout the random walk. It is not sensitive to N for larger values of S.Small populations are genotypically monomorphic most of the time , and efflarge evolving populations. Such populations are polymorphic most of the time. To model their evolutionary dynamics, one needs to track every individual in the population, unlike for monomorphic populations. We determined the cumulative number of novel phenotypes that are mutationally accessible to a population of metabolic networks evolving on (and restricted to) a specific genotype network. This number can be determined by examining, for each generation, the neighborhood of each individual in the population, and by counting the total number of different novel phenotypes encountered. We simulated populations of 100 individuals evolving for 2000 generations (see Methods for details). Additional file S). Qualitatively, the figure resembles our observations for a single random walk  for low values of S, and it increases with robustness at higher values of S. Additional file Taken together, these observations show that the number of novel phenotypes accessible to a population does not increase monotonically with phenotypic robustness. It decreases with increasing robustness (decreasing S) influences genotypic diversity only weakly, and not in the same unimodal way as seen in Additional file We next examined two candidate explanations of this pattern. The first is that environmental demands and network size affect how rapidly a population can diversify on its genotype network, and thus also how many novel phenotypes it can access. To find out whether this diversification rate matters, we examined the average pairwise genotype distance of our evolving populations. The smaller this difference is, the more slowly a population diversifies. Figure N and S. Figure N and S (20 metabolic networks for 10 different sets of sulfur sources at each S). The vertical axis of this figure shows the mean and standard deviation of the number of unique novel phenotypes in the neighborhood of the examined networks. It shows similar unimodal characteristics as the data in Figure S < 20. In this regime, larger metabolic networks have more novel phenotypes in their neighborhood than smaller networks. For S > 20, the dependency on metabolic network size disappears and the number of accessible novel phenotypes declines again. In sum, the different accessibility of novel phenotypes in evolving populations, at least qualitatively, emerges from how novel phenotypes are distributed in genotype neighborhoods, and how this distribution depends on S and N.The second candidate explanation is that the patterns of Figure maxD. For example, networks with as few as 200 reactions can show max >D0.7, meaning that they share fewer than 30 percent of their reactions. A second similarity regards phenotypic innovations, genotypes whose phenotypes allow viability on novel sulfur sources. The neighborhoods of two genotypes 1 Gand 2 Gtend to contain very different phenotypic innovations, even if 1 Gand 2 Gare only moderately different. Both features, taken together, facilitate the exploration of novel phenotypes. They would allow a population of organisms (networks) to explore different regions of genotype space, preserving their phenotype while exploring many novel phenotypes.The genotype-phenotype map we characterized here shows both similarities and differences to previously characterized such maps -6. One sA major difference to previously studied genotype-phenotype maps regards the relationship between a phenotype's robustness to mutation and a population's ability to explore novel phenotypes. In macromolecules, this relationship appears to be positive: Greater robustness facilitates innovation ,8. AlthoS of sulfur sources on which a network is already viable. If we took this exponential decrease into account, for example by determining the cumulative fraction instead of the number of novel phenotypes accessible to evolving population, this fraction would decrease with increasing S.In sulfur metabolism, we do not see this relationship. Robust phenotypes in this context are characterized by viability on few sulfur sources. They are less easily disrupted through eliminations of individual reactions. We found that the number of phenotypic innovations such phenotypes can access in their neighborhood - through changes of single reactions - is highest at intermediate robustness, that is, for phenotypes viable on approximately 20 out of 60 carbon sources we examined.  This phenomenon cannot solely be explained by the evolutionary dynamics of evolving populations, partly because populations whose members have intermediate robustness do not spread fastest through genotype space. Instead, the phenomenon is a simple consequence of how many novel phenotypes occur in the neighborhoods of individual genotypes. This number peaks for genotypes whose phenotypes have intermediate robustness. It shows the same qualitative dependence on robustness as the number of novel phenotypes accessible to populations. Thus, in this case, population dynamics do not dominate the process of novel phenotype exploration. We note that the total number of possible novel phenotypes decreases exponentially with the number These observations raise the question whether they are unique to sulfur metabolism or whether they occur in other metabolic systems. As we stated earlier, part of our motivation to study sulfur metabolism was to avoid the much larger number of reactions of carbon metabolism, which render population approaches like ours computationally intractable. Nonetheless, very limited analyses for carbon metabolism are possible. Additional file Taken together these analyses show that the organization of different phenotypes in genotype space can differ greatly among different classes of biological systems, such as proteins and metabolic networks. And these differences can affect the ability of a system to explore novel phenotypes in this genotype space.A third class of analyses regards features that have not been studied previously, partly because they are unique to metabolic systems and our representation of them. One of them regards the analysis of networks with different sizes (numbers of reactions). Our genotype representation can accommodate and allows us to compare networks of different sizes, whereas commonly used representations of other systems - molecules or regulatory circuits - cannot. For example, proteins of different length form genotype spaces of different dimensions, making their comparison challenging .When analyzing metabolic networks of different sizes, we found that populations of small networks can explore fewer novel phenotypes Figure . This obAnother analysis focusing on network sizes is our characterization of minimal metabolic networks, networks in which all reactions are essential. While the process of genome and metabolic network reduction leading to small networks has been studied for specific biological networks , our appN and viability on S sulfur sources, the maximum genotype distance within a genotype network is well approximated by one minus the fraction of superessential reactions in minimal metabolic networks. These are reactions found in all minimal networks viable on a given number of sulfur sources. We currently have no mechanistic explanation for this relationship and it, also, remains a subject for future work.Properties of minimal networks are also useful in explaining the maximal genotype distance in a genotype network. For example, for metabolic networks of a given size Studies like ours have several limitations. One of them is that we focus on biomass synthesis phenotypes, and not on other aspects of metabolism, such as secondary metabolite production. The reason is that biomass synthesis has the most immediate impact on an organism's survival. Other limitations include that the addition and deletion of reactions may have effects on fitness even if they do not affect biomass synthesis, that our knowledge of the reaction universe is limited, and that we face uncertainty about the biologically most important sulfur sources, about thermodynamic properties of individual metabolic reactions, and about the role of cellular compartmentalization in guiding sulfur metabolism. Our study, even though it uncovers generic features of genotype-phenotype maps with demonstrated relevance for evolutionary adaptation and innovation in other biological systems ,23, is tWe demonstrate that metabolic networks in sulfur metabolism with the same phenotype form large genotype networks that reach far through metabolic genotype space. How far they reach through this space is a linear function of the number of super-essential reactions specific to the environment. We show that the robustness of metabolic networks depends on the size of a metabolic network, on the average size of minimal networks viable in a given environment, and on how rapidly the proportion of essential reactions decrease with increasing network size. The neighborhoods of two metabolic networks on the same genotype network typically contain different novel phenotypes. In evolving populations of metabolic networks, robustness facilitates the discovery of novel phenotypes only up to some modest value of robustness, beyond which populations discover fewer novel phenotypes. The difference in the role of robustness in the evolution of metabolic networks compared to its role in the evolution of macromolecules shows that phenotypic innovations may not occur according to the same principles in all biological systems.http://www.chem.qmul.ac.uk/iubmb/enzyme/ which includes all enzymes with known classification [To obtain the global set of reactions involving sulfur-containing metabolites that can be present in the metabolic networks we studied, we used data from the LIGAND database of the Kyoto Encyclopedia of Genes and Genomes  . The LIGfication .2H6(CH2)n), or, similarly, general polymerization reactions that were of the form An+ B → An+1, because their abstract form makes them unsuitable for stoichiometric analysis, (ii) reactions involving glycans, because of their complex structure, (iii) reactions that were not stoichiometrically or elementally balanced, and (v) reactions involving complex metabolites without chemical information about their structure.Specifically, we used the REACTION and COMPOUND sections of the LIGAND database to construct our global reaction set. From this dataset we pruned (i) all reactions involving general polymer metabolites of unspecified numbers of monomer units (CE. coli metabolic network model (iJR904) [In addition, we merged all the reactions existing in the (iJR904)  that invE. coli iJR904 metabolic model [viable in a given environment if it can sustain a biomass growth rate greater than 1.0 × 10-3. In essence, the approach we take is equivalent to asking whether all the necessary sulfur containing biomass precursors are synthesizable given a metabolic network in a specified environment. Flux balance analysis relies on linear programming [http://www.ilog.com/) and CLP  to solve the associated linear programming problems.Flux balance analysis is a computational method used to find a set of fluxes through all metabolic reactions that maximize biomass production in a given metabolic network, assuming it is in a steady state . This asic model . This bigramming  to compusole source of sulfur. These 124 sources were all the sulfur containing metabolites in the 1221 reactions of our global reaction set. We provided any metabolite not containing sulfur in the environment, in effect making it a rich environment limited by sulfur containing metabolites only. Also, we allowed cells to secrete all metabolites. We define a metabolic phenotype as the set of environments (each with a different sole sulfur source) in which a metabolic network is viable. The environmental demands imposed on a metabolic network correspond to the set of sulfur sources that the metabolic network must at least be viable in.We here considered 124 different environments that differed in the chemical compound that could serve as the We define a reaction as essential if its removal from a metabolic network renders the metabolic network inviable on at least one of the sulfur sources that it had previously been viable on. We called a reaction super-essential if it occurred in all minimal metabolic networks generated under a given set of environmental demands.X of sulfur sources. To this end, we first created a binary vector of length 124 (each of whose entries corresponds to one sulfur source), initialized all its entries to the value zero, and then randomly changed X of these entries to one. These entries represent the set of sulfur sources on which we required our metabolic networks to be viable.We generated random viable metabolic networks as follows. First, we generated a random environmental demand, that is, we required viability in some given number N reactions as follows. We started from a metabolic network that contained all 1221 reactions  and sequentially removed randomly chosen reactions, while ensuring viability on the set of X sulfur sources chosen previously, until we had reached a network with the target number N of reactions.We then generated random viable metabolic network of We define a minimal metabolic network as a network where not a single reaction can be removed without destroying viability. To generate a (random) minimal metabolic network we used the same procedure until no reactions could be removed without destroying viability.We generated random walks for metabolic networks of given reaction numbers N and viability on a given number of sulfur sources by first generating a random metabolic network of this size, as just described. We then generated a series of steps ("mutations") in metabolic genotype space, each one either an addition or a deletion of a reaction. After each step, we recomputed the phenotype of the metabolic network. If the metabolic network was still viable on the same set of sulfur sources, we accepted the mutation and proceeded to the next mutation; if not, we rejected the mutation and repeated the process from the metabolic network prior to the mutation. We continued the resulting random walk for 10 000 accepted mutations. We kept the size of the metabolic network in the narrow interval  by ensuring that accepted mutations alternated between reaction additions and deletions.In a variation on this procedure, we also carried out forced random walks through genotype space. Their aim was to obtain metabolic networks that are as different (in terms of genotype distance) as possible from the initial metabolic network. In a forced random walk, we required that any reaction addition did not involve a reaction that had been part of the initial network at the start of the walk.Populations where the product of population size and mutation rate is much greater than one are polymorphic most of the time, and show evolutionary dynamics different from those of small populations . To undeJFMR and AW designed the study. JFMR carried out all computational analysis. JFMR and AW participated in data analysis and manuscript writing.Random metabolic networks required to be viable on a given number of sulfur sources are generally viable on more sulfur sources. Average number of sulfur sources that random metabolic networks are actually viable in, for varying environmental demands S, and varying metabolic network size N. The figure demonstrates that random metabolic networks required to be viable on a given number S of sulfur sources (as generated by the procedures described in Methods) are generally viable on more than S sulfur sources. Each data point represents an average over 200 random metabolic networks . Error bars correspond to one standard deviation.Click here for fileNumber of essential reactions decreases with metabolic network size. Number of essential reactions found in random metabolic networks of different size and for different environmental demands (S). Each data point represents an average over 200 random metabolic networks .Click here for fileCumulative number of novel phenotypes encountered in the neighborhoods of all evolving metabolic networks in a large population. The results are plotted for populations of metabolic networks of different sizes and subject to different environmental demands. Each data point represents an average over 200 simulations, 20 simulations for 10 different sets of environmental demands with the same number S of sulfur sources. Each population consisted of 100 individual metabolic networks.Click here for fileNumber of novel phenotypes as a function of robustness, metabolic network size and environmental demands. Three dimensional plots of number of novel phenotypes found in the (1-mutant) neighborhood of random metabolic networks versus robustness of the metabolic networks and (A) metabolic size N or (B) different environmental demands S. Each data point is an average over 200 metabolic networks .Click here for fileCumulative number of novel phenotypes found in the neighborhood of large and small evolving populations of metabolic networks viable in carbon sources. Plot of the cumulative number of novel phenotypes found in the neighborhood of (A) large and (B) small evolving populations of metabolic networks required to be viable in different number of carbon sources. (C) Number of novel carbon utilization phenotypes found in the neighborhood of random metabolic networks. Metabolic networks in these simulations had 931 reactions, the same as the size of the E. coli iJR904 model [04 model ,25.Click here for file


An algorithm called FBA-Gap is presented here that aids the construction of a working model based on plausible modifications to a given list of reactions that are known to occur in the organism. When applied to a working model, the algorithm gives a hypothesis concerning a minimal medium for sustaining the cell in culture. The utility of the algorithm is demonstrated in creating a new model organism and is applied to four existing working models for generating hypotheses about culture media. In modifying a partial metabolic reconstruction so that biomass may be produced using FBA, the proposed method is more efficient than a previously proposed method in that fewer new reactions are added to complete the model. The proposed method is also more accurate than other approaches in that only biologically plausible reactions and exchange reactions are used.Constraint-based metabolic models are currently the most comprehensive system-wide models of cellular metabolism. Several challenges arise when building an  Flux balance analysis (FBA) is the use of a linear program (LP) to model the flow of metabolites through the network of reactions in a cell . FBA simin silico cell to send flux through the biomass reaction.Several related challenges are encountered in the building of metabolic reconstructions. To apply FBA to a constraint-based model, both a reaction network  and an objective  need to be specified. Currently, complete reaction networks for organisms are not known. There may be reactions in a cell that must be active for the production of biomass that have not been cataloged in biological databases or documented in the literature. Another challenge is modeler error; the modeler can mistakenly omit a reaction or transport process that is necessary for the production of biomass. Aside from establishing a model that can produce biomass, a common difficulty in using FBA models is that of finding a culture medium that can allow the Several methods for restoring functionality in broken FBA models, those incapable of a desired level of flux through the biomass reaction, have been previously proposed. GapFind  is a proSeveral investigators have proposed methods for filling gaps in metabolic networks outside of the FBA paradigm, including searching through a network of metabolites and reactions for logically possible paths , 14 and metabolite to refer to any molecule whose concentration is of interest, including byproducts of metabolism, coenzymes, and protons. Let vj be the flux through reaction j, for each j ∈ R, which is the number of times that a reaction occurs per unit time. Let Sij be the stoichiometric coefficient for metabolite i in reaction j, for each i ∈ M and j ∈ R, with the convention that Sij is negative for molecules i that are reactants for reaction j, positive for metabolites i that are products for reaction j, and 0 otherwise. Metabolites may participate in a unidirectional or reversible exchange reaction. For our purposes, it will be helpful to distinguish source reactions from escape reactions and assign variables bisrc and biesc for the fluxes through these reactions. We wish to restrict transport fluxes to zero for any metabolite unless its concentration changes in the cell due to transport processes. The conservation of mass for metabolite i may be stated as follows:R may include a  biomass reaction which reflects the objective of the cell in terms of which metabolites are emphasized for production or consumption by other processes. The objectiveA fundamental assumption of FBA modeling is that metabolites remain at constant concentration within the cell. Throughout this paper, we use the term S. cerevisiae and E. coli models [vj/vk ≥ 4, the model indicates that a mechanism for maximizing biomass production exists wherein reaction j is at least 4 times as active as reaction k. If we generate . Whereas FBA can be considered a generalized maximum flow on a hypergraph, consider an analogy with maximum flows on graphs . Intuitii ∈ R. Then a minimum-cost set of exchange reactions for which a minimum threshold of flux through the biomass is attained can be determined by solving the following mixed-integer program:Lbiomass, is specified in the set of flux lower bounds. The first constraint ensures that a valid flux distribution is derived, that is, the mass balance constraints are satisfied. The last two constraints ensure that if the flux along a exchange reaction is positive, then an appropriate cost is enforced. The remaining constraint contains bounds for the reactions fluxes. Solving . The selection of exchange metabolites that are most biologically plausible and/or furthest from the biomass reaction is ensured by a cost structure that is described in the next section.Letprogram:min  where R⊆2M|| × 2M|| is the directed hypergraph associated with the reaction network for an organism, where each reaction corresponds to a hyperarc  as follows. For every hyperarc r ∈ R with tail nodes Tr and head nodes Hr, and for every i ∈ Tr and j ∈ Hr, there is an arc ∈ℛ (G from the metabolite to a biomass reactant (product). This distance measure does not work well because, for example, a large proportion of reactions in a cell involve cofactors such as ATP. Every metabolite is either involved in a reaction where ATP is produced or consumed or will be near such a reaction by this distance measure. Therefore, every metabolite will appear to be near the biomass reaction.First, we assign costs to extracellular metabolites. Extracellular metabolites are substances that are either postulated to exist in the culture medium or are secreted by the cell. Adding a source reaction for such a metabolite is plausible if experimental culture media that support growth are likely to contain the substance, and adding an escape reaction is plausible if the cell likely secretes the metabolite. A low cost of 1 is assigned for biologically plausible exchange reactions for extracellular metabolites, and a cost of 20 is assigned for implausible exchange reactions for extracellular metabolites (Table S1). Therefore, up to 20 plausible exchange reactions will be selected before 1 implausible exchange reaction. The costs for artificial exchange reactions for internal metabolites are assigned based on the distance of a metabolite to the biomass reaction. Distance to the biomass reaction is defined as follows. Assume for the moment that all stoichiometric coefficients are 1. Let hyperarc . Define ∈ℛ . An intui, j) in ℛ by d  = deg (i), where deg (i) is the degree of i. Note that the degree of node i in G is precisely the number of reactions in which metabolite i participates. The distance of a metabolite to the biomass reactants disrc is the length of the shortest directed path in G to a biomass reactant, which can be determined by applying Dijkstra's algorithm [diesc. Let dmaxsrc (dmaxesc) be the maximum distance among all metabolites with a directed path to the biomass reactants (products) in G. To penalize the source transport reactions that are near the biomass reactants, we define the cost for internal metabolite i to be dmaxsrc − disrc + 20. The penalty of 20 in the cost formula ensures that the cost of an artificial exchange reaction for an internal metabolite is at least as high as the cost for an exchange reaction for an extracellular metabolite. Escape reactions are penalized in an analogous fashion. Dijkstra's shortest path algorithm is polynomial time and is computationally easy for the networks considered here. The computational complexity of the proposed method is dominated by solving instances of , then the selected exchange reactions give an indication of the location of gaps in the reaction network. The modeler can then consult the appropriate diagram in a publicly available biochemical pathway database, for example, KEGG , BioCyc B, F, G, H, and I will not be selected as having source reactions, and metabolites A, B, C, D, E, and H will not be selected as having escape reactions because they are infinite distance from the biomass reactants and products, respectively. The instance of  with a time limit of 600 seconds.When using GapFind/GapFill in these experiments, GapFind is applied to find all nonproducible metabolites in a model, and GapFill is applied to each nonproducible metabolite to find reactions to add so that the metabolite is produced. If GapFill is infeasible, an exchange reaction is added for that metabolite. Integer programming instances for all methods are solved using Gurobi . GapFill adds 550 reactions from the reaction database and 182 exchange reactions for metabolites. 4, FDP, O2, NADPH, Gly, PRPP, ATP, and Acetyl-CoA. The escape reaction is for cytoplasmic CO2, and the added reaction is the peroxidative reaction catalyzed by catalase (2H2O2 → 2H2O + O2).The integer program for GapReed terminates at the time limit of 600 seconds. Source exchange reactions are added for nine cytosolic metabolites, one escape exchange reaction is added, and one reaction from the reaction database is added. The source exchange reactions are for cytoplasmic Gln-L, SOC. neoformans than these two established methods. GapFind/GapFill and GapReed are more conducive to an automated implementation than FBA-Gap, but in this example, one can see some of the pitfalls of an automated approach. GapFind/GapFill adds many internal reactions and exchange reactions for cytosolic metabolites so that there is a high probability that implausible reactions are present in the final model. Further, GapFind/GapFill requires significantly more computation time. GapReed adds exchange reactions for more implausible cytosolic metabolites than FBA-Gap. A hybrid computational/manual curation approach such as FBA-Gap is able to derive a biomass-producing model with higher-confidence reactions for our partial reconstruction for Trypanosoma cruzi [, Bacillus subtilis [, Heliocbacter pylori (iIT341 GSM/GPR) [Escherichia coli (iJR904 GSM/GPR) [FBA-Gap is applied to four existing models with exchange reactions removed. The metabolic reconstructions used in this experiment are for ma cruzi , Bacillusubtilis , HeliocbGSM/GPR) , and EscGSM/GPR) . The resT. cruzi is a protozoan parasite of humans that causes Chagas disease. The reconstruction is a multi-compartment model of central metabolism for T. cruzi. FBA-Gap selects biologically realistic source and escape reactions. The source reactions correspond to the transport of extracellular metabolites that are plausible constituents of a culture medium for sustaining T. cruzi and the escape reactions correspond to metabolites that are likely secreted by T. cruzi. B. subtilis is a Gram-positive bacterium found in soil. As with the T. cruzi model, FBA-Gap selects biologically realistic exchange reactions for production of biomass. For the E. coli reconstruction, one undesirable escape reaction is selected (clpn_ecc) that is unique to the E. coli model. This metabolite occurs in only one reaction in the model and is a reactant in the biomass reaction. Therefore, a simple remedy is to increase the stoichiometric coefficient in the biomass reaction. For the h. pylori reconstruction, a single implausible escape reaction is selected for rhcysc. Upon investigation of the network around this metabolite, we discover that there are reactions converting rhcysc to dhptdc and dhptdc to hmfurnc, but there are no reactions consuming hmfurnc. There are no reactions consuming hmfurnc in our universal database, so we can either add it as a reactant in the biomass reaction or add an escape reaction to remove it from the cell.The hypothesized culture media are summarized in Table S2. B. subtilis model (Table S9). Solving  from the C. neoformans. FBA-Gap produces a more accurate reconstruction than an application of existing methods for filling gaps and requires less computation time. However, in contrast to other methods, FBA-Gap also involves manually selecting and approving which reactions to add to a model so that the overall time may be longer. As noted by Latendresse et al. [This paper presents an optimization-based method for “debugging” metabolic reconstructions called FBA-Gap. We demonstrate the effectiveness of the procedure in helping to find gaps in a model for e et al. , a fullye et al. , a more B. subtilis, T. cruzi, H. pylori, E. coli, and C. neoformans are in Table S2. For the C. neoformans model, the _rst step of FBA-Gap yields that the metabolites in Table S3 are an in_nite distance from the biomass reaction. The results of three rounds of gap analysis are in Tables S4-S7 below. Each table contains the transport reactions selected, and the action taken to avoid high-cost transports in the next round of gap analysis. In general, reactions are added only if they are in the MetModel GUI database and if KEGG specifies that a gene encodes the appropriate enzyme in the organism. The reactions in the final working model for C. neoformans is in Table S8. Table S9 contains reactions deleted from the B. subtilis model, and Table S10 contains the results of applying FBA-Gap to the broken model.The Supplementary Material contains a proof that FBA-Gap is NP-Complete and tables related to computational results. A list of low-cost exchange reactions is in Table S1. Results from the first round of FBA-Gap for Click here for additional data file.Click here for additional data file.


Biochemical networks are used in computational biology, to model mechanistic details of systems involved in cell signaling, metabolism, and regulation of gene expression. Parametric and structural uncertainty, as well as combinatorial explosion are strong obstacles against analyzing the dynamics of large models of this type. Multiscaleness, an important property of these networks, can be used to get past some of these obstacles. Networks with many well separated time scales, can be reduced to simpler models, in a way that depends only on the orders of magnitude and not on the exact values of the kinetic parameters. The main idea used for such robust simplifications of networks is the concept of dominance among model elements, allowing hierarchical organization of these elements according to their effects on the network dynamics. This concept finds a natural formulation in tropical geometry. We revisit, in the light of these new ideas, the main approaches to model reduction of reaction networks, such as quasi-steady state (QSS) and quasi-equilibrium approximations (QE), and provide practical recipes for model reduction of linear and non-linear networks. We also discuss the application of model reduction to the problem of parameter identification, via backward pruning machine learning techniques. During the last decades, biologists have identified a wealth of molecular components and regulatory mechanisms underlying the control of cell functions. Cells integrate external signals through sophisticated signal transduction pathways, ultimately affecting the regulation of gene expression, including that of the signaling components. Metabolic functions are sustained and controlled by complex machineries involving genes, enzymes, and metabolites. The genetic regulations result from the coordinate effect of many, mutually interacting genes. These regulations involve many molecular actors, including proteins and regulatory RNAs, which form large, intricate networks.In vivo values of the parameters depend on crowding and heterogeneity of the intracellular medium, and can be orders of magnitude different from what is measured in vitro. Furthermore, learning models from data suffer for non-identifiability and over-fitting problems. Thus, model reduction is an avoidable step in the study of large networks, allowing to extract the essential features of the model, that can then be identified from data. Model reduction in computational biology should have several features.Current dynamical models of cellular molecular processes are small size networks. These small scale models, that are subjective simplifications of reality, can not take into account the specificities of regulatory mechanisms. New methods are needed, allowing to reconcile small scale dynamical models and large scale, but static, network architectures. The main obstacle to increasing the size of dynamical networks is the incomplete information, on the parameters and on the mechanistic details of the interactions. First of all, model reduction should cope with parametric incompleteness and/or uncertainty.A certain class of reduction methods are parameter independent and automatically comply with this specificity. In biochemical networks, the number of possible chemical species grows combinatorially due to numerous possibilities of interactions between molecules with multiple interaction sites. The exact lumping methods  and averaging approximations.The iterative methods of IM aim to find a slow low dimensional IM, containing the asymptotic dynamics  was proposed by Bodenstein  and was Another method allowing to simplify multiscale dynamics is averaging. This idea can be tracked back to Poincaré's perturbative treatment of the many body problem in celestial mechanics , to stochastic gene expression. In computational biology, model reduction should thus cope not only with deterministic, but also with stochastic and hybrid models. The need to reduce large scale stochastic models is acute. Indeed, stochastic simulation algorithm  are non-linear functions of the concentrations. For instance, the mass action law reads R+j(c) = k+j ∏icjiαi, R−j(c) = k−j ∏icjiβi, in which case Pi(c) is a multivariate polynomial on the concentrations cj.Dynamics of non-linear networks in homogeneous isochoric systems (fixed volume) is described by a system of differential equations:Ai, edges correspond to reactions Ai → Aj with kinetic constants kji > 0.Monomolecular reaction networks are the simplest reaction networks. The structure of these networks is completely defined by a digraph, in which vertices correspond to chemical species The kinetic equation isK:Krk = λkrk, and lkK = λklk.The solutions of (3) can be expressed in terms of left and right eigenvectors of the kinetic matrix k is the inverse of a timescale of the network. A reduced network having solutions of the type (4), with eigenvectors rk, lk, and eigenvalues λk approximating the eigenvectors and the eigenvalues of the original network is called a multiscale approximation.Each eigenvalue λi, j) ≠  one of the relations kji << kj′i′, or kji >> kj′i′ is satisfied.We say that the network constants are totally separated if for all  having the lowest label. These integer labels also indicate the timescales of the processes modeled by the network reactions.The reduced network is not always a subgraph of the initial graph. It is obtained from this integer labeled digraph by graph re-writing operations, that can be generically described as pruning and pooling. Two types of pruning operations are of primary importance  If one has one node from which leave more than one edge, then all the edges are pruned with the exception of the fastest one (lowest integer label). This operation corresponds to keeping the dominant term among the terms cikij consuming a species Ai, and reduces the node outdegree to one. The same principle can not be applied to reduce the indegree, because which production term is dominant among kijcj, j ∈ , depends not only on kij but also on the concentrations cj.Rule b) Cycles with separated constants can be transformed into chains, by elimination of the slowest step. This can be justified intuitively by topology, because any two nodes of a cycle are connected by two paths, one containing the slowest step and the other one not containing the slowest step. The latter shortcuts the former.However, a combination of rules a) and b) is not allowed to prune slow reactions leaving a cycle and further transform the cycle into a chain by eliminating the limiting step. Indeed, the total mass of such cycles is slowly decaying because of outgoing reactions. Pruning the slow reactions that leave a cycle would keep the total cycle mass constant and produce the wrong long time approximation. In this case, pooling operations are needed:Rule c) Glue each cycle in the pruned system into a new vertex and transform the network of all initial reactions into a new one. The concentration of this new component is the sum of the concentration of the glued vertices. Reactions to the cycles transform into reactions to the correspondent new vertices (with the same constants). To transform the reactions from the cycles, we have to calculate the normalized quasi-stationary distributions inside each cycle (with unit sum of the concentrations in each cycle). Let for the vertex Ai from a cycle this concentration be coi. Then the reaction Ai → Aj with the constant kji transforms into the reaction from the new (“cycle”) vertex with the constant kjicoi. The destination vertex of this reaction is Aj if it does not belong to a cycle of the pruned system, it is the correspondent glued cycle if it includes Aj and does not include Ai and the reaction vanishes if both Ai and Aj belong to the same cycle of the pruned system.After pooling we have to prune (Rule a) and so on, until we get an acyclic pruned system. Then the way back follows: we have to restore cycles and cut them (Rule b).lk, rk belong to the sets {0, 1}, and {0, 1, −1}, respectively. The 0−1 property of eigenvectors has a non-trivial consequence. On the timescale tk = (λk)−1, the reduced digraph behaves as an effective reaction (single step approximation). The effective reaction receives  the mass coming from the species with coordinate 1 in lk (pool) and transfers it (during a time tk) to the species with coordinate 1 in rk. The successive single step approximations of an acyclic deterministic digraph are illustrated in Figure In more detail, the graph re-writing operations, are described in the Appendix and illustrated in Figure Monomolecular networks with separation represent instructive examples where reduction and qualitative dynamics result from the network topology and from the orders of magnitude of the kinetic constants. This type of models can be used in computational biology to reduce linear subnetworks or even binary reactions for which one reactant is present in much larger quantities than the other (pseudo-monomolecular approximation).As argued by a few authors, total separation could be a generic property of biochemical networks  = aαxα, where xα= x1α1x2α2 … xnαn. Monomials with positive coefficients aα>0, become linear functions, log M = log aα + <α, log(x)>, by this transformation.Let us consider multivariate monomials rj=1 νj(R+j(c) − R−j(c)) = ∑Aα ∈ aαcα, A ⊂ ℕn.There is a straightforward way to use the logarithmic transformation from tropical geometry in order to obtain approximations of dynamical networks of the type (2). Let us suppose that reaction rates are polynomial functions of the concentrations , such that ∑c) = , si = sign and ai,αmax, αmax∈ Ai denotes the coefficient of a monomial for which the maximum occurring in (5) is attained.We call tropicalization of the smooth ODE system (2) the following piecewise-smooth system:Aα ∈ aαcα, the max-plus polynomialThe tropicalization associates to a polynomial ∑Pτ(c) is not smooth is called tropical manifold  is a piecewise smooth model  with polynomial rates is separated, if the monomials composing the rates are separated almost all the time on a trajectory, or, equivalently, almost everywhere in phase space . Separation of non-linear models results either from separated kinetic constants, or from separated species concentrations, or both. In the next section, we discuss some examples when the tropicalization provides useful approximations of smooth non-linear networks.Two simple methods are principally useful for model reduction of non-linear models with multiple timescales: the quasi-equilibrium (QE) and the quasi-steady state (QSS) approximations. As discussed in Gorban et al. ; Gorban Etot=[E]+[ES], is much lower than the total concentration of substrate, therefore, the complex ES is a low concentration, fast species. The complex concentration is slaved by the concentration of S, meaning that the value of [ES] almost instantly relaxes to a value depending on [S]. The simplified mechanism correspond to pooling the two reactions of the mechanism into a unique irreversible reaction k1 [S] (Etot − [ES]QSS) = (k−1 + k2) [ES]QSS. From this, it follows that R = k2Etot[S]/(km + [S]), where km = (k−1 + k2)/k1.The QSS approximation, proposed for this system by Briggs and Haldane, considers that the total concentration of enzyme, fast, reversible reaction. The simplified mechanism corresponds to a pooling of species. Two pools, Stot = [S] + [ES], and Etot = [E] + [ES] are conserved by the fast reversible reaction, but only one, Etot is conserved by the two reactions of the mechanism. The pool Stot is slowly consumed by the second reaction and represents the slow variable of the system. The single step approximation reads k1 (Stot − [ES]QE) (Etot − [ES]QE) = k−1 [ES]QE. From this it follows that Etot << Stot, we obtain the original equation of Michaelis and Menten, The QE approximation considers that the first reaction of the mechanism is a One of the main difficulties to applying QE or QSS reduction to computational biology models is that QE reactions and QSS species should be specified a priori. For some models, biological information can be used to rank reactions according to their rates. For instance, one knows that metabolic processes and post-transcriptional modifications are more rapid than gene expression. However, this information is rather vague. In detailed gene expression models, some processes can be rapid, while others are much slower. Furthermore, the relative order of these processes can be inverted from one functioning regime to another, for instance the binding and unbinding rates of a repressor to DNA, can be slow or fast depending on various conditions. Even if some numerical approaches such as iterative IM, CSP, and ILDM propose criteria for detecting fast and slow processes, at present there is no general direct method to identify QE reactions and QSS species.Here we present two methods, based, the first one on singular perturbations, and the second on tropical geometry ideas, allowing to detect QE reactions and QSS species.slaved species. Given the trajectories c(t) of all species, the imposed trajectory of the i-th species is a real, positive solution c*i(t) of the polynomial equationPi is the i-th component of the rhs of (2). We say that a species i is slaved if the distance between the trajectory ci(t) and some imposed trajectory c*i(t) is small for some time interval I, supt∈I | log(ci(t)) − log(c*i(t))| < δ, for some δ>0 sufficiently small. The remaining species, that are not slaved, are called slow species.The first method uses simulation of the trajectories, therefore, it can only be applied to a fully parametrized model. However, in systems with separation, the sets of QE reactions and QSS species are robust, i.e., remain the same for broad ranges of the parameters. One can use imprecise parameters (resulting for instance from crude estimates or fitting) to compute these sets. The method starts by detecting Slaved species are rapid and are constrained by the slow species. The minimum number of variables that we expect for a reduced model is equal to the number of slow species. The slow species can be obtained by direct comparison of the imposed and actual trajectories. This method is illustrated for a model of NFκB canonical pathway in Figure k1 [S] [E] = (k−1 + k2) [ES]. For QE condition, the term k2 will be dominated by k−1. We call pruned version of Equation (7) the equation obtained after removing all the dominated monomials, in this case the equation k1 [S] [E] − k−1 [ES] = 0. When the pruned version is a combination of reversible reaction rates set to zero, then the slaved species satisfy QE conditions. Again, the comparison of monomials is possible for a fully parametrized model, however we expect this comparison to be robust for models with separation.There are two types of slaved species. Low concentration, slaved species satisfy QSS conditions. Large concentration, slaved species are consumed and produced by fast QE reactions and satisfy QE conditions. Because the reduction schemes are different in the two situations, it is useful to have a method to separate the two cases. Using the values of concentrations can work when concentrations are well separated, but may fail for a continuum of values. A better method is to identify which are the dominant terms in the Equation (7). Using again the example of Michaelis–Menten mechanism, the complex ES will be detected as slaved in both QSS and QE conditions. Equation (7) reads n − 1 dimensional (n is the dimension of the vector field) then the conditions for sliding modes read:f+, f− are the vector fields on the two sides of Σ and n+ = −n− are the interior normals.The second method to identify QE and QSS conditions, follows from the calculation of the tropicalization (5). This can be done formally and do not require simulation of trajectories and numerical knowledge of the parameters. Indeed, is was shown in Noel et al. that there is a relation between sliding modes of the tropicalized system (5) and the QSS or QE conditions. The system (5) belongs to the class of ordinary differential equations with discontinuous vector fields ,α1−α2> = log (|a1|/|a2|)}. The stability of the QE or QSS manifold implies the existence of a sliding mode of the tropicalization (5) along this hyperplane. This result suggests that checking the sliding mode condition (8) on the tropical manifold, provides a method of detecting QE reactions and QSS species.In Noel et al.  we have E and P and the dynamics can be described by two ODEs:To illustrate this method, let us use again the Michaelis–Menten example. In this case, two conservation laws allow elimination of two variables S and ES are tripods with parallel arms like in Figure AOB of the tropical manifold of the species ES. This subset is a global attractor of the tropicalized dynamics and represents a tropicalized version of the IM of the smooth system. If the initial data is not in this set, the tropicalized trajectory converges quickly to it and continues on it as a sliding mode. When k2 >> k−1, ES satisfies QSS conditions leading to the Michaelis–Menten equation. The arm AO of the tropical manifold of the species ES carry a sliding mode, has the equation k1Etot[S] = (k−1 + k2)[ES] >> k1[S][ES], and corresponds to the linear regime of the Michaelis–Menten equation. Similarly, the arm OB of the tropical manifold of ES has the equation k1Etot[S] = k1[S][ES] >> (k−1 + k2)[ES] and corresponds to the saturated regime of the Michaelis–Menten equation. When k2 << k−1, the tropical manifolds of the two species S and ES practically coincide. Both species are rapid and satisfy QE conditions, namely k1Etot[S] = k−1[ES] >> k1 [S][ES] on the arm AO, and k1Etot[S] = k1[S][ES] >> k−1[ES] on the arm OB.The tropical manifolds of the two species The tropicalization can thus be used to obtain global reductions of models. Even when global reductions are not possible , the tropicalization can be used to hybridize smooth models, i.e., transform them into piecewise simpler models (modes) that change from one time interval to another. These changes occur when the piecewise smooth trajectory of the system meets a hyperplane of the tropical manifold and continues as a sliding mode along this hyperplane or leaves immediately the hyperplane. Hybridization is a particularly interesting approach to modeling cell cycle. Indeed, progression of the cell cycle is a succession of several different regimes (phases). This strategy is illustrated in Figure We have seen in section 3 that model reduction of monomolecular networks with total separation is based on graph rewriting operations.Similarly, QSS and QE approximations can be used to produce simpler networks from large non-linear networks. The classical implementation of these approximations leads to differential-algebraic equations. It is, however, possible to reformulate the simplified model as a new, simpler, reaction network. We showed in the previous section how to do this for the Michaelis–Menten mechanism under different conditions. In general, one has to solve the algebraic equations corresponding to QE or QSS conditions, eliminate (prune) QSS species and QE reactions, pool reactions (for QSS approximation) or species (for QE approximation), and finally calculate the kinetic laws of the new reactions.i of the reactions in the pool, ν = ∑i γi νi. If the reactions are reversible then the coefficients γi can be arbitrary integers, otherwise they must be positive integers. Reaction pools conserve certain species that where previously consumed or produced by individual reactions in the pools. These species were called intermediates in Radulescu et al. .The vectors Sf,In QSS conditions, reaction pools  are defined by vectors in the right kernel of According to the definition (11), a reaction pool does not consume or produce QSS species (these are intermediates). One can impose, like in Radulescu et al. , a minimAfter pooling, QE and QSS algebraic conditions must be solved and the rates of the new reactions calculated. The new rates should be chosen such that the remaining species and pools of species satisfy the simplified ODEs. The choice of the rates is not always unique , RelA (p65), RelB, NF-κB1 (p50 and its precursor p105) and NF-κB2 (p52 and its precursor p100). This generates a large combinatorial complexity of dimers, affinities, and transcriptional capabilities. IκB family comprises seven members in mammals . All these inhibitors display different affinities for NF-κB dimers, multiplying the combinatorial complexity. The activation of NF-κB upon signaling, occurs by phosphorylation by a kinase complex, then ubiquitination, and finally degradation of IκB molecules. The activation signal is transmitted by several possible pathways most of them activating the kinase IKK that modifies IκB. In the canonical pathway, one important determinant of IKK dynamics is the protein A20 that inhibits IKK activation. A20 expression is controlled by NF-κB. In order to cope with this complexity a model containing 39 species, 65 reactions, and 90 parameters was proposed in Radulescu et al. . Of courGraph rewriting was performed in a modular way, by applying the pruning and pooling rules to tightly connected submodels of the NF-κB network. The computation of the reaction pools was performed using Matlab and METATOOL (kp321+ [A20])) for the modulated inhibitor degradation, and kp115 [IkBa:p50:p65@csl]/((kp215 + [IkBa@csl])(kp315+ [A20])) for the release and translocation reaction.To give an example of modular reduction, let us consider the set of reactions involving six cytoplasmic located intermediates  and four terminal species . As can be seen from Figure Computational biology models contain mechanistic details that can not all be identified from available experimental data. Determining the parameters of such complex models could lead to overfitting, describing noise, rather than features of data, or can be simply meaningless, when model behavior is not sensitive to the parameters. Furthermore, many model identification methods (Golightly and Wilkinson, As discussed in Radulescu et al. , 2008; RUsing model reduction for determining critical features of the model has many advantages relative to numerical sensitivity studies (Rabitz et al., This methodology has been first proposed in Radulescu et al. . The modThe mathematical techniques described in this paper define strategies for the study of large dynamical network models in computational biology. Large networks are needed in order to understand context dependence, specialization, and individuality of the cell behavior. Extensive pathway database accumulation supports somehow the idea that biological cell is a puzzle of networks and pathways, and that once these are put together in a tightly bound, coherent map, the cell physiology should be unraveled by a computer simulation. Actually, confronting biochemical networks with real life is not an easy challenge. Model reduction techniques are needed to bring us one step closer to this objective, as these methods can reveal critical features of complex organizations.We have proposed that the ideas of limitation and dominance are fundamental for understanding computational biology dynamical models. The essential, critical features of systems with many separated time scales, can be resumed by a dominant, reduced, subsystem. This dominant subsystem depends on the order relations between model parameters or combinations of model parameters. We have shown how to calculate such a dominant subsystem for linear and non-linear networks. Geometrical interpretation of these concepts in terms of tropicalization provides a powerful framework, allowing to identify IM, QSS species, and QE reactions. We have also discussed how model reduction can be applied to backward pruning parameter learning strategies.Future efforts are needed to extend these mathematical ideas and model reduction algorithms and implement them into computational biology tools.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Caldicellulosiruptor saccharolyticus has the ability to produce hydrogen (H2) at high yields from a wide spectrum of carbon sources, and has therefore gained industrial interest. For a cost-effective biohydrogen process, the ability of an organism to tolerate high partial pressures of H2 (PH2) is a critical aspect to eliminate the need for continuous stripping of the produced H2 from the bioreactor.2 production in C. saccharolyticus can be sustained at PH2 up to 67 kPa in a chemostat. At this PH2, 38% and 16% of the pyruvate flux was redirected to lactate and ethanol, respectively, to maintain a relatively low cytosolic NADH/NAD ratio (0.12 mol/mol). To investigate the effect of the redox ratio on the glycolytic flux, a kinetic model describing the activity of the key glycolytic enzyme, glyceraldehyde-3-phosphate dehydrogenase (GAPDH), was developed. Indeed, at NADH/NAD ratios of 0.12 mol/mol (Ki of NADH = 0.03 ± 0.01 mM) GAPDH activity was inhibited by only 50% allowing still a high glycolytic flux (3.2 ± 0.4 mM/h). Even at high NADH/NAD ratios up to 1 mol/mol the enzyme was not completely inhibited. During batch cultivations, hydrogen tolerance of C. saccharolyticus was dependent on the growth phase of the organism as well as the carbon and energy source used. The obtained results were analyzed, based on thermodynamic and enzyme kinetic considerations, to gain insight in the mechanism underlying the unique ability of C. saccharolyticus to grow and produce H2 under relatively high PH2.Herein, we demonstrate that, under given conditions, growth and HC. saccharolyticus is able to grow and produce hydrogen at high PH2, hence eliminating the need of gas sparging in its cultures. Under this condition, it has a unique ability to fine tune its metabolism by maintaining the glycolytic flux through regulating GAPDH activity and redistribution of pyruvate flux. Concerning the later, xylose-rich feedstock should be preferred over the sucrose-rich one for better H2 yield. This makes the H2-generation reactions more energetically favourable [2 yields [2 yields reported to date, approaching the theoretical maximum, were obtained with (hyper)thermophiles [In the continuous quest for an economically competitive biohydrogen production plant, it is important to obtain as high Hpossible . The H2 f hexose , whereasl hexose . One sucvourable  and has,2 yields ,8. Indeemophiles -11.2 is known to have an inhibitory effect on growth and its own production in a variety of microorganisms, including (hyper)thermophiles [2 yield is made possible by keeping the PH2 in the fermentation vessel sufficiently low. Normally, this is ensured by continuous stripping of the H2 from the production broth using an inert gas, such as N2 or He [2 might offer an economic alternative [2 [2 increases the osmolality of the fermentation broth, ultimately reducing the growth of the H2-producing organism[2 yields have simply to be obtained at high PH2.Since Hmophiles ,8, maximN2 or He . HoweverN2 or He . Insteadernative  as it isative [2 . However organism. To avoiCaldicellulosiruptor saccharolyticus is a strict anaerobic, extreme thermophilic bacterium that is able to produce nearly stoichiometric amounts of H2 from glucose [C. saccharolyticus has the unique ability to co-metabolize a wide spectrum of carbohydrates including both pentoses and hexoses [ glucose  and sucr glucose . In addi hexoses ,18, and  hexoses -22. The  hexoses  facilita2 yields can only be achieved when acetate is the main metabolic by-product, since the formation of more reduced products, such as lactate and ethanol, drains electrons from H2 production. In this work, we evaluated the influence of elevated PH2 on growth, the extent of lactate formation and accordingly H2 yields, by C. saccharolyticus in batch as well as carbon-limited continuous cultures, with glucose as the main carbon and energy source. The effect of PH2 on C. saccharolyticus metabolism on pentoses (xylose) was also evaluated in batch cultures and was compared with previous results on disaccharides , and is compared with other related organisms. The obtained results are analyzed, based on thermodynamic considerations, to understand the mechanism underlying the unique ability of C. saccharolyticus to grow and produce H2 under relatively high PH2.High Hsucrose; ). We demC. saccharolyticus DSM 8903 was purchased from the Deutsche Sammlung von Mikroorganismen und Zellkulturen . A modified DSM 640 medium [0 medium  was used2 and supplemented with an anoxic solution of cysteine-HCl at a final concentration of 1 g L-1 to render the medium completely anaerobic. For continuous cultivations, the bioreactor was started to be fed with fresh medium at the end of the logarithmic growth phase of the culture, having an identical composition to the batch start-up medium, except for cysteine-HCl  at the required dilution rate (D). Steady states were assessed after at least 5 volume changes based on the criteria of constant H2 and CO2 production rates and constant biomass concentration. Glucose was used as a primary substrate in all batch and continuous experiments at an initial concentration of 5 g/L, if not stated otherwise.Cultures were grown in a jacketed, 3-L bioreactor equipped with an ADI 1025 Bio-Console and an ADI 1010 Bio-Controller  at a working volume of 1L, either in batch or continuous mode. The pH was maintained at 6.5 ± 0.1 at 70°C by automatic titration with 4 M NaOH. The temperature was thermostatically kept at 70 ± 1°C and the stirring rate was set to 250 rpm. A condenser with 5°C cooling water was fitted to the bioreactor's headplate. Prior to inoculation, the medium was sparged with N-1 N2 for continuous removal of produced H2 (Case I); no gas sparging, with the bioreactor's gas outlet open leading to higher concentrations of H2 in the headspace at 1 bar (Case II); no gas sparging, with the bioreactor's gas outlet closed allowing H2 to accumulate and increasing the total pressure in the bioreactor (Case III). Gas samples from the headspace for H2 and CO2 determination and culture samples for monitoring growth, substrate consumption and product formation were regularly withdrawn during fermentation. At steady states, samples were taken for determining the NADH/NAD ratio and cell dry weight (CDW) and anaerobic culture samples for enzyme activity measurements as described previously [-1 and 0.15 h-1 in duplicate under both, 'Case I' and 'Case II', conditions.Three different experimental designs were applied: continuous flushing with 100 mL mineviously . Continu2 and H2 by gas chromatography, using a dual channel Micro-GC , as previously described [620) using a U-1100 spectrophotometer . CDW was determined by filtration as previously described [2SO4 (0.6 ml min-1) as the mobile phase. The column was equipped with a refractive index detector .Headspace samples were analyzed for COescribed . The resescribed . Glucose2/H2/CO2 atmosphere (85/10/5 v/v). Cell suspensions were centrifuged outside the glove box for 5 min at 5,000 × g and 4°C, after the addition of sodium dithionite at a final concentration of 5.2 mg L-1 to ensure complete anaerobiosis. The cell pellets were resuspended in a reaction buffer  [g  and the resulting CE was either used directly or stored under anaerobic conditions at -20°C until use. For determination of GAPDH kinetics, the CE was freed from compounds with a MW below 5 kDa using a PD10 column (Sigma-Aldrich), as previously described [Cell extracts (CE) were prepared anaerobically in duplicates using cells harvested from continuous cultures. All cell manipulations were carried out in an anaerobic glove box  with a N pH 7.2)  Cells weescribed .K0.5 for the substrates GAP and NAD+ were determined by using seven different GAP concentrations and varying the NAD+ concentration. The Ki for NADH was determined by using four different NAD+ concentrations and varying NADH concentration.All enzyme activity measurements were carried out as described previously , with th-1. Protein concentration in the CE was determined according to Bradford (1976), with bovine serum albumin as a standard.Background reactions for the assays were determined by replacing the substrate with the reaction buffer. One unit of enzyme activity (IU) is defined as the amount of enzyme that catalyzes the conversion of 1 μmol of substrate per min. Specific activities are expressed as IU (mg protein)The intracellular concentrations of NADH and NAD were determined by a cyclic assay as described earlier ,24, withSamples were collected in screw-cap microcentrifuge tubes containing ice-cold chloroform and immediately frozen into liquid nitrogen. Samples were stored at -80°C until further analysis. During the sample preparation cells were not separated from the medium to avoid the loss of ATP and/or PPi due to possible leakage during centrifugation, as previously observed by Bielen et al . MoreoveSamples for PPi measurement were treated with ATP-sulfurylase  in the presence of excess Adenosine-5'-phosphosulfate to produce ATP from PPi  and the 2 productivity (mM h-1) and cumulative H2 formation  were calculated in two different ways depending on the experimental design. All calculations were based on the ideal gas law using H2 and CO2 concentration in the headspace. For Case I (sparging with N2) the calculations were based on the flow rate of the influent N2 gas and the percentages of H2 and CO2 in the effluent gas, as no other gases were detected, whereas for Case II (no sparging) the flow rate of the effluent gas was measured by the water displacement method with CO2-saturated water to avoid any further CO2 to dissolve. It was assumed that CO2 in the effluent gas did not dissolve in the CO2-saturated water; therefore the actual dissolved CO2 concentration was not determined. CO2-saturated water was prepared by stripping the boiling water with 100% CO2, cooling it down simultaneously and was kept cold throughout the experiment. At the steady state, the flow rate of the effluent gas was determined by measuring the volume of the effluent gas collected between two time points. Thus, H2 productivity and CHF were calculated based on hydrogen concentration in the effluent gas and the flow rate of the effluent gas.H-1.h-1), i.e. qNADH produced, qNADH used, qNADH available and qpyruvate, were estimated as described previously [The intracellular specific productivities  was calculated based on the equation previously described [The biomass yield per mol of ATP  in equilibrium was estimated according to Henry's law:The dissolved HKH is Henry's constant (mM/bar) and is dependent on the temperature according to:where K1 is KH at T1 = 298 K , ΔH (J/mol) is the enthalphy at standard conditions and R (8.314 J/mol/K) is the gas constant.where + was determined by fitting the Michaelis-Menten type kinetic equation to the obtained data [The affinity constants for the substrates GAP and NADned data :V is the reaction rate, Vmax is the maximum rate of the reaction, GAP Kand NAD Kare the affinity constants for GAP and NAD+, respectively, and α is a constant representing any interaction between NAD and GAP binding to the enzyme. However, from analysis of our data α was not significantly higher than zero and was, therefore, the term "α× GAPK" was excluded from the equation.where + concentrations and varying NADH concentrations. The type of inhibition kinetics was visualized by fitting equations for i) sigmoidal competitive inhibition, ii) uncompetitive inhibition iii) mixed inhibition, or iv) linear competitive inhibitionThe inhibition kinetics of NADH was determined by using four different NADNADH Kis an inhibition constant, to the experimental data [where tal data .Data obtained from the inhibition kinetics of NADH was also used to study the effect of NADH/NAD (redox ratio) on GAPDH activity by assuming redox ratio as a substrate. Hill type kinetic equation was fitted to the data :R Kis an affinity constant and h is the Hill coefficient of cooperatively.where S is the redox ratio, Vmax, α, GAPK, NADK, NADHK, R Kand h) was based on non-linear regression using the Surface Fitting Tool (sftool) or curve fitting tool (cftool) in MATLAB (R2009a), which also provides a statistical analysis. Model discrimination was based on the goodness of fit, which was evaluated by the 95% confidence bounds for the fitted parameters and by the square of the multiple correlation coefficients (R2).Parameter estimation .Consistent with previous findings , acetate2 Figure . LactatePH2 was 30 kPa, thus about 2-fold higher than the previously quoted critical PH2 for lactate formation . The H2PH2 clearly influenced the overall lactate yield, as the final lactate concentration was 5-fold higher when the culture was not sparged with N2  [Ki = 1.7 mM), therefore antagonizing the stimulating effect of elevated NADH/NAD ratios [C. saccharolyticus are correlated to the growth rate, as the PPi concentration is highest (4 ± 2 mM) during exponential growth, and decreased seven folds during the transition to the stationary phase [-1) ratios. However, when the growth rate decelerates the concomitant decrease in PPi levels enables LDH to become sensitive to an increase in the NADH/NAD ratio [PH2 (< 30 kPa), as long as the cells are able to sustain high PPi levels through maintaining a high growth rate.This behaviour is consistent with the outcome of another study on metabolic shifts in olyticus , which dse (LDH) . The kinD ratios . The PPiry phase . ConsequAD ratio . TherefoPH2 on C. saccharolyticus at a controlled physiological state. In the absence of gas sparging, the PH2 reached 67 kPa after about 10 volume changes at D = 0.05 h-1 without any washout of the culture indicating that C. saccharolyticus can withstand higher PH2 than was previously reported [PH2 in this case was the redirection of the pyruvate flux, i.e. 38% and 16% of the flux at the pyruvate node was directed to lactate and ethanol, respectively, whereas 0.5% and 4.1% of corresponding fluxes were observed in sparged cultures at similar dilution rates (Table glucose) was not reduced at this high PH2 (Table 2 sparging at a higher D (0.15 h-1) and the culture washed out at PH2 of 67 kPa. These results are in line with previous findings of supersaturation of hydrogen around the cell surface at high productivities due to mass transfer limitation [2,aq) and gaseous H2, the H2,aq should be 0.24 μM at 67 kPa in the headspace and 70°C (Eq. 2 and 3), hence well below the critical H2,aq concentration for growth  but retained at low D (0.05 h-1) strongly indicates that at a lower growth rate, the H2 productivity is in the same range as compared to the mass transfer rate such that H2,aq < H2,aq crit. In contrast, at the higher growth rate, the hydrogen productivity exceeds the mass transfer rate by far, especially under non-sparging conditions, resulting in supersaturation of hydrogen and thus extensive growth inhibition [Carbon-limited chemostat cultures were used to investigate the effect of reported . A noticmitation . Under t 2.2 mM; . Howeverfer rate . Therefohibition .2-sparging conditions, the dilution rate also had an effect on product distribution. At D = 0.15 h-1, the H2 yield was lower than at D = 0.05 h-1 , which agrees with the findings of de Vrije et al [D = 0.15 h-1, compared to a 95% flux at D = 0.05 h-1. In addition, as previously reported [D  showed that there is no interaction between NAD and GAP when binding to the enzyme. NADH inhibited the reaction in a competitive manner , based on Dixon and Cornish-Bowden plots . The activity of GAPDH appeared to follow Hill kinetics with respect to changes in NADH/NAD ratio . Moreover, increase in the NADH/NAD ratio up to one, was unable to fully inhibit the enzyme  and Cl.utylicum . Howeverulyticum , for whiolyticus ,37.Finally, the activity of GAPDH was not found to be influenced by ATP, ADP or PPi over the physiological range of metabolite concentrations (1-10 mM).PH2 can potentially inhibit H2 formation through product inhibition of the hydrogenase-catalyzed NADH oxidation [C. saccharolyticus in the absence of sparging, NADH and NAD levels were determined at steady state conditions with and without sparging  by regulating the activity of GAPDH and/or ii) by redirecting pyruvate flux to more reduced products.High xidation , which cxidation . To inveng Table . Interesng Table  at whichng Table . These rpyruvate = f(vGAPDH). The activity of GAPDH, in turn, is a function of NADH/NAD ratio (vGAPDH = f(NADH/NAD); Figure -1; Table Under given conditions, the overall glycolytic flux can be measured as pyruvate flux. Hence, it could be argued that, glycolytic flux is a function of the activity of GAPDH , which is eight folds lower at high PH2 ::PH2 is dE° is the midpoint reduction potential, F is the Faraday's constant, R is the ideal gas constant and T is the absolute temperature (K).where T. thermohydrosulfuricus strain, which is more tolerant to H2 than the wild-type strain, possessed a GAPDH which could tolerate approximately twice the amount of NADH concentrations as the GAPDH of its wild-type variant [Therefore, lower NADH/NAD ratios make the hydrogenase reaction more energetically favourable, which was confirmed experimentally by Veit et al . Consist variant .PH2 for hydrogen production at 70°C should be 12 Pa in the headspace (Eq. 7) and thus H2-generation should not be possible to occur spontaneously at 67 kPa. Yet, C. saccharolyticus cultures were able to produce H2 at this high PH2.At NADH/NAD ratios of 0.12 mol/mol Table  measuredC. saccharolyticus possesses two distinct hydrogenases, one NADH-dependent Fe-only hydrogenase (Csac_1860-1864) and one ferredoxin (Fd)-dependent, membrane-associated NiFe-hydrogenase  is close to that of H2 . Given d enzyme ), the re the PH2 .T. maritima using NADH and Fdred simultaneously in a bifurcating manner. This novel bifurcating hydrogenase could therefore catalyze the unfavourable oxidation of NADH to H2 by using the exothermic oxidation of Fdred to drive the reaction. It is noteworthy that the sequence of the Fe-only hydrogenase in C. saccharolyticus (Csac_1860-1864) is similar to the bifurcating hydrogenase in T. maritima . HoweveC. saccharolyticus increased with the growth rate  even at higher measured LDH activity in this condition , while lactate was formed on sucrose when H2 accumulated beyond 10-20 kPa [2 but also from obtaining an extra ATP. This is in accordance with previous work showing how the glycolytic flux is significantly increased when xylose is used as carbon source compared to that of sucrose [The Hm Figure . The res sucrose . This leel et al . The fer sucrose . The ace0-20 kPa . Every m sucrose . High th sucrose ).C. saccharolyticus has the attractive property of producing high H2 yields under ideal conditions. When the PH2 rises it has the ability to maintain glycolytic flux by regulating GAPDH. Required GAPDH activity is attained by keeping the NADH/NAD ratio relatively low through redistributing its metabolism towards more reduced end products, including lactate and ethanol. The results herein reveal that these redistributions are not solely dependent on the PH2, but also on the growth state of the organism and the carbon source fermented. Although ethanol is produced, lactate remains the main alternative for C. saccharolyticus for reoxidizing NADH. For an economically attractive industrial application of C. saccharolyticus, hydrogen yields need to be kept maximized, for which metabolic shift to lactate should be kept at bay. In addition, H2 production should be achieved preferably without the need for sparging gas to prevent central costs for the gas-upgrading process [PH2 should be set so as not to adversely affect the growth rate or biomass yield of the organism or to enhance lactate formation. This should be combined with a careful selection of the feedstock, based on the type of substrates, and operating at adequately low osmotic pressures [PH2 on growth and lactate formation than the former. If H2 production is possible on a xylose-rich lignocellulosic feedstock and accomplished at high yields at high PH2, it is definitely a critical step further towards a cost-effective biohydrogen process. process . A critiressures . Thus, aD: dilution rate; h-1; qacetate: specific formation/consumption rate;mmol (gCDW)-1 h-1; qethanol: specific formation rate of ethanol; mmol (gCDW)-1 h-1; qlactate: specific formation rate of lactate; mmol (gCDW)-1 h-1; qNADH: produced specific formation rate of NADH; mmol (gCDW)-1 h-1; qNADH: used specific formation rate of NADH used for lactate and ethanol; mmol (gCDW)-1 h-1; qglucose: specific consumption rate of glucose; mmol (gCDW)-1 h-1; qpyruvate: specific formation rate of intracellular pyruvate; mmol (gCDW)-1 h-1; Yx/ATP: energetic biomass yield; gCDW molATP-1; μ: specific growth rate; h-1; PH2: partial H2 pressure; kPa; Ki: inhibition constantThe authors declare that they have no competing interests.KW planned the content of the article. KW also planned and preformed the batch experiments and was involved in the planning of the continuous cultures experiments and enzyme kinetics experiments. SP planned and performed the continuous cultures, enzyme kinetics and metabolite analysis. KW and SP both wrote a part of the paper. EvN was involved in the planning of the experiments and supervised the processes. EvN also critically reviewed the text. All authors have read and approved the manuscript.