As more complete genome sequences become available, bioinformatics challenges arise in how to exploit genome sequences to make phenotypic predictions. One type of phenotypic prediction is to determine sets of compounds that will support the growth of a bacterium from the metabolic network inferred from the genome sequence of that organism.Escherichia coli K–12 MG1655 from the EcoCyc database. The program automatically partitioned the nutrients within these sets into 21 equivalence classes, most of which correspond to compounds serving as sources of carbon, nitrogen, phosphorous, and sulfur, or combinations of these essential elements. The nutrient sets were predicted with 72.5% accuracy as evaluated by comparison with 91 growth experiments. Novel aspects of our approach include (a) exhaustive consideration of all combinations of nutrients rather than assuming that all element sources can substitute for one another (b) leveraging the notion of a machinery-duplicating constraint, namely, that all intermediate metabolites used in active reactions must be produced in increasing concentrations to prevent successive dilution from cell division, (c) the use of Satisfiability Modulo Theory solvers rather than Linear Programming solvers, because our approach cannot be formulated as linear programming, (d) the use of Binary Decision Diagrams to produce an efficient implementation.We present a method for computationally determining alternative growth media for an organism based on its metabolic network and transporter complement. Our method predicted 787 alternative anaerobic minimal nutrient sets for Our method for generating minimal nutrient sets from the metabolic network and transporters of an organism combines linear constraint solving with binary decision diagrams to efficiently produce solution sets to provided growth problems. Approximately 75% of microbial organisms are unculturable (cannot be grown in the laboratory) even as we can fully sequence their genomes ,2. DeterEscherichia coli metabolic network could be “fed” the constituent compounds of M9 minimal medium, and the expectation would be that all the biomass compounds should be present in the final, fixed set of compounds generated via forward propagation.Given the high cost of evaluating laboratory growth conditions and the relative abundance of powerful genome sequencing resources, it makes sense to ask whether we can use the metabolic network inferred from an organism’s genome sequence to predict the media that will support the growth of the organism. We have previously shown that the biochemical reactions and metabolic pathways of an organism can be inferred from its annotated genome -5. We ha“Omnis cellula e cellula” (“Every cell from another cell” — Francois-Vincent Raspail) T (using the transpose convention for representing column vectors), where r1 is the rate of reaction (1) and r2 is the rate of reaction (2). The rate of production of compounds by the system is given by the column vector p=Mr.Here, N and a set p), as follows: Given a putative nutrient set ith compound is in N then we require pi>0.1. If the ith compound is not in N then we require pi≥0.2. If the N={A,F}. The compound B is consumed by reaction (1) with rate r1 and created by reaction (2) with rate r2 so it has a net production of −r1+r2 and thus B yields a constraint: In our example  Similar analysis yields the constraints C, D, and E, respectively.for compounds r1≥0 and r2≥0. We say that N is a steady-state nutrient set if there exists a vector r that satisfies the above constraints. In our example, r1=r2=k for any k>0 satisfies all the constraints. All the generated constraints are linear; thus, checking whether Because reactions are not allowed to run in reverse, we must add the additional constraints that aBased on a simple molecule-counting argument and linear algebra, we make the following claim relating the steady-state model to experimental observations.Claim 1. Assume the setincludes all reactions available to the organism. This set may also include extraneous reactions that are not actually available to the organism, due to errors in the available data. Assume that setonly contains compounds that the organism must produce to grow . Then the steady-state model over-approximates observable behaviors in the following sense: If the steady-state model predicates that some setof transportables is not a nutrient set then organism will be unable to grow on nutrient setNin the laboratory.Justification 1. For a contradiction, suppose we observe our organism to grow on N in the laboratory. Because everything in N at its disposal, it must have found a set of fluxes for B and non-negative net production of each compound not in N. However, because our system of linear constraints does not have a solution with putative nutrient set N, such set of fluxes does not exist.perfect data assumption. Though possibly unrealistic in practice, unless we are studying modeling methods that explicitly model errors and omissions in the data, making formal comparisons without an assumption of this kind is difficult on paper.Notice that although we need the set N then although there exists a flux that produces N then it is arithmetically impossible for the organism to grow on N. However if growth is indeed observed in the laboratory, then barring experimental error, at least one of our initial assumptions about the completeness of Informally, Claim 1 says that under the perfect data assumption, the steady-state model can produce only one-sided errors: false positives. If it predicts growth on a putative nutrient set b to the original cell, then all of the intermediate metabolites that were used along active pathways to produce compounds of N must also be duplicated ; in essence a dividing cell must at the very least be able to duplicate the active part of its metabolic machinery in addition to producing The steady-state model described above is somewhat unsatisfactory. We have assumed a set C is used as a reactant in a reaction with nonzero flux and C must have strictly positive net production, thus explicitly requiring that more of C will be produced as the organism grows and divides.Informally, we account for the need to duplicate the active part of the metabolic machinery by adding additional constraints to the steady-state model to require that if a compound Ij. Then for a given flux r, C is clearly used if and only if there exists i∈Ij such that ri>0. Since we have constrained the rates to be non-negative, that is equivalent to the test How we frame this constraint mathematically is rather subtle. Suppose for some compound Cj is given by the linear combination pj. We would like to require pj to be strictly positive whenever sj is strictly positive. The question is how to frame this as a linear constraint. Suppose the net production of pj≥αsj for some fudge factor α, thus constraining pj to be strictly positive when sj is strictly positive. The problem is in determining what fudge factor α to use, since too large a value might lead to an unsatisfiable constraint even though there exists a flux producing a positive amount of Cj.One approach is to require that pj≥sj/(sj+1) would work because sj/(sj+1) is always less than one and thus any flux producing a positive amount of Cj can be “scaled up” to satisfy this constraint. Unfortunately, however, when multiplied out, this constraint turns out to be quadratic.Requiring Our solution is to relax the requirement that our linear system be a conjunction of linear constraints and instead allow a monotone Boolean combination of linear constraints. Thus, we add the constraint Cj is positive, or all the reactions that use it as a reactant have zero rate. We call such constraints “make it if you use it” constraints. A nutrient set solution that satisfies these additional constraints will be called a machinery-duplicating nutrient set.That is, either production of feasibility  of such systems falls beyond the capabilities of a regular Linear Programming (LP) package and instead requires the use of a more recent development in computer science called a Satisfiability Modulo Theories (SMT) solverT. First, because C∉N, the net production of C is constrained to be non-negative, and thus r1−r2≥0. Likewise, because B∉N, the net production of B is constrained to be non-negative, and thus r2−r1≥0. Combining these two inequalities we can deduce that r1=r2. Furthermore, because we must make biomass compound E at a strictly positive rate, we have r2>0. Now the “make it if you use it” constraints come into play. Because r2>0 we are required to make C at a strictly positive rate and thus r1−r2>0. But this contradicts our previous deduction that r1=r2. □strictly more accurate than the steady-state model, in the following sense: Neither model can predict a false negative (Claims 1 and 2). However, the following relation exists between false positives predicted by the two models:Under the perfect data assumption we claim that the machinery-duplicating model is Claim 3. Under the perfect data assumption: 1. There exists a dataset 2. There does not exist a dataset Justification 3. Part (1) follows from Lemma 2 because any negative prediction by the machinery-duplicating model must be a true negative by Claim 2 and thus the positive prediction by the steady-state model must be a false positive. Part (2) follows directly from Lemma 1 as the machinery-duplicating model can never predict a positive when the steady-state model predicts a negative.Because the machinery-duplicating model is a theoretically more accurate model than the steady-state model, we consider only the machinery-duplicating model for the rest of this paper, except briefly in Discussion, where we compare our constraint-based modeling techniques to related work.The constraint systems described above handle all reactions as unidirectional. In practice, some metabolic reactions are reversible and will flow in either direction depending on the needs of the metabolic network as a whole. One way to model this situation is to replace a bidirectional reaction with a pair of complementary unidirectional reactions. This approach has the advantage of conceptual simplicity, but having a pair of reactions requires two variables rather than one. Although replacing a full range variable with a pair of non-negative variables is a common approach in naïve expositions of linear programming, SMT solvers such as Yices  and, indSuppose, instead that a bidirectional reaction is handled by removing the non-negativity constraint on its variable. This requires revising the growth constraint (3) to account for reactions running backward.Uj be the set of indices of reactions that have Cj as a reactant or a product. We now enforce a “make it if you mention it” constraint: For some compound This expression yields a constraint that is symmetric with respect to reactants and products.Theorem 1. For bidirectional reactions, the “make it if you mention it” constraint (4) is equivalent to the “make it if you use it” constraint (3).Proof. Clearly constraint (4) is at least as strict as constraint (3). To see that it is not more strict, consider a feasible solution to a system where bidirectional reactions are represented by pairs of unidirectional reactions and the original constraint (3) is enforced on reactants. For some reaction R with a positive reaction rate in the original system, the new constraint (4) enforces the additional requirement that pk>0 for each product compound Ck. If there is some reaction R′ with positive reaction rate that uses Ck, then the original constraint (3) for R′ subsumes this new requirement. Otherwise, because R produces Ck with a positive rate and it is not used by another reaction, the system as a whole must produce Ck with a positive rate satisfying the new requirement. □Impossible compounds and Useless compounds.The inputs for a constraint-based model can be simplified to remove compounds that will not be involved in any potential solutions. The two major classes of compounds that can be removed in this way are For clarity we define the simplification rules on a unidirectional system where any bidirectional reaction can be split into a pair of unidirectional reactions. Complementary reactions in the simplified system can then be turned back into bidirectional reactions before constructing the system of linear constraints.C is Impossible if it is not a potential nutrient and there is no reaction to make it. Such compounds can be deleted, together with any reactions that mention them. Because no reaction has C as a product, any reaction that mentions it must use it as a reactant. Any reaction that uses C as a reactant could not have a positive rate without violating the non-negativity condition for non-nutrient compounds. The deletion of reactions may enable more compounds to be recognized as impossible, so this search for impossible compounds must be iterated to fixed-point .A compound C is Useless if it has no downstream biomass compounds. The search for useless compounds proceeds by finding the complement set of Useful compounds. The biomass compounds are considered useful by definition. A non-biomass compound is considered useful if it is a reactant for a reaction that produces a useful compound. This test is iterated on the compounds not currently classified as useful until a fixed point is reached, such that all compounds not currently classified as useful have been checked without adding one of them to the useful set. At this point, all compounds not in the useful set are considered useless. Such compounds are eliminated from the reactions containing them, as are any reactions whose products are all useless. This leaves the possibility that a reaction will become unbalanced by losing one or more products without being deleted. However, from a constraint-solving viewpoint, this is simply the removal of a redundant non-negativity constraint.A compound N of transportables is a nutrient set by checking the satisfiability of a Boolean combination of linear inequalities. For a transportable N can be represented by a Boolean value, true or false; likewise, the prediction of whether N is a nutrient set, determined by checking the satisfiability of the linear system constructed for N, can also be represented by a Boolean value, true or false. We are now interested in systematically generating all minimal nutrient sets with respect to our scheme and it is perhaps not altogether surprising that we can exploit well-understood technology for computing with Boolean functions to assist us.We have presented a scheme for inferring whether a given set monotonicity and arises in our scheme because adding a transportable to a nutrient set removes one or more constraints from the linear system. More generally, monotonicity will arise whenever all negative effects  are ignored. Note that without monotonicity, the notion of a minimal nutrient set is much more subtle.However, the technique that we present is quite general and, mathematically, depends on just one property of nutrient sets: adding a transportable to a nutrient set produces another nutrient set. This property is called Our technique is based on translating the problem of computing minimal nutrient sets into the language of Boolean algebra. We then use a novel algorithm for computing a representation of all minimal nutrient sets. This new algorithm is built on top of standard tools for computing with Boolean functions.In the following subsections, we introduce all of the concepts needed and sketch the basic technique. But, the reader should be aware that obtaining reasonable runtime performance on realistic data sets, such as EcoCyc, requires a number of algorithmic refinements and implementation details  that are beyond the scope of this paper.true, false} of Boolean values by B and the set of vectors of Boolean values of length n by Bn. We now give some elementary definitions concerning functions on Booleans.In the following discussion, we denote the set {Definition 1. Given a Boolean functionf:Bn→B, a vectorv∈Bnis animplicantoffif and only iff(v)=true.Definition 2. A Boolean function ofnvariables,f:Bn → B, ismonotoneif and only if for anyv∈Bnsuch thatf(v) = true, making a new vectorv′fromvby converting afalsecomponent totrueguarantees thatf(v′)=true.Definition 3. Given a monotone Boolean functionf:Bn→B, a vectorv∈Bnis aprime implicantoffif and only ifvis an implicant off; and1. for every vectorv″constructed from v by converting a true component to false, we have f(v″) = false.2. N of transportables true or false result, depending on whether or not the system of constraints constructed for N is satisfiable.The method of constructing Boolean combinations of linear constraints proposed above defines a function Bn in the following way. We pick some linear ordering on vN∈Bn where the ith component of vN is true if N contains the ith member of false otherwise.Suppose nutset becomes a monotone Boolean function nutset:Bn→B, and the minimal nutrient sets that we seek correspond exactly to the prime implicants of nutset.Under this change of representation, f:Bn→B, solely by evaluating f on chosen inputs without making assumptions about how f is defined.We now consider the problem of computing all the prime implicants of an arbitrary monotone Boolean function v such that f(v)=true. The obvious approach to finding a single prime implicant is to systematically go through the components of v, setting true components to false whenever this can be done without f becoming false. This procedure turns out to be a key step in our algorithm which we call minimization of v with respect to f.Suppose we have some vector Definition 4. Letf:Bn→Bbe a monotone function andv∈Bnbe a vector such thatf(v)=true. We define the following procedure for minimizingvwith respect tof:We keep a vector variableu= that we initialize tov.•Forifrom1up tonifui=true, then•Iff=truethen setui:=false, and continue with the updated vector.Otherwise we continue with the value ofuunchanged.Returnuas a minimization ofvwith respect tof.Theorem 2. The result of the minimization procedure onvreturns a prime implicant off.Proof. Note that u=v initially and v is an implicant of f by definition. At each step we update the value of u only if the new value is also an implicant of f so the result of the minimization procedure must be an implicant of f. To see that it must be a prime implicant, suppose it were not. Thus, some component ui currently set to true could be changed to false and the new vector would also be an implicant. But in this case we would have set ui to false when it was its turn to be considered in the loop because f is monotone. □f is not false everywhere, we can find a first prime implicant by starting with the all- true vector and applying the minimization procedure. The tricky part is finding subsequent prime implicants.Assuming that f, the problem of deciding if the set is complete is known to be coNP-complete even when f is explicitly given in some quite natural representations [f is only accessed by evaluation, is unlikely.Given a set of prime implicants of a monotone Boolean function ntations , so an ef is the function nutset, defined by our linear constraint system instantiated from the EcoCyc dataset.Although the problem of finding the complete set of prime implicants appears theoretically intractable, we can solve it on the instance we care about, namely, where Our method is to compute successive prime implicants using minimization, where at each step we look for a new starting point based on previously found prime implicants that guarantees minimization will find a prime implicant not previously seen.f:Bn→B we want to test if further prime implicants exist.Given that we have found one or more prime implicants of some monotone Boolean function Definition 5. Given a collection of vectorsv1,…,vk∈Bn, a vectoru∈Bnis achoice vectorforv1,…,vkiff for alli∈{1,…,k}, u∧vi≠, where ∧ denotes point-wise conjunction.true components with each of the original vectors. Let Bn that corresponds to the subset relation and let u by ¬u.Informally a choice vector is a vector that shares one or more Theorem 3. Given a monotone Boolean functionf:Bn→Bwith prime implicantsv1,…,vk∈Bn, there exists another prime implicant iff there exists a choice vectoru∈Bnforv1,…,vksuch thatf(¬u)=true.Proof. Suppose there exists a choice vector u∈Bn for v1,…,vk such that f(¬u)=true. We can get a prime implicant v′ by minimizing ¬u. Furthermore, v′ cannot coincide with a known prime implicant vi because u shares a true component with vi and thus ¬u has false in some component where vi has true and thus v′ will have false because minimization never converts false to true.f has a prime implicant v′ that does not coincide with one of v1,…,vk∈Bn. Now for each vi there must exist some component where vi has true and v′ has false, because if v′ has true in each component that vi has true, either v′=vi (a contradiction) or v′ cannot be a prime implicant (another contradiction). Thus, ¬v′ is a choice vector for v1,…,vk. □In the other direction, suppose v1,…,vk∈Bn there are many choice vectors and searching among them for a choice vector u such that f(¬u)=true is prohibitively expensive.The problem is that for a given collection of vectors Definition 6. Given a collection of vectorsv1,…,vk∈Bn, a vectoru∈Bnis aminimal choice vectorforv1,…,vkiffuis a choice vector forv1,…,vk; and1. there is nosuch thatu′is a choice vector forv1,…,vk.2. Theorem 4. Given a monotone Boolean functionf:Bn→Bwith prime implicantsv1,…,vk∈Bn, if there exists a choice vectoruforv1,…,vk∈Bnsuch thatf(¬u)=truethen there exists a minimal choice vectorsuch thatf(¬u′)=true.Proof. If u is not already a minimal choice vector then there must exist a minimal choice vector false in u must also be false in u′. Thus evey component that is true in ¬u must also be true in in ¬u′. Since f(¬u)=true and f monotone it follows that f(¬u′)=true. □Thus we can limit our search to minimal choice vectors.v1,…,vk has at least one true component in common with each of v1,…,vk. Let Ti be the set of indices of the true components of vi.Recall that a choice vector for x=∈Bn we can determine if it has at least one true component in common with vi by forming the disjunction: Given a vector true component in common with each of v1,…,vk by forming the conjunction of disjunctions:  and we can determine if it has at least one g:Bn→B thus defined is necessarily monotone as no negations are involved. Thus the choice vectors of for v1,…,vk correspond to the vectors x that make g(x)=true . The function f we still need to examine the prime implicants u of another monotone function g to find one on which f(¬u)=true. At first sight it might appear that we have come full circle and are back where we started, trying the find the prime implicants of a monotone Boolean function. However, recall that f is considered to be a black box and can be accessed only by evaluating it on each input vector whereas g is defined as a conjunction of disjunctions formed from previously computed prime implicants of f. As we will see, this symbolic representation is much more amenable to prime implicant extraction.In order to compute a new prime implicant of a monotone function Binary Decision Diagram (BDD) is a popular data structure for representing and manipulating Boolean functions [true terminal.The unctions ,13. Althunctions ,15. The g. We can construct the BDD for g by incremental updates each time that we find a prime implicant of f. However, to find the prime implicants of g at any given point, we construct a new BDD for the function pig:Bn→B defined by Recall that the search space has been restricted to minimal choice vectors or, equivalently, the prime implicants of pig(x)=true if g(x)=true and for all x′ formed by changing a true component in x to false, g(x′)=false. The new BDD is constructed by applying standard BDD operations to the BDD for g.Intuitively, u of g by enumerating the implicants of pig which is done by tracing the paths in the BDD for pig from the root node to the true terminald. As soon as we find u such that f(¬u)=true, we can stop, find a prime implicant of f by minimizing ¬u, update g with the new prime implicant, and start over. If we cannot find such a u in the implicants of pig we are done.We can systematically enumerate the prime implicants E. coli, a classification of nutrient compounds into equivalence classes that correspond to biological intuitions. Specifically, computed nutrient equivalence classes often contain all compounds that supply one element .How can we help a biologist user interpret a collection of hundreds or thousands of computed minimal nutrient sets? At least in the case of EcoCyc, we observe that the complete collection of predicted minimal nutrient sets has a very regular structure, and that elucidating this structure yields both a compact representation of the large collection of predicted minimal nutrient sets and, in many cases in Definition 7. Given a collectionof nutrient sets, we want to capture the notion of two transportablesbeing equivalent ifc1can always substitute forc2in any nutrient set wherec2occurs and vice versa.Formally, we sayare equivalent with respect toif and only ifFor allsuch thatc1∈N:; and1. For allsuch thatc2∈N:2. equivalence relation on the compounds occurring in members of This relation is trivially reflexive and symmetric, and can easily be shown to be transitive. It is therefore an representative compound. Given some N′ by replacing each compound c∈N by the representative compound of the equivalence class of c. Because of the mutual substitutability of compounds within an equivalence class, N′ must necessarily be a member of N′ the canonical form of N (given our choice of representative compounds).For each equivalence class of compounds we can choose a If we convert each Of course, the question naturally arises: What is the connection between our original collection of minimal nutrient sets exactly encode N can be obtained from N′ by substituting for each c∈N′ some compound from the equivalence class of c. Conversely if N by substituting for each c∈N some arbitrary compound from the equivalence class of c then N must be a member of The answer is that zero loss of information.Thus, we have a very elegant compression scheme that reduces the size of our collection of predicted minimal nutrient sets and at the same time increases the comprehensibility of our results with generic reactions whose substrates include metabolite classes to capture the broad substrate specificity of their catalyzing enzymes. For example, EcoCyc contains four enzymes that are described as “sugar phosphatase” (E.C. 3.1.3.23), for which the official substrate is “sugar phosphate” and the product is “sugar”.The metabolic reaction sets found in Pathway/Genome Databases such as EcoCyc include many instantiated reactions containing solely metabolite instances. For each compound class in the left and right sides of generic reactions, the software generates new potential reactions by substituting for the compound classes all combinations of instances of those classes. Relationships between a compound class and its instances are stored explicitly in the EcoCyc compound ontology. New reaction equations are added to For each generic reaction, our software generates the set of corresponding E. coli under anaerobic conditions. The E. coli constraint-based model used for this work was obtained from the manually curated EcoCyc database [E. coli biochemical reactions This section describes the inputs we provided to the minimal nutrient prediction algorithm to compute the minimal nutrients of database . The setFour hundred and forty one EcoCyc generic reactions with classes yielded at least one instantiated reaction. Furthermore, unbalanced reactions were removed programmatically from To refine and correct the reactions in the model, over the course of this work, numerous changes were made to EcoCyc as a result of our analysis of executions of the minimal nutrient algorithm. They included fixing erroneous compound structures and reaction equations, adjusting the protonation state of the compounds and reactions to pH 7.3, adding missing reactions, reversing reaction directions, changing reactions from unidirectional to reversible or from reversible to unidirectional, and adjusting cell compartment assignments of reactions. In addition, we added compound instances or reclassified existing instances under appropriate classes in our compound ontology, to allow more instantiations of generic reactions to be inferred.in vitro characterization of an enzyme catalyzing the reaction. We found that some false positive predictions were apparently utilizing this reaction in the physiologically implausible reverse direction. Changing PYRUVFORMLY-RXN to unidirectional, in accordance to usage of the reaction in two in vivo pathways, suppressed several false positive predictions and increased the overall accuracy from 67.0% to 72.5%.One interesting example was the reaction PYRUVFORMLY-RXN, which was labelled as reversible in EcoCyc, due to a literature reference describing the O2) was not supplied as a nutrient.A set of 111 transportable metabolites is plate . T also E. coli transporters. Some such executions have terminated, predicting approximately 8,500 minimal nutrient sets. An execution based on the current metabolic network in EcoCyc has not terminated after two months of run time. Runs larger than the 111 transportable metabolites cannot be validated because of a lack of experimental data.We have tried running the algorithm with all metabolites transportable by known The set of biomass metabolites used in our model was similar to that used in . It contnutset whether or not E. coli would grow on a given We ran the BDD-based minimal-nutrient-set-generation algorithm using the machinery-duplication constraint model on EcoCyc data to predict at each evaluation of a Given the combinatorial process by which nutrient sets are constructed from individual nutrients, this abundance of minimal nutrient sets was not surprising. However, this large solution set does not lend itself to evaluation and validation of the results, especially via laboratory experiment. To facilitate human comprehension and testing of our predictions, we used the notion of equivalence between compounds with respect to a collection of nutrient sets  to factor the set of compounds occurring in predicted minimal nutrient sets into equivalence classes. By picking a representative compound within each such equivalence class and discarding minimal nutrient sets that contain equivalence-class members other than those chosen representatives, we obtained a reduced group of representative minimal nutrient sets from which each original minimal nutrient set could be generated by the appropriate substitution of equivalent compounds Table . The redWe determined 21 nutrient equivalence classes  technology [We compared our predictions against published data on anaerobic chnology .PMs evaluate the metabolic activity of an organism on multiple distinct sets of nutrients in parallel, allowing high-throughput analysis. Although PM technology measures respiration rather than growth, it usually represents a reasonable proxy for growth.One important limitation of PM data is that it typically tests a single element axis at a time. For example, one PM 96-well plate tests a wide array of carbon sources while providing fixed sources of all other elements. In contrast, some of our computed minimal nutrient sets include single metabolites that are predicted to source multiple elements.NPM consisting of four metabolites, each of which sources one of the elements C, N, P, or S. If an exact match of NPM can be found with one of the predicted minimal nutrient sets, Npred, then we count this predicted nutrient as a correct prediction (true positive). Because our method predicts some nutrients that provide more than one element, we also count subset matches as true positives, i.e., Npred⊂ NPM. As an example, alpha-D-glucose-1-phosphate occurs in a predicted nutrient set together with ammonium and sulfate. However, this nutrient set does not exactly match any PM nutrient set, because every PM well in the carbon-source plate also includes phosphate as a separate metabolite. But our method predicted that alpha-D-glucose-1-phosphate can also serve as a phosphorous source, and that it is thus redundant to add phosphate explicitly to the nutrient set. By allowing subset matches, we can correctly score the Npred consisting of alpha-D-glucose-1-phosphate, ammonium, and sulfate as a true positive.The anaerobic PM data we had access to  tested sNPM demonstrated growth experimentally and matches an Npred, we score a true positive prediction; if no matching Npred was found, we score a false negative. If NPM demonstrated no growth, and did not match any predicted nutrient set, a true negative is scored; if it does match an Npred, then a false positive is scored. A table with all results is provided in Additional file If an When evaluated in this way, the overall prediction accuracy of our method was 72.5% based on 91 experimental data points Table . The incE. coli. For some nutrient in these nutrient sets, no known transport reactions or consuming metabolic reactions could be found in the literature. If these six false negative predictions are removed from the comparison, the prediction accuracy of our method was 77.6%.Six of the eight false-negative predictions are due to missing knowledge regarding the fate of the nutrients in The increasing ease with which complete genomes can be sequenced should be accompanied by the ability to make predictions about the growth requirements of the corresponding organisms. We have previously shown that the metabolic network and transporter suite for a given organism can be inferred from its annotated sequence. We have shown here that using such databases to predict a large number of nutrient sets that should support growth of the organism is both possible and practical. These predictions can be distilled into a testable set of compound equivalence classes.As explained earlier , given a complete set of reactions, and a set of biomass compounds without extraneous members, our model for predicting whether an organism can grow on a putative nutrient set can produce only false positives; false negatives are impossible. Thus, when used to compute minimal nutrient sets, our method can find sets of compounds that allow the possibility of growth (based on molecule counting) but offers no guarantee that such growth can actually occur in reality. This occurs because of a number of potential negative effects on growth that our model does not attempt to account for.E. coli with glucose as the carbon source suppresses expression of several nitrogen-assimilation enzymes. As a consequence, even though the metabolic network may suggest that glucose and certain nitrogen sources should be able to work together to provide the organism’s carbon and nitrogen needs, genetic regulatory interactions mean that these combinations will prove inviable in the laboratory.Our method cannot predict the relative concentrations of nutrients, because our method does not account for enzyme reaction rates or regulation at the level of enzyme abundance and activity. Thus, some of the predicted minimal nutrient sets that work on a “parts” level by providing increasing amounts of all the correct metabolites would possibly not lead to viable growth. For example, it is known that growth of In the future, extending our method to incorporate many of these factors may be possible, but we can also learn a great deal from the differences that we see between our current minimal nutrient predictions and experimental reality.and a toxin. “The dose makes the poison,” and even typical nutrients such as glucose are naturally toxic at high concentrations. Other nutrients, however, have a surprisingly narrow gap between viability and toxicity. Thus, predicting a growth solution that is both correct and potentially difficult to apply in the laboratory is possible. The upshot of this effect is that in many cases a laboratory researcher operating without the guidance of a prediction might accidentally discard interesting, experimentally useful growth conditions based on a test that was performed using nutrient concentrations outside this “viable band”.A special and fascinating case of this difference between prediction and reality is toxicity. It is possible for a nutrient to simultaneously be a correct growth solution any of the predicted nutrients may be a sign that a signal should be sought.One key caveat about predicting “growth” for an organism based on its metabolic network arises from an increasing pool of experimental evidence that many microbes will grow only in the presence of signaling molecules. In nature, many microbes can thrive only in the presence of appropriate quorum-sensing signals from their community. When these signals are absent, they will fail to grow despite the presence of all required nutrients . AlthougThe main axes of differentiation between various nutrient set prediction methods are the 1. Mathematical model used to define growth2. Algorithmic solving technique used to find solutions that fit that definition3. Procedure for enumerating all possible sets of minimal growth mediaThere are different notions in the literature for how a nutrient set is defined to support growth.The simplest definition is based on reachability — a nutrient set supports growth if there is a path from available nutrients to every biomass metabolite . In thisM is the stoichiometric matrix and r is a vector of reaction fluxes, then FBA defines r to be a steady state of the network if Mr=0. The set of reactions includes uptake reactions that encode the availability of given nutrients. Furthermore, a special reaction that uses all metabolites required for biomass production is also added to the set of reactions. In FBA, the given nutrient set is said to support growth if there is a solution r for reaction fluxes such that Mr=0 and the growth reaction has nonzero flux.The more commonly used definition of growth is based on flux-balance analysis (FBA), which is a classical approach for performing structural  analysis of metabolic networks . If M isD, exist, then Mr=0 forces the flux on all reactions that produce D to be zero. We now illustrate this scenario. Recall Example 1 from Methods; here we have two reactions: In our approach, we also use a different definition where we require a net positive production (rather than zero) for every metabolite that is involved in a reaction with nonzero flux. There are two reasons for considering this alternate formulation. First, FBA is highly sensitive to missing reactions in the metabolic network. For example, if no reactions that use a metabolite, say E is the sole biomass compound. We now add the following exchange reactions,  and A,F are available as nutrients and E is a biomass compound that we need to synthesize. Because D is not consumed by any reaction, it follows that the flux on the first reaction must be zero and that all steady-state fluxes must be zero. . Thus, FBA will conclude that no steady-state solutions exist because the model is missing some reactions. If we add dummy reactions that consume compounds such as D (that are consumed by no other reactions in the model), then FBA is more likely to generate steady-state solutions. This shortcoming of standard FBA is overcome by having a manual curation step that adds (dummy) import, export, or spontaneous, reactions [Mr=0 to Mr≥0 in our approach partly solves the problem of missing reactions. Specifically, we do not need dummy export reactions  because D can have a net positive production in a solution of our constraints. that capture the information that eactions . The genThe second reason for proposing an alternate definition of growth concerns the case when the metabolic network has cycles, a common scenario. As we claimed earlier , a growing and dividing cell must be able to duplicate the metabolic machinery it uses to grow on a given nutrient set, and this is not accounted for in FBA. In our approach, cycles are handled by introducing disjunctive constraints. A side effect of our solution is that each individual constraint in our approach is a disjunction of linear inequalities. In contrast, in flux-balance analysis, each individual constraint is a linear equation or linear inequality. Table Although the FBA approach does not account for possible problems induced by cycles, it seems to give good results. An interesting problem for future work is to understand what features of a metabolic network suppress the effects of cycles on the space of solutions.When plain reachability is used to define growth, a simple forward propagation procedure — based purely on qualitative reasoning — suffices for deciding if a given medium supports growth . Such a Mr=0. In our approach, we generate disjunctions of linear constraints, and hence we cannot use LP solvers. We instead use modern and highly efficient solvers, called Satisfiability Modulo Theory (SMT) solvers [nutset.Flux-balance analysis uses standard Linear Programming (LP) solvers for finding the maximum flux for the biomass generation reaction subject to the constraint  solvers -25. Not The problem of enumerating all minimal nutrient sets has not been widely studied. Handorf et al.  and CottA→B, B→A, they will conclude that the network can synthesize B starting from an empty bag of nutrients. The scalability of the approach on large reaction networks, such as from EcoCyc, is a concern: Cottret et al. [Cottret et al.  perform t et al.  show thac1 and c2 and two nitrogen sources N1 and N2, with compound M representing biomass. Consider the reactions: Feist et al.  and Marac1 and c2 cannot substitute for one another. The single-element variation method might choose nutrient set {C1,N1} as its seed nutrient, and vary the N source to produce nutrient set {C1,N2}. If this nutrient set failed to support growth, the method would erroneously conclude that N2 could never function as a nitrogen source.Clearly, Another problem with the single-element variation method is that it assumes exactly one nutrient is needed for each element, which might be false. Consider a metabolic system that is configured such that one set of nutrients can supply nitrogen to amino acids only, and an orthogonal set of nutrients can supply nitrogen to nucleotides only, with no possible flow of nitrogen between the amino acids and the nucleotides.E. coli transportable instance metabolites would expand to on the order of 107 potential four-compound combinations . If we consider that two distinct compounds might source C together, on the order of 109 five-compound combinations would be obtained. Since in general an organism might require more than one source of a given element, we cannot decide a priori the upper bound on the number of nutrients to consider.One might argue that such metabolic systems have never been observed in the natural world, so why should we build algorithms to detect them? We argue the converse: that if we do not build algorithms to detect them, we will never discover them from sequenced genomes, and given the incredible diversity of nature, such systems may well exist. For example, many genome sequences are in hand for parasitic microbes that have lost major components of their metabolic machinery. By using a novel algorithm built on top of a classical data representation (Binary Decision Diagrams), we can systematically search an otherwise intractably large space without making any assumptions about the independence of elemental sources. The full pool of 236 known Other efforts to use genome-scale metabolic models to determine minimal nutrient sets include -30. Eachall minimal nutrient sets. We demonstrate that the computation is feasible for large genome-scale models.In contrast to all work described above, we present a technique for computing Moreover, our approach for enumerating all nutrient sets is generic — it is independent of the underlying definition of growth and the solver used.Unlike FBA, our approach does not need an objective function because it is not based on solving an optimization problem.The various methods described above, including our approach, predict growth on minimal nutrient sets based on structural analysis of the metabolic network. All these methods are limited by the accuracy of the compounds and reactions modeled in the metabolic network, by their list of transportable nutrients, and by the specification of biomass compounds.E. coli K–12 MG1655 under anaerobic conditions. These minimal nutrient sets were predicted with 72.5% accuracy as evaluated by comparison with data from 91 growth experiments.We have described a method for computing alternative minimal growth media for an organism based on its metabolic network and transporter complement. The method combines linear constraint solving with binary decision diagrams. Whereas previous approaches to this problem did not consider all possible combinations of nutrients, our method does consider all such possibilities. Previous approaches assumed that all element sources are independent from one another, and that one source of each element only is required, whereas we show that in general these assumptions are invalid. Science is unlikely to detect organisms whose metabolic networks violate those assumptions unless we have computational methods that do not depend on those assumptions. A key aspect of our approach is the machinery-duplicating constraint, namely that all metabolites used in active reactions must be produced in increasing concentrations to prevent cell divisions from diluting these metabolites to the point that they are not available to the cell’s metabolic network. We validated our method by predicting alternative minimal nutrient sets of E. coli to aid researchers who are trying to study uncultivatable pathogens and environmentally sampled organisms, and to develop effective synthetic biology platforms. The ability to rapidly sequence and annotate such research targets must necessarily be complemented by the ability to quickly address potentially enormous research challenges such as “How do we grow this organism?” into tractable questions.The method that we present in this paper must next be applied beyond Clear areas for future enhancements to our method exist, beginning with developing a better understanding of the source(s) of the differences between our predictions and biological reality. The goal will be to develop tools to identify when discrepancies represent a problem with the method or a potential area of new study. These improvements will be built on the back of enhancements that make the method itself more computationally efficient, opening up the opportunity to include knowledge of regulation, metabolite concentrations, and other factors that will become more readily available as new high-throughput methods are developed.aNote that this cannot be formulated as a theorem since a theorem can only state properties about models of the real world rather than about the real world itself. Justification of claims in this section tacitly rely on a model of the world that is often implicitly assumed in biology and where the notion of discrete biochemical reactions makes sense. In particular organisms are assumed to be composed of molecules and a molecule is considered to be a discrete assemblage of atoms. Molecules are only transformed by biochemical reactions and those reactions must be balanced with respect to counts of each kind of atom. Atoms themselves are assumed to be indivisible, immutable, and conserved. We will maintain this fiction for the rest of this paper.bWe make the simplifying assumption that a given cell divides into two daughter cells that are identical at the molecular level. Of course in practice this is extremely unlikely; however, we really only require that the daughter cells are sufficiently similar in terms of their molecular composition. This vague notion of sufficiently similar could be made precise by the development of a formal mathematical measure of fission similarity based on the exact molecular composition of the cells in question and then we could formally prove the validity of our model for organisms that have fission similarity above a certain threshold. However, since determining the exact molecular composition of a given cell is well in advance of current experimental technique, the development of such a mathematical theory at the present time seems to us to be superfluous, and is certainly beyond the scope of this paper.cThe word feasibility is the standard terminology in the field of Linear Programming, while the word satisfiability is the standard terminology in the field of Computational Logic, where SMT solvers were developed.dBecause pig encodes the prime implicants of a monotone function it can never happen that we have two prime implicants that differ only on the value of a single component. Consequently, every variable must occur as the label for some node on every path from the root node to the true terminal, which slightly simplifies the extraction of the implicants of pig.The authors declare that they have no competing interests.SE designed and implemented the minimal-nutrients algorithm and authored sections of the manuscript. MK contributed to the design of the algorithm; he ran the algorithm and analyzed its results; he updated the EcoCyc metabolic network to incorporate new information that was found as a result of analyzing algorithm runs; he authored sections of the manuscript. AGS contributed to the design of the algorithm and the analysis of its results, and authored sections of the manuscript. AT contributed to the design of the algorithm and authored sections of the manuscript. IMK contributed to the analysis of runs and updating of the EcoCyc metabolic network. CT contributed to the design of the algorithm. PDK contributed to the design of the algorithm and the analysis of its results, and authored sections of the manuscript. All authors read and approved the final manuscript.The set of EcoCyc biochemical reactions, .Click here for fileThe set of transportable compounds, .Click here for fileThe set of biomass metabolites, .Click here for fileThe set of auxilliary compounds that must be present for the model to run.Click here for fileAll minimal nutrient sets computed by our program.Click here for fileThe nutrient equivalence classes.Click here for fileThe reduced minimal nutrient sets.Click here for fileGrowth status for each compound.Click here for file


Genome-scale metabolic network models have contributed to elucidating biological phenomena, and predicting gene targets to engineer for biotechnological applications. With their increasing importance, their precise network characterization has also been crucial for better understanding of the cellular physiology.framework for network modularization and Bayesian network analysis (FMB) to investigate organism’s metabolism under perturbation. FMB reveals direction of influences among metabolic modules, in which reactions with similar or positively correlated flux variation patterns are clustered, in response to specific perturbation using metabolic flux data. With metabolic flux data calculated by constraints-based flux analysis under both control and perturbation conditions, FMB, in essence, reveals the effects of specific perturbations on the biological system through network modularization and Bayesian network analysis at metabolic modular level. As a demonstration, this framework was applied to the genetically perturbed Escherichia coli metabolism, which is a lpdA gene knockout mutant, using its genome-scale metabolic network model.We herein introduce a After all, it provides alternative scenarios of metabolic flux distributions in response to the perturbation, which are complementary to the data obtained from conventionally available genome-wide high-throughput techniques or metabolic flux analysis. Genome-scale metabolic network models have proved to be useful in aiding biological and biotechnological research by providing large-scale predications, and their scope of applications is still expanding . They prReconstruction of metabolic network models has been accompanied by systematic characterization of their network properties for appropriate analysis of the prediction outcomes . Some noClustering has succeeded in revealing key components and their correlations, contributing to our understanding of biological systems through a systemic concept of module ,17. Moducausal relationship, in the form of a directed acyclic graph, and has successfully been applied to reverse engineering of various biological networks from large datasets, including transcriptome and proteome data [BN analysis is increasingly adopted to extract useful information from messy high-throughput biological data ,24. It iome data -27. BN aframework for network modularization and Bayesian network analysis (FMB) in order to investigate the effects of specific perturbation on metabolism and to explore other complementary features of the perturbed cell in the context of metabolism, which other currently available high-throughput techniques do not provide. This framework analyzes a biological system subjected to the specific perturbation by clustering reactions of similar functions (or metabolic flux variation pattern), and graphically displaying direction of influences among these clusters as a result of the specific perturbation. For this, FMB employs constraints-based flux analysis, hierarchical clustering and BN analysis in sequence. For a demonstration, FMB was applied to the genetically perturbed Escherichia coli metabolism using its genome-scale metabolic network model [With these tools, we herein conceptualized a rk model . The fincore reactions were selected, which are most influenced by specific perturbation, and yet almost always active in the observed samples of either control or perturbed condition [representative reaction from each module, which is the most influential one in the module, leaves adequate number of reactions that can be subjected to BN  [13C-based MFA are considered to be more reliable than those purely predicted by constraints-based flux analysis. Reactions constrained with experimental data are listed in Table Generation of genome-scale metabolic flux data in the first step of the FMB aims at two purposes: the first is to calculate reliable genome-scale metabolic fluxes under specific conditions, and the other is to get a large number of data samples for statistically reliable conclusions through BN analysis. First, fluxes for the branching reactions were constrained with metabolic flux data from is (MFA)  in order13C-based metabolic flux and cell culture data constraining the reactions, and constraints-based flux analysis was iteratively performed with these randomly adjusted experimental constraints in order to generate a large number of data samples; as a result, constraints-based flux analysis produces a different solution every time it is performed, satisfying the requirement that statistically reliable BNs demand many observations of the target system (see Methods). Infeasibility of optimization problem from applying too many constraints was handled with least absolute deviation method (see Methods). In this study, a total of 2,000 samples were generated at the end, each 1,000 sets of data from the wild-type (control condition) and its genetic mutant (perturbed condition), respectively. The resulting matrix, N-by-M, is named flux matrix in this study, where N is the number of reactions and M is the number of samples; M is 2,000 in this study , 6 (chorismate mutase), 9 (asparagine synthetase), 12 (phosphate transporter), 18 (NAD kinase) and 29  Figure . Unlike r weight . AppearalpdA [E. coli resulted in increased biomass formation, which, in some part, experimentally supports the correlation between cluster 1 and 29 [lpdA knockout mutant cannot efficiently generate energy for biosynthesis of biomass components, compared to the wild-type, due to the blocked pyruvate dehydrogenase that hinders further oxidation of glucose for oxidative phosphorylation . First, reactions involved in clusters 1 to 44 had average essentiality of 0.744 and 0.748 for the wild-type and y Figure , indicatclusters . On the Taken together, FMB provides theoretical evidences for the possible correlations among metabolic modules in response to the perturbation, as exemplified by cluster 1, 29, and 44 in this study. This study would consequently reveal additional pictures of the cellular physiology that could complement conventionally available biological information.FMB developed in this study groups functionally similar reactions, and subsequently investigates the causal patterns among metabolic clusters in response to specific perturbation. This newly generated knowledge of causal patterns enables us to capture significant changes in metabolism at the level of metabolic modules, which distinguishes itself from conventional MFA. Hence, it could be used as another tool for examining the microbial physiology in addition to currently available genome-wide high-throughput techniques . FurtherOverall scheme and rationale for each step of FMB are elaborated in Results and Discussion Figure  while itE. coli was used [In this study, a previously reported genome-scale metabolic network model of was used . This mowas used  and GAMSlpdA mutant (perturbed condition), separately. For application of FMB to E. coli metabolism, 13C-based metabolic flux data were adopted from Li et al. [lpdA mutant (perturbation) for genetic perturbation. All constraints were from the continuous culture of E. coli at dilution rate of 0.2 h-1 under the defined minimal medium with glucose  and i et al.  for wildconv) and experimental fluxes with randomly assigned errors (kv and bv), namely 13C-based metabolic flux and cell culture data, for the reactions catalyzed by the aforementioned enzymes, such that metabolic fluxes were reconciled to give realistic metabolic flux distributions. It should be noted that random errors are required to be within the standard error of the experimental measurements. This whole procedure is mathematically formulated as follows:The simulation by optimization often produces infeasible solutions when a large number of constraints are used . There cS is a stoichiometric matrix that consists of metabolite i in reaction j (i x j), and v is a vector showing the flux of reaction j (j x 1). α and β are lower and upper bounds for the flux of reaction j. K, a subset of J, is a set of reactions constrained with 13C-based metabolic flux data, and B, the other subset of J, is a set of reactions constrained with the rates of cell growth and metabolite secretion from the continuous cell culture. conv is a vector of fluxes calculated from constraints-based flux analysis whose experimental values are available. 13Cv is a vector of intracellular fluxes obtained from 13C-based MFA, and fmtv is a vector of the cell growth rate and glucose uptake rate from the continuous cell culture. kv and bv are constrained to be randomized values, being equivalent to 13Cv and fmtv with randomly added errors, respectively. Finally, random is a function that generates a random number according to normal distribution with mean μ and standard deviation σ. In this study, mean was set to the flux value from 13C-based MFA, 13Cv, or the rates of cell growth and glucose uptake, fmtv, and standard deviation was differently set for each reaction according to the measurement error reported in the literature [lpdA mutant (perturbed condition), yielding a total of 2,000 sets of data. Flux data generated for both conditions were merged into a flux matrix, so that the directionality of causal relationships can properly be inferred by BN analysis [where terature . Above oanalysis .thm column) in the reduced flux matrix was compared to that of its following sample (m+th1 column), and ‘1’ is given in the position  of flux-pattern matrix if the flux value of the reaction in  is increased in  of the flux matrix. ‘-1’ is given if the flux value is decreased in the following sample, and ‘0’ is given for the same flux values in the thm and m+th1 sample  into flux-pattern matrices, adjoining flux-pattern matrices and hierarchical clustering were conducted in Matlab (Mathworks). For hierarchical clustering, the “clusterdata” command was used. Euclidean distance, average linkage and cutoff value of 1.0 were used as calculation options.iX is conditionally independent of the set of all its non-descendents, given its parents iPa in the graph [BN is a probabilistic graphical model, which represents the causal relationships among variables, and it is important that its graph be acyclic. Nodes in the probabilistic graphical model correspond to random variables, which represent the fluxes of biochemical reactions in metabolic network, and its directed arcs from a variable to another indicate the direction of influences between them. In particular, node at tail of the directed arc is called “parent” and node at head of the arc is called “descendent.” Such statistical dependence is characterized by Markov condition, which states that each variable he graph . Based oOne of the goals of this study is to infer such BNs from metabolic flux profiles. All the BN analyses were performed with commercially available software BayesiaLab  ,45. FirsI is represented as:BN analysis and MI were employed to identify a representative reaction for each metabolic module. BN analysis was first applied to reactions in each module, so that BN of each module was inferred, namely local scale BN. MI was then employed to assess the degree of influence that each reaction exerts on other reactions. MI is a concept adopted in information theory, which calculates the mutual dependence between two random variables . The higp is a joint probability mass function, and p(x) and p(y) are marginal probability mass functions. Let y be a child node and x be its parent node. Then, the MI between x and y describes the mutual influence between the two. Specifically for this work, we extended the concept of MI to TMI [I to TMI ,32 to meI to TMI . To ident is the target node and rn1,…, are the remaining nodes in the BN. This procedure was repeated by selecting other remaining reactions as the target node and calculating the TMIs likewise. At the end, a reaction with the highest TMI value was identified as the representative reaction. Representative reactions determined from each local scale BN were then subsequently subjected to another BN analysis altogether to infer causal relationships among metabolic fluxes on a global scale. From this procedure, global scale BN is completed, consisting of representative reactions for each metabolic module, thereby explicitly revealing possible influences existing among them.g∆/g), wherein g∆ and g are the calculated cell growth rates of the wild-type and gene knockout mutant, respectively [Essentiality for each reaction was obtained by calculating the cell growth rate with that reaction removed using constraints-based flux analysis. The calculated cell growth rate for each reaction was then scaled (= 1 – ectively . EssentiBN: Bayesian network; FMB: framework for network modularization and Bayesian network analysis; MDL: minimum description length; MFA: metabolic flux analysis; MI: mutual information; PEP: phosphoenolpyruvate; TMI: total mutual information.The authors declare that they have no competing interests.TYK and SYL conceived the study and directed the project. HUK implemented the framework, and HUK, TYK and SYL analyzed the data. All authors read and approved the final manuscript.Table S1. Detailed information of clustered core reactions and their duplicate reactions in E. coli metabolism perturbed with lpdA knockout.Click here for fileFigure S1. Local scale Bayesian networks of clusters from wild-type and lpdA mutant of E. coli.Click here for file


Human Recon 1) coupled with abundant high-throughput data now makes possible the reconstruction of tissue-specific metabolic models. However, the number of available tissue-specific models remains incomplete compared with the large diversity of human tissues.Human tissues perform diverse metabolic functions. Mapping out these tissue-specific functions in genome-scale models will advance our understanding of the metabolic basis of various physiological and pathological processes. The global knowledgebase of metabolic functions categorized for the human genome . mCADRE is able to infer a tissue-specific network based on gene expression data and metabolic network topology, along with evaluation of functional capabilities during model building. mCADRE produces models with similar or better functionality and achieves dramatic computational speed up over existing methods. Using our method, we reconstructed draft genome-scale metabolic models for 126 human tissue and cell types. Among these, there are models for 26 tumor tissues along with their normal counterparts, and 30 different brain tissues. We performed pathway-level analyses of this large collection of tissue-specific models and identified the eicosanoid metabolic pathway, especially reactions catalyzing the production of leukotrienes from arachidnoic acid, as potential drug targets that selectively affect tumor tissues.This large collection of 126 genome-scale draft metabolic models provides a useful resource for studying the metabolic basis for a variety of human diseases across many tissues. The functionality of the resulting models and the fast computational speed of the mCADRE algorithm make it a useful tool to build and update tissue-specific metabolic models. Metabolic dysfunction has been implicated in a wide variety of human diseases such as obesity, diabetes, inborn errors of metabolism, neurodegenerative diseases, and cancer. The recent reconstruction of genome-scale models of human metabolism ,2 providMYC oncogene-induced liver tumors show increased glutamine uptake, while MYC-induced lung tumors show glutamine secretion31]. All ded from . To faciRecon 1), with most models ranging from 1000 to 1300 reactions  and undergo significant de novo fatty-acid synthesis [FASN has been identified as a drug target in many tumors [ACS), which acts downstream of FASN and converts long-chain fatty acids to acyl-CoA. Fatty acid activation is a critical step in several lipid metabolic pathways, including phospholipid and triacylglycerol biosynthesis. Some genes in this pathway  are overexpressed in certain types of cancer and inhibition of these genes induced apoptosis in cancer cells [Additionally, tumors overexpress fatty-acid synthase  and compared values for the above pathways in brain vs. non-brain and tumor vs. normal tissues. As shown in Additional file As a comparison, we also calculated the enrichment statistic for these brain and tumor enriched pathways using only expression data. Using the gene-reaction-pathway annotation from individual reactions that occur significantly more frequently in tumor tissue models than in normal tissues. Interestingly, the top most differentially included reactions  metabolic model with the existing reduced kidney metabolic model (kidneyReduced) from Chang et al.[kidneyTSEM is a genome-scale metabolic model with 1416 reactions, kidneyReduced aims to capture the core kidney metabolic phenotype and only includes 443 reactions. First, we compared the 578 and 34 gene-associated reactions unique to each model that also have protein staining evidence  capable of genome scale tissue-specific metabolic network reconstruction was published. This method was used to build genome-scale metabolic networks for 69 human cell types and 16 cancer types, collectively referred to as the Human Metabolic Atlas (HMA) . The funThere are certain important differences between the two methods. INIT primarily uses the evidence from the Human Protein Atlas (HPA) as input, but can also incorporate gene expression and metabolomics data; algorithm parameters  are also optimized for HPA data . mCADRE We validated mCADRE by showing that the mCADRE-built liver model can simulate a wide range of liver metabolic functions  built by mCADRE and the Human Metabolic Atlas built by INIT. Overall, TSEM included 126 models and HMA included 85 models. HMA included 16 cancer models while TSEM included 26 cancer models; 11 are shared. There are 100 and 69 normal tissue/cell types in TSEM and HMA respectively. Overall, the two collections shared 21 normal tissues (counting multiple cell types of the same tissue in HMA as one single tissue), with 30 unique to HMA and 79 to TSEM. One main difference is the coverage of tissues in the brain. While HMA included 8 models covering 3 brain structures (each brain structure has 2 or 3 cell-type specific models), TSEM included 30 models covering 30 distinct brain structures. Thus, the two collections of tissue-specific metabolic models are largely complimentary, with TSEM covering many more tissues.Large amounts of data have accumulated in public data repositories, characterizing the molecular phenotype of a variety of human tissue and cell types across a wide range of pathophysiological states . Howeverbetween different tissues and cell types play important roles in health and disease [Importantly, metabolism is under extensive transcriptional regulation . Mutatio disease -55, and  disease ,19. The The majority of the automated reconstruction pipeline in mCADRE, including the MAS5 detection call, is implemented in Matlab, and the pipeline produces genome-scale draft metabolic models from raw expression intensity files. To validate this pipeline, we built a liver model with MAS5 as the binarization method and compared it to a liver model constructed with MBA. Aside from the hepatic functional testing of liver models, all steps described below were subsequently applied to generate context-specific models for 126 different human tissues. Note that the Gene Expression Barcode project already used the barcode method to produce binarized transcriptomic data for the 126 tissues, so MAS5 was not used in this case. As the barcode binarization tends to be more stringent in calling a gene expressed than MAS5 , the resHuman Recon 1) to a context-specific subset; the SBML file for Recon 1 was obtained from the BiGG database [g∈G in samples n∈N for a selected context or phenotype is represented as the expression matrix XG|×|N||, where Xg,n = {0,1} represents the presence of gene g in sample n; for our liver training data, |G| = 20,283. There are 54,613 probe sets on the Affymetrix U133Plus2 platform . Only probes that can uniquely map to a single gene are retained; these probes map to 20,283 unique genes. When multiple probes map to the same gene, the maximum expression value is used.This new method uses gene expression microarray data as input evidence to prune a generic model ) to deterministically evaluate which reactions to keep or remove when pruning to get a context-specific network. We first calculate the expression-based evidenceEx(r) for all reactions to provide an overall ranking and to divide reactions into core and non-core sets. Next, the network topology of the generic model is used to calculate the connectivity-based evidenceEc(r) for each non-core reaction; this provides a second level of evidence when determining the order of reactions to remove during pruning. Finally, if expression- and connectivity-based evidence is insufficient to determine the rank of a reaction, evidence based on confidence level in the generic model El(r) is considered.For each reaction U(g) for each gene g:After binarizing the input data, we first quantify how often a gene is expressed across samples of the same context; this is the ubiquity score Ex(r) for reaction r is a function of how often its associated genes gr∈Gr are expressed in the selected context, as measured by the ubiquity score:This score ranges from 0 (not expressed in any context samples) to 1 (ubiquitously expressed in context samples). According to gene-reaction rules, ubiquity scores for metabolic genes are mapped to corresponding reactions. That is, the expression-based evidence Gr and Ex(r), denoted by f, is a composite of the Boolean gene-reaction rules defined in the generic model: AND is replaced with MIN, while OR is replaced with MAX, following ref [Ex(r) also ranges from 0 to 1, indicating how likely the reaction is to be present in the selected context. The high-confidence core set of reactions is then defined as those with Ex(r) > 0.9 when building liverCADRE with MAS5 call binarization, and Ex(r) > 0.5 when building the 126 tissue models with the Barcode binarization. Higher cutoff is used for MAS5 call binarized data, as it is less stringent than Barcode in calling a gene expressed. Reactions with Ex(r) = 0 are defined as the negative reaction set: these reactions have strong evidence of not being active in the tissue context.The relationship between the ubiquity scores of wing ref  . This score is particularly designed to rank non-gene-associated reactions, which account for 40% of all reactions in Recon 1, and by definition, will not be in the core because they are not associated with expression data. The connectivity-based evidence for each non-core reaction accounts for both the expression-based evidence and connectedness of all adjacent reactions (core or non-core). Using the stoichiometric relationships defined in the S matrix, we can describe whether any two reactions in the generic model are connected  with the binary adjacency matrix AR|×|R||. Specifically, Ai,j = {0,1}, where 1 indicates reaction i is connected to reaction j.For non-core reactions, we use network topology to define a secondary metric called influence I(r) of each reaction as its normalized connectedness to all adjacent reactions. That is, for each reaction r,We consider the outgoing r exhibits influence on all other reactions j∈R/r that is inversely proportional to the number of reactions to which it is connected. Furthermore, we measure the weighted influence WI(r) = Ex(r)×I(r) such that r will exhibit stronger influence on connected reactions if it was found to have strong expression-based evidence; reactions with Ex(r) = 0 thus have no weighted influence on adjacent reactions.In this way, Ec(r) as the net incoming weighted influence to reaction r from all other reactions j∈R/r:Finally, we define connectivity-based evidence rj is connected to a highly expressed reaction ri that has few other connections, this provides strong support for its inclusion in the context-specific model. Conversely, if a core reaction ri is connected to many other reactions, then it is less clear whether any particular connected non-core reaction rj is the one that functions in a pathway with ri in the pruned network; as such, the resulting connectivity-based evidence for rj will be lower.If a non-core reaction Human Recon 1. The confidence level evidence El(r) for a reaction ranges from 1 (in silico modeling evidence only) to 3 ; midlevel scores (2) indicate some physiological evidence, or experimental support from a related organism, and a score of 0 indicates that the reaction was not evaluated. Importantly, these confidence scores represent evidence for the generic model, not for the specific context, and thus are considered as a tertiary measure of evidence for non-core reactions.Confidence scores indicate the level of biological evidence associated with each reaction, as determined during manual curation of the generic metabolic model—in this case, Ex(r) =0) are needed to enable flux through the high confidence core reactions; (ii) by removing the high confidence core reactions, more non-core reactions (including those in the negative reaction set) will be removed. Consistency of the core reaction set is confirmed by calculating the maximum and minimum flux for each reaction, and ensuring that at least one is non-zero. As the naïve implementation of flux variability analysis (FVA) is extremely slow, we adapted the checkModelConsistency module described by Jerby et al. in [After defining the high-confidence core and ranking all non-core reactions, our algorithm attempts to sequentially remove each non-core reaction, starting from those ranked at the bottom (lowest evidence). The selected reaction will be removed only if (i) the core set of reaction remains consistent; and (ii) removal does not prevent model from producing any key metabolites. Reactions in high-confidence core set can only be removed when (i) reactions in the negative reaction set  is unconstrained, except ammonia: as ammonia detoxification is an important hepatic function, only ammonia influx is allowed. The simulation objective function is to maximize the uptake of the amino acid being tested. Using FVA, if the model can allow for finite urea efflux and urea efflux. The ethanol detoxification test is the same as ammonia detoxification test, except that no urea efflux is required and ethanol is constrained to be influx.Similarly, in the ammonia detoxification test, only glucose uptake is allowed, and the objective is to maximize ammonia uptake. This test is passed if the model can allow for finite ammonia influx In the glucogenic test, the model is only allowed to uptake the glucogenic substrate being tested while all other organic compounds, including glucose, are constrained to be efflux only. Ammonia is only allowed to be influx, and urea is only allowed to be efflux. The simulation objective is to maximize glucose secretion. This test is passed if the model can allow for finite glucose efflux. Glucogenic substrates tested are the 18 glucogenic amino acids , lactate, pyruvate, and glycerol.Recon 1 lacks a reaction accounting for the formation of glycogenin, the primer for glycogen synthesis, so a sink reaction for glycogenin is added to all the liver models to allow for glycogen synthesis.In growth simulation, the widely-used RPMI-1640 tissue culture medium was used. Detail of medium composition is in Additional file The authors declare that they have no competing interests.YW and NDP designed research; YW and JAE performed research; YW, JAE, and NDP analyzed data; and YW, JAE, and NDP wrote the paper. All authors read and approved the final manuscript.Codes and input data to run mCADRE in a zipped file.Click here for fileAll 126 TSEM metabolic model files in a zipped file.Click here for fileTable S2. Includes a sensitivity analysis of the inactivated core to non-core ratio. Table S3. Includes GEO accession numbers for microarray samples used to build liverCADRE. Table S4. Includes the coverage and functionality comparison of liverCADRE, the original liver, MBA model, and the new liver MBA model (based on the same training data as liverCADRE). Table S5. Includes the GEO accession numbers for microarray samples used in independent comparison of liverCADRE and liverMBA. Table S6. Includes the Human Protein Atlas staining data for 560 metabolic genes. Tables S7-S9. Include the detail results of liver metabolic function tests. Table S10. Includes the biomass composition. Table S11. Includes the RPMI-1640 medium. Table S12. Includes the enrichment of pathways listed in Table Table S13. Includes comparison of protein staining evidence for unique gene associated reactions in kidneyReduced and kidneyTSEM. Table S14. Includes kidneyTSEM simulation results of 41 renal metabolic functions. Table S15. Includes kidneyTSEM simulation of effects of genetic perturbations on renal metabolic functions.Includes a list of universally important metabolites that must be produced from glucose. Click here for fileInconsistent reactions in each of the 126 TSEM models.Click here for fileliverCADRE in SBML format.Click here for file


Performance of industrial microorganisms as cell factories is limited by the capacity to channel nutrients to desired products, of which optimal production usually requires careful manipulation of process conditions, or strain improvement. The focus in process improvement is often on understanding and manipulating the regulation of metabolism. Nonetheless, one encounters situations where organisms are remarkably resilient to further optimization or their properties become unstable. Therefore it is important to understand the origin of these apparent limitations to find whether and how they can be improved. We argue that by considering fitness effects of regulation, a more generic explanation for certain behaviour can be obtained. In this view, apparent process limitations arise from trade-offs that cells faced as they evolved to improve fitness. A deeper understanding of such trade-offs using a systems biology approach can ultimately enhance performance of cell factories. Escherichia coli, by far the most widely studied microorganism, is an excellent work-horse for the production of several high value products , fungi (Aspergilli) and mammalian cell lines, each utilized for the production of a wide range of products that are directly or indirectly an inherent part of our daily lives.Among the several microorganisms used in the food and biotechnology industry, ts Table . Other wNot all of these organisms had the complete set of desired traits to start with. Multiple methods are employed to obtain the preferred properties, including evolutionary engineering, classical mutagenesis and screening, rational and reverse metabolic engineering, global transcription machinery engineering or genetic modification Nevoigt , and morfitness. Most often, the strain properties necessary for industrial production processes are not the same as those that enable the cell to attain maximal fitness. Hence, identifying the selection pressures and strategic decisions that microorganisms can make, will help in tuning their environment so as to align their cellular objectives with the production process objective, and ensure constancy in biotechnological applications.Industrial strains need to have and retain the required properties to maintain high production rates. However, the one process that none of these strains can evade is their evolution, governed by their “fitness” in the respective environments. Microorganisms are subject to selection and the selection pressure is often on specific growth rate. In a fermentor the fastest growing strain produces the most progeny and therefore is likely to invade most of the population. How well microorganisms flourish in terms of competing with other strains, is called their The end result of microbial physiology is a direct consequence of adaptations that improve fitness, which can be mimicked in silico by adopting some optimality criterion for a microorganism in its environment. The premise of this approach is that cells will adapt, often surprisingly fast, and move towards some optimal fitness if cultivated under constant conditions. Such an in silico optimality approach has been used frequently over the years, and is often also disputed: microorganisms might not be optimal for specific tasks. At the end of this section, we will show a counterexample of this optimality assumption.E. coli were explained by optimization of growth within stoichiometric constraints  and some capacity constraints  to bound fluxes through the network. New approaches which apply additional constraints routed in physics and chemistry have to be used to understand metabolic strategies that FBA cannot explain. Beg et al.  for the E. coli strain communities. Their results suggest that in certain communities, cross-feeding enhances the growth rate of participating species. The main issues in all of these approaches, that are currently being actively investigated, are how to balance fluxes that are catalyzed by species with different abundances in the population, and what would be a realistic objective for such a community. In summary, multispecies metabolic modelling is an emerging field that aims to quantify metabolic interactions, identify trade-offs and to provide insights into the impact of different substrate availability on species abundance in microbial communities. Some powerful approaches are starting to develop and are getting ready for use in biotechnological applications.Microorganisms seldom live in isolation and analysing single species metabolic networks in isolation provides little insight into microbial interactions in communities. Consequently there have been recent efforts to model competition, co-existence, and strain and species interactions using multispecies stoichiometric metabolic modelling. Zomorrodi and Maranas  recentlyLactococcus lactis excretes an extracellular protease to degrade milk proteins into free utilizable peptides, a feat required when the peptides in the environment are insufficient for growth. Under these conditions, one would intuitively expect this trait to be selected for. However, this protease is extracellular and the peptides produced do not merely benefit the cell secreting the protease, but in part also diffuse away from it, becoming accessible to neighbouring cells. To grow well, it would indeed be beneficial if all cells produce this protease, but imagine a scenario where one cell does not. This “cheater” cell will still consume peptides released by neighbouring cells but will have more resources  available for growth and reproduction. This, on average, will lead to more progeny and a spread of the protease-negative trait in the population. In fact, it was shown experimentally that this leads to a population that completely loses the protease-positive trait and depending on the conditions, grows much slower  stable. In one such example, constraints (to obtain a certain flux some intracellular space is required) and limits (there is a limited amount of intracellular space), which necessitate a choice between oxidative phosphorylation and aerobic glycolysis and this we call a trade-off. Essentially, trade-offs call for a choice between two incompatible features, either of which if chosen, automatically leads to forfeiting the other. There are several biological examples of trade-offs: cells can invest in growing bigger or producing new cells, cells can be optimized for their current environment, or be prepared for possible future changes, just to mention a few.In the previous section we discussed a modelling framework (FBA) using empirically derived uptake flux constraints and additionally an intracellular space constraint. The latter results in a shift, from efficient use of potential chemical energy in the substrate through oxidative phosphorylation to inefficient use through aerobic glycolysis, in a model of human cancer cells  metabolism during fast growth. Above a critical growth rate and corresponding glucose concentration, 2 for fluffy breads. However, during the initial start-up or growth phase of the fermentation process as well as for starter culture suppliers, the aim is to maximize biomass production without compromising adequate functionality of the resulting strain. Thus growing fast with high biomass yields versus achieving high levels of end products represents a trade-off.Trade-offs in industrial processes are not uncommon either, the most classic one being the choice between batch and continuous fermentation. Batch fermentations bear a lower contamination risk and a higher cost due to additional cleaning cycles, whereas continuous fermentations offer the advantages of steady-state operation, longer runs with shorter downtimes, better product consistency, easier process control, and steady utility demands  or the negative impact of efficient metabolism. The following explanations all assume a trade-off between growth yield and growth rate. Subsequently, if the selection pressure acts on growth rate, only inefficient pathway usage is expected to prevail, simply because it is faster. Under such presumptions, the use of Modelling efforts show that the existence of spatial structure in a population (due to incomplete mixing or biofilm formation) can select for efficient metabolism , the self-replicator model predicts that at low substrate concentrations efficient metabolism leads to a higher growth rate, and at high substrate concentrations inefficient metabolism leads to a higher growth rate (Fig. Molenaar et al.  hypothesate Fig. g. This aE. coli using various network and optimization tools to link protein investment and transcriptional regulation of pathways. With this integrative approach they identified and suggested an evolutionary trade-off between protein investment and rapid response time. From the industrial perspective, there have been quite a number of successes in systems metabolic engineering combining systems biology, synthetic biology and evolutionary engineering principles (Lee et al. It remains a challenge to validate or falsify the hypotheses described in the previous section. Many of them look at only a specific aspect of metabolism. Nevertheless, these hypotheses call for an integrative approach, since fitness-associated costs are a systems property and cannot be inferred by studying a single component in isolation. Even then, efforts to approximate the costs of protein synthesis (Dekel and Alon E. coli (Kotte et al. E. coli as it has been studied for decades and can boast of a rich source of detailed knowledge, unlike other microorganisms. This necessitates multi-level omics studies in the latter to be able to investigate them with realistic models. There is hope that we can translate such kinetic models developed for model organisms to less-well studied organisms through what we have called comparative systems biology (Levering et al. Finally and ultimately, a systems biology approach should connect environmental conditions to genes, transcriptional regulation, transcription factor interactions and protein production to metabolism in a single model. One such example exists that proposes cell regulation via flux sensing metabolites in We have discussed industrially-relevant examples of metabolic shifts exhibited by organisms, summarized the underlying regulatory mechanisms, emphasized the existence and role of trade-offs in these metabolic choices, and scrutinized various hypotheses and their pitfalls in explaining the fitness advantage of metabolic shifts. Systems biology, we believe, is the best approach we currently have to tackle such complexities of cell factories. Nevertheless, one must proceed with caution in the midst of current high-throughput data generation methods and avert sinking in oceans of data by regularly stepping back to recapitulate the greater objective. We firmly believe that the functional perspective, i.e. the contribution of the observed adaptive mechanisms to fitness, in the light of constraints and trade-offs, provides a powerful context to our understanding of the physiology of microbial cell factories. We are still quite at the tip of the iceberg but with constant consolidated systems biological efforts we can aim to reach a deeper understanding that will guide future major innovations in biotechnology and medicine.