We consider the possibility of engineering metabolic pathways in a chassis organism in order to synthesize novel target compounds that are heterologous to the chassis. For this purpose, we model metabolic networks through hypergraphs where reactions are represented by hyperarcs. Each hyperarc represents an enzyme-catalyzed reaction that transforms set of substrates compounds into product compounds. We follow a retrosynthetic approach in order to search in the metabolic space (hypergraphs) for pathways (hyperpaths) linking the target compounds to a source set of compounds.To select the best pathways to engineer, we have developed an objective function that computes the cost of inserting a heterologous pathway in a given chassis organism. In order to find minimum-cost pathways, we propose in this paper two methods based on steady state analysis and network topology that are to the best of our knowledge, the first to enumerate all possible heterologous pathways linking a target compounds to a source set of compounds. In the context of metabolic engineering, the source set is composed of all naturally produced chassis compounds (endogenuous chassis metabolites) and the target set can be any compound of the chemical space. We also provide an algorithm for identifying precursors which can be supplied to the growth media in order to increase the number of ways to synthesize specific target compounds.We find the topological approach to be faster by several orders of magnitude than the steady state approach. Yet both methods are generally scalable in time with the number of pathways in the metabolic network. Therefore this work provides a powerful tool for pathway enumeration with direct application to biosynthetic pathway design. Metabolism is the process of synthesis and degradation of molecules occurring in living organisms. Metabolism is generally represented as a network where metabolites are interconnected by reactions. In order to give a functional description of metabolism, metabolic networks are often decomposed into separated parts, called metabolic pathways. The description of metabolism through metabolic pathways is useful, even though any division in pathways is arbitrary, because it helps in modeling and understanding the behavior of the full network. A metabolic pathway can be defined as a coherent set of enzyme-catalyzed biochemical reactions by which a living organism transforms a set of source compounds into a set of target compounds. By regulating enzyme and protein synthesis, living organisms can adapt to different environments. This model of metabolism as composed by independent metabolic pathways is simplistic, since pathways are nested and interdependent. In fact, metabolism is a complex system and pathways interact with each other.E. coli K-12, as shown in Figure The aim of the work presented here is to find all the viable sets of heterologous enzymes, which can produce a predefined target compound when added to the pool of endogenous enzymes of a given organism. Our method enables a metabolic engineer to find all heterologous metabolic pathways producing a target compound, for instance liquiritigenin (a plant secondary metabolite with therapeutic applications), from the endogenous metabolites of As depicted in Figure We have already discuss elsewhere the first and third step ,2, i.e. S) representing the network (cf. [v) satisfying Sv = 0. One notices that extreme pathways and elementary modes while representing pathways  do not directly enumerate all pathways linking a source set to a target set of compounds. However as shown in the subsection "Enumerating pathways using the steady state approach" one can construct stoichiometric matrices where input fluxes are added to the set of source compounds and outgoing fluxes are associated to the target and heterologous coproducts such that the extreme pathways and elementary modes enumerated from these matrices do correspond to all pathways linking the source set to the target.Different mathematical models that describe metabolism have been proposed . Aside from the fact that integer linear programs suffer from computational complexity (MILP is an NP-hard problem) all the above methods search for at most k optimized (shortest) pathways and do not guarantee a full enumeration of the possibilities. In our methods the optimal pathways are computed in a post process by ranking the pathways that have been enumerated. Our approach allows one to decouple enumeration from optimization, and thus to plug any optimization criteria, including nonlinear functions and not only target yield or pathway length . Recently, some of the above topological methods have been benchmarked against the integer linear programming technique mentioned above [There exist standard graph-based methods to search and eventually enumerate pathways in metabolic networks, but these methods including PathFinding -16 and Pet al. ,15 cofacent work ,19, whiced above  to searcAs reviewed above, while there are methods and theoretical results to enumerate elementary modes or extreme pathways and graph based techniques to search for pathways in a given metabolic network, to the best of our knowledge there is no known methods to directly enumerate pathways in the context of metabolic engineering, that is, to enumerate all the pathways encompassing all substrates and products necessary and sufficient to produce a given set of target compounds from a given set of source compounds. In the present paper we address that specific problem and present two methods one based on elementary modes (steady state approach) and one based on a direct enumeration algorithm . In order to address this problem we need, in addition, to consider the problem of determining supplement molecules, i.e. metabolites that the organism cannot synthesize, but which can be added to the growth media in order to increase the number of viable pathways; and bootstrap molecules, i.e. metabolites which are required fist in order to be produced . While iE. coli. We "experimentally" probe the computational complexity of the steady state and topological approaches for a series of networks of growing sizes and discuss the theoretical complexity results of the topological approach, which are provided in Appendices A and B.The paper is divided as follows. In the Methods section we first provide some definitions, then outline our algorithms to solve the pathway enumeration problem with both the steady state approach and the topological approach. The problem of finding and enumerating all the pathways going from a large source  to a target chosen by the user is considered. All the algorithms presented for the topological approach  have polynomial worst-case running time, the algorithm for enumeration is a polynomial time per output algorithm on some classes of hypergraphs. We also provide algorithms to determine supplements, which are metabolites that the organism cannot synthesize, but which can be added to the growth media in order to increase the number of viable pathways. Furthermore, an analysis of pathways containing supplements allows finding out pathways that contain bootstrap molecules. In the Results and Discussion section we illustrate our algorithms with the enumeration of the possible pathways to synthesize more than 5000 compounds in E. coli, our methods can be applied to any chassis organism and more generally to any source set (for instance a set of nutrients or a set of abundant currency metabolites).While we illustrate our two methods for the production of heterologous compounds using as a source set all the endogenous metabolites of et al [In the context of metabolic engineering, metabolic networks have been represented as directed graphs . In sucIn the present paper to palliate the limitations of the directed graph model, we represent networks as directed hypergraphs. The first examples of modeling through hypergraphs can be found in . In a hyDefinition 1 (Hypergraphs and hyperarcs).A directed hypergraph is a pair = {v1, v2 ..., nv} is the set of vertices and E = {e1, e2,..., me} i is an ordered pair ei is the set of hyperarcs. A hyperarc e=  of disjoint subsets of V.i Xis also called the tail of i eand the set i Yis called the head, with reference to the graphical representation of arcs (directed edges) and hyperarcs as arrows.The set i ereturns its tail X(ie) ⊂ V. Analogously we use We denote by Definition 2 (Reactions and networks).In a metabolic network each vertex corresponds to a metabolite and each hyperarc corresponds to a reaction. A metabolic network of m metabolites and n reactions can be represented with a m × n stoichiometric matrix Sj , where the rows correspond to the m metabolites and the n columns to the reactions. A reaction j is represented by the column vector S= T where sij is the stoichiometric coefficient of metabolite i in reaction j. Reactants have negative coefficients and products have positive coefficients.X, Y) is ordered so to make the distinction between reactants and products. In this representation reactions are irreversible. Many biochemical reactions can be considered as irreversible, since in organisms the homeostatic equilibrium is often strongly polarized. Nonetheless, metabolic network may comprise reversible reactions, and we model these reactions by introducing both hyperarcs: , and .Examples of hypergraph, network, and stochiometric matrix are given in Figure E of hyperarcs defines a hypergraph ε = (∪e∈EX(e) ∪ ∪e∈EY (e), E). By abuse of the terminology we denote by E the hypergraph corresponding to the set E of hyperarcs and all the heads and tails of the hyperarcs in E. The following definition for hyperpaths is borrowed from Nielsen et al. [Hyperpaths, a generalization of simple paths in graphs where cycle free paths going from one vertex to another, are used to represent pathways. A hyperpath connects a source set of vertices to a target set of nodes. Two examples of hyperpaths are given in Figures n et al. .Definition 3 (Hyperpaths).P of P in a hypergraph A hyperpath P going from a source subset ⊆ V, EP ⊆ F of the hyperarcs EP with the following propertiesE, such that there is an ordering .• • a is not a hyperpath because neither reaction R1 nor R2 can happen until the other does not start. The definition (3), though complex, is computationally tractable, meaning that the time required to determine if a hypergraph is a hyperpath is proportional to the number of reactions. A polynomial time algorithm to determine if a hypergraph is a hyperpath is given in [FindAll presented below can also be used for that purpose. In fact, as discussed below, if the set of reactions returned by FindAll From the point of view of metabolism, the first condition corresponds to the requirement that reactants of reactions participating in the hyperpath can be produced without the presence of the reaction itself. Hyperpaths defined in this manner represent a metabolic route from the source to the target. According to definition (3) the hypergraph of Figure given in , the algThe metabolic network described by a hypergraph has to be as comprehensive as possible, containing every known enzyme-catalyzed reaction occurring in organisms. We say that a hyperpath produces a set of target metabolites if it contains all those target elements. A set of target compounds is said to be reachable from a given source, or linked to the source, if there is at least one hyperpath producing the targets.We are interested in the enumeration of pathways leading to the production of a desired compound. Hyperpaths do not generally give the best representation of pathways because hyperpaths can contain reactions not necessarily linking the target to the source. Minimal hyperpaths, cf. definition (4), are an appropriate representation of pathways since they contain only the essential reactions linking the source to the target.V ⊆ V' and E ⊆ E'. For instance the hyperpath of Figure In the definition given below, we say that a hyperpath Definition 4 .A hyperpath  with target TP is said to be minimal if it has no proper subsets with the same target.The target is disconnected from the source if a reaction is removed from a minimal hyperpath. In this sense minimal hyperpaths cannot be reduced. From a metabolic engineering perspective the concept of minimal hyperpath is useful as it defines the minimum set of reactions necessary to produce a target heterologous compounds, and consequently the minimum set of enzymes needed to be inserted into the chassis organism where the compound is going to be produced.In the following we define Supplements for a target are molecules whose presence in the source set increases the number of pathways for target production. Finding supplements is an important improvement when exploring ways to produce the target, since they make possible new pathways.FindSupp, an algorithm that returns the supplements.For each target of interest one can look for vertices that once inserted in et al [An analysis of pathways containing supplements allows to find out pathways containing bootstrap molecules, i.e. metabolites that are needed in reactions producing compounds afterwards used for the production of the bootstrap molecules. As a matter of fact, many pathways can be made viable once bootstrap molecules become available in the metabolic network (a concept introduced in ). Looselet al  stated tIn steady state, all possible pathways in a metabolic network are by definition stoichiometrically balanced, i.e. all metabolites produced from the source set must be consumed except for those that are target products. Extreme pathways and elementary modes are two methods that compute the set of independent non-decomposable pathways in the network that generate all feasible steady state solutions in the flux space. They do not directly enumerate all pathways linking a source set to a target set of compounds. However, one can construct stoichiometric matrices where input fluxes are added to the set of source compounds and outgoing fluxes are associated to the target and heterologous co-products such that the extreme pathways and elementary modes enumerated from these matrices can be used to generate all pathways linking the source set to the target.vP for the hyperpath where components Pj vcorresponding to those reactions in the pathway Given a hyperpath P Tis defined as stoichiometrically balanced if the rows corresponding to each metabolite i v∈ V that are obtained from the product of the stoichiometric matrix S and the associated flux vector vp verify:A hyperpath S additional columns corresponding to input reactions (reactions with no substrate that produce the source set PT). These auxiliary reactions, even if non-properly balanced in terms of the law of conservation of mass, are useful in order to define completely the problem in a compact manner:A way to introduce the constraint on input and output metabolites in the previous equation is by adding to the stoichiometric matrix v. Since in our hypergraph definition all reactions are irreversible, the set of pathways solving Equation 3 computed by both extreme pathways and elementary modes are identical . FurthePT, we need to constrain the computation of elementary modes only to those that have non-zero fluxes for heterologous reactions. Efficient solutions to this problem have been considered in the divide-and-conquer approach [V does not belong to the source set S is defined by reactions in In order to determine all stoichiometrically balanced heterologous pathways approach ,25 by rev for Equation 3, we are interested in enumerating all minimal hyperpaths from T on the hypergraph given by T are given by those cycle-free solutions in v containing only reactions linking the source to the target. Since any feasible flux pattern v is a superposition of elementary modes with non-negative coefficients [T is a subset of the elementary modes producing T that are solution of Equation 3. Namely, any feasible solution generated from the elementary modes will contain at least as many reactions as the ones that are in those elementary modes that form its basis. Therefore no additional minimal hyperpaths can be generated in this case by superposition of elementary modes.Finally, from the computed set of solutions ficients , the setFindAll that allows to find FindAll, by explicitly constructing the ordered set F in definition (3), provides a proof of the tractability of the problem of checking if a hypergraph is a hyperpath. Moreover FindAllF permits to prune the original hypergraph enabling a faster enumeration algorithm.The algorithm Minimize, when called on the output of FindAll, returns, if exists, a minimal hyperpath linking a given target to the source. These algorithms are the main components of the algorithm enumerating the pathways FindPath described next. Then we present FindSupp an algorithm to enumerate supplements.As presented below the algorithm n = |V|, m = |E| and let Let FindAll returns all the reactions that can contribute to the production of any element in FindAll is a linear algorithm in the number of vertices, hyperarcs and in the total coordination; the complexity is O(n + m +Σv∈V |X -1(v) |+ |Y-1(v)|) that is bounded by O(n + m + n · m). Therefore, such algorithm can be applied to the hypergraph E' is the set of reactions returned by FindAll. In the context of metabolic engineering FindAll returns all the compounds that can be produced from a given set of source compounds and reactions. For instance, using FindAll with all know metabolic reactions one can determine all the compounds that can be produced from the metabolites of E. coli.The algorithm Algorithm FindAll  ← X(r)2.    end for3. 4. 5. F ← {∅}6. while V ≠ {∅}7. i be an element of V8.    let V ← V \ i9.    D ← D ∪ i10.    for all r ∈ H such that i ∈ x(r):11.    x(r) ← x(r) \ i12.       if x(r) = {∅}13.       F ← {F, r}14.          for all j in Y (r) and not in D:15.          V ← V ∪ j16.             end for17.          end if18.       end for19.    end while20. output:F   D be the union of the source set and of the heads of all the reactions output in F by FindAll. The correctness of the algorithm above is given by the following claims: every element in D is the target of some hyperpath or is part of the source, and every vertex in D. For the first claim we can give a constructive proof by using the output vector F, the second claim is proved by contradiction.Let D is reachable from the source is given constructively by the ordered set F returned by the algorithm. In fact at each step F is a hyperpath. This claim can be proved by induction on the steps of the algorithm, each time a hyperarc r is appended to F (line 14) the tail X(r) is contained in D (hyperpath by inductive hypothesis) and if a vertex j is added to D (line 10) it means that it was previously added to V (line 16) and thus it was in the head Y (E) of some hyperarc already in the hyperpath.• The proof of the fact that every element in D there would be a hyperpath linking it to the source. In such hyperpath let consider the first (according to the order given by the definition) reaction r whose X(r) belongs to D and such that one of the elements of Y (r) does not. Consider among x(r) the last one that has been inserted into the set V ; after its removal from X(r) this set becomes empty and the elements of Y (r) are inserted into V (line 16) and then in D (line 10), which is a contradiction.• The second claim can be proved by contradiction: if an element of D and every hyperarc is an element of F. Thus the algorithm FindAll provides an effective pruning of the original hypergraph: in kFk or vertices not in From the above statements follows that each vertex appearing in a hyperpath having as source FindAll algorithm as presented above returns in polynomial time a hyperpath valid for each target vertex in FindAll is polynomial, the use of it does not affect the complexity analysis of the algorithms making use of its output.Notice that FindAll. Namely, given a hyperpath S to T, it is always possible to find a minimal hyperpath Minimize Remark that a minimal hyperpath going to a specific target can be easily extracted from the hyperpath output of Minimizef Rsubset of T and a source S. If T to S the empty set is returned, otherwise a hyperpath contained in f Rand linking T to S is returned. In particular, if f Ris empty, the output of Minimize is the minimal hyperpath going from S to T, provided it exists. Minimize returns a hyperpath obtained by removing all inessential hyperarcs except for the ones in fR. In the context of metabolic engineering, pathways containing a small number of heterologous reactions are generally preferred, since they are easier to engineer in the host organism. Therefore, given two pathways that produce the same target, where one is subset of the other, the one requiring the smaller number of heterologous reactions has to be selected. This is the reason that makes relevant to obtain minimal hyperpaths from generic hyperpaths.Algorithm Minimize 1. P' ← P2. if T ⊄ k Y ∪(kF)3. 4. else5. for all r in 6.    if r not in fR7.       F ← FindAll8.          if T ⊂ ∪k Y (kF)9.          10.             end if11.          end if12.       end for13.    end if14. output:   0 Pnot containing r does not produce the target. The worst-case time for this algorithm is Minimize returns a minimal hyperpath The proof of correctness of this algorithm is simple and is based on the fact that Minimize is the minimal constrained hyperpath problem: the problem of finding if a minimal hyperpath from a given source to a given target, containing the hyperarcs in f Rexists. Notice that Minimize, although linked to this problem does not solve it. In fact, if the output of Minimize is an empty set then there are no minimal hyperpaths satisfying the constraints; however if the output is a minimal hyperpath then obviously a minimal constrained hyperpath exists; and finally, if the output is a non-minimal hyperpath then we do not know if a minimal hyperpath satisfying the constraints exists or not.A related problem to Below we will discuss why we are interested in algorithms for the minimal constrained hyperpath problem, while in Appendix A.2 we show that in general the problem is NP-complete (reduction to 3-SAT).f Rand nR) of the original hypergraph. These sets are used during the enumeration process, f Ris a set of hyperarcs that must be present in the enumerated hyperpath and n Ris the set of hyperarcs that must not be part of the enumerated hyperpath. The problem of finding a solution in one of the parts is addressed at each iteration and if a solution is found the part is divided in finer parts. This process is repeated until all the minimal hyperpaths have been found.The basic idea behind the enumeration algorithm presented below is to introduce an iterative refinement of partitions of the space of feasible solutions i.e. of the space of hyperpaths and in each part to look for a solution. In our implementation, a part is defined by two sets of reactions  on the hypergraph A minimal hyperpath Once R4,• the hypergraphs which do not contain R4 and do not contain R3,• the hypergraphs which do contain R4 and R3.• the hypergraphs which do contain R4 is removed, v8 is disconnected from the source. The second set contains a solution and thus has to be partitioned. The third set contains only one minimal pathway  is partitioned in two parts defined by new sets of constraints. The way the partition is done is explained in detail in algorithm FindPath and gives two non overlapping sets:The minimal hyperpath in the second set is found by running R4 and do not contain R3 and R7.• the hypergraphs which do contain R4 and R7 and do not contain R3.• the hypergraphs which do contain v8: once R3 and R7 are removed, node v8 is disconnected from the source. The second one only contains the second and last minimal hyperpath: the one consisting of hyperarcs R7, R4. The algorithm here sketched is based on the fact that all minimal hyperpaths are found once the problem of finding a minimal hyperpath has been solved for each part of the partition.The first of these sets does not contain hyperpaths going to FindPath, which enumerates all the minimal pathways and does not output duplicate hyperpaths. Precisely FindPath  returns a set of hyperpaths containing all the minimal hyperpaths in T to fR. FindPath  returns all the minimal hyperpaths from the source T in The enumeration procedure is performed by the algorithm FindPath works for the enumeration of the pathways of liquiritigenin is given in Figure FindPath either a new hyperpath is found and then FindPath is executed with new constraints, or there are no new hyperpaths and the branching process is stopped. The new constraints sets f R', nR' for a new call of FindPath are obtained by incrementing the sets fR, n Rof the father process. Given an order for the hyperarcs of the hyperpath nR' relative to the child process is constructed by incrementing n Rby one element r belonging to f R' is constructed by incrementing f Rby all the hyperarcs coming before r. For each element in n Ra child process is called.A schematic representation of how FindPath  returns all the minimal hyperpaths from the source T in FindPath returns all the metabolic pathways for the production of the target compounds.Algorithm FindPath 2. En ← ∅3. if 4. 5.    6.       for all k in |F|,..., 1}7.       r = kF8.          if r not in f R:9.          10.             f R← f R∪ r11.             end if12.          end for13.       end if14.    output:En   FindPath is done according to the order given by line 6 where the hyperarcs are ordered so that at least one of the head vertices of each hyperarc is a tail vertex of some previous reaction. Such an ordering is always possible since indPath is an algorithm that iteratively calls itself, see line 10. Note that even if n Ris not explicitly defined in FindPath, it is constructed implicitly when at line 10 of FindPath is called on the smaller graph The loop at line 7 of Minimize while running in polynomial time can return a non-minimal hyperpath. An algorithm always returning minimal hyperpaths cannot be polynomial since the problem of finding a minimal hyperpath containing a set f Rof hyperarcs is an NP-complete problem as showed in Appendix A.2. However, in many practical instances (for instance when hyperarcs only have one head node), the algorithm Minimize returns a minimal constrained hyperpath. As a matter of fact, for all the enumeration studies we have so far carried out, we observed that the output obtained by Minimize when called by the algorithm FindPath introduced above was always a minimal hyperpath. Nonetheless, a characterization of hard instances of the minimal constrained hyperpath problem is given in Appendix.Let us note that the output of the enumeration is not always composed of minimal hyperpaths. This is due to the fact that the algorithm Provided a given metabolic network and a set of source compounds  it may not be possible to link all the metabolites of the network to the source set. When a target compound is not accessible from the source set, one can consider the possibility of inserting into the metabolism of the organism some precursors so that the target becomes reachable. In practice such a task can be carried out through the enrichment of the growth media. More generally, the insertion of supplements can be used even when the target compound is reachable in order to access to new pathways for the production of the target.T be any compound T. Below we give the algorithm FindSupp finding the supplements for the production of a given target. In Figure Algorithm FindSupp WishList ← T1. D ← {∅}2. while WishList \ D ≠ {∅}3. i be an element of WishList \ D4.    let D ← D ∪ i5.    Aux ← {∅}6.    for all reactions r with i ∈ Y (r)7.    8.       end for9.    WishList ← WishList ∪ Aux10.       end while11. 12. 13. output:Dv2, v3 of Figure FindPath algorithm.Bootstrap molecules relative to a source Algorithm FindBootstraps (Given a hypergraph B of bootstrap nodes)input:1. 2. 3. for all r in 4. r' ← (X (r) \ D, Y (r) \ D)5.    if Y (r') ≠ {∅}:6.    7.       end if8.    end for9. while exists v in 10. for all r' containing v:11.    r' ← (X(r'), Y(r') \ v)12.       if (Y(r') = {∅}) or (v ∈ X(r')):13.       14.          end if15.       end for16.    end while17. 18. output:   BFindBootstraps algorithm is linear in the number of vertices, hyperarcs and in the total coordination. Remark that the set The × 2.66 Ghz Quad-Core Intel Xeon, 16 GB. All the algorithms of the topological approach were implemented in Python. For the steady state approach we used two software products, one computing elementary modes and the other extreme pathways. Elementary modes were computed by using the MATLAB interface to the Java implementation of efmtool Version 4.7.1 [E. coli was chosen as source organism. The hypergraph E. coli and 5571 heterologous compounds. Each sampled hypergraph FindBootstraps defined in Methods and added to the growth media.To evaluate the performance of our topological approach , we have compared running times of this approach with the running times of the steady state approach presented in the Methods section. All our tests were run on a Mac Pro server 2 database . E. coliThe observed trend is consistent in the three cases, as it is shown in Figure 2 ~ 103-fold slower than FindPath. The execution times are on average of 0.0136 ± 0.0051 s per output with the FindPath algorithm, of 0.218 ± 0.042 s with elementary modes, and of 2.84 ± 1.24 s with extreme pathways. The observed execution times mean that for the compound with the highest number of pathways in our tests, the anti-diabetic drug acarbose containing 1513 pathways, enumerating the pathways with elementary modes takes more than 329.83 seconds, while it can be computed in 20.58 seconds with FindPath. We have measured running times per output and memory usage as function of input size and output size (see Table O(n) for both input size and output size. In the case of running times, FindPath and efmtool have running times per output approximately constant in function of the size of the input stoichiometric matrix S of 3.1 × 10-2 and 1.6 × 10-1 seconds, respectively, being ExPA the less efficient code with a constant time of 4.2 × 10-1 seconds and a linear growth of 1.6 × 10-7 seconds per input. Similar values were obtained depending on the size of the output, although the scaling in this case for ExPA was of 5.7 × 10-3 seconds per output. Regarding memory usage, ExPA and FindPath are less demanding, especially for output size, with linear growths of 3.9 and 54 kB per output, respectively, while efmtool required significantly more memory allocation, 3.3 × 104 kB per output.We also find that FindPath has the fastest execution time, the algorithm required less than 5 min to compute the full network. Elementary modes has an execution time approximately 10-fold slower than FindPath. The computation of extreme pathways, in turn, is the slowest one, being between 10To the best of our knowledge, the computational complexity of enumerating elementary modes on networks comprising irreversible reactions is up-to-date unknown . In AppeE. coli. As shown in Figure To further probe the computational complexity of FindPath we computed distribution of the run time for each of the 2338 heterologous targets that are linked to FindPath algorithm is ranked depending on this cost estimation. Once a cost function to minimize is introduced, the search for an optimal pathway can be formulated as a shortest path problem. The shortest path problem for weighted graphs consists of finding a path going from a given source vertex to a given target vertex while minimizing a cost function given by the sum of the costs of the arcs involved in the path. In order to define a shortest-hyperpath problem we need a definition of cost for hyperpaths in a weighted hypergraph . A natural generalization of the cost function for paths in graphs to hypergraphs is the sum of the costs of the hyperarcs contained in the hyperpath. If on the one hand this generalization seems to be natural, on the other hand the two problems have, however, not the same complexity. In fact the shortest-hyperpath problem with this cost function is known to be an NP-hard problem (see Appendix A.1).Another aspect of pathway design is the definition of a cost function that estimates limiting effects on production efficiency by multiple factors associated with genetic and metabolic engineering of the pathway, such as gene heterogeneity, metabolite toxicity, or steady-state fluxes. In order to facilitate the designer in the selection of the best synthetic pathways to be implemented each hyperpath enumerated by our ia, tails of a hyperarc j ewhose cost is jw, then the cost of the hyperpath containing all the hyperarcs in ∪iAi and j eis not equal to i Ais non trivial.The reason why the algorithms commonly used for graphs cannot be easily adapted for hypergraphs is that given the costs O(m · (n + ln m)) given in [B-hyperpaths. In fact, this algorithm does apply to minimal hyperpaths given in definition (4).Provided an additive cost function as defined in , findinggiven in , which fk solutions of a combinatorial problem, but it cannot be extended to the k-shortest problem on hypergraphs not even with an additive cost function (cf. definition in Appendix A.1), since when splitting the problem into a constrained part and a free one (by using the sets n Rand f R) we get a problem that is known to be NP-hard (see the proof in [k-shortest hyperpath on hypergraphs whose hyperarcs have only one head vertex).Generally speaking, the strategy we used in the enumeration algorithm (iterative partition of the space of feasible solutions) is often used to solve the problem of finding the first proof in  where ank-shortest path problem might be solvable efficiently if one is able to develop a specific cost function making the problem tractable. However such an artifact cost function may not be necessarily appropriate to practical problems, such as the production of heterologous targets using metabolic engineering. As an alternative we have proposed in this paper an enumeration algorithm, which systematically enumerates all pathways linking source compounds to target compounds. In all the cases that we have so far processed for metabolic networks, our algorithm runs in polynomial time per output, and therefore pathways can be ranked by the designer based on their own user-defined cost functions.Therefore, the In summary, the methods presented in this paper provide metabolic engineers with powerful tools that extend the toolbox for heterologous biosynthetic pathway design. Besides pathway enumeration of biosynthetic routes for a given target product, our methods have several other possible applications. For instance, they can be used in combination with gene deletion strategies in order to determine pathway manipulations leading to overproduction of the target compound. Another application is in biodegradation and bioremediation, where our algorithms would need to be slightly modified in order to reverse the pathway search so that it can identify degradation routes for a given compound, while the underlying structure of the algorithms remains still valid. Finally beyond metabolism, our algorithms could also be utilized in the context of chemical synthesis to enumerate all the possible routes linking a target molecule to a source set of starting reactants, enabling the search for the best routes in terms of production costs.http://bioretrosynth.issb.genopole.fr/tools/metahype See details in Additional file A web server is available: JLF and PC designed the algorithms and the experiments. DF implemented the algorithms, and wrote the proof of the NP-completeness for the minimal constrained hyperpath problem. SBP wrote the example of pathway enumeration. PC implemented the web service prototype, performed the experiments and analyzed the results. All authors wrote the manuscript. All authors read and approved the final manuscript.i esuch that |Y (ie)| = 1) and the hyperpath was ill-defined. In fact the given definition by these authors does not permit to say if some hypergraphs := 0 is additive if W (x) is the minimum over all the arcs x of x e+ f(iW) where i of x ein f is an increasing monotone function of its argument and f(iW) i ≥ W∀i. Remark that the cost of a hyperpath determined with an additive cost function in general is not given by the sum of the costs of the hyperarcs.In order to define a shortest path problem that can be solved polynomially by a variant of Dijkstra algorithm, the additive cost functions have been introduced in  for the n variables i σand m clauses j Xconsisting of the problem of deciding if there exists an assignment of Boolean values to the i σsuch that all the clauses are satisfied. For each boolean variable i σcontained in at least one clause introduce one hyperarc i εwith the head of each i εhaving two vertices Y(iε) = {vi+, vi-}. For each clause j Xconsider a vertex j νand seven hyperarcs (each one corresponding to boolean assignment of the three variables satisfying the clause jX). A boolean assignment is a triple a1, a2, a3 of boolean values. Let these hyperarcs be μj1,..., μj7 and let the head Y(jkμ) of a hyperarc jk μcorresponding to the combination a1, a2, a3 of the boolean variables σj1, σj2, σj3 be Y(jkμ) = {vj1a1, vj2a2, vj3a3, jν}. Now let the tails of each hyperarc introduced be connected to the source nodes. And let consider a node T being the product of the reaction R having as substrates the vertices j νand the heads {vi+, vi-} of the hyperarcs iε.Consider a 3-SAT instance concerning i εare mandatory, and the target is T.Given the hypergraph described above (whose size is linear in the size of the underlying 3-SAT problem) consider the minimal constrained hyperpath problem where all the hyperarcs M, solution of this problem, for each i consider vi+, vi-} belonging to the head of one or more of the μ arc in μ hyperarc in i εwould be superfluous). The boolean assignments j Xamong the ones whose head only contains j νand vertices A solution of this problem gives in linear time a solution for the underlying SAT problem, which makes the problem of minimal constrained hyperpath an NP-complete problem. In fact, given a minimal hyperpath ε1, ε2, ε3 are parts of the hyperpath. Each solution corresponds to a valid boolean assignment of the variables σ1, σ2, σ3.In Additional file Minimize solves the minimal constrained hyperpath problem, characterizing these instances helps to individuate which hypergraphs are expected to give an output only containing minimal hyperpaths.On many hypergraphs the algorithm enumerating the pathways only returns minimal hyperpaths, this is the case for the metabolic networks that we analyzed in the main sections of this paper. In this section we give a characterization of the hypergraphs where the algorithm Y(fR) be the set of all the metabolites produced by reactions in fR, the mandatory reactions: fR, n Rwe say that the well-separation condition holds if for every reaction Y(r) of products of r either is a subset of Y(fR) or does not contain elements of Y(fR). If the well-separation condition holds for a hypergraph fR, the algorithm Minimize returns a minimal hyperpath solving the minimal constrained hyperpath problem if a solution exists.Let f Rin a hypergraph whose reactions have one only product, as the hypergraphs defined in [Minimize returns a minimal hyperpath solving the minimal constrained hyperpath problem even if the well-separation condition holds on the pruned graphs obtained by keeping from the original hypergraph only the reactions belonging at least to one hyperpath linking the target to the source and only the nodes being tail of such reactions.The well-separation condition holds for every choice of fined in . If on tExamples of hard instances can be found among the ones used for the proof of NP-completeness. In general, hard instances This is the case for nested networks as the one in Additional file v8 is a simple problem, but finding new ones gets more and more involved. This is a consequence of the fact that the nodes v1, v2, v3, v4, v5 can be produced by different choices of the reactions R1, R2, R3, R4, R5 and each of these reaction has more of one product susceptible to participate to the production of the target.Finding one minimal hyperpaths leading to the production of Figure S1. Distribution of graph hierarchies  in heterologous metabolic networks (0.169 ± 0.11) in comparison with the graph hierarchy of central (0.032), nucleotide (0.027), lipid (0.051), and amino acid (0.030) metabolic networks in E. coli.Click here for fileFigure S2. Performance comparisons for FindPath, ExPA and efmtools for run time per output and memory use in function of size of input and size of output.Click here for fileAn example of pathway enumeration with bootstraps. An example of pathways enumeration with Findpath using bootstraps.Click here for fileAvailability and requirements. Description of the available MetaHype web server for running the algorithms for KEGG compounds.Click here for fileFigure S8. Minimal constrained hyperpath problem reduction of a 3-SAT formula.Click here for fileFigure S9. Instance of minimal constrained hyperpath problem.Click here for file


The use of computational models in metabolic engineering has been increasing as more genome-scale metabolic models and computational approaches become available. Various computational approaches have been developed to predict how genetic perturbations affect metabolic behavior at a systems level, and have been successfully used to engineer microbial strains with improved primary or secondary metabolite production. However, identification of metabolic engineering strategies involving a large number of perturbations is currently limited by computational resources due to the size of genome-scale models and the combinatorial nature of the problem. In this study, we present (i) two new bi-level strain design approaches using mixed-integer programming (MIP), and (ii) general solution techniques that improve the performance of MIP-based bi-level approaches. The first approach (SimOptStrain) simultaneously considers gene deletion and non-native reaction addition, while the second approach (BiMOMA) uses minimization of metabolic adjustment to predict knockout behavior in a MIP-based bi-level problem for the first time. Our general MIP solution techniques significantly reduced the CPU times needed to find optimal strategies when applied to an existing strain design approach (OptORF) , and identified strategies for producing compounds where previous studies could not . Additionally, we found novel strategies using SimOptStrain with higher predicted production levels (for succinate and glycerol) than could have been found using an existing approach that considers network additions and deletions in sequential steps rather than simultaneously. Finally, using BiMOMA we found novel strategies involving large numbers of modifications (for pyruvate and glutamate), which sequential search and genetic algorithms were unable to find. The approaches and solution techniques developed here will facilitate the strain design process and extend the scope of its application to metabolic engineering. Metabolic engineering of microbial strains has been of great interest for producing a wide variety of chemicals including biofuels, polymer precursors, and drugs. While conventional metabolic engineering approaches often focus on modifications to the desired and neighboring pathways, recent developments in computational analysis of metabolic models allow identification of genetic modifications needed to improve production of biochemicals Escherichia coliSaccharomyces cerevisiaeTo avoid this computational challenge, approaches like flux balance analysis (FBA) A number of bi-level strain design approaches use mixed-integer programming (MIP) to efficiently identify the mutations needed to achieve the highest production rates, including OptKnock, OptStrain, OptReg, OptForce, and OptORF. These bi-level MIP approaches consist of an ‘outer’ problem and an ‘inner’ problem, where the outer problem optimizes an engineering objective function and the inner problem optimizes a cellular objective function. Frequently, the inner problem is FBA which is a linear programming (LP) problem. In MIP-based approaches, the inner FBA problem is converted into optimality constraints by formulating a dual LP of FBA and enforcing strong duality In this study, we report two new MIP-based bi-level strain design approaches and solution techniques to improve their runtime performance. First, we present SimOptStrain which simultaneously considers gene deletions in a host organism and reaction additions from a universal database such as KEGG Second, we present a new quadratic bi-level MIP approach, BiMOMA, to identify gene deletions for improving biochemical production when MOMA is used as an inner problem see . MOMA haThese bi-level computational approaches lead to MIP formulations that become intractable when the number of allowed modifications is large. Pre-processing and heuristic algorithms have been used to improve tractability We recently developed a bi-level optimization approach (OptORF) which uses metabolic and transcriptional regulatory models to find metabolic and/or regulatory gene perturbation strategies First, we tightened the bounds on a subset of variables in the dual LP of FBA by examining its feasible region. Similar to FBA, the dual LP often has alternate optimal solutions due to the redundancy in metabolic networks. In a bi-level problem, any optimal solution of the dual LP will provide a feasible solution to the bi-level problem without affecting solutions of the primal LP since the primal-dual LP pair is only connected via strong duality. Therefore, we can obtain a valid solution of the dual LP among alternate optimal solutions by minimizing the norm of the dual variables subject to the dual LP constraints and optimal objective function value. We focused on dual variables corresponding to the reaction removals, and sampled their values using 1,000,000 samples of 10 random gene knockouts. We initially tested different sample sizes and numbers of gene knockouts and found the results were consistent above ∼100,000 samples and ∼5 gene knockouts. Therefore, we collected 1,000,000 samples, which was computationally tractable, and 10 gene knockouts, which was the maximum number allowed for the case studies in this work.For each sample, we randomly choose 10 genes and solve FBA where the reactions corresponding to the 10 genes are removed via gene to protein to reaction (GPR) associations. If the FBA problem is feasible and biomass production is positive, we then minimize the Euclidean norm of the dual variables for reaction removals in the dual LP while the objective function is constrained to be equal to the optimal biomass production value. This process is repeated 1,000,000 times to sample the values of dual variables for removed reactions in different modified network structures.iAF1260 metabolic model of E. coliSecond, we applied a penalty (α) for each additional gene deletion in the outer objective function to create a trade-off between biochemical production and the required number of genetic modifications (Equation A.1). This penalty results in selection of strategies with fewer modifications among solutions with equal production and reduces the solution time.Third, as other studies have done k) is used as a starting point for the next problem (pk+1). Unlike a local search where the next solution is constrained to keep parts of the previous solution, here the next solution is not at all constrained by this starting point, but it facilitates the search by providing a good feasible solution that can be used to prune large numbers of suboptimal solutions. The successive runs improved solver stability for some difficult cases.Fourth, we used an iterative algorithm by solving successive problems to optimality with increasing numbers of allowed gene deletions (k) where the solution from the previous problem . No solutions were found for 2 and 10 gene deletions because deleting an additional gene (over a 1 and 9 deletion strategy) did not increase TMP more than 0.5%, the penalty for the additional deletion. We subsequently limited the size of search tree (using the nodelim CPLEX option) to 104 nodes in order to evaluate if we could identify solutions of high quality faster using multiple runs, in this case lowering the penalty . This resulted in the same optimal solutions as those found in α = 0.5% TMP case for k = 1, 3, 4, 8, and 9, near optimal solutions that were still within 4% TMP of the optimal solutions for k = 5 to 7, and new solutions for k = 2 and 10 due to the smaller penalty. Overall, the process took less than 1 hour to find all 10 strategies. We did not observe any cases where bounding the dual variables prevented us from finding the global solutions (for k = 1 to 4) or affected the predicted growth and production rates for all k, which was confirmed by solving just the FBA inner problems after the deletions were identified. We also performed a sensitivity analysis on these bounds on dual variables by collecting optimal OptORF solutions for k = 1 to 4 with increasing restrictions on the bounds on dual variables  was unable to find better deletion strategies (4 seconds). These algorithms found strains with lower acetate production .To compare the proposed MIP techniques to local search methods, we implemented and modified the Genetic Design through Local Search algorithm GDLS  to use grategies . This isrategies . For exaWe additionally used OptORF to find high production strategies for metabolites that were previously found to be difficult to couple to biomass production under glucose and/or xylose aerobic conditions  [30]. Th−1. We used a metabolic model of E. coli with GPR associations To demonstrate the benefit of considering gene deletions and non-native reaction additions simultaneously, we applied the SimOptStrain approach to succinate and glycerol production under glucose aerobic conditions with a minimum growth of 0.1 hE. coli model. Therefore, the previous multi-step OptStrain procedure would not identify any non-native reactions from the KEGG database to be added to the host model. However, when we explored the simultaneous deletion of genes in E. coli model and addition of non-native reactions from KEGG database using SimOptStrain, we were able to identify non-native reactions which can significantly improve the amount of succinate produced when the E. coli mutant strains achieve their maximum growth rate  lowered fluxes in the pentose phosphate pathway and increased fluxes in the TCA cycle thereby improving succinate production . Without the addition of the identified non-native reactions, we found that the best strategies using only gene deletions resulted in very low glycerol yields  is deleted. These adjusted fluxes can be predicted by finding solutions that are closest to the wildtype flux distribution. We used a metabolic model of E. coli with GPR associations −1, and bounds on the dual variables  were used for all cases. First, we identified the best strategies for 1 to 10 deletions (k = 1 to 10) using a local search with search size of 1, which is equivalent to a sequential search  for cases when FBA does not correctly predict by-product secretion. Improvements in the universal reaction database are still needed, particularly with respect to reaction reversibility which affects constraint-based model predictions We further expanded the application of the solution techniques to BiMOMA, the first mixed-integer programming approach that uses MOMA In summary, we developed two new bi-level strain design approaches using mixed-integer programming. The developed approaches could be useful particularly for identifying novel metabolic engineering strategies to improve production of non-native secondary metabolites. We also presented mixed-integer programming solution techniques based on concepts from duality to effectively identify genetic perturbation strategies, within a reasonable amount of time even for a large number of perturbations. The MIP techniques were successfully applied to existing strain design approaches as well as new approaches developed in this study. They will likely improve the efficiency of other bi-level problems as well, including model identification, synthetic lethal identification, and objective function prediction Figure S1Analysis of dual variables for reaction removals using dual LP of FBA in different media conditions. Results from sampling of dual variable values are shown for (A) glucose aerobic, (B) glucose anaerobic, (C) xylose aerobic, and (D) xylose anaerobic conditions. The top plots show for each reaction the average of positive dual variable values (downward triangle) and negative dual variable values (upward triangle) observed over different samples, and their respective standard deviations (error bars). The averages and standard deviations were calculated for positive and negative values separately, and zero values were excluded from these statistical calculations. The bottom plots show the maximum (downward triangle) and minimum (upward triangle) of observed dual variable values for each reaction across the 1,000,000 samples of 10 gene knockouts in each condition.(PDF)Click here for additional data file.Figure S2Analysis of dual variables for reaction removals using dual QP of MOMA in different media conditions. Results from sampling of dual variable values are shown for (A) glucose aerobic, (B) glucose anaerobic, (C) xylose aerobic, and (D) xylose anaerobic conditions. The top plots show for each reaction the average of positive dual variable values (downward triangle) and negative dual variable values (upward triangle) observed over different samples, and their respective standard deviations (error bars). The averages and standard deviations were calculated for positive and negative values separately, and zero values were excluded from these statistical calculations. The bottom plots show the maximum (downward triangle) and minimum (upward triangle) of observed dual variable values for each reaction across the 1,000,000 samples of 10 gene knockouts in each condition.(PDF)Click here for additional data file.Figure S3Sensitivity analysis of the bounds on dual variables. Optimal solutions were collected with no bounds or different values of bounds on dual variables for (A) acetate production using OptORF and (B) pyruvate production using BiMOMA, respectively.(PDF)Click here for additional data file.Table S1List of reaction changes to the universal database. This table includes the list of reactions removed in this study from the original universal database (reported in (XLSX)Click here for additional data file.Text S1Detailed formulations of the bi-level strain design approaches used in this study. Complete formulations of single-level transformed bi-level strain design approaches are included for (A) OptORF without regulatory considerations, (B) SimOptStrain, and (C) BiMOMA.(PDF)Click here for additional data file.Dataset S1iJR904 model in SBML format. The file contains details for the iJR904 model in SBML format. Most of the compound abbreviations used in this file differ from those in the original iJR904 publication (XML)Click here for additional data file.Dataset S2Universal database in SBML format. The file contains details for the universal database used in the SimOptStrain simulations. This database was modified slightly from the original published database (reported in (XML)Click here for additional data file.


Metabolomics is a relatively new “omics” platform, which analyzes a discrete set of metabolites detected in bio-fluids or tissue samples of organisms. It has been used in a diverse array of studies to detect biomarkers and to determine activity rates for pathways based on changes due to disease or drugs. Recent improvements in analytical methodology and large sample throughput allow for creation of large datasets of metabolites that reflect changes in metabolic dynamics due to disease or a perturbation in the metabolic network. However, current methods of comprehensive analyses of large metabolic datasets (metabolomics) are limited, unlike other “omics” approaches where complex techniques for analyzing coexpression/coregulation of multiple variables are applied. This paper discusses the shortcomings of current metabolomics data analysis techniques, and proposes a new multivariate technique (ADEMA) based on mutual information to identify expected metabolite level changes with respect to a specific condition. We show that ADEMA better predicts De Novo Lipogenesis pathway metabolite level changes in samples with Cystic Fibrosis (CF) than prediction based on the significance of individual metabolite level changes. We also applied ADEMA's classification scheme on three different cohorts of CF and wildtype mice. ADEMA was able to predict whether an unknown mouse has a CF or a wildtype genotype with 1.0, 0.84, and 0.9 accuracy for each respective dataset. ADEMA results had up to 31% higher accuracy as compared to other classification algorithms. In conclusion, ADEMA advances the state-of-the-art in metabolomics analysis, by providing accurate and interpretable classification results. Metabolomics is an experimental approach that analyzes differences in metabolite levels detected in experimental samples. It has been used in the literature to understand the changes in metabolism with respect to diseases or drugs. Unlike transcriptomics or proteomics, which analyze gene and protein expression levels respectively, the techniques that consider co-regulation of multiple metabolites are quite limited. In this paper, we propose a novel technique, called ADEMA, which computes the expected level changes for each metabolite with respect to a given condition. ADEMA considers multiple metabolites at the same time and is mutual information (MI)-based. We show that ADEMA predicts metabolite level changes for young mice with Cystic Fibrosis (CF) better than significance testing that considers one metabolite at a time. Using three different datasets that contain CF and wild-type (WT) mice, we show that ADEMA can classify an individual as being CF or WT based on the metabolic profiles . Compared to other well-known classification algorithms, ADEMA's accuracy is higher by up to 31%. Metabolomics has emerged as a new “omics” platform in the last two decades with significant improvements in precision and sample throughput in the fields of analytical chemistry and mass spectrometry. Emergence of metabolomics has resulted in the creation of large datasets consisting of diverse classes of metabolites from multiple metabolic pathways. Metabolomics has been used to detect biomarkers of disease or drug-related changes between the control and experiment groups in a vast array of topics, such as Cancer Univariate and multivariate statistical approaches have been used to analyze metabolites to determine meaningful biomarkers of disease presence/severity or treatment effectiveness. Univariate techniques include correlation/partial correlation analysis variable importance in projection scores in PLS-DA) to find a small number of components that explain the variance best, they are not a good fit for this use. Instead, researchers use univariate techniques to locate significant changes per metabolite between the variable and the control. Then, they map these changes onto a metabolic network in order to detect pathways with increased/decreased flux based on the significances of increases/decreases, and the number of metabolites that are significantly changed in a detected pathway It is important for domain scientists to see how each metabolite level changes with respect to a given condition , in order to hypothesize about the metabolic alterations in the variable group. Since multivariate techniques truncate variables  is the first work that aims to find whether a predefined set of genes are enriched in a group of experiments with a condition up or down (two discrete bins). Finally, the algorithm determines that mice in the variable group have higher levels of A, B, C, and decreased levels of D as compared to mice in the control group.In light of the limitations of the current approaches and motivated by the combinatorial approach used for gene expression analysis More specifically ADEMA has the following steps: (i) discretize (bin) metabolite observations using B-Spline curves, (ii) identify the related subsets of metabolites out of the observed metabolites by generating the Elementary Flux Modes (EFM) To evaluate ADEMA, a Cystic Fibrosis (CF) dataset See  that conIn this section, we describe how each subcomponent of ADEMA works. Please see All animal care and use was approved by the Institutional Animal Care and Use Committee of Case Western Reserve University.Mutual information works with discrete values, whereas metabolite measurements are continuous real numbers. Therefore, to work with mutual information, one needs to discretize (bin) real values into discrete bins. In this subsection we discuss the existing methods employed in the literature and the reasoning behind picking a B-spline based strategy.There are two types of methods in the literature to estimate probability densities out of continuous data: Parametric and Non-parametric methods l, estimates a density for each observation x, by counting the number of points in the window, weighted by their distances using a pre-selected kernel x value in y values found using the B-spline curves would sum up to 1. In comparison, the histogram-based approach would divide the range  in There are two non-parametric approaches in the current literature. The first one is kernel density estimation (KDE), which, given a window length M and k. M denotes the number of bins. k, k ∈ , denotes the number of bins that an observation can be assigned to. Given M and k, the so-called “knot” vector t of length M+k+1 is defined as follows:P, and we have n observed metabolites, {1, m2, .., mnm}, for each individual s in P. Let s[jm] be the value of jth metabolite for individual s, where j ∈ . Note that the domain of z in jm. Hence, we use the linear transformation defined in jmin and mjmaxm are the minimum and maximum values observed in the population for jm respectively.i is denoted as s, the joint probability for any subset of metabolites to be in the given bins is found by multiplying probabilities of each metabolite in the subset to be in the corresponding bins. Once all metabolite measurements are associated with the corresponding bins, next step in the algorithm is to find related metabolites to be considered together.In this paper, we make use of B-spline-based binning. The use of B-spline functions in our problem formulation requires two parameters, ctor (2)Assume thn metabolites and M bins, there are nM possible combinations of metabolites and their corresponding levels (bins) as each metabolite can be in M different levels. As the method suffers from the curse of dimensionality, the subsets of metabolites to be considered together must be chosen carefully. Next we discuss three strategies to select related subsets of metabolites.ADEMA is a multivariate method that considers multiple metabolites at a time to capture interdependencies between molecules. There are two extremes. One can (i) calculate expected levels per metabolite, but then would miss the dependencies between metabolites themselves , or (ii) look for expected states of all observed metabolites together , but, this time, would unnecessarily consider metabolites that are not related at the same time. Moreover, for associated with the reactions in an elementary mode as a subset. In our context, association for a metabolite with a reaction means being a substrate or a product of that reaction. Note that elementary modes might still contain metabolites in the order of O(n). Should this be the case, we break down EFMs into pieces using a predefined threshold that limits the maximum number of metabolites that can exist in a subset.Metabolic networks provide a good understanding of the dependencies between metabolites by defining producer-consumer relationships. Elementary Mode Analysis The second strategy for related metabolite subset selection aims to group metabolites that are close to each other in the metabolic network. For each metabolite, we construct a subset that contains all metabolites within one-hop distance to that metabolite . The origin metabolite itself is also added to the set. Note that in the case of hub metabolites, the number of metabolites within a subset can still be large; thus, we apply the threshold strategy used for EFMs for this strategy as well. In contrast with the first approach, which locates the related metabolites using elementary fluxes that traverse the network, this approach disregards fluxes at the steady state, and focuses purely on topological closeness in order to determine the subsets.The third strategy is to randomly pick distinct metabolites to form related subsets of metabolites. This strategy is used as a baseline strategy, assumes no prior metabolic network knowledge, and disregards all flux or topology based relationships among metabolites. The number of metabolites is again limited by a threshold. One advantage of the third strategy is that it can be used when there is no or limited knowledge about the metabolic network or when the network is very complex or large.In the experimental evaluation, we compare performances of the three strategies in terms of the classification performance of ADEMA and report our findings on threshold selection and its effect on the algorithm efficiency. After the related subsets of metabolites are determined and the observations are discretized, the algorithm measures how informative the determined subsets are about the class variable (CF vs. WT) using mutual information.CF or WT genotype is reflected by discretized measurements (See Binning Observations Subsection) of subsets of metabolites (See Selecting Related Metabolites Subsection).Mutual Information (MI) is an information theoretic technique to determine linear or non-linear statistical dependencies of variables. In our case, we would like to determine how much X, the entropy of X is denoted as H(X). It is defined as in p(x) denotes the probability of observing x ∈ X.X given Y, accounts for the uncertainty of X when Y is known, and is derived as in I can be defined as the reduction in the uncertainty of a random variable when the other random variable is known , a real value in the range , is zero when observing one random variable does not give us any more information about the other.Sub of observed metabolites, we are interested in the reduction of the uncertainty of the class variable C, given the binned versions of observations for Sub, namely MSub|| possible combinations ino in Sub| where each o[jm] ∈  represents the bin for metabolite jm ∈ Sub. s(o)p is the probability of observing the bin combination o for individual s, and found in accordance with s(o)p. p(o) is the probability of observing o in population P and is found as shown in C is a binary random variable, which can be either the control group or a variable group. Note that we take the liberty of using a binary random variable C for the sake of clarity, and our method can be generalized to beyond C being binary. As we compare wild-type mice with mice with Cystic Fibrosis disorder in the WT and a variable group as CF. Each combination o contributes to WT and CF  uses B-splines and (ii) attaches weights to each bin combination even when a combination has a low probability to occur. This enables ADEMA to use these substates for classification purposes instead of training a third party classifier. We exploit the following theorems.MI is based on Shannon Entropy, which measures the uncertainty associated with a random variable. Given a discrete random variable nown See . I,Please see WT and CF. We are seeking (i) the expected metabolic state occurs in CF, but not in WT and (ii) the expected metabolic state that is to occur in WT but not in CF. In order to do so, we classify each CF and WT, respectively, based on Sub. We find one expectation for CF and one for WT using sets o among all other combinations that are informative about the class variable C. Note that, o is multiplied by the associated probability, and the resulting sum is rounded to the nearest integer value.Sub1. We assume there are two bins (e.g. either up or down). Hence, there are 8 possible combinations of ups and downs for these 3 metabolites. In this hypothetical example, we determine that combinations o2, o3, o4, and o7 are WT-specific, and combinations o1, o5, o6, and o8 are CF-specific (using ClassifyCombination function). Then per group (CF vs. WT), each combination is weighed by the marginal information it provides and summed up to find the aggregation , the other is forced to be less than that . As stated before, our goal is to locate the expected metabolite levels for on using . Final miSub, ADEMA performs the following task of unifying results found per EFM. First, for each observed metabolite jm, it finds all Subi such that jm ∈ Subi. We name this set of subsets of metabolites as jm for each class as shown in CF-specific and a WT-specific level for each metabolite. As shown in the figure each metabolite subset contains a different combination of metabolites. For each metabolite, all subsets, which include that metabolite are determined. Then, each subset votes for the final prediction of the level of the metabolite. Predictions are weighed by the ratio of MI provided by the subset divided by MI provided by all subsets. Thus, the more informative the subset is, the more decisive its prediction is.As explained earlier, ADEMA may obtain more than one subset of metabolites using EFMs. After obtaining CF with respect to WT as the distance between Finally, ADEMA finds the change in o as an indicator of either the variable or the control based on To summarize, ADEMA first classifies each bin combination CF and WT as defined in x to be classified is binned using the same M and k values as the training sample.In this section, we show how ADEMA can be used as a classifier. The algorithm starts by generating the expected states for o, x is classified using the following function, where S is the set of all subsets of metabolites considered:x to be in the combinations that are indicative of WT and CF, weighted by the marginal information per combination, in a manner very similar to calculating the expected states as in WT-indicative” states, then x is classified to be WT, and, otherwise, it is classified as CF.After the binning is done, for each bin combination in vivo datasets as analyzed in Bederman et. al.Linoleic Acid (C18:2ω6 (LA)), Arachidonic Acid (C20:4ω6 (AA)), Linolenic Acid (C18:3ω6 (ALA)), Eicosatetraenoic Acid (C20:4ω3 (ETA)), Eicosapentaenoic Acid (C20:5ω3 (EPA)) and Docosahexaenoic Acid (C22:6ω3 (DHA)), and (ii) non-essential fatty acids: Decanoic Acid (C10:0), Dodecanoic Acid (C12:0), Tetradecanoic acid (C14:0), Palmitic Acid (C16:0), Palmitoleic Acid (C16:1), Stearic Acid(C18:0) and Oleic Acid (C18:1). There are 7 CF and 9 WT mice in Alanine, Glycine, Valine, Leucine, Isoleucine, Proline, Urea, Serine, Threonine, Aspartate, Methionine, Glutamine, Oxo-proline, L-Phenylalanine, Tyrosine, Lactate, Glycerol, Succinate, Fumarate, β-alanine, Malate, PEP, alphaGP, Glucose, Citrate, Pantothenic acid, Uridine, and Inosine. There are 12 WT and 10 CF mice in this dataset.We have used three In this section we explain how we have applied ADEMA to the datasets described above. We implemented ADEMA in C# language and .NET Framework 4.0. All tests were performed on a Dell PowerEdge R710 Server with two Intel® Xeon® quad processors and 48 GB main memory, running the Windows Server 2008 operating system.M, k and the maximum number of metabolites in a subset (maxSub) such than 1≤maxSub≤7,2≤k≤3 and 3≤M≤6. We selected the following <M,k,maxSub> combinations per dataset as they provide the best accuracy: <6,3,8> for 3-week-old dataset, <3,3,7> for 6-week-old data set and <6,2,6> for the liver profile. Picking the best performing parameters with respect to the classification performance is also employed in the literature As described in the The next step in the algorithm is to select the related sets of metabolites. We employed all three strategies described in the et. al.To obtain the EFMs we used the YANA software package For the neighborhood approach, we obtained 1-neighborhood of each metabolite and constructed the metabolite subsets. For the fatty acid data, we obtained 11 subsets  and, for the liver profile, we obtained 22 subsets of metabolites. Two of the subsets contained more than 8 metabolites; thus they were broken into two pieces to obtain 24 subsets in the end. Finally, to test the random strategy, we generated 4 random subsets for the fatty acid data and 123 subsets of metabolites for the liver profile each of which have less than 8 metabolites.th amino acid position of the protein This section applies ADEMA to experimental metabolomics data on CF and wildtype mice, evaluates the results, and validates the approach. Cystic Fibrosis is an autosomal disorder caused by mutations in cystic fibrosis trans-membrane conductance regulator (CFTR), with the symptoms of respiratory and pancreatic dysfunction and low body-mass index. The most common mutation, F508del, results in deletion of a phenylalanine at 508CF and WT mice cohorts , it has been determined that CF mice had significantly lower de novo lipogenesis . DNL was 75% lower in CF mice as compared to WT. This implies that the flux through the DNL pathway (Decanoic Acid - Stearic Acid) was drastically reduced. DNL pathway. It is not entirely clear why DNL rates were markedly decreased in CF mice; however, Bederman et al. found significantly decreased food intake in 3 week old mice (CF mice consume 50% less food) suggesting that carbohydrate/insulin activation of DNL pathway can be delayed in 3-week-old CF mice CF mice have significantly decreased adipose tissue stores and delayed growth overall as adults. Also, gene expression analysis shows that the ELOVL6  gene expression was down by 3-fold in CF mice. Similarly, the gene SCD1 which expresses the enzyme that converts (desaturates) Palmitic Acid to Palmitoleic Acid and Stearic to Oleic Acid is down by 22-fold in CF mice. These changes are marked in cues for reaction activity. Here, by considering the gene expression levels together with the reduction in DNL activity, it is safe to assume that the reactions are downregulated in the CF mice compared to WT mice.We show where the DNL pathway fits in the big picture in DNL pathway (Decanoic Acid to Stearic Acid) other than an increase for Dodecanoic Acid (marked with dark grey). These conclusions do not comply with the data or with the gene-expression-level-based expectation noted above. One would expect a drastic change on the metabolite values as the evidence suggests that there is a substantial alteration on the pathway flow. The only point, which is in line with the independent study, is that there are decreases of Palmitoleic Acid and Oleic Acid levels which agree with drastically low SCD1 levels.CF mice with respect to WT mice found by ADEMA. We set M = 6, k = 3 maxSubset = 8, which provide the best classification performance as shown in Classification Performance Section below. Unlike the results in CF mouse, which supports the independent results. ADEMA's prediction shows that Dodecanoic Acid and Tetradecanoic Acid are increased. The increases in Dodecanoic Acid and Tetradecanoic Acid can be explained by a downstream effect of Stearic Acid and Palmitic Acid that lead to accumulation of these two metabolites as they are no longer consumed . Finally, ADEMA predicts that all metabolites in essential fatty acid elongation pathways Linolenic Acid to Docosahexaenoic Acid and Linoleic Acid to Arachidonic Acid are decreased. When metabolites are analyzed one by one, one would argue that there are no significant changes, which would lead to a different conclusion than the independent study. ADEMA provides a more consistent scenario, where the main products of the pathway are all decreased and lead to the accumulation of the precursors.To present the classification performance of ADEMA See , we makeCF or WT with an accuracy of 1 in p-values of 3*10−4, 6.3*10−3 and 1.323*10−5 for CF- and WT- specific metabolite levels, which are also used to obtain et al.Next, we compare the accuracy of our classifier with other non-linear classifiers from the literature: PLS-DA, Random Forest, SVM, AdaBoost, and Neural Network. For PLS-DA, MetaboAnalyst's implementation is used CF), the majority voting obscures the interpretability of the final result. Finally, AdaBoost tries to improve the performance of the underlying classifier by reassigning weights to the misclassified examples in the previous iterations. AdaBoost is an optimization algorithm that relies on another classification algorithm, and the interpretability of the result depends on the underlying algorithm; in the end, the output is a binary decision. In comparison, ADEMA outputs expected levels, and outputs a snapshot of the metabolic changes that have led to the classification conclusion. This way, ADEMA lets researchers to hypothesize on the metabolic activity that distinguishes variable from the control. Once again, classification scheme for ADEMA uses the same WT- or CF-specific combinations that have been found to predict the expected levels as shown in From in silico dataset using the kinetic model of simplified Glycolysis , 3 Phosphoglycerate, Pyruvate, Acetaldehyde, and External Acetaldehyde.To further validate the expected levels found by ADEMA, we generated an ycolysis . We usedM,k,maxSubset> combination based on the classification performance. YANA returned a single EFM , which is then broken into two subsets.Observations are discretized using B-splines as described before. We picked <6,2,8> as the <Fructose 1,6 Bisphosphate levels. On the other hand, ADEMA predicted an increase for all metabolites in the variable group with respect to the control group. As the initial input to the metabolic network was increased  in the variable group, the expectation is to observe an increase in the metabolic activity along the network and increased metabolite concentrations. However, student's t-test cannot detect any significant changes between two groups for ATP and M, k, the number of subsets, and the subset size increase. Parameter k increases the time requirement because of the recursive computations shown in For the picked parameters described at the beginning of this section, ADEMA took 17 seconds for M and k values using M and k increase, the computation time increases exponentially. Although this may raise a question on the applicability of ADEMA in a more general setting with networks of larger sizes and increased number of EFMs, next we show that ADEMA's parameters can be relaxed to trade accuracy for time. M,k,maxSubset> combinations tested on 6,2,6>. This particular test took ∼66 seconds as shown in 4,2,5> resulted in 0.77 accuracy which is also significant at the 0.05 level, and took only ∼2 seconds. Thus, the execution time of the algorithm can be limited by relaxing the parameters, while still providing statistically significant classification performance. The accuracy/time performance tables for In WT group and 5 simulations with high concentrations for “Ps_f” (>2800) which represent the CF group. Same is done for “Src” in the second model (low concentrations <6 and high concentrations >20). Again, per simulation, we obtained 75 values per metabolite for 75 time points and averaged them into a single representative amount. For both datasets we tested ADEMA's classification scheme using LOOCV. ADEMA was able to achieve perfect accuracy for both datasets, and took only 0.96 and 0.86 seconds on average, respectively. To further validate that ADEMA can be applied on large-scale networks we have tested the algorithm on two in silico datasets generated for models Bungay2003_ Thrombin_ Generation ADEMA is a new framework that identifies expected level changes for metabolites with respect to a condition. For each related set of metabolites, it calculates the mutual information between each combination of discretized levels of the metabolites in that set, and the class variable. We have shown how each combination can be classified as being informative in terms of the variable group or the control group, and have used this information to calculate the expected levels per class variable. ADEMA also presents a scheme to use expected levels to classify individuals with unknown class labels. We have shown that the expected metabolite level changes calculated by ADEMA conform to flux measurement results and the gene expression analysis done on 3-week-old CF mice. We have also shown that ADEMA's classification performs more accurately than five other well-known classification techniques by up to 31%. Unlike all other classification techniques, ADEMA's classification results are also interpretable. That is, ADEMA provides an explanation of the classification result by outputting the expected level changes, along with the prediction. We think that this feature is very important for metabolomics researchers who attempt to capture a snapshot of the metabolism, and understand the differences between the two groups.ADEMA attempts to minimize the loss of biological information contained in a metabolic profile. Preservation of information is particularly important when a disease causes subtle changes in metabolite levels, i.e., changes that are insignificant at a single metabolite level, but significant when taken together with other metabolite levels.In terms of Cystic Fibrosis, our hope is for ADEMA to contribute to the biomarker potential of dyslipidemia in Cystic Fibrosis. Fatty acid profiles are currently used as outcome measures in clinical trials for CF patients; the use of ADEMA would maximize the amount of information obtained from fatty acid profiles, improving the outcome measure sensitivity. Metabolite profiles are useful in the treatment of other diseases as well. For instance, comprehensive serum fatty acid profiles are used to diagnose and monitor individuals with inborn errors of mitochondrial fatty acid oxidation and peroxisomal disorders We foresee that there is room for improvement in ADEMA on selecting the relevant subsets of metabolites. Rather than relying on the existing knowledge of relations between metabolites, one can search for signatures ADEMA fills an important gap in the metabolomics literature because it provides an analysis of non-linear dependencies among multiple metabolites, and derives an expectation of changes with respect to a condition. This is a question that all “omics” platforms seek an answer for, and the need for techniques that embrace transcriptomics, proteomics and metabolomics data is substantial. ADEMA has no metabolite-specific dependencies other than the use of EFMs, and it can easily be incorporated to other high-throughput techniques.Dataset S1Metabolite measurements for 3-week-old mice. This data is referred as 3 week data in the text and contains blood measurements for metabolites of DNL pathway.(DOC)Click here for additional data file.Dataset S2Metabolite measurements for 6-week-old mice. This data is referred as 6 week data in the text and contains blood measurements for metabolites of DNL pathway.(DOC)Click here for additional data file.Dataset S3Liver profile for adult mice. This data is referred as liver profile in the text and contains blood measurements for metabolites of DNL pathway.(DOC)Click here for additional data file.Dataset S4In Silico Dataset generated using Wolf2000_Glycolytic_Oscillations Model. We have generated the following data by running 10 distinct simulations on Wolf2000_Glycolytic_Oscillations using different initial concentrations for Glucose. For each metabolite in each experiment we have obtained 75 values (there were 75 time points) and averaged them to obtain a representative value. We assumed variable group had higher (>10) initial Glucose concentrations and control group had low (<6) Glucose concentrations.(DOCX)Click here for additional data file.Figure S1YANA screenshot of the network created to obtain EFMs for  In this figure blue circles represent internal metabolites and pink circles represent external metabolites. External metabolites are not considered in the analysis, but they are input to specify the entrance and exit points to the network. Rectangles represent reactions that relate metabolites. These reactions are “abstract” reactions that might contain one or more reactions. This network represents the DNL pathway and was used to obtain the EFMs.(DOC)Click here for additional data file.Figure S2YANA screenshot of the network created to obtain EFMs for  Colors and shapes representing entities are same as in (DOC)Click here for additional data file.Figure S3Example that shows basic calculations done for ADEMA. Given one individual per class and two measured metabolites, ADEMA generates 4 possible metabolite combinations and based on the probabilities obtained using B-spline curves (in this case estimates) expected levels per group are found. ADEMA first classifies bin combinations as WT- and CF-specific to conclude that ↑↑ are the expected levels for CF and ↓↓ are the expected levels for WT.(DOC)Click here for additional data file.Table S1Accuracy results for different M,k and max subset size parameters for (DOC)Click here for additional data file.Table S2Accuracy results for different M,k and max subset size parameters for (DOC)Click here for additional data file.Table S3Time results (secs) for different M,k and max subset size parameters for (DOC)Click here for additional data file.Table S4Time results (secs) for different M,k and max subset size parameters for (DOC)Click here for additional data file.Text S1Proof for Theorem 1.(DOC)Click here for additional data file.