Flow analysis of metabolite fragments for flux

estimation

Ari Rantanen, Juho Rousu, Esa Pitkanen, Hannu Maaheimo1, Esko UkkonenFirstname.Lastname@cs.Helsinki.Fi,

Hannu.Maaheimo@vtt.Fi

Dept. of Computer Science and HIIT Basic Research UnitUniversity of Helsinki, Finland1VTT Biotechnology, Finland

Abstract. Flux estimation using isotopic tracers is currently the onlymethod that can give quantitative estimated of the activity of metabolicpathways. The success of these methods, however, is intimately depen-dent on the quality and amount of data on isotopomer distributions ofintermediate metabolites.

In this paper we present a novel method for discovering sets of metabolitefragments that always have identical isotopomer distributions, regardlessof the velocities of the reactions in the metabolic network. Our prelim-inary computational experiments indicate that the sets can help in thepropagation of isotopomer data in the network and thus ease the re-quirements posed to measurement technology. The sets have potential ofbeing useful in other tasks of flux estimation, too.

1 Introduction

The goal of metabolic flux analysis is to discover the steady state conversionvelocities of metabolites to others through chemical reactions catalyzed by theenzymes of an organism. Information about reaction rates, or fluxes, constitutesan important aspect of the physiological state of the cell that can be harnessedin many different applications ranging from pathway optimization in metabolicengineering [13] and from characterization of the physiology of an organism [8]to more efficient drug design for human diseases such as cancer [3].

The basic method to analyze the flux distribution is to rely only on mea-surements of extracellular fluxes and stoichiometric description of the metabolicnetwork of an organism [16, 17]. This kind of analysis can give interesting infor-mation about the feasible fluxes but can not in general discover the completeflux distribution. Separate fluxes for forward and backward reactions and thefluxes of cycles and alternate pathways between metabolites remain unknown[20].

More information about the fluxes can be deduced by conducting isotopomertracer experiments where the cell is fed with a mixture of natural and 13C-labeled nutrients. The fate of the 13C atoms can be observed by measuring the

isotopomer distributions1of metabolic products and intermediates with NMR [15]or mass spectrometry [4, 21]. Based on these observations more constraints tothe relative fluxes of reactions producing same metabolites can be obtained. Thismethodology has been successfully applied in numerous cases to explicitly solve(some key) fluxes in specific metabolic network and experimental conditions ofinterest [14, 9, 11, 13, 5].

The prominent computing method for estimating the flux distribution of anarbitrary metabolic network is based on iteration where a generated flux distri-bution is iteratively improved until the corresponding isotopomer distributionfits well-enough with the measurements [19, 11, 18, 12, 22].

Recently Rousu et al. [10] proposed a general direct flux estimation methodthat first propagates the measurement information in the metabolic networkand then augments the stoichiometric constraints to the fluxes with generalizedisotopomer balances.

In this work-in-progress paper we present a method for discovering the setsof fragments of metabolites that always have identical isotopomer distributions,regardless of the flux distribution. The method is based on the flow analysis of thefragments in the metabolic network. Information about the sets of fragments withidentical isotopomer distributions can be useful in many subtasks of metabolicflux estimation, including calculability analysis, estimation and controlling theerror and design of optimal labeling of nutrients. Interestingly the sets can alsobe used to improve the propagation of measurement data in the flux estimation.Our initial results show that this improvement can be significant in real metabolicnetworks.

2 Metabolic networks and carbon maps

A metabolic network is a formal model of the flow of atoms between metabo-lites (chemical compounds) in a metabolic system of metabolic reactions. Eachreaction takes some molecules of certain metabolites and transforms them intomolecules of some other metabolites. The available molecules of a metaboliteconstitute a pool, uniformly spatially mixed with the pools of the other metabo-lites. All reactions are continuously running in parallel, consuming and producingmolecules uniformly in the associated pools. Reaction rates and pool sizes maydynamically change. In this paper, however, we are analyzing the steady states,with invariant rates and sizes over time. Some metabolites are thought externalto the system in the sense that there is an incoming or outgoing flow of moleculesbetween the outside world and the corresponding pool. The other metabolitesare called internal.

1 By different isotopomers of a metabolite we mean molecules having specific com-bination of 12

C and 13C atoms in different positions of the carbon chain of the

metabolite. For example, pyryvate CH3COCOOH has three carbon and 23 = 8 dif-ferent isotopomers. Isotopomer distribution of the metabolite then gives the relativeconcentrations of different isotopomers.

When modeling the flow of atoms, we restrict the consideration to the carbonatoms of the metabolites; the measured data will concern the carbons. Eachmetabolite is treated simply as a set of its uniquely named carbon locations. Forexample, pyryvate CH3COCOOH has three such locations.

In general, let M1, . . . , Mm be the metabolites of the system. Each Mi iswritten as Mi = (ci1, . . . , ci|Mi|) where the cij ’s are the carbon locations of Mi.This gives a template for the molecules of Mi.

A metabolic reaction is represented as ρ = (α, λ) where α = (α1, . . . , αm) ∈Zm gives the stoichiometric coefficients αi ∈ Z of the reaction, and λ is thecarbon mapping describing the transitions of carbon atoms in a reaction event.Coefficients αi indicate the usage of different metabolites in an event of ρ: Foreach αi < 0, an event consumes |αi| molecules of Mi and for each αi > 0 anevent produces αi molecules of Mi, and for each αi = 0, metabolite Mi does notparticipate in ρ. Metabolites Mi with αi < 0 and αi > 0 are called the reactantsand products of ρ, respectively. Mapping λ is a one-to-one relation between thecarbon locations of the reactant and product molecules of ρ such that (c, c′) ∈ λ

indicates that ρ takes the carbon in reactant molecule location c to productmolecule location c′.

Note that when all stoichiometric coefficients are in {−1, 0, 1}, the carbonmapping between molecular locations reduces to the corresponding one-to-onemapping between the carbon locations in the metabolite templates themselves.As this simplifies many considerations, we assume in this paper that all reactionshave such a simple stoichiometry. Then we may also use the function notationλ(F ) to denote the carbon mapping image of any set F of reactant carbon loca-tions. Generalization of our results to the unrestricted case is possible, however.Symmetric molecules introduce carbon mappings that are not one–to–one. Thesecan be handled, too, but we omit them here.

Summarized, we define a metabolic network as a triple G = (M,ME,R)where M = {M1, . . . , Mm} is the set of the metabolites, ME ⊆ M is the setof the external metabolites, and R = {ρ1, . . . , ρn} is the set of the reactionsof G. Each metabolite consists of carbon locations Mi = (ci1, . . . , ci|Mi|). Eachreaction ρj is of the form ρj = (αj , λj) where αj = (αj1, . . . , αjm) gives thestoichiometric coefficients (in {−1, 0, 1}) and λj the one-to-one carbon mappingof ρj .

The state of the network G is described by fixing the reaction velocitiesvj ≥ 0 for each reaction ρj ∈ R as well as the velocities βj of the external fluxesimporting (βj ≥ 0) or exporting (βj < 0) molecules to or from the metabolitepool of Mj . An external velocity βj can be non-zero only for external metabolitesMj ∈ ME. Hence the state is given by vectors V = (v1, . . . , vn) and B =(β1, . . . , βm).

A velocity vj gives the number of reaction events of ρj per time unit. Asthe stoichiometry is assumed simple, vj gives in fact the number of moleculesof each metabolite that are consumed/produced in a time unit by the reaction.The velocities are often called the fluxes of the network. The external velocities

βj similarly give the number of molecules transported to/from the metabolitepool.

Fig. 1 depicts an example metabolic network. The metabolites are A, B, C,D, and E, and the reactions (the reactants on the left-hand side and the productson the right-hand side of the arrow)

ρ1 : A → B + C ρ2 : B + C → D ρ3 : D + C → E

Metabolites A and E are external. The arrow entering A illustrates the externalinflow of A and the arrow leaving E the external outflow of E. The carbonlocations and carbon maps are not shown in Fig. 1.

ρ1

C

D

E

B ρ2

ρ3

A

Fig. 1. Example metabolic network

A state (V, B) of a metabolic network G is a steady state, if the sizes of themetabolite pools stay invariant when the network is continuously running withthe given velocities (V, B). Hence in a steady state the incoming and outgoingfluxes for each metabolite pool should be in balance, the balance equations formetabolites Mi yielding a linear system

n∑

j=1

αjivj = βi (i = 1, . . . , m) (1)

The balance equations (1) of our example network in Fig. 1 are

v1 = v2 v1 = v2 + v3 v2 = v3

for the internal metabolites B, C, and D, respectively. (One might notice thatthis network can not have a non–trivial steady state in which all internal fluxeswould be positive.)

From balance equations system (1) one can completely solve the fluxes V ifthe external fluxes B are known from measurements and the metabolic networkhas a tree-shaped topology, which has no cycles or alternative pathways betweenmetabolites. Unfortunately, real metabolic networks are not tree-shaped. Thus,the system (1) is left under-determined which means that for some fluxes weonly can give linear constraints rather than a point solution. This problem canbe tackled with isotopic tracing experiments, as described in the next section.

3 Flux analysis using isotopomer data

Isotopic tracing measurements can be used for tackling the problem of under-determination of the system (1). The metabolic system is fed with metabolites

that are mixtures of different isotopic variants of the original metabolite. Thevariants, called isotopomers, have in certain locations the carbon isotope 13Cinstead of the standard 12C.

To define isotopomers formally, let (c1, . . . , ck) be the carbon locations of ametabolite M , and let b = (b1, . . . , bk) ∈ {0, 1}k be a binary sequence of lengthk. Then a molecule of M belongs to the b–isotopomer of M if the molecule hascarbon 13C in its carbon location ci when bi = 1 and carbon 12C when bi = 0,i.e., the 1’s of sequence b give the labeling pattern by the 13C. We denote theb–isotopomer by M(b).

We also need isotopomers restricted to some fragments of M , defined bysubsets F ⊆ {c1, . . . , ck} of the carbon locations of M . Let F = (f1, . . . , fh)be such a fragment, and let b = (b1, . . . , bh) ∈ {0, 1}h. Then a molecule of M

belongs to the F (b)–isotopomer of M if the 13C labeling of locations in F followthe pattern b, i.e., if ci = fj for some j, then the molecule has 13C in locationci if bj = 1 and 12C if bj = 0. A fragment F of M is denoted as M |F and theF (b)–isotopomer as M |F (b).

The isotopomer distribution

D(M) = (P (M(0, . . . , 0)), P (M(0, . . . , 0, 1)), . . . , P (M(1, . . . , 1))) ∈ [0, 1]2|M|

of metabolite M gives the relative abundances P (M(b)) of each isotopomer

M(b) in the pool of M . Hence∑(1,...,1)

b=(0,...,0) P (M(b)) = 1. Isotopomer distribution

D(M |F ) of fragment M |F is defined analogously. Hence P (M |F (b1, . . . , bh))gives the relative abundance of molecules having carbon pattern (b1, . . . , bh) inlocations belonging to F , i.e.,

P (M |F (b1, . . . , bh)) =∑

(a1,...,ak)|F=(b1,...,bh)

P (M(a1, . . . , ak))

where (a1, . . . , ak)|F denotes the restriction of binary sequence (a1, . . . , ak) tolocations that belong to F .

In order to use the isotopomers to improve flux estimation, we need to makesome further assumptions about the metabolic system. First, we assume thatthe isotopomer pools are completely mixed with another and that the reactionsdraw their reactant molecules independently, uniformly randomly according theirisotopomer distributions. Second, we assume a stronger form of a steady state,called the isotopomeric steady state. In such a state, in addition to the metabolitepools as a whole, their isotopomer distributions stay constant over time.

Given the above assumptions we can write a version of (1) separately foreach isotopomer of the metabolite, or, in general, for any linear combination ofisotopomers [10]. If the pathways leading to the junction manipulate the carbonsof the metabolite differently, often at least some of the equations are linearlyindependent. Consequently, the fluxes will be better determined.

Denote by Mij the subpool of the pool Mi that contains the molecules pro-duced by the inflow from reaction ρj . In particular, let Mi0 denote the subpoolproduced by the external inflow of Mi. All outflows from Mi use the resulting

mixture pool which we denote by Mi. For convenience we will use the subpoolnotation Mij also for an outflow to reaction ρj and for an external outflow al-though there are no separately identifiable outflow subpools. In the outflow caseMij just refers to the entire pool of Mi.

Using the relative abundances of the b–isotopomer in the flows adjacent toMi, the equation (1) gets the form

n∑

j=1

αjivjP (Mij(b)) = βiP (Mi0(b)) (2)

This can be written not only for a single b–isotopomer but for any fixed com-bination of them. For a Mi|F (b)–isotopomer which is a union of some disjointfull–length isotopomers we thus obtain

n∑

j=1

αjivjP (Mij |F (b)) = βiP (Mi0|F (b)) (3)

In practice, measurements are not available for each metabolite, and theisotopomer distributions of measured metabolites and fragments maybe onlypartially determined. To counteract this problem, in [10] a method was de-veloped where NMR or MS measurements, given as linear combinations dj =∑

b sbP (Mij(b)) of isotopomers could be propagated up and downstream in themetabolic network towards junctions. This propagation resulted in generalizedisotopomer balance equations

m∑

j=1

αjivjdj = βid0 (4)

for the junctions.A shortcoming of the above method is that the propagation of information

stops at the nearest junction metabolite. Hence, for example, if in between twojunctions there are no measurements, (4) cannot be formed and fluxes aroundthat junction remain undetermined.

In the next section we develop a data-flow analysis method for finding metabo-lite fragments (of different metabolites) that are equivalent in the sense that theyhave the same isotopomer distribution, irrespective of the fluxes. Whenever twosuch fragments are on opposite sides of a junction, any isotopomer informationknown for one fragment can be propagated to the other. This leads to moreefficient and accurate propagation of isotopomer data.

4 Finding equivalent fragments

Metabolite fragments M |F and M ′|F ′ are equivalent in network G, denotedM |F ≡ M ′|F ′, if there is a permutation of the carbon locations of F ′ such thatin all isotopomeric steady states of G

D(M |F ) = D(M ′|F ′′)

where F ′′ is the fragment obtained from F ′ by applying the permutation.By analyzing the flow of carbon atoms locally, within one reaction, it is easy

to find equivalent fragments, namely the fragments that the reaction does notsplit. We start with a basic result concerning fragments in inflow subpools.

Lemma 1. Let M |F be a fragment of a reactant metabolite M of a reactionρj = (αj , λj). If λj(F ) is a fragment of a product metabolite Mi of ρj , thenM |F ≡ Mij |λj(F ).

Proof. Reaction ρj samples its input pools randomly. Hence M |F is taken accord-ing to the distribution D(M |F ) and, as the atoms of F go through ρj together,its image λj(F ) still has the original isotopomer distribution of M |F in the in-flow subpool Mij of Mi, independently of the rest of the molecule it belongs to.We only need to permute the carbon locations of λj(F ) according to the carbonmap λj to get the components of the distribution in the right order. �

If Mi has only one inflow, then Mij = Mi in Lemma 1, and we have M |F ≡Mi|λj(F ).

We call the metabolites with more than one inflow the junction metabolites ofthe network. To analyze fragment equivalence in junctions, we check whether ornot a fragment originates from a common source along all routes to the junction.On each route, the carbons should stay together such that no intermediate frag-ment is cleaved into parts that belong to more than one molecule. If such cleavingtakes place, the fragment will not necessarily carry its original isotopomer dis-tribution because the different molecules carrying the subfragments are sampledindependently by the reactions.

Technically, the fragment flow graph F(G) = (V, W ) of the metabolic networkG = (M,ME ,R) is a directed graph defined as follows. The set V of the nodesconsists of

(n1) the root node ∆;(n2) all fragments (i.e., subsets of carbon locations) of all metabolites of G;

and(n3) for every fragment F of every external metabolite in ME , there is an

additional ’external’ node FE .The set W of the arcs contains(a1) a (directed) arc (F, F ′) whenever λj(F ) = F ′ for the carbon mapping

λj of some reaction ρj ∈ R;(a2) an arc (FE , F ) for each fragment F of an external input metabolite and

an arc (F, FE) for each fragment F of an external output metabolite; here weassume that this division of ME to input and output metabolites is given;

(a3) an arc (∆, F ) for all nodes F , F 6= ∆, such that the number of arcs fromnon–external nodes to node F is by the above rule (a1) less than the number ofreactions that produce the metabolite of F ; and

(a4) an arc (∆, FE) for all external nodes FE ; note that no arcs by rules (a1)- (a3) are entering external nodes FE .

As the nodes of F(G) are metabolite fragments, the nodes will sometimes becalled also fragments.

A node F ′ of F(G) weakly dominates another node F if all directed pathsfrom the root ∆ to F must go through F ′. Node F ′ is an immediate weakdominator of F , iff (i) F 6= F ′; (ii) F ′ weakly dominates F ; (iii) F ′ does notweakly dominate any other weak dominator of F .

The weak dominator tree of F(G) is a tree with the same node set V as F(G)and with an arc from each node to its immediate weak dominator. The weakdominator tree can be constructed in linear time [6]; in our implementation wehave used the O(n log n) algorithm of [7] where n is the number of nodes.

To properly trace the flow of carbon atoms, the weak dominance (which infact is the standard dominance relation of data flow analysis [1]) is not enoughas it ignores the effect of the carbon maps. We say that a node F ′ of F(G)dominates another node F if F ′ weakly dominates F and the carbon map fromF ′ to F , which is induced by some directed path in F(G) from F ′ to F , is thesame for all such paths, i.e., every path from F ′ to F gives the same mapping ofthe carbons of F ′ to the carbons of F . Immediate domination and dominator treeare defined as the corresponding weak versions. Note that for nodes representingone–carbon fragments the dominance and the weak dominance coincide.

The following lemma gives a procedure for converting a weak dominator treeinto a dominator tree.

Lemma 2. Let E be the immediate weak dominator of F . If the immediate weakdominator of each one–carbon subfragment of F is some one–carbon subfragmentof E, then E is the immediate dominator of F . Otherwise ∆ is the immediatedominator of F .

Proof. Omitted in this draft. �

Theorem 1. If nodes F = M |F and F ′ = M ′|F ′ belong to the same subtreeof the root ∆ in the dominator tree of F(G), then M |F ≡ M ′|F ′ under thepermutation of carbon locations of F ′ that is induced by the combined carbonmaps from the root of the subtree to F and F ′.

Proof. Let Y be the set of the nodes of the common subtree of F and F ′, andlet H ∈ Y be the root. As H dominates all Y , any path in F(G) from outsideY to a node of Y must go via the root H . Hence, if (E ′, E) is an arc of F(G)for some E ∈ Y then also E ′ ∈ Y or E = H .

By the construction of the dominator tree, the combined carbon map fromH to any E ∈ Y is uniquely defined and one-to-one. Hence it induces a one-to-one mapping between any two fragments E and E ′ in Y . In what follows, theequivalence will be proved with respect to this permutation which we implicitlyassume when comparing isotopomer distributions of different fragments.

Assume that the metabolic network is in some isotopomeric steady state withpositive reaction velocities. Let I be the isotopomer distribution of the (total)inflow to H ; this is the distribution of the external inflow if H is an external nodeand otherwise it is the sum distribution given by the flows from outside Y to H .To derive a contradiction, assume that some fragment E ∈ Y has isotopomerdistribution I ′ 6= I . Then we can select E and t such that the difference between

the tth component p of I and the tth component p′ of I ′ is largest possible andp 6= p′. As the reaction velocities are positive, there is a positive inflow, possiblyvia several reactions, to the pool of E from the subpool of H created by theinflow from outside with distribution I . From the selection of E it follows thatin the inflows to the pool of E, the tth component is ≥ p′ for all inflows or ≤ p′

for all inflows; Here we also need the above remark that all inflow to E is fromnodes in Y . But then p′ can be a weighted average of the tth components, asrequired by balance (3), only if p′ = p, contradicting our assumption. �

Fig. 2 gives an example network with carbon maps. Five subtrees of thedominator tree are also shown, representing equivalence classes of fragments.The trees for the subfragments of the fragments in the largest tree are omittedas they are similar to that tree.

5 7 8

2

6

C − C

C − C − C

1

3 6

2

4 5

7

8

9

10

9C − CC C

CC − C

4

10

C − C − C

3

1

Fig. 2. Example network with carbon maps; the subtrees of the corresponding domi-nator tree on the right.

We next utilize the dominator analysis to analyze when balance equations(3) can help solving the fluxes at metabolite Mi.

If Mi is not a junction metabolite (i.e., it has only one inflow), equations (3)and (1) do not differ as there is only the isotopomer distribution of the singleinflow available. In this case (3) does not help.

Junctions are more interesting:

Theorem 2. Let F be a fragment of a junction metabolite Mi. If F has a dom-inator in the fragment graph F(G), then equation (3) for F does not differ fromequation (1) for Mi.

Proof (sketch). For each reaction ρj that produces Mi, each fragment Fj =λ−1

j (F ) (inverse image of F in a reactant pool of ρj) has the same dominatorsubtree as F and hence is by Theorem 1 equivalent to F under the permutationinduced by combined carbon maps. By Lemma 1, Mi|F ≡ Mij |λj(Fj) = Mij |Fwhere Mij is the subpool of Mi for the inflow from ρj . But then all isotopomer

abundances P in equation (3) are equal under the permutation induced by car-bon maps, and we are back to (1). �

Theorem 2 tells us that dominated fragments are not useful in estimatingfluxes. However, we can still use domination information to improve flux estima-tion. The key is to consider fragments M |F that are not dominated but whosepreimages λj(M |F ) in the inflows have equivalent fragments with measured iso-topomer information:

Theorem 3. Let F be a non–dominated fragment of Mi, and assume that weknow (by measurement) the isotopomer distributions D(Mk|Hk) of some frag-ments Hk. This data can be utilized to write equation (3) for the fragment F , ifF is equivalent to some Hk and for each reaction ρj that produces Mi, fragmentλ−1

j (F ) is equivalent to some Hk.

Finally we sketch an alternative way to construct the dominator subtree andthe corresponding equivalent fragments for a given fragment F only. The abovemethod did this for all possible F at the expense of using the full fragment flowgraph F(G) which can be quite large: for each metabolite M there are O(2|M |)nodes in F(G).

It is sufficient to build the fragment flow graph and dominator tree only forfragments of size ≤ 2. Let T be the resulting dominator tree. To find dominancesubtree for a fragment F larger than 2, take some subfragment E = (a, b) ⊂ F ofsize 2. Let TE be the subtree of the root of T that contains E. Let E ′ = (a, b′) ⊂F be another subfragment which shares carbon location a with E. Then thedominator subtree TE∪E′ for E ∪ E′ can be build from the trees TE and TE′

with a simple intersection operation. The operation compares the two trees usingthe natural correspondence of nodes (the corresponding nodes have non–emptyintersection) and arcs, and retains the nodes and arcs that according to thiscorrespondence are present in both. Next we take another pair E ′′ = (a, b′′) ⊂ F ,and intersect TE∪E′ by TE′′ and so on, until we have TF .

The tree TF can be constructed from T in this way in time O(|F |m) where m

is the number of metabolites: The intersection operation has to be done O(|F |)times, and each operation obviously takes time proportional to the sizes of thetrees intersected. As the dominator subtrees contain at most one fragment fromeach metabolite, the trees are of size O(m), and the time bound follows.

5 Experiments

We tested our method for finding flux invariant fragments with the model ofcentral carbon metabolism of Saccharomyces cerevisiae containing glycolysis,penthose phosphate pathway and citric acid cycle. Carbon mappings were pro-vided by the ARM project [2]. The network consisted of 35 metabolites and 37reactions of which five were bidirectional. Cofactor metabolites were excludedfrom the analysis. The only carbon source of the network was glucose. There werefive external products in the model. Eleven of the metabolites were produced bymore than one reaction and thus formed junctions.

Our analysis discovered 12 nontrivial equivalence sets having more than onemember. In our method for flux estimation [10] the propagation of isotopomerinformation always stops to junction metabolites meaning that the equivalence offragments residing in different sides of a junction remain undetected. Thus withthe method of [10] there in general exists greater number of smaller equivalencesets. The size distributions of the equivalence sets obtained with the presentflow analysis and with the method of [10] are given in Table 1. The flow analysisuncovered ten dominated maximal fragments from six different junction metabo-lites. Two junctions were totally dominated by another metabolite and thereforethese fluxes cannot be solved using isotopomer balances. The equivalence setscovered 110 carbon atoms of total amount of 193 carbons in the network. Dom-inated fragments of junction metabolites contained 21 carbons.

We also compared the efficiency of the propagation of isotopomer informa-tion by flow analysis against the method of [10]. We selected randomly anduniformly a set of metabolites considered as measured and tested how manycarbon locations could in the best case get isotopomer information from themeasured metabolites with both methods. Technically the number of locationsequals the total number of carbons in all equivalence sets having at least onefragment from the measured metabolite. The sampling was repeated 1000 timesfor each number of selected metabolites. The mean and standard deviations ofthe number number of carbon locations reached are given in Fig. 3. The compu-tational experiment shows that when the amount of measurable metabolites islimited, as usually is the case in real experiments, our new method can propagateisotopomer information more efficiently than the older one.

Visualizations of the metabolic network used in the experiments and theequivalence sets discovered are available at http://www.cs.helsinki.fi/group/sysfys/.

# of fragments 2 3 4 5 6 7 8 9 10 13 14 Σ

# of subtrees (flow analysis) 2 3 1 1 2 1 1 1 12# of subtrees (method of [10]) 6 10 3 1 1 1 1 22

Table 1. Number of fragments in nontrivial equivalence sets in the model of centralcarbon metabolism of S. cerevisiae with and without utilizing the flow analysis.

6 Discussion

In this work-in-progress article we have presented a novel method for computingsets of fragments of metabolites having identical carbon isotopomer distribu-tions, regardless of the fluxes of the metabolic network. The analysis is basedon the flow analysis of the metabolic fragments. These sets can be utilized toimprove the propagation of the measurement data in our existing flux estimationalgorithm [10] and thus ease the requirements posed to measurement technol-ogy. In general, better propagation of measurement data means that the same

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

180

200

# of measured metabolites

# of

car

bons

hav

ing

isot

opom

er in

form

atio

n

with flow analysiswithout flow analysis

Fig. 3. Efficiency of propagation of isotopomer information with the flow analysismethod of this article and with the method of [10]. Y-axis gives the mean number ofcarbon locations that can get isotopomer information from the metabolites randomlyselected as measured. Total lengths of the symmetric error bars equal two standarddeviations.

information about the fluxes can be obtained by measuring fewer number ofmetabolites. Our preliminary experiments suggest that the improvement can besubstantial. In near future we will integrate the discovery of the sets of frag-ments with identical isotopomer distributions to the implementation of our fluxestimation method, together with proper handling of symmetrical metabolitesand unrestricted stoichiometric coefficients.

The isotopomer equivalence sets ca also be useful in modeling of measurementerror as the isotopomer distributions of the fragments in the same equivalenceclass should be equal. By comparing measured isotopomer distributions of frag-ments of the same set one can study the repeatability and the effect of metaboliteconcentration to the accuracy of the measurements. On the other hand largerthan expected deviation in one or few isotopomer distributions in the set mightindicate erroneous topology in the model of the metabolic network.

Another possible application is in the design of optimal labeling of externalsubstrates of the network. If we are interested in relative fluxes in some specificjunction metabolite, it is not worthwhile to spend expensive labels to carbonsending up in the dominated fragment of this metabolite. Also, with the help ofequivalence classes one can think of an algorithm that selects the minimal set of(fragments of) metabolites whose isotopomer distribution should be measuredin order to gain as much information about the fluxes as possible.

The dominator analysis can also be used to enumerate useful targets forMETAFoR analysis [14].

Acknowledgements. We thank Markus Heinonen and Arto Akerlund for theircontribution to the experimental work reported in this paper. This work hasbeen supported by the SYSBIO programme of Academy of Finland, NEOBIOprogramme of TEKES and by Marie Curie Individual Fellowship grant HPMF-CT-2002-02110.

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