The framework for ¹³C metabolic flux analysis presented in this article rests on the assumption that the metabolic network is in metabolic and isotopomeric steady state. In the metabolic steady state, metabolite balance, or mass balance

∑ j = 1 n α i j v j = β i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaabCaeaacqaHXoqydaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabdAha2naaBaaaleaacqWGQbGAaeqaaOGaeyypa0JaeqOSdi2aaSbaaSqaaiabdMgaPbqabaaabaGaemOAaOMaeyypa0JaeGymaedabaGaemOBa4ganiabggHiLdaaaa@3ED5@

holds for each metabolite M_i. Here, β_i is the measured external in flow (β_i < 0) or external out flow (β_i > 0) of metabolite M_i. From balance equations (1) defined for every metabolite M_i we will obtain a metabolite balancing, or stoichiometric equation system.

In isotopomer steady state, for each isotopomer b of each metabolite M_i in the metabolic network the following isotopomer balance holds:

∑ j = 1 n α i j v j P M i j ( b ) = β i P M i 0 ( b ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaabCaeaacqaHXoqydaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabdAha2naaBaaaleaacqWGQbGAaeqaaOGaemiuaa1aaSbaaSqaaiabd2eannaaBaaameaacqWGPbqAcqWGQbGAaeqaaaWcbeaakiabcIcaOiabdkgaIjabcMcaPiabg2da9iabek7aInaaBaaaleaacqWGPbqAaeqaaOGaemiuaa1aaSbaaSqaaiabd2eannaaBaaameaacqWGPbqAcqaIWaamaeqaaaWcbeaakiabcIcaOiabdkgaIjabcMcaPaWcbaGaemOAaOMaeyypa0JaeGymaedabaGaemOBa4ganiabggHiLdGccqGGUaGlaaa@504B@

For metabolic flux analysis (1) and (2) bear a fundamental difference: the former cannot be used to estimate fluxes of alternative pathways producing M_i while the latter can. However, using (2) is not in general admissible in practice: typically abundances P_M(b) of isotopomers are not fully determined from the measurements, and we need to settle for some constraints to the distribution D_M. A crucial building block of our framework is the representation of isotopomer measurements as systems of linear equations (c.f. 21)

∑ b s b h P M ( b ) = d h , h = 1 , ... , r MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaabuaeaacqWGZbWCdaWgaaWcbaGaemOyaiMaemiAaGgabeaakiabdcfaqnaaBaaaleaacqWGnbqtaeqaaOGaeiikaGIaemOyaiMaeiykaKIaeyypa0Jaemizaq2aaSbaaSqaaiabdIgaObqabaGccqGGSaalcqWGObaAcqGH9aqpcqaIXaqmcqGGSaalcqGGUaGlcqGGUaGlcqGGUaGlcqGGSaalcqWGYbGCaSqaaiabdkgaIbqab0GaeyyeIuoaaaa@474D@

where s_bh is the weight of isotopomer b in the h'th constraint, d_h is a value derived from isotopomer measurements, and r is the total number of constraints. We call (3) isotopomer constraints. For a k carbon metabolite, 2^k linearly independent isotopomer constraints – one for each isotopomer – are necessary and sufficient to constrain the isotopomer distribution D_M to a point solution. A set of isotopomer constraints has a natural matrix representation SD_M = d, where S = (s_bh)_b,h is a 2^k by r matrix, where 1 ≤ r ≤ 2^k is the number of isotopomer constraints (the trivial constraint Σ_bP_M(b) = 1 by definition always holds).

The use of (3) follows from the simple observation that isotopomer balance (2) implies that each linear combination of isotopomers is balanced. Thus, we can write a new balance equation that constrains the fluxes producing M_i as soon as we know the value of the same linear combination of isotopomer abundances for each subpool of M_i. We have

∑ j = 1 n α i j v j d i j = β i d i 0 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaabCaeaacqaHXoqydaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabdAha2naaBaaaleaacqWGQbGAaeqaaOGaemizaq2aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGH9aqpcqaHYoGydaWgaaWcbaGaemyAaKgabeaakiabdsgaKnaaBaaaleaacqWGPbqAcqaIWaamaeqaaaqaaiabdQgaQjabg2da9iabigdaXaqaaiabd6gaUbqdcqGHris5aOGaeiilaWcaaa@47CE@

where each dij=∑bsbPMij(b) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemizaq2aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGH9aqpdaaeqaqaaiabdohaZnaaBaaaleaacqWGIbGyaeqaaOGaemiuaa1aaSbaaSqaaiabd2eannaaBaaameaacqWGPbqAcqWGQbGAaeqaaaWcbeaakiabcIcaOiabdkgaIjabcMcaPaWcbaGaemOyaigabeqdcqGHris5aaaa@3FAD@ is a linear combination of the form (3), with coefficients s_b that do not depend on j, i.e. they are the same for each subpool M_ij. We call (4) a generalized isotopomer balance.

In the following, we will show by examples how MS and NMR data can be represented as isotopomer constraints. Let us first consider mass isotopomer distributions obtained from MS. Here we omit discussion on practicalities such as corrections for natural abundances of ¹³C isotopes (c.f., 636) and concentrate on the conceptual level. For example, the +2 mass isotopomer of a three carbon metabolite M satisfies

P_M(+2) = P_M(011) + P_M(110) + P_M(101)

which conforms to (3) by taking s_011,2 = s_110,2 = s_101,2 = 1, and s_b,2 = 0 otherwise. Similarly, the coefficients s_bk can be derived for all mass isotopomers +k, k = 0, ..., 3.

Isotopomer data originating from Tandem MS, or MS-MS, falls into the same representation. Consider, for example, a fragment M|F of a three-carbon metabolite, containing the first and second carbon of M. The following equation holds for the mass isotopomer M|F(+1):

P_M|F(+1) = P_M|F(01) + P_M|F(10).

The equation can be written in terms of the precursor M, but the exact form of the equation depends on the mode of tandem MS. If the full scan mode is used, we obtain

P_M|F(+1) = P_M(010) + P_M(011) + P_M(100) + P_M(101),

as all precursor molecules M that have exactly one carbon among the first and second location contribute to the fragment mass isotopomer M|F(+1). On the other hand, in the daughter-ion scanning mode a single mass isotopomer, for example M(+2), is selected as the precursor. Then we obtain

P_M|F(+1) = P_M(011) + P_M(101),

as the precursor must always have two ¹³C atoms in total. We refer the reader to 636 for further details. Also NMR ¹³C isotopomer measurements, where relative intensities of different combinations of ¹³C and ¹²C atoms that are coupled to an observed ¹³C atom are measured, can be expressed as linear combinations of isotopomer abundances. For example, for a three-carbon metabolite M, the following constraints to DMi MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiraq0aaSbaaSqaaiabd2eannaaBaaameaacqWGPbqAaeqaaaWcbeaaaaa@2FC8@ can be inferred for the labeling pattern 010:

P M ( 010 ) ∑ b 1 , b 3 ∈ { 0 , 1 } P M ( b 1 1 b 3 ) = d ( 010 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqcfa4aaSaaaeaacqWGqbaudaWgaaqaaiabd2eanbqabaGaeiikaGIaeGimaaJaeGymaeJaeGimaaJaeiykaKcabaWaaabeaeaacqWGqbaudaWgaaqaaiabd2eanbqabaGaeiikaGIaemOyai2aaSbaaeaacqaIXaqmaeqaaiabigdaXiabdkgaInaaBaaabaGaeG4mamdabeaacqGGPaqkaeaacqWGIbGydaWgaaqaaiabigdaXaqabaGaeiilaWIaemOyai2aaSbaaeaacqaIZaWmaeqaaiabgIGiolabcUha7jabicdaWiabcYcaSiabigdaXiabc2ha9bqabiabggHiLdaaaOGaeyypa0JaemizaqMaeiikaGIaeGimaaJaeGymaeJaeGimaaJaeiykaKcaaa@5307@

where d(010) is the measured intensity. Rewriting the above as

d ( 010 ) ⋅ ∑ b ∈ { 110 , 011 , 111 } P M ( b ) = ( 1 − d ( 010 ) ) ⋅ P M ( 010 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemizaqMaeiikaGIaeGimaaJaeGymaeJaeGimaaJaeiykaKIaeyyXIC9aaabuaeaacqWGqbaudaWgaaWcbaGaemyta0eabeaakiabcIcaOiabdkgaIjabcMcaPiabg2da9iabcIcaOiabigdaXiabgkHiTiabdsgaKjabcIcaOiabicdaWiabigdaXiabicdaWiabcMcaPiabcMcaPiabgwSixlabdcfaqnaaBaaaleaacqWGnbqtaeqaaOGaeiikaGIaeGimaaJaeGymaeJaeGimaaJaeiykaKcaleaacqWGIbGycqGHiiIZcqGG7bWEcqaIXaqmcqaIXaqmcqaIWaamcqGGSaalcqaIWaamcqaIXaqmcqaIXaqmcqGGSaalcqaIXaqmcqaIXaqmcqaIXaqmcqGG9bqFaeqaniabggHiLdaaaa@5F8B@

and denoting s_b = d(010), for b ∈ {110, 011, 111}, s₀₁₀ = d(010) - 1 and s_b = 0 for b ∈ {000, 100, 001, 101}, the above can be seen to conform to (3). Similar derivation can be used for other isotopomer signals present in the NMR spectrum to obtain the corresponding isotopomer constraints.

In our computational framework, it will be necessary to project the measurement data obtained for a metabolite M to its fragments M|F and vice versa. In this projection, we want to avoid any unnecessary loss of measurement information, that is, we want to obtain as many linearly independent constraints to the isotopomer distribution of F as possible. For example, if we have measured that a two-carbon metabolite M has the isotopomer distribution

P_M(00) = 0.2, P_M(01) = 0.3, P_M(10) = 0.4, P_M(11) = 0.1

and we need to know the isotopomer distribution of the fragment M|F consisting of the first carbon of M, we should obtain

P F ( 0 ) = P M ( 0 ∗ ) = P M ( 00 ) + P M ( 01 ) = 0.5 , P F ( 1 ) = P M ( 1 ∗ ) = P M ( 10 ) + P M ( 11 ) = 0.5. MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6A08@

For the general model of isotopomer measurements (4) the projection of measurement information from a metabolite to its fragments can be done by the techniques of linear algebra introduced in 21. We recapitulate the techniques in the following.

Recall the general form of isotopomer measurement SD_M = d, where S denotes a matrix with 2^k columns, one column for each isotopomer b of k-carbon metabolite M, and each row h of S corresponds to a measured isotopomer constraint (3). The rows of S span a subspace S⊆IM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8NeXpLaeyOHI0Sae8heHK0aaSbaaSqaaiabd2eanbqabaaaaa@3BBC@ in a 2^k dimensional vector space IM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8heHK0aaSbaaSqaaiabd2eanbqabaaaaa@37E0@ spanned by all possible isotopomer distributions D_M.

Also the metabolite fragments are naturally represented as vector subspaces. Let U_F denote a matrix with also a column for each isotopomer M(b) and a row for each isotopomer F(b') of M|F, that is,

U F ( b ′ , b ) = { 1 if  b j = b ′ j for all carbon positions  j ∈ F k 0 otherwise . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7503@

The rows of U_F span another subspace UF⊆IM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8hfXx1aaSbaaSqaaiabdAeagbqabaGccqGHgksZcqWFqessdaWgaaWcbaGaemyta0eabeaaaaa@3D0B@. Any isotopomer distribution D_M|F lies in this subspace, and hence also any isotopomer constraint S_FD_M|F = d for fragment M|F necessarily lies in the same subspace.

In conclusion, the available information about D_M is given as its linear projection onto S MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8NeXpfaaa@376D@, and anything we can express about D_M|F in terms of isotopomer constraints is contained within UF MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8hfXx1aaSbaaSqaaiabdAeagbqabaaaaa@38B2@. Thus, any isotopomer constraint for D_M|F that we can derive from the measurements can be expressed in terms of the vector space intersection S∩UF MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8NeXpLaeyykICSae8hfXx1aaSbaaSqaaiabdAeagbqabaaaaa@3C2B@.

Thus, to obtain isotopomer constraints for fragment M|F from a measurement SD_M = d, we need to compute the vector space intersection YM|F=S∩UF MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8hgXN1aaSbaaSqaaiabd2eanjabcYha8jabdAeagbqabaGccqGH9aqpcqWFse=ucqGHPiYXcqWFueFvdaWgaaWcbaGaemOrayeabeaaaaa@4306@ and project the measurement to Yi,F MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8hgXN1aaSbaaSqaaiabdMgaPjabcYcaSiabdAeagbqabaaaaa@3AF5@. This can be done by standard linear algebra (c.f. 21). This process gives us as output isotopomer constraints of the required form

Y_FD_M|F = d_F.

Finally, transforming a fragment constraint Y_FD_M|F = d_F into an isotopomer constraint SD_M = d is easy: we postmultiply the fragment constraint with the matrix U_M|F : S = Y_FU_M|F and d = d_FU_M|F.

The goal of the flow analysis 27 is to partition the fragments of the metabolites in the network to equivalence classes such that fragments in the same equivalence class have identical isotopomer distributions in every steady state. This can be guaranteed if a fragment is produced from another a such a manner that the carbons within the fragment never depart from each other regardless of the pathway that is being used.

Formally, we say that fragment F' dominates fragment F if the following conditions are met

1. F and F' have the equal number of carbons;

2. all carbons of F originate always from the carbons of F';

3. carbon of F' stay connected to each other via all pathways from F' to F;

4. composite carbon mappings are the same in all pathways from F' to F.

Intuitively, a dominated fragment (F) is always produced from its dominator (F') without manipulating the carbon chain of the fragment. Thus, isotopomer distribution of the dominated fragment does not contain any information about the metabolic fluxes. For a fragment F that has no dominators, the transitive closure of the domination relation corresponds to the class of equivalent fragments in the network.

The simplest example of fragment equivalence is the one between a substrate M_k and product M_i in a single reaction ρ. If the atoms in M_i|F originate from M_k|F', then the fragments M_k|F' in the subpool M_ij produced by reaction ρ_j, are equivalent with the fragment M_i|F (Figure 1). Furthermore, if metabolite M_i has only one producing reaction ρ_j, isotopomer distributions of subpool M_ij and M_i coincide. Thus, if fragment M_i|F is produced from a single fragment M_k|F' of some substrate M_k of ρ_j, F and F'are equivalent. By transitivity, all fragments in the linear pathway are equivalent.

More complicated case of fragment equivalence is found when a fragment of a junction metabolite is dominated by an upstream fragment (Figure 3). In 27 we show that the the classes of equivalent fragments corresponding the conditions (1–4) can be efficiently computed. Very brie fly, first the metabolic network is transformed to a fragment flow graph that connects substrate metabolite fragments to their product fragments for each reaction in the network. Then, the dominator tree 3738 of the fragments in the fragment flow graph is constructed. It turns out that the subtrees of this dominator tree correspond to the required fragment equivalence classes (see Figure 4).

Fragment equivalence classes have many uses 27. Most importantly, measured isotopomer constraints to fragment F can be directly propagated to another fragment F' in the same equivalence class, by applying the joint carbon mappings between F and F'. This helps in the construction of generalized balance equations (4) where the same isotopomer information is required for each subpool of junction metabolites.

A complementary property to fragment equivalence ¹³C is the statistical independence of fragment isotopomer distributions. Intuitively, if two fragments of the same metabolite are statistically independent, new isotopomer constraints to the union of them can be obtained by taking a product of the isotopomer distributions of the independent fragments.

More formally, the basic question is on what conditions the distribution D_M|E∪F of union of two fragments will necessarily have the form of a product distribution

D_M|E∪F = D_M|E ⊗ D_M|F

where ⊗ denotes the tensor product consisting all terms of the form P_E∪F(b) = P_E(b')·P_F(b"), where b' (resp. b") ranges over all isotopomers of E (resp. F), and b is the isotopomer of M|E ∪ F formed by joining E and F.

The utility of fragment independence is in that it gives us constraints to the isotopomer distributions that are complementary to the isotopomer constraints (3) obtained from the measurements.

In general, two criteria need to be satisfied for statistical independence of two fragments M|E and M|F. First, the fragments need to be structurally independent, meaning that along all pathways producing the metabolite, at some point all carbons of the fragments have resided in different metabolite molecules. This property can be defined in recursive manner. Fragments M|E and M|F are structurally independent if for all carbon pairs (a, b), a ∈ E and b ∈ F, for all reactions ρ producing M, it holds that

• a and b originate from different reactants of ρ, or

• a and b originate from the same reactant M', and the reactant fragments M'|F_a and M'|F_b, where F_a = λρ−1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4UdW2aa0baaSqaaiabeg8aYbqaaiabgkHiTiabigdaXaaaaaa@3153@(a), F_b = λρ−1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4UdW2aa0baaSqaaiabeg8aYbqaaiabgkHiTiabigdaXaaaaaa@3153@(b), are structurally independent.

The second necessary condition is that the two fragments need to be dominated by some other metabolite fragments in the network. This will make the fragment distributions flux invariant. Together, the two criteria guarantee (6) to hold.

A simple case of statistical independence of fragments is a (subpool) product metabolite M_i of a single reaction ρ_j, where the fragments M_i|E and M_i|F are disjoint and originate from different reactants. The fragments are structurally independent (by originating from different reactants) and are dominated (by reactant fragments of ρ_j). Hence (6) holds. The underlying assumption here is that enzymes pick their reactants independently and randomly from the available pools. This case of statistical independence of fragments is depicted Figure 1, where white and grey fragments of D-fructose 1,6-biphosphate are statistically independent (in the subpool of reaction ρ_j).

Another simple example is a junction metabolite M_i that has two or more producers with associated subpools M_ij. If M_ij|E and M_ij|F are structurally independent in all subpools, M_i|E and M_i|F are structurally independent as well. If M_i|E and M_i|F are dominated by some fragments in the network, all subpools have the same distribution which takes the form of (6). Without dominance the distribution DMi|E∪F MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiraq0aaSbaaSqaaiabd2eannaaBaaameaacqWGPbqAaeqaaSGaeiiFaWNaemyrauKaeyOkIGSaemOrayeabeaaaaa@3510@ will in general be a flux-dependent mixture of product distributions DMi|E∪F=∑jvjDMij|E⊗DMij|F MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiraq0aaSbaaSqaaiabd2eannaaBaaameaacqWGPbqAaeqaaSGaeiiFaWNaemyrauKaeyOkIGSaemOrayeabeaakiabg2da9maaqababaGaemODay3aaSbaaSqaaiabdQgaQbqabaGccqWGebardaWgaaWcbaGaemyta00aaSbaaWqaaiabdMgaPjabdQgaQbqabaWccqGG8baFcqWGfbqraeqaaaqaaiabdQgaQbqab0GaeyyeIuoakiabgEPielabdseaenaaBaaaleaacqWGnbqtdaWgaaadbaGaemyAaKMaemOAaOgabeaaliabcYha8jabdAeagbqabaaaaa@4E38@. This case of statistical independence is depicted in Figure 5.

A generalized form of (6), useful for propagation of isotopomer constraints, is derived as follows. Assume independent fragments M|E and M|F of metabolite M and isotopomer constraints S_E∪FD_M|E∪F = d_E∪F, S_ED_M|E = d_E and S_FD_M|F = d_F, where S = S_E ⊗ S_F. Now, the the h'th constraint for fragment E and g'th constraint for fragment F, written as Σ_as_ah,EP_M|E(a) = d_h,E and Σ_cs_cg,FP_M|F(c) = d_g,F.

Multiplying the constraints, and denoting s_b = s_ah,E·s_cg,F where b is the isotopomer consistent with fragment isotopomers a and c, we get the following equation

d h , E ⋅ d g , F = ( ∑ a s a h , E P M | E ( a ) ) ⋅ ( ∑ a s c g , F P M | F ( a ) ) = ∑ a s b P M | E ∪ F ( b ) = d l , E ∪ F , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7DA1@

for the l'th constraint for E ∪ F. The equations of the above kind can be concisely written in terms of tensors:

d = SD_M|E∪F = (S_E ⊗ S_F) D_M|E ⊗ D_M|F = d_E ⊗ d_F.

From above, if two of the three vectors d, d_E, d_F are known, the remaining unknown one can be solved.

We note in passing that computing constraints to the metabolite given constraints to the fragments is straightforward application of (7).

In our framework, statistical independence of fragments has two uses. We apply it

1. to compute isotopomer constraints for the union of independent fragments, given isotopomer constraints to its independent fragments, and

2. to compute isotopomer constraints for an independent fragment given isotopomer constraints to the other fragments and the metabolite as a whole.

In both cases making use of (6) gives us a larger set of constraints than the vector space and flow analysis approach alone.

Next we describe how (7) generalizes the basic measurement propagation step of the traditional metabolic flux ratio analysis 31. In the basic case, the flux ratios are solved for two competing pathways p and q, which p cleaves a certain carbon-carbon bond b of junction M while the q preserves b intact from the external substrate. (See Figure 6 for an example). This serves also as an example of applying (7) to compute isotopomer constraints for the union of independent fragments.

When a uniformly labelled substrate is used, the labelling degree of every carbon in the network is the same (and known a priori) when the system reaches isotopomeric steady state. Thus, the isotopomer distribution DFp MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiraq0aaSbaaSqaaiabdAeagnaaBaaameaacqWGWbaCaeqaaaWcbeaaaaa@2FC8@ of a two-carbon fragment F (metabolite M_F in Figure 6) containing bond b can be computed by (7) for pathway p cleaving b, while for pathway q, DFq MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiraq0aaSbaaSqaaiabdAeagnaaBaaameaacqWGXbqCaeqaaaWcbeaaaaa@2FCA@ can be propagated from the external substrate (metabolite M_E in Figure 6) using the fragment equivalence classes of the previous section. If we are able to measure (constraints to) the isotopomer distribution of the mixed pool F, we can then automatically derive a generalized isotopomer balance corresponding the manually derived ratio. To use (7) to compute isotopomer constraints for an independent fragment from the known isotopomer constraints to the other independent fragment and to the whole metabolite is complicated by the incompleteness of the measurement data: an arbitrary measurement SD_M|E∪F = d might not be directly representable via a tensor product S = S_E ⊗ S_F. Instead, we need to first compute isotopomer subspaces for known isotopomer constraints where (7) can be applied.

The detailed description of this technique is rather technical and omitted from this article. Here we give an example of the technique (See Figure 7). We assume that we know the mass isotopomer distributions DMim MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiraq0aa0baaSqaaiabd2eannaaBaaameaacqWGPbqAaeqaaaWcbaGaemyBa0gaaaaa@312C@ of metabolite M_i (metabolite M₁ in Figure 7) and DMi|Em MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiraq0aa0baaSqaaiabd2eannaaBaaameaacqWGPbqAaeqaaSGaeiiFaWNaemyraueabaGaemyBa0gaaaaa@33BF@ of fragment M_i|E. We furthermore assume that M_i|E and M_i|F are independent. From this information the mass isotopomer distribution of DMi|Em MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiraq0aa0baaSqaaiabd2eannaaBaaameaacqWGPbqAaeqaaSGaeiiFaWNaemyraueabaGaemyBa0gaaaaa@33BF@ can be solved. To be exact, DMi|Em MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiraq0aa0baaSqaaiabd2eannaaBaaameaacqWGPbqAaeqaaSGaeiiFaWNaemyraueabaGaemyBa0gaaaaa@33BF@ can be solved from the system containing an equation

P M i ( + p ) = ∑ k + l = p P E ( + k ) ⋅ P F ( + l ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiuaa1aaSbaaSqaaiabd2eannaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabgUcaRiabdchaWjabcMcaPiabg2da9maaqafabaGaemiuaa1aaSbaaSqaaiabdweafbqabaGccqGGOaakcqGHRaWkcqWGRbWAcqGGPaqkcqGHflY1cqWGqbaudaWgaaWcbaGaemOrayeabeaakiabcIcaOiabgUcaRiabdYgaSjabcMcaPaWcbaGaem4AaSMaey4kaSIaemiBaWMaeyypa0JaemiCaahabeqdcqGHris5aaaa@4C87@

for each mass isotopomer p of M_i. To see that (8) conforms to (7), we denote the measured mass isotopomer distribution DMim MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiraq0aa0baaSqaaiabd2eannaaBaaameaacqWGPbqAaeqaaaWcbaGaemyBa0gaaaaa@312C@ = S·D(M_i) of M_i by d_M (i.e. rows of coefficient matrix S correspond to different mass isotopomers of M_i) and the measured mass isotopomer distributions of E and F by d_E and d_F. Let |E| = 2, |F| = 1 and |M_i| = 3, thus M_i = E ∪ F. We have

S E = ( 1 0 0 0 0 1 1 0 0 0 0 1 ) , S F = ( 1 0 0 1 ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4uam1aaSbaaSqaaiabdweafbqabaGccqGH9aqpdaqadaqaauaabeqadqaaaaqaaiabigdaXaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabigdaXaqaaiabigdaXaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabigdaXaaaaiaawIcacaGLPaaacqGGSaalcqWGtbWudaWgaaWcbaGaemOrayeabeaakiabg2da9maabmaabaqbaeqabiGaaaqaaiabigdaXaqaaiabicdaWaqaaiabicdaWaqaaiabigdaXaaaaiaawIcacaGLPaaacqGGUaGlaaa@470E@

with the tensor product

S E ⊗ S F = ( 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 ) , and  S = ( 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4uam1aaSbaaSqaaiabdweafbqabaGccqGHxkcXcqWGtbWudaWgaaWcbaGaemOrayeabeaakiabg2da9maabmaabaqbaeqabyacaaaaaaqaaiabigdaXaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabigdaXaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabigdaXaqaaiabicdaWaqaaiabigdaXaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabigdaXaqaaiabicdaWaqaaiabigdaXaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabigdaXaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabicdaWaqaaiabigdaXaaaaiaawIcacaGLPaaacqGGSaalcqqGHbqycqqGUbGBcqqGKbazcqqGGaaicqWGtbWucqGH9aqpdaqadaqaauaabeqaeGaaaaaaaeaacqaIXaqmaeaacqaIWaamaeaacqaIWaamaeaacqaIWaamaeaacqaIWaamaeaacqaIWaamaeaacqaIWaamaeaacqaIWaamaeaacqaIWaamaeaacqaIXaqmaeaacqaIXaqmaeaacqaIWaamaeaacqaIXaqmaeaacqaIWaamaeaacqaIWaamaeaacqaIWaamaeaacqaIWaamaeaacqaIWaamaeaacqaIWaamaeaacqaIXaqmaeaacqaIWaamaeaacqaIXaqmaeaacqaIXaqmaeaacqaIWaamaeaacqaIWaamaeaacqaIWaamaeaacqaIWaamaeaacqaIWaamaeaacqaIWaamaeaacqaIWaamaeaacqaIWaamaeaacqaIXaqmaaaacaGLOaGaayzkaaGaeiOla4caaa@8AEB@

As the two matrices are not the same (7) is not directly applicable. However, by summing up the second and the third rows and the fourth and the fifth rows of S_E ⊗ S_F we obtain S. Intuitively, this means that Equation (7) can be applied to compute DM|Fm MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiraq0aa0baaSqaaiabd2eanjabcYha8jabdAeagbqaaiabd2gaTbaaaaa@322E@, when we take into account that the isotopomer constraints corresponding both second and third rows of S_E ⊗ S_F contribute to mass isotopomer M_i(+1), while the isotopomer constraints corresponding fourth and fifth rows of S_E ⊗ S_F contribute to mass isotopomer M_i(+2). (From the definition of the tensor product we see that, for example, the second row of S_E ⊗ S_F corresponds to isotopomer constraints P_E(00)·P_F(1) = P_E(+0)·P_F(+1) and the third row corresponds to the constraints P_E(01)·P_F(0) + P_E(10)·P_F(0) = P_E(+1)·P_F(+0), thus validating our intuitive observation.) When the similiar information for all rows of S_E ⊗ S_F is collected to a linear equation system, we will obtain the following constraints to the mass isotopomer distribution DM|Fm MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiraq0aa0baaSqaaiabd2eanjabcYha8jabdAeagbqaaiabd2gaTbaaaaa@322E@ (which in the case of one-carbon-fragment M|F coincides with the isotopomer distribution D_M|F):

( P E ( + 0 ) 0 P E ( + 1 ) P E ( + 0 ) P E ( + 2 ) P E ( + 1 ) 0 P E ( + 2 ) ) D m ( M | F ) = d M = ( P M i ( + 0 ) P M i ( + 1 ) P M i ( + 2 ) P M i ( + 3 ) ) = S D M i , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaeWaaeaafaqabeabcaaaaeaacqWGqbaudaWgaaWcbaGaemyraueabeaakiabcIcaOiabgUcaRiabicdaWiabcMcaPaqaaiabicdaWaqaaiabdcfaqnaaBaaaleaacqWGfbqraeqaaOGaeiikaGIaey4kaSIaeGymaeJaeiykaKcabaGaemiuaa1aaSbaaSqaaiabdweafbqabaGccqGGOaakcqGHRaWkcqaIWaamcqGGPaqkaeaacqWGqbaudaWgaaWcbaGaemyraueabeaakiabcIcaOiabgUcaRiabikdaYiabcMcaPaqaaiabdcfaqnaaBaaaleaacqWGfbqraeqaaOGaeiikaGIaey4kaSIaeGymaeJaeiykaKcabaGaeGimaadabaGaemiuaa1aaSbaaSqaaiabdweafbqabaGccqGGOaakcqGHRaWkcqaIYaGmcqGGPaqkaaaacaGLOaGaayzkaaGaemiraq0aaSbaaSqaaiabd2gaTbqabaGccqGGOaakcqWGnbqtcqGG8baFcqWGgbGrcqGGPaqkcqGH9aqpcqWHKbazdaWgaaWcbaGaemyta0eabeaakiabg2da9maabmaabaqbaeqabqqaaaaabaGaemiuaa1aaSbaaSqaaiabd2eannaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabgUcaRiabicdaWiabcMcaPaqaaiabdcfaqnaaBaaaleaacqWGnbqtdaWgaaadbaGaemyAaKgabeaaaSqabaGccqGGOaakcqGHRaWkcqaIXaqmcqGGPaqkaeaacqWGqbaudaWgaaWcbaGaemyta00aaSbaaWqaaiabdMgaPbqabaaaleqaaOGaeiikaGIaey4kaSIaeGOmaiJaeiykaKcabaGaemiuaa1aaSbaaSqaaiabd2eannaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabgUcaRiabiodaZiabcMcaPaaaaiaawIcacaGLPaaacqGH9aqpcqWGtbWucqWGebardaWgaaWcbaGaemyta00aaSbaaWqaaiabdMgaPbqabaaaleqaaOGaeiilaWcaaa@8744@

which is equal to (8).

Isotopomer tracer experiments using less common carbon sources can be very costly because of the prices of purposefully labelled substrates. Thus, it is very useful to be able to first conduct in silico calculability analysis to find out, whether it is even in principle possible to obtain the required flux information from the tracer experiment. By analyzing the fragment equivalence classes, it is relatively easy to perform this kind of "structural identifiability analysis" (cf. 3940 for global isotopomer balancing), that is, to discover the set of junction metabolites for which the flux ratios can be calculated (in the best case) from the given measurement data: it is enough to check what type of isotopomer constraints

∑ b s b i j P M i j ( b ) = d i j , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaabuaeaacqWGZbWCdaWgaaWcbaGaemOyaiMaemyAaKMaemOAaOgabeaakiabdcfaqnaaBaaaleaacqWGnbqtdaWgaaadbaGaemyAaKMaemOAaOgabeaaaSqabaaabaGaemOyaigabeqdcqGHris5aOGaeiikaGIaemOyaiMaeiykaKIaeyypa0Jaemizaq2aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGGSaalaaa@43C8@

can be propagated to each subpool M_ij of junction metabolites M_i from the measured metabolites (we need to know only coefficients s_bij, not the isotopomer abundances d_ij). Then, by applying the techniques of computing vector subspace intersection described above, we can compute the maximal number of linearly independent constraints obtainable for the flux ratios of each junction. Thus, it is possible to check before costly and time-consuming wet lab experiments, whether the experiments even have potential to answer the biological questions at hand. The results of the calculability analysis tell which flux ratios are in principle determinable, given the labelling of external substrates, topology of the metabolic network and the available measurement data. It then depends on the actual flux distribution and the accuracy of the measurements, whether these ratios can be reliably determined from the experimental data.

The aim of the propagation of measurement data is to infer from the isotopomer constraints of measured metabolites as many isotopomer constraints as possible to unmeasured metabolites. As a rule of thumb, more constraints the unmeasured metabolites will get more generalized isotopomer balance equations (4) bounding the fluxes can be written.

Fragment equivalence classes can be utilized in the measurement propagation: from isotopomer constraints known for fragment M_i|F isotopomer constraints for other fragments M_l|F_k in the equivalence class of F can be easily computed. The process is the following:

1. Before measurements are propagated from fragment M|F of measured metabolite M to other fragments in the equivalence class of F, isotopomer constraints to F are computed from the constraints measured to the whole metabolite M by using the vector space projection techniques (see Section Projection of isotopomer measurements to fragments).

2. The fragment constraints are propagated to all fragments F' that have been found equivalent to F via the flow analysis technique.

This requires mapping of isotopomers of F to isotopomers of F' by applying the carbon mappings of the reactions along any pathway between F and F'.

3. After the propagation of measurement data inside the fragment equivalence classes, new isotopomer constraints for independent fragments of the same metabolite can be derived, as described in Section Independence analysis of fragments.

Steps 2 and 3 can be iterated until no new isotopomer constraints to the fragments are discovered.

After the propagation step, we have some isotopomer constraints SijDMij=dij MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4uam1aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqWGebardaWgaaWcbaGaemyta00aaSbaaWqaaiabdMgaPjabdQgaQbqabaaaleqaaOGaeyypa0JaeCizaq2aaSbaaSqaaiabdMgaPjabdQgaQbqabaaaaa@3A8B@ for each subpool j of every junction metabolite M_i. (For non-junction metabolites, isotopomer balance equations do not contain any additional flux information compared to the mass balances.) In the best case we know complete isotopomer distribution DMij MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiraq0aaSbaaSqaaiabd2eannaaBaaameaacqWGPbqAcqWGQbGAaeqaaaWcbeaaaaa@3125@, in the worst case we have only trivial constraints stating that the sum of relative abundances of all isotopomers equals one.

Next, a linear equation system containing flux constraints obtained from mass balances (1) and generalized isotopomer balances (4) is constructed.

However, the isotopomer constraints of different subpools do not yet conform to (4) as the matrices S_ij are not necessarily the same.

Thus we still need to compute a common subspace Yi=∩jSij MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8hgXN1aaSbaaSqaaiabdMgaPbqabaGccqGH9aqpcqWIPissdaWgaaWcbaGaemOAaOgabeaakiab=jr8tnaaBaaaleaacqWGPbqAcqWGQbGAaeqaaaaa@4187@ (Sij MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8NeXp1aaSbaaSqaaiabdMgaPjabdQgaQbqabaaaaa@3A51@ is spanned by the rows of S_ij) of the isotopomer constraints known for each subpool M_ij and project subpool constraints SijDMij=dij MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4uam1aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqWGebardaWgaaWcbaGaemyta00aaSbaaWqaaiabdMgaPjabdQgaQbqabaaaleqaaOGaeyypa0JaeCizaq2aaSbaaSqaaiabdMgaPjabdQgaQbqabaaaaa@3A8B@ to Yi MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8hgXN1aaSbaaSqaaiabdMgaPbqabaaaaa@3900@.

This can be done with the same techniques that were previously applied to project measured isotopomer information of a metabolite to its fragments. Let Y_i be a matrix with row space Yi MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8hgXN1aaSbaaSqaaiabdMgaPbqabaaaaa@3900@. After the projection we obtain isotopomer constraints YiDMij=zij MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemywaK1aaSbaaSqaaiabdMgaPbqabaGccqWGebardaWgaaWcbaGaemyta00aaSbaaWqaaiabdMgaPjabdQgaQbqabaaaleqaaOGaeyypa0JaeCOEaO3aaSbaaSqaaiabdMgaPjabdQgaQbqabaaaaa@3966@ for each subpool M_ij (See Figure 8 for an example).

Now the isotopomer constraints of all the subpools lie in the same subspace of IMi MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8heHK0aaSbaaSqaaiabd2eannaaBaaameaacqWGPbqAaeqaaaWcbeaaaaa@3973@ and we are ready to write the system of generalized isotopomer balance equations (4) for every junction M_i:

∑ j = 1 n α i j v j z i j = β i z i 0 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaabCaeaacqaHXoqydaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabdAha2naaBaaaleaacqWGQbGAaeqaaOGaeCOEaO3aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGH9aqpcqaHYoGydaWgaaWcbaGaemyAaKgabeaakiabhQha6naaBaaaleaacqWGPbqAcqaIWaamaeqaaaqaaiabdQgaQjabg2da9iabigdaXaqaaiabd6gaUbqdcqGHris5aOGaeiilaWcaaa@482E@

that is,

A i v = [ α 1 i ( z 1 i ) 1 ⋯ α n i ( z n i ) 1 ⋮ ⋱ ⋮ α 1 i ( z 1 i ) r ⋯ α n i ( z n i ) r ] ⋅ [ v 1 ⋮ v n ] = g i , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7C5E@

where g_i = β_iz_i0.

The ratios of (forward) fluxes producing M_i can be computed by solving the corresponding Equation (11) augmented with a constraint that fixes the out flow from M_i to equal 1. Thus, we obtain flux ratios of junction metabolites without manual derivation of ratio equations, without nonlinear optimization and without knowing intake and outtake rates of external metabolites or biomass composition.

In addition, when the equations (11) of all junction metabolites are combined with the mass balances (1) of non-junctions, we obtain a linear equation system

A v = ( A 1 ⋮ A m ) ⋅ [ v 1 ⋮ v n ] = [ g 1 ⋮ g m ] = g MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyqaeKaeCODayNaeyypa0ZaaeWaaeaafaqabeWabaaabaGaemyqae0aaSbaaSqaaiabigdaXaqabaaakeaacqWIUlstaeaacqWGbbqqdaWgaaWcbaGaemyBa0gabeaaaaaakiaawIcacaGLPaaacqGHflY1daWadaqaauaabeqadeaaaeaacqWG2bGDdaWgaaWcbaGaeGymaedabeaaaOqaaiabl6UinbqaaiabdAha2naaBaaaleaacqWGUbGBaeqaaaaaaOGaay5waiaaw2faaiabg2da9maadmaabaqbaeqabmqaaaqaaiabhEgaNnaaBaaaleaacqaIXaqmaeqaaaGcbaGaeSO7I0eabaGaeC4zaC2aaSbaaSqaaiabd2gaTbqabaaaaaGccaGLBbGaayzxaaGaeyypa0JaeC4zaCgaaa@50B7@

constraining the fluxes v of the network that contains a block (junctions) or a row (non-junctions) A_k for each metabolite M_k. Measured external fluxes and other known constraints, such as the composition of biomass can also be added to (12) as additional constraints. Additional constraints, like ones derived from gene regulatory information 41 or from thermodynamic analysis of metabolism 424344 can also easily be included to (12).

If (12) is of full rank, the whole flux distribution can be solved with standard linear algebra 45. Also, more complex, nonlinear methods can be applied to model the effect of experimental errors to the estimated flux distribution 20. In a common case where the system is of less than full rank, a single flux distribution can not be pinpointed without additional constraints. Instead, (12) defines the space of feasible flux distributions, that are all equally good solutions. In that case we can apply techniques developed for the analysis of stoichiometric matrices to determine as many fluxes as possible 46 from (12). More generally, by linear programming we can obtain maximum (resp. minimum) values for each flux v_i:

For each  v i : max ⁡ v i s .t . A v = g v i m i n ≤ v i ≤ v i m a x ∀ v i ∈ v , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6A0B@

where vimin MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemODay3aa0baaSqaaiabdMgaPbqaaiabd2gaTjabdMgaPjabd6gaUbaaaaa@32F5@ and vimax MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemODay3aa0baaSqaaiabdMgaPbqaaiabd2gaTjabdggaHjabdIha4baaaaa@32F9@ are predetermined minimum and maximum allowable values for v_i

Furthermore, it is possible to search for in some sense optimal flux distribution – for example a flux distribution maximizing the production of biomass – from the feasible space defined by (12) by linear programming techniques of flux balance analysis 134748. In that case, isotopomer data constrain the feasible space more than the stoichiometric information would alone do, thus possibly allowing more accurate estimations of the real flux distribution.