A procedure for the estimation over time of metabolic fluxes in scenarios where measurements are uncertain and/or insufficient
 Francisco Llaneras^{1}Email author and
 Jesús Picó^{1}
DOI: 10.1186/147121058421
© Llaneras and Picó; licensee BioMed Central Ltd. 2007
Received: 24 May 2007
Accepted: 30 October 2007
Published: 30 October 2007
Abstract
Background
An indirect approach is usually used to estimate the metabolic fluxes of an organism: couple the available measurements with known biological constraints (e.g. stoichiometry). Typically this estimation is done under a static point of view. Therefore, the fluxes so obtained are only valid while the environmental conditions and the cell state remain stable. However, estimating the evolution over time of the metabolic fluxes is valuable to investigate the dynamic behaviour of an organism and also to monitor industrial processes. Although Metabolic Flux Analysis can be successively applied with this aim, this approach has two drawbacks: i) sometimes it cannot be used because there is a lack of measurable fluxes, and ii) the uncertainty of experimental measurements cannot be considered. The Flux Balance Analysis could be used instead, but the assumption of optimal behaviour of the organism brings other difficulties.
Results
We propose a procedure to estimate the evolution of the metabolic fluxes that is structured as follows: 1) measure the concentrations of extracellular species and biomass, 2) convert this data to measured fluxes and 3) estimate the nonmeasured fluxes using the Flux Spectrum Approach, a variant of Metabolic Flux Analysis that overcomes the difficulties mentioned above without assuming optimal behaviour. We apply the procedure to a real problem taken from the literature: estimate the metabolic fluxes during a cultivation of CHO cells in batch mode. We show that it provides a reliable and rich estimation of the nonmeasured fluxes, thanks to considering measurements uncertainty and reversibility constraints. We also demonstrate that this procedure can estimate the nonmeasured fluxes even when there is a lack of measurable species. In addition, it offers a new method to deal with inconsistency.
Conclusion
This work introduces a procedure to estimate timevarying metabolic fluxes that copes with the insufficiency of measured species and with its intrinsic uncertainty. The procedure can be used as an offline analysis of previously collected data, providing an insight into the dynamic behaviour of the organism. It can be also profitable to the online monitoring of a running process, mitigating the traditional lack of reliable online sensors in industrial environments.
Background
Fostered by the importance of studying the cell metabolism under a systemlevel approach [1, 2], the set of metabolic pathways of organisms of interest are assembled in metabolic networks [3, 4]. If it is assumed that the intracellular metabolites of a network are at pseudo steadystate, mass balances around each metabolite can be described by means of a homogeneous system of linear equations [5]. These equations can be considered as stoichiometric constraints. Then, the constraints imposed by enzyme or transport capacities and thermodynamics (e.g. irreversibility of reactions) can be incorporated to the system [6]. Thereby the imposed constraints define a space where every feasible flux distribution lives [7]. Since the metabolic phenotype can be defined in terms of flux distributions through a metabolic network, this space represents (or at least contains) the set of feasible phenotypes [8]. The environmental conditions given at a certain time instant will determine which of these flux distributions corresponds to the actual one [9].
Coupling constraints with experimental measurements
Experimental measurements of fluxes can be incorporated as constraints, in order to determine the actual flux distribution or at least to reduce the space of possible flux distributions. However, it must be taken into account that measurements are not invariant constraints, but specific condition constraints [8]. There are several methodologies that use this approach with different purposes: estimate the nonmeasured fluxes, predict flux distributions, investigate the cell behaviour or monitor bioprocesses.
Metabolic Flux Analysis (MFA) provides a methodology to uniquely determine the actual flux distribution by using a metabolic network and a set of measured fluxes [5]. It has been intensively used in recent years with successful results [10–13]. As it can only consider stoichiometric constraints, a considerable number of fluxes need to be measured to determine the rest of the fluxes. Unfortunately, the available measurements are often insufficient [14].
The Flux Balance Analysis (FBA) can be used to predict metabolic flux distributions [15, 16]. Firstly, a constraintbased model is defined as a set of invariant constraints: stoichiometrics, thermodynamics, etc. Then, only a few specific condition constraints (usually substrates uptakes) are imposed. Subject to these constraints, which define a region of possible flux distributions, an optimal flux distribution is calculated using linear programming. Yet, the optimal solution may not correspond to the actual flux distribution. It must be hypothesized that i) the cell has identified the optimal solution, ii) the objective sought by the cell is known, and iii) it can be mathematically expressed. However, FBA predictions based on different objective functions (e.g. maximize growth) are consistent with experimental data [17–19].
Estimating the evolution over time of flux distributions
Typically, calculation of a flux distribution (e.g. with MFA or FBA) is done under a static point of view: the measured fluxes are assumed to be constant. That means that the obtained flux distribution will only be valid during a certain period of time, while the environmental conditions and the cell state remain steady (e.g. during the growth phase). However, if these conditions change along time, as it happens in an actual culture, the flux distribution will change. The estimation of the flux distribution over time can be useful to investigate the dynamic behaviour of the microorganism or to monitor the progress of industrial fermentations [20]. In [21], the classical FBA is extended to predict the dynamic evolution of flux distributions. In [22], an approach based on elementary modes and the assumption of optimal behaviour is used to estimate the flux distributions of Corynebacterium glutamicum at different temporal phases of fermentation. Elementary modes are also employed in [23], where the cell life is decomposed in a succession of phases, and then the timevarying intracellular fluxes are obtained by switching the flux distributions calculated at each phase. In [24], online MFA is successfully applied to quantify coupled intracellular fluxes. Takiguchi et al. [25] use a similar approach to recognize the physiological state of the cells culture. They also show how this information can be used to improve Lysine production yield. Very recently [26] has presented an online estimation of intracellular fluxes applying MFA to an overdetermined metabolic network.
To calculate the succession of flux distributions, it is usually assumed that intracellular fluxes are in quasisteady state within each measurement step. However, that does not mean that the intrinsic dynamic nature of the cultivation is being disregarded. Instead, the intracellular fluxes will follow the change of environmental conditions as mediated by the measured fluxes (e.g. substrate uptakes). Hence, steady states may undergo shifting from one state to another depending on the evolution of the measured fluxes [27]. Such assumption has been successfully applied in the works cited in above and in the development of several dynamic models [23, 28–32]. This approach makes it possible to study the dynamic behaviour of the organism, without considering the still not wellknown intracellular kinetics.
Using the flux spectrum approach to estimate the fluxes
MFA can be successively applied to estimate the evolution of a flux distribution over time. However, this approach has three main difficulties: i) It cannot be used when measurements are scant (i.e. when the system is underdetermined). This happens very often due to the lack of measurable fluxes. ii) The uncertainty of the measured fluxes cannot be considered. Not only gross errors may appear which could be managed only in case there are redundant measured fluxes but also most sources of measurements are intrinsically uncertain and the propagation of this uncertainty to the estimated fluxes is not controlled, and iii) only equalities can be used as constraints. For instance, reversibility constraints or maximum flux values cannot be taken into account. FBA solves the first difficulty and provides a framework to deal with the other ones. But the use of FBA in this context could be problematic due to the appearance of a timevariant metabolic objective [22]. For these reasons, the procedure introduced in this work uses the Flux Spectrum Approach (FSA) [33]. It is a variant of MFA that includes some characteristics of FBA (e.g. it is not restricted to stoichiometric constraints) and provides some additional benefits (e.g. it allows to consider measurements uncertainty). The use of FSA will make it possible to face the difficulties described above without assuming an optimal behaviour of the organism.
Although FSA is capable of considering a wide range of constraints, in this work we will only use stoichiometric relationships and simple thermodynamic constraints (reactions directions), and we assume them to be known a priori. However, it must be noticed that the incorporation of thermodynamic constraints based on measurements or estimations of the standard Gibbs free energy change of reactions is capturing attention in recent times. A genomescale thermodynamic analysis of Escherichia coli has been recently carried out [34]. Kümmel et al. have introduced an algorithm that based on thermodynamics, network topology and heuristic rules automatically assigns reaction directions in metabolic models such that the reaction network is thermodynamically feasible [35]. Interestingly, the reaction directions obtained can be incorporated as constraints before using FSA. Standard Gibbs free energy changes have been also used to incorporate thermodynamic realizability as constraint for FBA [36] or in an analogous manner to FSA, and to develop a new form of MFA with the capability of generating thermodynamically feasible fluxes [37].
The objectives of this article are twofold: first, introduce a procedure for the estimation of the metabolic fluxes over time by using a metabolic network as a constraintbased model and a reduced set of measurable species. This procedure is capable of coping with lack of measured species and with its intrinsic uncertainty, thanks to the use of the Flux Spectrum Approach (FSA). Second, illustrate this procedure with a real example: the estimation of nonmeasured fluxes during a cultivation of CHO cells in batch mode in stirred flasks.
Results and discussion
Procedure overview
It is sometimes overlooked that extracellular fluxes are not directly measured. Instead, the concentrations of a set of species are measured (step 1), and those data are converted to flux units or measured fluxes (step 2). The importance of a good conversion should not be disregarded: error in the measurements of concentrations may be amplified through the conversion, incorporated into the measured fluxes, and then propagated to the estimation of the nonmeasured fluxes. To minimize this hitch, the conversion should be done carefully. Afterwards, the nonmeasured fluxes can be estimated by coupling the metabolic network and the measured fluxes (step 3). This has been done before by means of the MFA methodology [24–26]. Yet, this approach has certain limitations. We will overcome some of them using FSA.
It must be remarked that the procedure can be used in two main scenarios: as an offline analysis of previously collected data or as an online monitoring of an industrial process. The structure of the procedure and its fundamental step (step 3) are exactly the same in both cases. Nevertheless, there are several differences concerning step 2. These differences will be briefly described along the article and illustrated in an additional file [additional file 3].
Preliminaries: choice and analysis of the metabolic network
A metabolic network can be represented with a stoichiometric matrix S, where rows correspond to the m metabolites and columns to the n fluxes. Assuming that the intracellular metabolites are at pseudosteady state, material balances around them can be formulated as follows [38, 39]:
S·v = 0
where v is a flux distribution. Assuming that S has full row rank, the number of independent equations is m. As typically n becomes larger than m, the system (1) is underdetermined (nm degrees of freedom). That means that there is not a unique flux distribution fulfilling (1), but an infinite number of feasible flux distributions. In order to determine which of these feasible flux distributions is the current one, the constraints imposed by the measured fluxes will be incorporated latter on it will be shown that other constraints, for example the reversibility constraints, can be added.
Thereby, when choosing the metabolic network to be used through the procedure, it must be taken into account that its degree of detail needs to be compatible with the number of available measurements i.e. the available measurements must be sufficient to offset the underdeterminacy of the network. In order to study this, we can analyze the system formed by the stoichiometric constraints given by (1) and the constraints imposed by the measured fluxes. This system which constitutes the fundamental equation of MFA can be obtained making a partition between measured (subindex m) and nonmeasured or unknown fluxes (subindex u):
S_{ u }·v_{ u }= S_{ m }·v_{ m }
System Determinacy and Calculability of Fluxes
System (2) is determined when there are enough linearly independent constraints for uniquely calculate all nonmeasured fluxes v_{ u }; i.e., when rank(S_{ u }) = u (u is the number of nonmeasured fluxes). On the contrary, when rank(S_{ u })>u, the system is classified as underdetermined because at least one nonmeasured flux, and probably most of them, is non calculable [14]. If the system is underdetermined, the traditional MFA methodology cannot be used to calculate the nonmeasured fluxes. Fortunately, the use of FSA may provide an estimation of the nonmeasured fluxes even in this situation. However, it must be taken into account that the likelihood of obtaining a precise estimation increases as the underdeterminancy reduces, as the set of flux distributions compatible with the measured values will be smaller.
System Redundancy and Consistency of Measurements
System (2) is redundant when some rows in S_{ u }can be expressed as linear combinations of other rows; i.e., when rank(S_{ u })<m. This can lead to an inconsistent system if the vector v_{ m }contains such values that no v_{ u }exists that exactly solves (2). Therefore, when the system is redundant, the inconsistency of the measurements can be checked and its importance can be estimated (see methods). Unfortunately some measured fluxes have no impact on the consistency of the system, so they cannot be considered in the analysis of consistency. These fluxes are called nonbalanceable. The balanceable fluxes can be detected as explained in [14], and they should be adjusted (or balanced) in case the system is inconsistent (see methods). All these methods are commonly applied when MFA is used [12, 24, 26]. They can also be used within our procedure, but in addition the use of FSA provides new methods to deal with inconsistency as it will be shown in a subsequent section.
Step 1: Getting experimental measurements of species
There are several alternatives to measure the concentration of species e.g. online sensors, isotopic tracer experiments or laboratory procedures but providing a detailed description of each one is out of the scope of this work. In any case, it must be remembered that the more measurements are available, the more nonmeasured fluxes may be accurately estimated. However, it is necessary to be prepared to overcome a lack of measurements, especially when the procedure is done online (due to the lack of reliable online sensors).
Step 2: conversion of measured concentrations in measured fluxes
where ξ is the specie concentration, v_{ ξ }its flux (substrate uptake or product formation), X the biomass concentration,D the dilution term and F_{ ξ }the net exchange of the specie with the outside. Notice that this equation is only valid for extracellular species; however, the biomass growth and the mass balance around an internal metabolite not assumed to be at pseudosteady state can be represented in a similar way [40, 41].
It is possible to calculate v_{ ξ }as a function of ξ, X, D, F_{ ξ }and dξ/dt. But this presents two main difficulties: i) approximate a derivative (directly or indirectly) and ii) deal with the presence of errors and noise in the measurements of the concentration ξ [42]. The underlying problem is how precision can be combined with robustness with respect to measurement errors. The most straightforward approach is to approximate the derivative with a simple method (e.g. Euler or RungeKutta methods) and then solve (3) [42]. Very often this straight approximation needs to be combined with the use of filters to eliminate or at least to reduce the presence of noise. This approach provides very good results when centred methods can be used to approximate the derivative and to filter the resultant signal, i.e. when the whole procedure is done offline, or when it is done online but certain delay in the calculation of the fluxes is allowable (i.e. when past, ki, and future information, k+i, is available for the calculation of v_{ ξ }(k)). Furthermore, there are methods especially aimed to the online approximation of the derivative. If the noise signal is well characterized (e.g. the frequency band or a stochastic feature is known) a linear differentiator [43] or even a Luenberger observer may be used [44]. If nothing is known on the structure of the signal, then sliding mode techniques are profitable. For example, the method introduced in [45] combines exact differentiation for a large class of input signals with robustness with respect to any small noises. Finally, there are other approaches to calculate the extracellular fluxes that avoid the approximation of the derivative, as for example the use of extended Kalman filters [26, 46] or the observers based on concepts from nonlinear systems theory, such as the high gain estimators described in [40, 47]. These methods do not use future information because they are aimed to the online operation mode.
The importance of the use of filters should be remarked: not only the signal of measured concentrations should be filtered to reduce its noise, but also the calculated extracellular fluxes may be filtered to get a smooth signal. Filters based on the moving average will be used in this work since they are simple and versatile. Basically, the filtered value at time k is calculated by averaging the values of the original signal within a time window. There are several versions that differ in the time window used (backward or centred) and in the distribution of weight over the averaged values (uniform or exponential). Interestingly, this kind of filters has already been successfully applied to the calculation of metabolic fluxes [42].
Step 3: estimation of the nonmeasured fluxes using FSA
Finally, the measured fluxes obtained in step 2 are coupled with known biological constraints in order to estimate the nonmeasured fluxes (Figure 1). Basically, this implies that a solution for system (2) has to be found at each time instant k. Traditionally MFA was successively applied with this purpose. Unfortunately, as mentioned in the background section, this has some limitations which become especially critical if the procedure is done online, due to the traditional lack of reliable online sensors. To overcome them, the Flux Spectrum Approach (FSA) will be used instead in the third step of our procedure.
Comparison between MFA and FSA
Traditional Metabolic Flux Analysis (MFA)  Flux Spectrum Approach (FSA)  

Flux estimation  Consistency  Flux estimation  Consistency  
Determined  Yes.  Consistency check χ^{2}.  Yes.  Consistency check χ^{2}. 
Redundant  Flux adjustment.  Considers uncertainty.  Flux adjustment or use of a band of uncertainty.  
Detects sensitivity problems.  
Determined  Yes.  No.  Yes.  Detect some inconsistencies. 
Not Redundant  No.  Considers uncertainty.  
Detects sensitivity problems.  
Underdetermined  No.  Consistency check χ^{2}.  Yes (not guaranteed).  Consistency check χ^{2}. 
Redundant  Flux adjustment.  Considers uncertainty.  Flux adjustment or use of a band of uncertainty.  
Detects sensitivity problems.  
Underdetermined  No.  No.  Yes (not guaranteed).  Detect some inconsistencies. 
Not Redundant  No.  Considers uncertainty.  
Detects sensitivity problems. 

It makes it possible to consider the uncertainty of experimental measurements and even qualitative knowledge (e.g. maximum values of certain fluxes). Hence, if measurements uncertainty is indeed present and it is well characterized, the estimation of nonmeasured fluxes will be more reliable (Figure 3E). FSA provides not only a prediction of the fluxes, but also an indication of the reliability of this prediction.

It considers the reversibility constraints of certain fluxes. This provides an estimation of the nonmeasured fluxes even when measurements are insufficient, i.e. when (2) is underdetermined (Figure 3A). This estimation will be precise if the degree of underdeterminancy is limited and there are irreversible fluxes. On the contrary, the estimation could be poor and some intervals may be unbounded. The reversibility constraints will also restrict the intervals of the estimated fluxes when uncertainty is considered (Figure 3C). Finally, the reversibility constraints can also provide a means to detect inconsistencies even when the system is not redundant (Figure 3F).

It provides a straight method for coping with inconsistency: a band of uncertainty is used instead of adjusting the inconsistent measurements. As any inconsistent set of measurements is necessarily uncertain, it seems reasonably to define a band of uncertainty around the measured values trying to enclose nearby consistent sets of measurements. Thus, every consistent set of measurements enclosed by the band will be taken into account in the estimation of the nonmeasured fluxes (Figure 3B). Furthermore, the band size needed to find the nearest consistent flux distribution gives an indication of the degree of inconsistency.
Additional advantages arise when FSA is used in a successive way to estimate the temporal evolution of the metabolic fluxes:

It may detect sensitivity problems. Assume that a band of uncertainty is being used and that the measured fluxes change smoothly over time. If the interval of values for an estimated flux is strangely large at a certain instant k, it indicates that a slight change in the measured fluxes has a big effect over the estimated flux, i.e. that a sensitivity problem exists (Figure 3D). Thereby, an analysis of sensitivity is incorporated in the estimation procedure.

The peak values at certain time instants k which may appear when MFA is used are avoided with FSA. These peaks are consequence of slight errors in the measurements (which are common due to the lack of reliable sources of measurements and due to the uncertainty of the conversion of concentration data into measured fluxes). Since FSA considers a band of uncertainty around the measured values, it avoids, or at least reduces, this phenomenon.

The estimation given by FSA for a certain flux at time k (an interval of possible values), combined with the inspection of past and future estimations and with our qualitative knowledge about cell behaviour, may be used to hypothesize which of the possible temporal evolutions corresponds to the actual one. That is to say, the richness of the estimation given by FSA makes it possible to exploit our qualitative knowledge to support certain hypothesis without being confused by measurements uncertainty.
Application: estimation of the fluxes during a cultivation of CHO cells
The threestep procedure described in the previous section is now applied to a real problem taken from the literature: the estimation of the intracellular fluxes of CHO cells cultivated in batch mode in stirred flasks. The available experimental data are the typical data measured offline (accurate measurements of the concentration of a few species but with a low sample rate), and therefore this example will be approached assuming that the procedure is done offline. This assumption is important during the second step of the procedure, and for this reason an example has been included in the Additional File 3 that illustrates the differences between the online and the offline operation modes. However, hereinafter we will pay special attention to the third step of the procedure because it is the most important one. In particular, the benefits provided by the use of FSA will be compared with those obtained with the wellestablished MFA methodology, which is the basis of most of the similar procedures [24–26]. This comparison illustrates the advantages of the new estimation procedure in three different scenarios:
S1. When measurements are almost sufficient. The number of measured fluxes is almost sufficient when there are enough to determine all the nonmeasured fluxes but there are not redundant measurements (i.e. when the system (2) is determined and not redundant).
S2. When measurements are sufficient, i.e. when the measured fluxes are enough to determine the nonmeasured fluxes and there are also redundant measurements (the system (2) is determined and redundant).
S3. When measurements are insufficient. The number of measured fluxes is insufficient when there are not enough to determine all the nonmeasured fluxes (i.e. when the system (2) is underdetermined and not redundant).
For completeness, the most uncommon case (when the system is underdetermined but redundant) is illustrated with a toy example in an appendix [Additional File 2]. In the three scenarios, the intrinsic uncertainty of the measured fluxes is taken into account.
Metabolic network of CHO Cells
These constraints can be represented with a 4×18 matrix S_{ ξ }fulfilling (11). Then, (11) and (1) can be joined to define an extended homogeneous system of linear equations (see methods). The extended system has 16 metabolites (mx) and 22 reactions (nx).
The mathematical model, formed by the stoichiometric matrixes S and S_{ξ} , is given in a Matlab script and a standard SBML file [see Additional File 4].
Step 1: getting experimental measurements of species
Step 2: conversion of measured concentrations in measured fluxes
Step 3 (S1): estimation of fluxes if measurements are almost sufficient and uncertain

The estimated values at time 24 h an 168 h for fluxes v_{ 8 }, v_{ 9 }, v_{ 10 }, v_{ 11 }, v_{ 12 }and v_{ 21 }seem unreasonable: the measured fluxes evolve in a smooth way, but these fluxes show peak values.

The estimated fluxes v_{ 8 }, v_{ 9 }and v_{ 10 }do not fulfil the reversibility constraints (they are not considered by MFA).

MFA assumes that there is not any kind of error in the measurements, which is unlikely, and therefore the estimated fluxes are unreliable.
A new estimation has been done at time 24 h, where the measured values for fluxes v_{ 1 }and v_{ 6 }are slightly modified (+2% and 5% respectively). In a similar way, a new estimation at time 168 h assumes a slight variation of the measured values for v_{ 1 }and v_{ 6 }(0.05 and +0.05 mM/(d·10^{9}·cells), respectively). As it can be observed in Figure 7 (red crosses), the peak values in fluxes v_{ 8 }, v_{ 9 }, v_{ 10 }, v_{ 11 }, v_{ 12 }and v_{ 21 }are eliminated or reduced, while the values of the rest of nonmeasured fluxes remain almost unchanged. This demonstrates that the peak values at times 24 h and 168 h could be caused by slight errors in the measured fluxes. The same issue is illustrated with figure A1 (Additional File 7). Hence, the main weakness of MFA in the determined case is pointed out: the effect of slight errors in the measured fluxes is not under control. These slight errors will exist in virtually all the measured fluxes (none sensor has a precision of 100%). Moreover, even the conversion of the measured concentrations into measured fluxes may introduce slight errors. For this reason, the fluxes estimated with MFA are unreliable in this scenario.
With this expression the relative error (relErr) will be considered when the measured value is high, and the absolute one (absErr) when it is near to zero (see figure A2 in the Additional File 7). If more information about the measurements sources were available, the range of uncertainty of each measured flux could be defined accordingly. For example, if a commercial sensor is employed, its technical specifications can be used to define the band.
The nonmeasured fluxes estimated with FSA when the band of uncertainty is considered and the reversibility constraints are incorporated are shown in Figure 7 (black intervals). If they are compared with those obtained when MFA was used, several conclusions can be pointed out:

The peaks at time 24 h an 168 h for fluxes v_{ 8 }, v_{ 9 }, v_{ 10 }, v_{ 11 }, v_{ 12 }and v_{ 21 }which appeared when MFA was used are avoided with FSA. As it was shown, when the measurements were slightly modified, these peakvalues were replaced by more sensible predictions. Since these modified measurements are included in the band of uncertainty, the obtained intervals for v_{ 8 }, v_{ 9 }, v_{ 10 }, v_{ 11 }, v_{ 12 }and v_{ 21 }contain the sensible predictions. In principle, the peakvalues would be within the intervals. However, a peak value could not satisfy the reversibility constraints and therefore it will not be considered a valid solution by FSA this is the case at k = 24 h.

The uncertainty of experimental measurements is nontrivially propagated to the nonmeasured fluxes. Hence, the use of FSA provides not only a prediction of the nonmeasured fluxes, but also an indication of the reliability of this prediction. For example, the predicted v_{ 8 }, v_{ 9 }and v_{ 10 }are highly influenced by measurements uncertainty, while v_{ 2 }, v_{ 4 }, and v_{ 5 }are quite insensitive. Although all fluxes can be determined, FSA highlights that the estimated values for v_{ 8 }, v_{ 9 }and v_{ 10 }are less reliable (or less precise) than those assigned to v_{ 2 }, v_{ 4 }and v_{ 5 }. This issue is more deeply analyzed in a subsequent section.

Reversibility constraints provide a method to detect inconsistencies. For example, it can be easily checked that the solution provided by MFA do not satisfy the reversibility constraints at 24 h (a negative value is given to the irreversible fluxes v_{ 8 }, v_{ 9 }and v_{ 10 }). This inconsistency is detected and avoided with FSA.

The underdeterminancy introduced as uncertainty in the measurements can be partially neutralized with the reversibility constraints. Hence, the estimated fluxes are more reliable but not necessarily highly imprecise.
This example shows that the procedure provides a reliable and rich estimation of the evolution along time of the nonmeasured fluxes when the measurements are only almost sufficient, i.e. when the system is determined but not redundant. In particular, the use of FSA in the third step of the procedure instead of the wellestablished MFA provides several benefits, thanks to taking into account the uncertainty of measurements and considering the reversibility constraints.
Step 3 (S2): estimation of fluxes if measurements are sufficient and uncertain
When the system (2) is determined and redundant, an estimation based on MFA will work as follows (approach 1): firstly, the importance of the inconsistency is checked and the measured flux values are adjusted; then, the pseudoinverse matrix is used to estimate the nonmeasured fluxes. These two properties checkable consistency and adjustable measurements are responsible of the success of MFA in this scenario. However, FSA provides a new approach (approach 2) which holds the property of checkable consistency, but replaces the adjustment of the measurements by the definition of a band of uncertainty. We will apply both alternatives to our example.
The system (2) was determined and not redundant when six fluxes were known. If another independent flux is measured, the system will be redundant because the rank of S_{ u }(15) will be less than m (16). Since no more fluxes were measured in [28], we will assume that the evolution of v_{ 21 }(CO2) has been measured we chose it because it is a wellknown extracellular flux. We assume that v_{ 21 }evolves smoothly and that its values are within the intervals estimated with FSA in the previous section. Hence, at each time instant k, except 24 h and 168 h, the values given by MFA in the previous section are used as measured values (they lay within the intervals). The values at 24 h and 168 h are calculated by the approximation of a spline curve (see Figure A3 in the Additional File 7).

All the consistent sets of measured values enclosed by the band of uncertainty are considered by FSA. That guarantees that the intervals obtained enclose the actual values of the fluxes if the band was correctly chosen. Contrarily, when MFA is used (approach 1), the actual values of the measured fluxes need to be exactly found to ensure that the estimations fit in with the actual fluxes. To illustrate this idea a consistent flux distribution within the band of uncertainty has been highlighted in Figure 8 (dotted line). This flux distribution corresponds to a set of measured values very near to the original ones; nevertheless the evolution of v_{ 8 }, v_{ 9 }, v_{ 10 }and v_{ 12 }is quite different to the estimation given by MFA. That proves that the values estimated with MFA may be deviated from the actual ones, even if there are only slight errors in the measured fluxes. Conversely, FSA shows that two qualitatively different interpretations of fluxes v_{ 8 }, v_{ 9 }and v_{ 10 }are possible: they can be stable around 0.6 or evolve from 0.2 to 0.7 mM/(d * 10^{9} cells). If there were other evidences supporting one alternative over the other one, we could hypothesize which of these two scenarios corresponds to the actual one. Hence, FSA not only reduces the number of wrong predictions, but may also provide a quantitative support for our qualitative knowledge.

Although there is a gross error in the measurements at 168 h, FSA finds at least one consistent set of measured values within the band of uncertainty (providing an error bound that complements the χsquare method). The estimations provided by FSA at 168 h seem sensible: the measurements are only slightly adjusted (the adjustment is limited by the band size) and the peak values are avoided. On the contrary, the fluxes estimated with MFA are very sensitive to the gross error. The value of v_{ 21 }is significantly changed by the adjustment method resulting in a peak. Moreover, this insensible peak also appears in the estimated values of v_{ 8 }, v_{ 9 }, v_{ 10 }, v_{ 11 }and v_{ 12 }. In fact, the fluxes calculated with MFA are generally discarded when the measurements fail the χsquare method.

When FSA is used, the uncertainty of experimental measurements is nontrivially translated to the nonmeasured fluxes. Again, FSA provides not only a prediction of the nonmeasured fluxes, but also an indication of the reliability of this prediction.
This example has shown that the procedure can be useful to estimate the evolution of the fluxes even when measurements are sufficient but uncertain, i.e. when the system is determined and redundant. Although this is the scenarios were the procedures based on the use of MFA are most successful, the use of FSA provides a more reliable estimation of the nonmeasured fluxes and offers an interesting approach to cope with inconsistency.
Step 3 (S3): estimation of fluxes if measurements are insufficient and uncertain
Comparison of different estimations of the nonmeasured fluxes
Ref (v_{1}, v_{6}, v_{7}, v_{19}, v_{20}, v_{22})  G (no v_{22})  F (no v_{20})  E (no v_{19})  B (no v_{6})  A (no v_{1})  C (no v_{7})  I (no v_{20}v_{22})  H (no v_{19}v_{22})  

Reactions ^{ d }  MI^{a} [^{b}]  MI [^{a}]  [%^{c}]  MI [^{b}]  [%]  MI [^{b}]  [%]  MI [^{b}]  [%]  MI [^{b}]  [%]  MI [^{b}]  [%]  MI [^{b}]  [%]  MI [^{b}]  [%] 
1: G→G6P  0.267^{ e }  0.267    0.267    0.267    0.267    x    0.267    0.267    0.267   
2: G6P→G3P+DAP  0.367  0.387  5%  0.628  71%  0.541  47%  0.398  8%  x    0.627  71%  0.628  71%  0.572  56% 
3: G6P→R5P+CO2  0.131  0.199  53%  0.526  303%  0.340  160%  0.131  0%  0.131  0%  0.401  207%  0.526  303%  0.383  193% 
4: DAP→G3P  0.367  0.387  5%  0.628  71%  0.541  47%  0.398  8%  x    0.627  71%  0.628  71%  0.572  56% 
5: G3P→Pyr  0.735  0.774  5%  1.256  71%  1.082  47%  0.795  8%  x    1.253  71%  1.256  71%  1.144  56% 
6: Pyr→L  0.475  0.475    0.475    0.475    x    0.475    0.475    0.475    0.475   
7: Pyr+Glu→A+aKG  0.100  0.100    0.100    0.100    0.100    0.100    1.488  inf  0.100    0.100   
8: Pyr→ACA+CO2  1.031  1.031  0%  1.562  51%  1.901  84%  x    x    0.957  7%  1.562  51%  1.906  85% 
9: Oxa+ACA→Cit  1.031  1.031  0%  1.562  51%  1.901  84%  x    x    0.957  7%  1.562  51%  1.906  85% 
10: Cit→aKG+CO2  1.031  1.031  0%  1.562  51%  1.901  84%  x    x    0.957  7%  1.562  51%  1.906  85% 
11: aKG→Mal+CO2  1.156  1.156  0%  1.604  39%  2.532  119%  x    x    1.443  25%  1.604  39%  2.530  119% 
12: Mal→Oxa  0.994  0.994  0%  1.398  41%  1.769  78%  x    x    1.093  10%  1.398  41%  1.769  78% 
13: Mal→Pyr+CO2  0.209  0.240  15%  0.352  68%  0.920  341%  0.209  0%  0.209  0%  0.903  332%  0.352  68%  0.918  340% 
14: Oxa+Glu→Asp+aKG  0.131  0.199  53%  0.526  303%  0.340  160%  0.131  0%  0.131  0%  0.401  207%  0.526  303%  0.383  193% 
15: Glu→aKG+NH4  0.150  0.182  21%  0.298  98%  0.870  479%  0.150  0%  0.150  0%  0.586  289%  0.298  98%  0.870  479% 
16: Q→Glu+NH4  0.117  0.145  23%  0.325  177%  0.553  372%  0.117  0%  0.117  0%  0.569  386%  0.325  177%  0.548  367% 
17: R5P+Asp+Q→Pu  0.104  0.277  165%  0.293  181%  0.200  91%  0.104  0%  0.104  0%  0.225  116%  0.526  404%  0.383  267% 
18: R5P+Asp+2Q→Py  0.078  0.132  69%  0.283  262%  0.177  126%  0.078  0%  0.078  0%  0.209  168%  0.263  237%  0.163  108% 
19:→NH4  0.141  0.141    0.141    1.419  904%  0.141    0.141    0.141    0.141    1.412  899% 
20:→Q  0.132  0.132    1.127  752%  0.132    0.132    0.132    0.132    1.107  737%  0.132   
21:→CO2  3.338  3.338  0%  4.770  43%  6.966  109%  x    x    3.843  15%  4.770  43%  6.966  109% 
22: PuPy (constraint)  0.100  0.354  254%  0.100    0.100    0.100    0.100    0.100    0.526  426%  0.383  283% 
Mean  0.554  0.587  39%  0.899  155%  1.138  196%  0.217  2%  0.156  0%  0.802  122%  0.927  180%  1.168  214% 
Measured fluxes [number]  6  5  5  5  5  5  5  4  4  
Estimated fluxes [number]  16/16  17/17  17/17  17/17  7/17  10/17  17/17  18/18  18/18  
<25% (·Ref)  12  0  0  10  7  5  0  0  
25–100% (<2·Ref)  3  11  8  0  0  4  11  7  
100–300% (2–4·Ref)  2  3  5  0  0  5  2  7  
>300% (>4·Ref)  0  3  4  0  0  3  5  4 
With four sets of 5 measurements (G, F, E and C) the evolution over time of all the nonmeasured fluxes can be estimated. Case G, where v_{ 22 }is not known, provides the best results. There is a mean interval increment of 39% with respect to the determined case and the increment is minor than 25% for 12 fluxes (out of 17). This case is depicted in Figure 9 (in green). The intervals are practically the same than in the determined case for most fluxes (v_{ 2 }, v_{ 4 }, v_{ 5 }, v_{ 8 }, v_{ 9 }, v_{ 10 }, v_{ 11 }, v_{ 12 }, v_{13}, v_{ 15 }and v_{ 21 }). Intervals for v_{ 3 }and v_{ 14 }are larger, but still accurate, and only the estimations of v_{ 16 }, v_{ 17 }and v_{ 18 }are highly imprecise. Moreover, the temporal evolution that can be characterized by using the middle point of the interval is almost the same than of the determined case even for these fluxes (see figure A4 in the Additional File 7). Case C, where v_{ 7 }is not measured, provides very good results. All fluxes are predicted (with a mean interval increment of 122%), the interval increment is minor than 25% for 5 fluxes and minor than 100% for 9 fluxes. Case F, where v_{ 20 }is not measured, provides good results too. There is a mean interval increment of 155% and the interval increment is minor than 100% for 11 fluxes (out 17). Case E, where v_{ 19 }is not measured, provides slightly worse results than F. With the other sets of 5 measurements (B and A), some nonmeasured fluxes cannot be estimated. Nevertheless the intervals of the fluxes that can be estimated (10 and 7 fluxes respectively) are exactly the same that in the determined case.
Two sets of 4 measurements have been studied (I and H). Case I, where v_{ 20 }and v_{ 22 }are not measured, provides remarkable results. There is a mean interval increment of 180% with respect to the determined case and the increment is minor than 100% for 11 fluxes (out 18). This case is depicted in Figure 9 (in blue). For most fluxes the intervals are similar to the determined case (v_{ 2 }, v_{ 4 }, v_{ 5 }, v_{ 8 }, v_{ 9 }, v_{ 10 }, v_{ 11 }, v_{ 12 }, v_{ 13 }, v_{ 15 }and v_{ 21 }). Intervals for v_{ 16 }and v_{ 20 }are larger but still useful, and only v_{ 3 }, v_{ 14 }, v_{ 17 }and v_{ 18 }are highly imprecise. Again, the temporal evolution of the estimated fluxes is similar to the determined case (see figure A4 in the Additional File 7).
This scenario has illustrated an important feature of the introduced procedure: it can estimate the evolution of the nonmeasured fluxes even when there is a lack of measurable species (i.e. the system is underdetermined) and the available measurements are uncertain.
Unbalanced propagation of measurements uncertainty
As it has been shown in previous sections, the uncertainty of the experimentally measured fluxes is not equally propagated to the estimated fluxes (i.e. not all the estimated fluxes are equally affected by measurements uncertainty). On the contrary, the structure of the metabolic network (the stoichiometric relations and the reversibility constraints) will determine how the uncertainty is propagated from the measured fluxes to the estimated ones.
Imprecision of the estimated fluxes caused by measurements uncertainty
Determined case  Determined/Redundant case  Comparative  

Max [^{ a }]  AIS [^{ a }]  AIS [%^{ b }]  Max [^{ a }]  AIS [^{ a }]  AIS [%^{ b }]  Diff. [^{ a }]  Diff. [%]  
v _{ 2 }  6,041  0,377  6,25%  6,032  0,321  5,32%  0,057  14,97% 
v _{ 3 }  0,853  0,129  15,12%  0,859  0,123  14,35%  0,006  4,41% 
v _{ 4 }  6,041  0,377  6,25%  6,032  0,321  5,32%  0,057  14,97% 
v _{ 5 }  12,081  0,755  6,25%  12,065  0,642  5,32%  0,113  14,98% 
v _{ 8 }  1,166  1,053  90,37%  0,715  0,231  32,32%  0,822  78,07% 
v _{ 9 }  1,166  1,053  90,37%  0,715  0,231  32,32%  0,822  78,07% 
v _{ 10 }  1,166  1,053  90,37%  0,715  0,231  32,32%  0,822  78,07% 
v _{ 11 }  3,769  1,180  31,30%  3,073  0,165  5,37%  1,015  86,02% 
v _{ 12 }  1,854  1,017  54,89%  1,263  0,241  19,05%  0,777  76,34% 
v _{ 13 }  1,813  0,209  11,52%  1,809  0,195  10,78%  0,014  6,58% 
v _{ 14 }  0,853  0,129  15,12%  0,859  0,123  14,35%  0,006  4,41% 
v _{ 15 }  1,113  0,150  13,52%  1,109  0,147  13,27%  0,003  2,11% 
v _{ 16 }  2,665  0,117  4,39%  2,668  0,114  4,26%  0,003  2,91% 
v _{ 17 }  0,426  0,101  23,64%  0,442  0,087  19,60%  0,014  14,10% 
v _{ 18 }  0,426  0,079  18,42%  0,417  0,063  15,17%  0,015  19,48% 
v _{ 21 }  8,698  3,407  39,17%          
Mean  0,699  32,31%  0,202  14,32%  0,497  71,09% 
Furthermore, the data given in Table 3 provides a quantitative indication of the benefits of incorporating a redundant measurement. When seven fluxes are assumed to be measurable instead of six, the estimations of the nonmeasured fluxes are more precise (the interval sizes are reduced around 71% on average). This is particularly important for those fluxes that were poorly estimated in the determined case (reductions of 78% for v_{ 8 }, v_{ 9 }and v_{ 10 }and 76% for v_{ 12 }).
Nonlinear propagation of measurements uncertainty
In the previous section we analyzed the unbalanced propagation of the uncertainty from the measured fluxes to the estimated ones. Herein we investigate some characteristics of this propagation and, in particular, the interrelation between the uncertainty of the different measured fluxes and their combined effect over the estimated fluxes.
Again, the time series of the five measured species (G, L, A, NH_{ 4 }and Q) have been used, under the assumption that the formation of purine and pyrimidine are equal (v_{ 22 }= 0). Then, 15·15 executions of the estimation procedure have been carried out with different degrees of uncertainty for the measured fluxes v_{ 1 }and v_{ 6 }(between ± 2% and ± 30%). Afterwards, the averaged interval size for each estimated flux was calculated. This makes it possible to analyze how the different combinations of uncertainty in v_{ 1 }and v_{ 6 }affect to the estimated fluxes.
On the one hand, the propagation of the uncertainty does not satisfy the principle of superposition. Let f(u_{ i }) be the interval size of a calculated flux when the degree of measurements uncertainty is u_{ i }, then: f (u_{1}) + f (u_{2}) ≠ f (u_{1} + u_{2}). To remark this, the result of summing up the independent effect of the uncertainty of v_{ 6 }and v_{ 1 }has been depicted in Figure 10 (black dots). Interestingly, if the uncertainty of one of the two measured fluxes is kept low then f (u_{1}) + f (u_{2}) > f (u_{1} + u_{2}); but if the uncertainty of both fluxes is increased, this is inverted: f (u_{1}) + f (u_{2}) <f (u_{1} + u_{2}). This implies that the net result of considering the uncertainty of two (or more) measured fluxes can not be predicted just by summing up the results of considering only the uncertainty of one measured flux at a time. On the contrary, the net result may be given by a complex nonlinear function, as it happens in the example of Figure 10, where:

The uncertainty in v_{ 6 }is always translated to the estimated v_{ 2 }(see right bottom figure). When v_{ 6 }uncertainty increases, the AIS of v_{ 2 }increases even if there is not v_{ 1 }uncertainty. However, the bigger v_{ 1 }uncertainty gets, the bigger the effect of adding v_{ 6 }uncertainty is.

When v_{ 6 }uncertainty is low, the addition of v_{ 1 }uncertainty has a low effect over the estimated v_{ 2 }(see right bottom figure) More precisely, the first small addition of v_{ 1 }uncertainty has a slight effect, but the subsequent additions do not (there is a saturation). However, the saturation limit increases with v_{ 6 }uncertainty, and, therefore, the more uncertainty there is in v_{ 6 }, the more important effect has addition of uncertainty in v_{ 1 }. In summary, v_{ 1 }uncertainty has not an important effect itself, but its combination with v_{ 6 }uncertainty boosts it.
On the other hand, as the superposition principle is not fulfilled the effect of the uncertainty of one measured flux over one estimated flux is not linear, i.e. f (k·u_{1}) ≠ k·f (u_{1}) For example, assume that the uncertainty of v_{ 6 }is fixed in 10% (fourth row in the right top figure). It can be observed that the effect of adding a first 4% of uncertainty to v_{ 1 }is higher than the effect of adding a second one. In fact, when the uncertainty of v_{ 1 }is bigger than 16%, the addition of more uncertainty has practically zero effect (there is a saturation phenomenon).
In the last two sections it has been shown that the relationship between the uncertainty of the measurements and the precision of the estimation is a complex one. On the one hand, the propagation of measurements uncertainty to each estimated flux will be different. On the other hand, the net effect of considering the uncertainty of two (or more) measured fluxes simultaneously does not correspond to the sum of the effects of considering the uncertainty of each measured flux one at a time. Finally, the effect of the uncertainty of one measured flux over one estimated flux is not linear.
Therefore, when the procedure introduced in this paper considers the propagation of the uncertainty from the measurements to the estimated fluxes, it provides nontrivial information.
Analysis of the effect of the uncertainty of each measured flux
 a)
Direct approach. Calculate the increase of the imprecision of the estimated fluxes when the uncertainty of one measured flux is increased. This calculation is repeated for each measured flux.
 b)
Indirect approach. Calculate the reduction of the imprecision of the estimated fluxes when the uncertainty of one measured flux is decreased. This is repeated for each measured flux. Notice that the effect of decreasing the uncertainty is not the inverse of increasing it, i.e. f (u + x).+ f (u + x) ≠ f (0).
The direct approach (a) informs about the effect of considering the uncertainty of each measured flux over the estimated ones (it is similar to a classical analysis of sensitivity). This information may be useful during the settingup of a process plant in order to choose the sources of measurements (the equipment and the protocols). Nevertheless, the indirect approach (b) is probably more promising. It calculates how much the imprecision of the estimated fluxes will be reduced, if we reduce the uncertainty of one of the measured fluxes. Given the characteristics of our current equipment and our measuring protocols (e.g. our sensors provide measurements with a ± 5% of uncertainty), we can calculate which of the measured fluxes should be more accurately measured in order to improve the precision of the estimations (e.g. using a more accurate sensor or taking redundant measurements).
More details about this analysis including the complete dataset are given in the Additional File 8. In addition, the results obtained with the direct analysis are also included there.
Conclusion
In this contribution we have presented a new procedure to estimate the temporal evolution of the metabolic fluxes. It copes with the intrinsic uncertainty of experimental measurements and with the lack of measurable species by means of the use of the Flux Spectrum Approach (FSA). The potential of the procedure has been demonstrated using a real problem: the estimation of the intracellular fluxes of CHO cells cultivated in batch mode in stirred flasks. Using this example, the benefits that the use of FSA brings to the whole procedure have been illustrated through a comparison with the use of Metabolic Flux Analysis (MFA), a wellestablished methodology that is the basis of related procedures [24–26]. When the available measurements are only almost sufficient (i.e. the system is determined but not redundant), the procedure provides a more reliable and richer estimation of the evolution of the fluxes, because it takes into account measurements uncertainty and it considers the reversibility constraints. It has been also shown that even when measurements are insufficient (i.e. the system is underdetermined), the procedure is capable of estimating the evolution of the nonmeasured fluxes. Finally, when measurements are sufficient (i.e. the system is determined and redundant), the procedure provides a reliable estimation of the nonmeasured fluxes because it considers measurements uncertainty and offers an interesting approach to cope with inconsistency.
The procedure to estimate the metabolic fluxes can be applied offline (with previously collected data), providing an insight into the timevarying behaviour of the organism. This can help in the understanding of its dynamic metabolic regulation and its adaptation to the environmental conditions. It can also be useful for physiological studies, strain characterization tasks, and to guide research to improve strain and processes. On the other hand, the procedure is a promising tool for online monitoring processes in industrial environments, where there is still a lack of reliable online sensors. The features of the procedure are especially suitable for this application.
In summary, it has been shown that the temporal evolution of nonmeasured fluxes can be estimated by using a set of measurable species and a set of known biological constraints. Moreover, the procedure proposed considers the intrinsic uncertainty of the experimental measurements and could be applied even if there is a lack of measurable species.
Methods
Calculation of the measured fluxes by approximating the derivative
The backward version does not introduce an intrinsic delay (the derivative of f(.) at k is calculated with the values of f(.) at k and k1), but the middle point provides a less noisy approximation. In any case, usually this approach needs to be combined with the use of filters.
Moving average filters
The key parameter of moving average filters is the size of the window (i.e., the number of averaged values of the original signal). The optimal size would be one observation in order to be as close as possible to the original signal. However, as noise rejection is desired, the window size needs to be increased. Hence, there is a tradeoff between sensitivity to noise and delay with respect to the original signal. A typical variant of these filters includes multiplying factors to give a different weight to each value within the time window (e.g. the exponential moving average).
Calculation of the measured fluxes with a nonlinear observer
where ξ_{ e }denotes the observed concentration of the extracellular specie and v_{ e }the observed flux. The unique adjustable parameter is θ. Not only these observers are proved to be stable, but also its asymptotic error can be made arbitrarily small by choosing sufficiently large values of θ. However, very large values need to be avoided in practice since the observer may become noise sensitive. Thereby the choice of θ represents a tradeoff between fast convergence (minor delay) and sensitivity to noise.
Flux spectrum approach
 1.
Impose the set of constraints given by (2) and the reversibility constraints for the irreversible reactions, v_{ i }≥ 0. Then, in order to consider the uncertainty of the measured fluxes, the unique value of each measured flux v_{ m }can be replaced by an interval [v_{m, min}, v_{m, max}]. Accordingly, each equation of (2) is substituted by two inequalities. The resultant constraints define a region where the actual flux distribution may live.
 2.Calculate the minimum and maximum values within the region for each nonmeasured flux v_{ uj }, by solving a set of min/max linear programming problems (n_{ u }minimizations and n_{ u }maximizations):$\begin{array}{l}\forall {v}_{uj},j=1,\dots ,nu\\ \begin{array}{cc}Min\left\{{v}_{uj}\right\}& subject:\end{array}\\ \begin{array}{ccc}{S}_{u}\cdot {v}_{u}\ge {\left\{{S}_{m}\cdot [{v}_{m}]\right\}}^{\mathrm{min}\phantom{\rule{0.25em}{0ex}}}& {S}_{u}\cdot {v}_{u}\le {\left\{{S}_{m}\cdot [{v}_{m}]\right\}}^{\mathrm{max}\phantom{\rule{0.25em}{0ex}}}& {v}_{i}\ge 0\end{array}\\ \begin{array}{cc}Max\left\{{v}_{uj}\right\}& subject:\end{array}\\ \begin{array}{ccc}{S}_{u}\cdot {v}_{u}\ge {\left\{{S}_{m}\cdot [{v}_{m}]\right\}}^{\mathrm{min}\phantom{\rule{0.25em}{0ex}}}& {S}_{u}\cdot {v}_{u}\le {\left\{{S}_{m}\cdot [{v}_{m}]\right\}}^{\mathrm{max}\phantom{\rule{0.25em}{0ex}}}& {v}_{i}\ge 0\end{array}\end{array}$(10)
where n_{ u }is the number of unknown fluxes, v_{ u }is the vector of nonmeasured fluxes, v_{i} represents each irreversible flux and [v_{ m }] represents the two vectors [v_{ m, min }, v_{ m, max }] formed with the uncertain representation of the measured fluxes. {S_{ m }·[v_{ m }]}^{min} and {S_{ m }·[v_{ m }]}^{max} are vectors formed with the maximum and minimum row values of the product S_{ m }· [v_{ m }]. To calculate each row, maximum and minimum values of v_{ m }need to be combined taking into account the signs of the elements of the corresponding row of S_{ m }(you can find a detailed description of the algorithm in the Additional File 5).
The obtained v_{ uj, min }and v_{ uj, max }for each nonmeasured flux define an interval bracketing its possible values: v_{ uj }(k) = [v_{ uj, min }, v_{ uj, max }].
Auxiliary: link between extracellular and intracellular fluxes
A set of ne extracellular fluxes can be linked with the intracellular fluxes by using a matrix S_{ ξ }fulfilling:
v_{ ξ }= S_{ ξ }·v
Hence, the extended system holds the structure of a homogeneous system of linear equations.
Auxiliary: inconsistency of the measured fluxes
Chisquare values for two confident levels
Degrees of freedom  90%  95% 

1  2.71  3.84 
2  4.61  5.99 
3  6.25  7.81 
4  7.78  9.49 
where R_{ r }^{#} denoted the Penrose pseudoinverse of the matrix R_{ r }. This equation provides the adjusted values for the balanceable fluxes and the original values for the nonbalanceable ones.
Auxiliary: Metabolic flux analysis
Finally, if system (2) is underdetermined, Metabolic Flux Analysis cannot be used. Only some fluxes may be uniquely calculable by using the method explained in [14].
List of abbreviations
 MFA:

Metabolic flux analysis
 FSA:

Flux spectrum approach
 FBA:

Flux balance analysis
 CHO:

Chinese hamster ovary (cells)
 AIS:

Average along time of the interval size of an estimated flux.
Declarations
Acknowledgements
This research has been partially supported by the Spanish Government (CICYTFEDER DPI200501180). The first author is recipient of a fellowship from the Spanish Ministry of Education and Science (FPU AP20051442).
Authors’ Affiliations
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