System (2) is determined when there are enough linearly independent constraints for uniquely calculate all non-measured fluxes v_u; i.e., when rank(S_u) = u (u is the number of non-measured fluxes). On the contrary, when rank(S_u)>u, the system is classified as underdetermined because at least one non-measured 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 non-measured fluxes. Fortunately, the use of FSA may provide an estimation of the non-measured 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 (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 non-balanceable. 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 122426. 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.

The metabolic network (Figure 4) has been taken from 48. The network describes only the metabolism concerned with the two main energetic nutrients, glucose and glutamine. Thus, the metabolism of the amino-acids provided by the culture medium is not included. Four pathways are considered: the glycolysis, the glutaminolysis, the TCA cycle and the nucleotides synthesis. All reactions are assumed to carry flux only in only one direction, except reactions 2, 4, 5, 6 and 7 that are reversible (e.g. when glucose is exhausted lactate and alanine are consumed instead of produced). The complete lists of species and reactions are given in the Additional File 1.

The mass balance around intracellular metabolites at pseudo-steady state is given by eq. 1 (the stoichiometric matrix S is given in the Additional File 1). There are 12 metabolites (m) and 18 intracellular fluxes (n). Therefore, the system is underdetermined and has 6 degrees of freedom. The extracellular fluxes v_G, v_L and v_A coincide with the fluxes -v₁, v₆ and v₇. Three equations that link v_NH4, v_Q and v_CO2 with the intracellular fluxes can be obtained by inspection of the metabolic network. Moreover, it is a natural assumption to consider that the formation of purine and pyrimidine nucleotides is the same. As a result, four equations are incorporated by the authors 48:

v N H 4 : v 19 = v 15 + v 16 − v Q : v 20 = v 16 + v 17 + 2 ⋅ v 18 v C O 2 : v 21 = v 3 + v 8 + v 10 + v 11 + v 13 v E x t r a : v 22 = 0 = v 17 − v 18 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakqaabeqaaiabdAha2TWaaSbaaWqaaiabd6eaojabdIeaiHqaaiab=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@8E44@

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].

The experimental data taken from 28 is given in Figure 5. The cell density (X) and the concentration of 5 extracellular species are measured; two substrates, glucose (G) and glutamine (Q), and three excreted products, lactate (L), alanine (A) and ammonia (NH4). This data was collected with a sample rate of 24 h. These measurements cannot be filtered because -due to the low sample rate- it is impossible to distinguish between noise and true changes of the signal.

The second step of the procedure is the conversion of the measured concentrations in measured fluxes. The measured fluxes (and the biomass growth) calculated with three different approximations of the derivative are depicted in Figure 6 (see methods). Since the procedure is being done off-line, a centred approximation is the most advisable choice. Therefore, the fluxes calculated with the middle point Euler approximation will be used hereinafter. We obtained similar results (not shown) when the complete example was done using a backward Euler approximation (which would be more suitable in case the procedure were done on-line). It is also remarkable that Figure 6 already gives the idea of uncertainty -differences between the conversions obtained with different methods are significant. In fact, the different conversions, along with the precision of the sensors and the protocols used to measure the concentration of species, could be used to characterize the uncertainty in the measured fluxes.

If the five measured fluxes are used (v₁ (G), v₆ (L),v₇ (A), v₁₉ (NH₄) and v₂₀ (Q)) and it is assumed that the formation of purine and pyrimidine nucleotides is the same (v₂₂ = 0), the rank of S_u (16) is equal to the number of unknown fluxes (22-5-1). Thereby the system (2) is determined but not redundant. In this case we could use MFA to determine the non-measured fluxes. More precisely, at each time instant k, the unique flux distribution fulfilling (2) can be obtained by using the inverse matrix of S_m (see methods). However, as it can be observed in Figure 7 (green solid line) the results obtained are not very satisfactory:

• The estimated values at time 24 h an 168 h for fluxes v₈, v₉, v₁₀, v₁₁, v₁₂ and v₂₁ seem unreasonable: the measured fluxes evolve in a smooth way, but these fluxes show peak values.

• The estimated fluxes v₈, v₉ and v₁₀ 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 v₁ and v₆ 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 v₁ and v₆ (-0.05 and +0.05 mM/(d·10⁹·cells), respectively). As it can be observed in Figure 7 (red crosses), the peak values in fluxes v₈, v₉, v₁₀, v₁₁, v₁₂ and v₂₁ are eliminated or reduced, while the values of the rest of non-measured 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.

The same scenario is now approached following the procedure introduced in this paper, i.e. using FSA instead of MFA in the third step. If uncertainty is not considered and all reactions are assumed to be reversible, FSA provides the same solution that MFA (results not shown). But it is possible to include the reversibility constraints for those reactions classified as irreversible. By using these constraints, FSA has detected a high inconsistency at 24 h and a lower one at 144 h (i.e. the region defined by the imposed constraints does not contain any solution at these time instants). It must be highlighted that the system is not redundant, so methods to check consistency based on redundancy cannot be used; however, FSA is detecting inconsistencies thanks to the reversibility constraints. Afterwards, it is also interesting to consider the intrinsic uncertainty of the measurements. We will define a band of uncertainty around the measured values, and then we will use FSA to estimate the non-measured fluxes. The most common ways to define a band of uncertainty are the use of a relative error around the measured values (e.g. of the 5%) and the use of an absolute one (e.g. 0.05 mM/(d•109•cells)). Herein, we use a mixed approach. For each measured flux v_m, at each time instant k, the band is defined as:

If  r e l E r r ⋅ v m > a b s E r r → b a n d = v m ± r e l E r r ⋅ v m Else → b a n d = v m ± a b s E r r MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@7B55@

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 non-measured 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 v₈, v₉, v₁₀, v₁₁, v₁₂ and v₂₁ -which appeared when MFA was used- are avoided with FSA. As it was shown, when the measurements were slightly modified, these peak-values were replaced by more sensible predictions. Since these modified measurements are included in the band of uncertainty, the obtained intervals for v₈, v₉, v₁₀, v₁₁, v₁₂ and v₂₁ contain the sensible predictions. In principle, the peak-values 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 non-trivially propagated to the non-measured fluxes. Hence, the use of FSA provides not only a prediction of the non-measured fluxes, but also an indication of the reliability of this prediction. For example, the predicted v₈, v₉ and v₁₀ are highly influenced by measurements uncertainty, while v₂, v₄, and v₅ are quite insensitive. Although all fluxes can be determined, FSA highlights that the estimated values for v₈, v₉ and v₁₀ are less reliable (or less precise) than those assigned to v₂, v₄ and v₅. 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 v₈, v₉ and v₁₀). 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 non-measured 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 well-established MFA- provides several benefits, thanks to taking into account the uncertainty of measurements and considering the reversibility constraints.

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 pseudo-inverse matrix is used to estimate the non-measured 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 v₂₁(CO2) has been measured -we chose it because it is a well-known extracellular flux. We assume that v₂₁ 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).

First of all, we apply the χ-square method to estimate the importance of the inconsistency at each time instant k (see methods). The data fails the consistency check at time 168 h, what indicates that the set of measurements contains gross errors at this point (see table A1 in the Additional File 7). Afterwards, we estimate the non-measured fluxes at each time instant k with the two approaches described above. In the first one, the measured values are adjusted to be consistent (as explained in methods). Then, the non-measured fluxes are estimated with MFA. In the second one, a band of uncertainty around the measured values is defined trying to enclose some nearby consistent sets of measured fluxes (the band is the same that in the previous section). Then, the non-measured fluxes are estimated with FSA. The results (shown in Figure 8) illustrate the benefits of using FSA in this scenario:

• 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 v₈, v₉, v₁₀ and v₁₂ 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 v₈, v₉ and v₁₀ are possible: they can be stable around 0.6 or evolve from 0.2 to 0.7 mM/(d * 10⁹ 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 χ-square 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 v₂₁ is significantly changed by the adjustment method resulting in a peak. Moreover, this insensible peak also appears in the estimated values of v₈, v₉, v₁₀, v₁₁ and v₁₂. In fact, the fluxes calculated with MFA are generally discarded when the measurements fail the χ-square method.

• When FSA is used, the uncertainty of experimental measurements is non-trivially translated to the non-measured fluxes. Again, FSA provides not only a prediction of the non-measured 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 non-measured fluxes and offers an interesting approach to cope with inconsistency.

Finally, it will be shown that our procedure can be used even when the available measurements are insufficient (i.e. when system (2) is underdetermined). In this situation procedures based on MFA cannot be applied, but the use of FSA allows our procedure to estimate the interval of possible values for each non-measured flux. In particular, the non-measured fluxes will be estimated by using different sets of 5 and 4 measured fluxes -remember that 6 were necessary to get a determined system. In all cases, uncertainty has been considered using the band described above. All results are given in Table 2 and two illustrative cases are depicted in Figure 9.