Table 1.

General characteristics of the obtained NLLSP solutions

Characteristic	Exp 1.1	Exp 1.2	Exp 2.1	Exp 2.2	Exp 3.6	Exp 3.1	Exp 4.1	Exp 4.2	Exp 4.3	Exp 5.1	Exp 5.2	Exp 5.3	Exp 5.4	Exp 5.5	Exp PLE_1 (5.1-5.5)
Number of measurements, W (a)	36	36	45	45	45	45	45	42	41	56	61	66	59	48	290
Number of free fluxes, p (b)	15	15	15	15	15	15	15	15	15	15	15	15	15	15	15
Degrees of the freedom, W-p	21	21	30	30	30	30	30	27	26	41	46	51	44	33	275
Termination tolerance, TT	1 × 10 −4	1 × 10 −6	1 × 10 −4	1 × 10 −6	1 × 10 −4	1 × 10 −4	1 × 10 −4	1 × 10 −4	1 × 10 −4	1 × 10 −4	1 × 10 −4	1 × 10 −4	1 × 10 −4	1 × 10 −4	1 × 10 −4
Minimal reached value of Ξ(Ξmin)	18.2	18.2	1.5 × 10 −3	1.5 × 10 −3	9.65	22.8	46.6	38.9	18.9	23.2	24.6	50.4	44.4	31.5	239
Maximal reached value of Ξ(Ξmax)	8.5 × 10 4	263.1	105.6	1.4 × 10 4	3.5 × 10 4	1.5 × 10 4	1.4 × 10 4	1.4 × 10 4	1.1 × 10 4	23.7	28.8	53.9	983	910.5	9.0 × 10 3
Number of ${\mathrm{\Xi }}_{\mathrm{k}}:{\Xi }_k<{\chi }_{0.025}^2\left(\mathit{W}-\mathit{p}\right)$	0	0	98	99	92	0	0	0	0	100	100	0	0	0	0
Number of ${\mathrm{\Xi }}_k:{\chi }_{0.025}^2\left(W-p\right)<{\mathrm{\Xi }}_k<{\chi }_{0.975}^2\left(\mathit{W}-\mathit{p}\right)$	82	85	0	0	6	96	26	47	97	0	0	100	99	99	97
Number of ${\mathrm{\Xi }}_k:{\chi }_{0.16}^2\left(\mathrm{W}-\mathrm{p}\right)<{\mathrm{\Xi }}_k<{\chi }_{0.84}^2\left(\mathit{W}-\mathit{p}\right)$	34	85	0	0	2	96	0	0	97	0	0	100	99	99	0
Number of trials with Ξk : Ξk = Ξmin (c)	26	77	6	51	28	11	4	4	7	94	89	96	96	82	91
$\mathrm{Null}\left({\mathbf{J}}_{\mathbf{f}}\left(\stackrel{⌢}{\mathbf{\theta }}\right)\right)$ (d)	Empty	Empty	Empty	Empty	Empty	Empty	Empty	Empty	Empty	Empty	Empty	Empty	Empty	Empty	Empty
Individual Residuals ∈ N(0, 1) (d, e)	Yes	Yes	No	No	Yes	Yes	No	No	Yes	Yes	Yes	Yes	Yes	Yes	Yes

To determine the global minimum of the NLLSP, 100 independent iterative trials were provided with an arbitrary set of initial free fluxes for all represented in silico LEs. All iterative trials were terminated successfully for each independent NLLSP. More detailed characteristics for all performed in silico experiments are listed in Additional file 3.

^(a)In the case of the SLE, the vector of the measured data consists of both measured MID values and measured effluxes. In the case of the PLE, the vector of the measured data consists of the measured MIDs and effluxes that were available for each SLE.

^(b)The measured effluxes are included in the set of free fluxes, constrained according to Eq. (S − 1.2.2) with experimentally determined parameters ${\mathbf{q}}_{\mathit{eff}}^{mea},{\mathbf{\sigma }}_{\mathit{eff}}^{mea}$.

^(c)The values of Ξ_k that were obtained during different iterative trials were assumed to be equal to the Ξ_min if the values were different from the Ξ_min by a value less than or equal to the TT level, i.e., Ξ_k − Ξ_min ≤ TT.

^(d)The analysis was performed for the NLLSP solution, which provided the minimal achieved value of Ξ, i.e., Ξ_min.

^(e)The individual variance-weighted residuals were recognized to be not distributed as N(0, 1) if both the Kolmogorov-Smirnov test, provided by MATLAB, rejected this hypothesis and the individual residual plot revealed unpredictable high values for the individual residual/group of the individual residuals or, in contrast, unpredictable small values for the primary part of the individual residuals.