Fig 1.
(A) Diagram of the 15-variable original model. The short lasting cytoplasmic complexes (IKK|IκBα) and (IKK|NF-κB|IκBα) are not depicted for the sake of simplicity. The bold arrows represent the fast dynamics. (B) Diagram of the reduced 6-variable model. NF-κB* denoting cytoplasmic (NF-κB|IκBα) complexes is not associated with the separate equation, since the amount of total NF-κB remains constant: NFκBn + NFκB* = 1.
Table 1.
Parameter values for the reduced and reduced fitted model.
Fig 2.
Dynamics of the original, reduced and reduced fitted models in the combination experiment defined in S1 Table.
Red dots indicate time points from which the nuclear NF-κB and total IκBα protein in silico measurements are used for fitting the reduced model to the original one. The time points for the remaining variables are given in S1 Table.
Fig 3.
Simulations of the original model for five different sets of parameters and the corresponding reduced model with refitted parameter values (see S3 and S4 Tables).
Simulations are performed for the combination experiment defined in S1 Table. For each parameter set the corresponding is computed based on trajectories of the 5 main model variables. Each color corresponds to a different parameter set; for each parameter set, a continuous line shows the original model trajectory, while a thinner dashed line shows the corresponding trajectory of the reduced model with refitted parameters.
Fig 4.
Structural identifiability analysis of the reduced and original models.
Singular values of the sensitivity matrices S for the original and reduced models. Graphs show all 23 scaled singular values obtained (for the combination experiment) for the original model vs. 13 singular values obtained for the reduced model. Singular values, arranged in descending order, in the original model reveal a clear gap, indicating that the corresponding sensitivity matrix is rank deficient and hence the original model is structurally non-identifiable.
Fig 5.
Linear identifiability analysis of the reduced model.
(A) Comparison of the three smallest scaled singular values computed for different protocols. (B) Smallest scaled norms of sensitivity vectors ||Sj||. (C) Three smallest scaled norms of perpendicular components . (D) Three largest scaled ratios of the parameter estimation error to experimental error Rj = ln(σlinear,j)/ln(σdata). The combination experiment includes the continuous and 5 pulsatile protocols. Scaling with
allows to compare the protocols having a different dimension of sensitivity vectors.
Fig 6.
Identifiability analysis of the reduced model—comparison of the protocols: Continuous, combination experiment and on–off.
(A) Scaled singular values. (B) Scaled norms of sensitivity vectors ||Sj||. (C) Scaled norms of perpendicular components . (D) Scaled ratios of parameter estimation error to experimental error ln(σlinear,j)/ln(σdata). Scaling with
allows for a comparison of the three protocols having a different dimension of sensitivity vectors.
Table 2.
Summary of the on-off protocol of the NF-κB system in WT and A20 KO cells used for in silico experiments.
A simple protocol with a continuous TNF stimulation (0–120min) followed by a TNF washout (120–720min) for which measured variables are: IKKa, nuclear NF-κB, A20, total IκBα protein denoted by IκBα* (IκBα* = IκBα + 1 − NFκB), and IκBα mRNA. For each variable the measurement times used for the sensitivity-based identifiability analysis and reduced model fitting are given.
Fig 7.
Practical identifiability of the reduced model based on Monte Carlo simulations.
(A) A comparison of 75% confidence ellipses (shown in black) computed for σdata = 1.3 in the linear sensitivity matrix based analysis with results from 50 Monte Carlo simulations for four values of σdata (1.0, 1.1, 1.2, 1.3). Shown are projections on 15 planes spanned by 6 parameters with the largest geometric standard deviations σcarlo,j (for σdata = 1.3). (B) The geometric standard deviation of parameter estimation error σcarlo,j obtained in Monte Carlo simulation for four values of geometric standard deviations of measurements, σdata (1.0, 1.1, 1.2, 1.3), is compared with the geometric standard deviation of parameter estimation σlinear,j obtained from the linear analysis for σdata = 1.3.
Fig 8.
Three types of the oscillatory responses of the reduced model to tonic TFN stimulation: Left subpanels show nuclear NFκBn(t) in response to tonic TNF stimulation started at t = 1 hour, whereas the right subpanels show trajectories projected on the (NFκBn, IκBα) plane.
(A) Damped oscillations observed for the nominal (fitted to the original model) parameters values (see Table 1), in particular a2 = 0.0762, c5a = 0.000058, i1a = 0.000595 (modified in panels B and C). The black dot on right subpanel shows the ‘TR = 1’ stable steady state. (B) Limit cycle oscillations of nuclear NFκBn(t) observed in simulations performed for a2 = 0.02, c5a = 0.00001, i1a = 0.0001 and other parameters unchanged. The orange dot on the right subpanel shows the ‘TR = 1’ unstable steady state surrounded by a stable limit cycle (black line). A numerically computed orbit (shown in red) approaches the stable limit cycle. (C) Periodic relaxation-like oscillations observed in simulations performed for a2 = 0.01, c5a = 0.00001, i1a = 0.0001 and other parameters unchanged. Again, the orange dot on the right subpanel shows the ‘TR = 1’ unstable steady state surrounded by a stable limit cycle (black line).
Fig 9.
Influence of A20 feedback on the reduced model responses to tonic TNF stimulation.
Left subpanels show nuclear NFκBn(t) in response to tonic TNF stimulation started at t = 1 hour, whereas the right subpanels show trajectories projected on the (NFκBn, IκBα) or (NFκBn, A20) planes. (A) Numerically computed solution for k2 = 0 (and other parameters unchanged, as in Table 1) implying absence of A20 mediated feedback (as in A20 KO cells). (B) Nearly-perfect adaptation observed in the model variant with significantly stronger negative feedbacks mediated by A20 and IκBα. Numerical simulations are performed for k2 = 3.57 and i1a = 0.01 (changed from the nominal values k2 = 0.0357 and i1a = 0.00595) and other parameters unchanged.