An iterative identification procedure for dynamic modeling of biochemical networks

Abstract

Background: Mathematical models provide abstract representations of the information gained from experimental observations on the structure and function of a particular biological system. Conferring a predictive character on a given mathematical formulation often relies on determining a number of non-measurable parameters that largely condition the model's response. These parameters can be identified by fitting the model to experimental data. However, this fit can only be accomplished when identifiability can be guaranteed.

Results: We propose a novel iterative identification procedure for detecting and dealing with the lack of identifiability. The procedure involves the following steps: 1) performing a structural identifiability analysis to detect identifiable parameters; 2) globally ranking the parameters to assist in the selection of the most relevant parameters; 3) calibrating the model using global optimization methods; 4) conducting a practical identifiability analysis consisting of two (a priori and a posteriori) phases aimed at evaluating the quality of given experimental designs and of the parameter estimates, respectively and 5) optimal experimental design so as to compute the scheme of experiments that maximizes the quality and quantity of information for fitting the model.

Conclusions: The presented procedure was used to iteratively identify a mathematical model that describes the NF-kappaB regulatory module involving several unknown parameters. We demonstrated the lack of identifiability of the model under typical experimental conditions and computed optimal dynamic experiments that largely improved identifiability properties.

Figure 1

Model building loop.

Figure 2

Model building procedure incorporating the proposed model identification scheme.

Figure 3

Minimum identifiability tableau for the generating series method. A cross in the coordinates (i, j) indicates that the corresponding non-zero generating series coefficient depends on the parameter θ_j. Green crosses represent those parameters that can be computed from a single equation of the system. Green circles correspond to those parameters that may be uniquely identified, i.e. only one solution exist. Red crosses represent possible identifiability problems, i.e. sets of parameters that require more than 2 equations to be identified if possible. Red boxes and arrows represent sets of equations that result in an unique solution for the parameters. Numbers represent the order in which the equations were solved.

Figure 4

The NF-κB module. Network model as in [9]. The notation corresponds to that used in the mathematical model. Kinetic constants are indicated in blue; T_R regards a logical function which is 1 when the signal is activated and 0 otherwise; k_v represents the nuclear-cytoplasmic volume ratio.

Figure 5

Identifiability tableau for the NF-κB model.

Figure 6

Ranking of parameters for the NF-κB example. Parameters are ordered by decreasing δ_msqr using the mean rank as reference.

Figure 7

Sensitivity analysis in the range (). δ^msqr measures for the different combinations of parameters and observables for the three different experiments.

Figure 8

Practical identifiability analysis for the full set θ. Illustrative examples of the histograms of the solutions achieved with the Monte-Carlo based approach for t₁, e_2a and k₂ under the experimental scheme ES1.

Figure 9

Experiments performed throughout the identification procedure.

Figure 10

Expected uncertainties for all parameters at the end of the identification procedure. Red line indicates the nominal value of the parameter, blue line indicates the mean value for the given experiment and yellow line indicates the estimated expected uncertainty.

Figure 11

Illustrative examples of the evolution of the robust uncertainty ellipses for several pairs of parameters. k_prod-k_deg the most correlated parameters in all experimental schemes, i₁-i_1a the less correlated parameters in ES1 and ES2 and k₁-k₂ the less correlated parameters in ES3.