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Parameter optimization in S-system models.

Vilela M, Chou IC, Vinga S, Vasconcelos AT, Voit EO, Almeida JS - BMC Syst Biol (2008)

Bottom Line: We tackle this challenge here through the parameterization of S-system models.It was previously shown that parameter identification can be performed as an optimization based on the decoupling of the differential S-system equations, which results in a set of algebraic equations.A procedure was developed that facilitates automated reverse engineering tasks for biological networks using S-systems.

View Article: PubMed Central - HTML - PubMed

Affiliation: Dept. Bioinformatics and Computational Biology, University of Texas M,D, Anderson Cancer Center, 1515 Holcombe Blvd, Houston, TX 77030, USA. mvilela@mdanderson.org

ABSTRACT

Background: The inverse problem of identifying the topology of biological networks from their time series responses is a cornerstone challenge in systems biology. We tackle this challenge here through the parameterization of S-system models. It was previously shown that parameter identification can be performed as an optimization based on the decoupling of the differential S-system equations, which results in a set of algebraic equations.

Results: A novel parameterization solution is proposed for the identification of S-system models from time series when no information about the network topology is known. The method is based on eigenvector optimization of a matrix formed from multiple regression equations of the linearized decoupled S-system. Furthermore, the algorithm is extended to the optimization of network topologies with constraints on metabolites and fluxes. These constraints rejoin the system in cases where it had been fragmented by decoupling. We demonstrate with synthetic time series why the algorithm can be expected to converge in most cases.

Conclusion: A procedure was developed that facilitates automated reverse engineering tasks for biological networks using S-systems. The proposed method of eigenvector optimization constitutes an advancement over S-system parameter identification from time series using a recent method called Alternating Regression. The proposed method overcomes convergence issues encountered in alternate regression by identifying nonlinear constraints that restrict the search space to computationally feasible solutions. Because the parameter identification is still performed for each metabolite separately, the modularity and linear time characteristics of the alternating regression method are preserved. Simulation studies illustrate how the proposed algorithm identifies the correct network topology out of a collection of models which all fit the dynamical time series essentially equally well.

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Error surfaces. a) Ten error surfaces associated with variable X1 of the 2-dimensional system were obtained using an exhaustive grid search covering 10 different initial guesses. b) Zooming in shows the composite contour map (level sets) of the error surfaces.
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Figure 3: Error surfaces. a) Ten error surfaces associated with variable X1 of the 2-dimensional system were obtained using an exhaustive grid search covering 10 different initial guesses. b) Zooming in shows the composite contour map (level sets) of the error surfaces.

Mentions: To explore the results of the proposed algorithm visually and to investigate patterns of convergence, we performed a grid search on the parameters of the 2-dimensional system (Equation 2). Specifically, we searched a 100 × 100 grid where each point represented the kinetic orders h11 and h12 over the range [-2.5, 2.0]. Correspondingly, 100 time points for X1 and X2 and its correspondent slopes S1 and S2 were generated by numerical integration of the 2-dimensional system (Equation 2) with X1(0) = 3 and X2(0) = 1 as initial conditions. Methods described in a later section were used on time series of X1 and X2 to calculate the regression matrix L, and for each given initial value of the rate constant β1 (uniformly spaced over the interval [1,6]) and for each point of the grid, the error surface for the variable X1 was constructed. The algorithm started with the degradation term for the first grid point using a given value for β1 and the time series points for X1 and X2. Subsequently, the production vector (Vp1 = [log(α1) g11 g12]) was obtained from the slope vector S1, the regression matrix L, and the degradation term DT1 in Equations (7)–(10). Once all parameter values for variable X1 in the production and degradation vectors were determined, the estimated slopes were calculated ( = PT1 - DT1) and the logarithm of the sum of the squared errors between these slopes and the target solutions was computed as . This process was repeated for all points on the grid such that an error surface resulted for each β1 value. In this manner, ten surfaces were constructed using different β values; they are shown superimposed in Figure 3.


Parameter optimization in S-system models.

Vilela M, Chou IC, Vinga S, Vasconcelos AT, Voit EO, Almeida JS - BMC Syst Biol (2008)

Error surfaces. a) Ten error surfaces associated with variable X1 of the 2-dimensional system were obtained using an exhaustive grid search covering 10 different initial guesses. b) Zooming in shows the composite contour map (level sets) of the error surfaces.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC2333970&req=5

Figure 3: Error surfaces. a) Ten error surfaces associated with variable X1 of the 2-dimensional system were obtained using an exhaustive grid search covering 10 different initial guesses. b) Zooming in shows the composite contour map (level sets) of the error surfaces.
Mentions: To explore the results of the proposed algorithm visually and to investigate patterns of convergence, we performed a grid search on the parameters of the 2-dimensional system (Equation 2). Specifically, we searched a 100 × 100 grid where each point represented the kinetic orders h11 and h12 over the range [-2.5, 2.0]. Correspondingly, 100 time points for X1 and X2 and its correspondent slopes S1 and S2 were generated by numerical integration of the 2-dimensional system (Equation 2) with X1(0) = 3 and X2(0) = 1 as initial conditions. Methods described in a later section were used on time series of X1 and X2 to calculate the regression matrix L, and for each given initial value of the rate constant β1 (uniformly spaced over the interval [1,6]) and for each point of the grid, the error surface for the variable X1 was constructed. The algorithm started with the degradation term for the first grid point using a given value for β1 and the time series points for X1 and X2. Subsequently, the production vector (Vp1 = [log(α1) g11 g12]) was obtained from the slope vector S1, the regression matrix L, and the degradation term DT1 in Equations (7)–(10). Once all parameter values for variable X1 in the production and degradation vectors were determined, the estimated slopes were calculated ( = PT1 - DT1) and the logarithm of the sum of the squared errors between these slopes and the target solutions was computed as . This process was repeated for all points on the grid such that an error surface resulted for each β1 value. In this manner, ten surfaces were constructed using different β values; they are shown superimposed in Figure 3.

Bottom Line: We tackle this challenge here through the parameterization of S-system models.It was previously shown that parameter identification can be performed as an optimization based on the decoupling of the differential S-system equations, which results in a set of algebraic equations.A procedure was developed that facilitates automated reverse engineering tasks for biological networks using S-systems.

View Article: PubMed Central - HTML - PubMed

Affiliation: Dept. Bioinformatics and Computational Biology, University of Texas M,D, Anderson Cancer Center, 1515 Holcombe Blvd, Houston, TX 77030, USA. mvilela@mdanderson.org

ABSTRACT

Background: The inverse problem of identifying the topology of biological networks from their time series responses is a cornerstone challenge in systems biology. We tackle this challenge here through the parameterization of S-system models. It was previously shown that parameter identification can be performed as an optimization based on the decoupling of the differential S-system equations, which results in a set of algebraic equations.

Results: A novel parameterization solution is proposed for the identification of S-system models from time series when no information about the network topology is known. The method is based on eigenvector optimization of a matrix formed from multiple regression equations of the linearized decoupled S-system. Furthermore, the algorithm is extended to the optimization of network topologies with constraints on metabolites and fluxes. These constraints rejoin the system in cases where it had been fragmented by decoupling. We demonstrate with synthetic time series why the algorithm can be expected to converge in most cases.

Conclusion: A procedure was developed that facilitates automated reverse engineering tasks for biological networks using S-systems. The proposed method of eigenvector optimization constitutes an advancement over S-system parameter identification from time series using a recent method called Alternating Regression. The proposed method overcomes convergence issues encountered in alternate regression by identifying nonlinear constraints that restrict the search space to computationally feasible solutions. Because the parameter identification is still performed for each metabolite separately, the modularity and linear time characteristics of the alternating regression method are preserved. Simulation studies illustrate how the proposed algorithm identifies the correct network topology out of a collection of models which all fit the dynamical time series essentially equally well.

Show MeSH
Related in: MedlinePlus