Fig 1.
Model of biological system at homeostatic state.
(A) Many biological systems behave around a certain stable state. Such stable behavior can be observed in many biological systems although the size of the stable region around the stable state differs by systems. (B) Assumed structure of biological stable region. If there are upward-convex regions in potential focused region, the size of the basin of attraction is limited, therefore, our concept is to diminish such regions in parameter searching. (C) Three factors are considered to search the parameter sets satisfying the BSR condition and incorporated in the objective function to minimize.
Fig 2.
Advantages of focusing BSR and framework of TEAPS.
(A) Advantages of focusing BSR were depicted. See detail in the text. (B) The framework of thorough exploration of parameter sets satisfying the BSR constraints, which we named TEAPS.
Fig 3.
TEAPS search expands the basin stability during its process and found the parameter sets satisfying the BSR condition.
(A) Three basic models were used for implementation of the BSR. (B) The estimated basin stability was increased by individual optimization process (gL-BFGS) of searching parameter sets satisfying the BSR condition using the TEAPS search algorithm. (C–E) Comparison of the parameter distributions found by the TEAPS and the brute-force searches as histograms or multidimensional density profiles for the models T1 (C), T2 (D) and T3 (E). The parameter sets satisfying the BSR condition were searched by TEAPS (upper left) or by the brute-force search (upper right), and the histograms of the distribution of the found parameter sets were compared. Density profiles of the obtained parameter sets were compared following the kernel density estimation (middle left: TEAPS, middle right: brute-force search). Two density profiles were overlaid (bottom).
Table 1.
Similarities between the distributions of the obtained parameter sets which met the BSR condition by TEAPS and the brute-force search.
Fig 4.
Performance assessment of TEAPS using various simple structure models.
(A) Additional five sample models to be prepared for performance test. (B–F) Comparison of the TEAPS results and the brute-force search results for the models T4 (B), T5 (C), T6 (D), T7 (E) and T8 (F). The parameter sets satisfying the BSR condition were searched by TEAPS (upper left) or by the brute-force search (upper right), and the histograms of the distribution of the found parameter sets were compared. Density profiles of the found parameter sets were compared following the kernel density estimation (middle left: TEAPS, middle right: brute-force search). Two density profiles were overlaid (bottom).
Fig 5.
Application of TEAPS for NF-κB signaling pathway model.
(A) The model structure of NF-κB signaling pathways. This signaling pathway includes feedback inhibition of NF-κB activation through IκBα and A20. (B) Time courses of NF-κB activation. The parameter set fitting to in vitro observation outputs a damped oscillation as is consistent with experimental observation (red), while the model outputs continuous oscillation when the parameter related to feedback inhibitions are changed (green). (C) The estimated basin stability for the reported parameter set (actual) and the parameter set outputting continuous oscillation (oscillation). (D) The estimated basin stability was increased by individual optimization process (gL-BFGS) when TEAPS were applied to the NF-κB model. (E) The distribution of parameter sets found by TEAPS for the NF-κB model. The found parameter values were broadly distributed (blue). For many parameters, the previously reported value (red) was included in the range found by TEAPS. (F, G) The distribution of the parameter values found by TEAPS in PC space. The reported values (red) were included in the found distribution for most axes, while the oscillative parameter set was outside of dominant distribution for later PCs. (F) The distributions for all PCs are shown. (G) The distributions of later PCs are enlarged.
Fig 6.
Utility of TEAPS in the analyses of arachidonic acid pathway model.
(A) The model containing arachidonic acid pathways in polymorphonuclear cells (PMN), endothelial cells (EC), and platelets (PLT) was reconstructed according to the previously reported model structure. The Michaelis-Menten constants were set to the same values if there were same reactions in different cells. The reactions which only appears in each cell type are colored in cyan (PMN), pink (EC), and brown (PLT). The reactions with black arrows are shared with at least two cell types. For modifications on reactions, lines ending with a circle indicate promoting effect on a reaction whereas lines ending with a bar indicate inhibitory effect. Modificative effects by metabolite but not enzymes are described as dashed lines. (B) The estimated basin stability was increased by individual optimization process (gL-BFGS) when TEAPS were applied to the arachidonic acid pathway model. (C) The parameter sets obtained for the arachidonic acid pathway model were subjected to the principal component analysis. To figure out the entire shape of the parameter distribution, the number of components was set to the number of parameters in the model. (D) The reactions with tight and robust regulation were searched. Each arrow corresponds to each parameter on every reaction. The reactions related with parameters that appeared as high contribution to PCA axes in which the parameter distributions are narrow are colored in red, while those to each of the PCA axes in which the parameter distributions are broad are colored in blue.
Fig 7.
Part of parameter sets obtained by TEAPS can reproduce the in vivo phenotype associated with the administration of NSAIDs.
(A, B) The situations where each NSAID was administered to suppress PGE2 levels in PMN to approximately 10% of control were compared with the previously reported article. Each blue point corresponds to a result by one parameter set obtained by TEAPS. Red asterisks indicate the reported values by the simulation results using the previously reported model. PGE2 level in PMN (A) and the ratio of PGI2 to TXA2 (PT ratio) (B), a marker of physiological output, are shown. (C) Patterning of pharmacological outputs by clustering analysis showed the difference of pharmacological actions. Clustering analysis was performed based on the pattern of the simulated PT ratios by the obtained parameter sets under each drug exposure. The COX-2 selectivity was figured out by the phylogenetic tree.