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Parameter estimation in systems biology models using spline approximation.

Zhan C, Yeung LF - BMC Syst Biol (2011)

Bottom Line: Moreover, the cores of our algorithms are LP and NLP based, which are flexible and consequently additional constraints can be embedded/removed easily.Eight system biology models are used for testing the proposed approaches.The proposed approaches have general application to identify unknown parameter values of a wide range of systems biology models.

View Article: PubMed Central - HTML - PubMed

Affiliation: Department of Electronic Engineering, City University of Hong Kong, PR China. zchoujun2@student.cityu.edu.hk

ABSTRACT

Background: Mathematical models for revealing the dynamics and interactions properties of biological systems play an important role in computational systems biology. The inference of model parameter values from time-course data can be considered as a "reverse engineering" process and is still one of the most challenging tasks. Many parameter estimation methods have been developed but none of these methods is effective for all cases and can overwhelm all other approaches. Instead, various methods have their advantages and disadvantages. It is worth to develop parameter estimation methods which are robust against noise, efficient in computation and flexible enough to meet different constraints.

Results: Two parameter estimation methods of combining spline theory with Linear Programming (LP) and Nonlinear Programming (NLP) are developed. These methods remove the need for ODE solvers during the identification process. Our analysis shows that the augmented cost function surfaces used in the two proposed methods are smoother; which can ease the optima searching process and hence enhance the robustness and speed of the search algorithm. Moreover, the cores of our algorithms are LP and NLP based, which are flexible and consequently additional constraints can be embedded/removed easily. Eight system biology models are used for testing the proposed approaches. Our results confirm that the proposed methods are both efficient and robust.

Conclusions: The proposed approaches have general application to identify unknown parameter values of a wide range of systems biology models.

Show MeSH
Cost function surface and contours. (Color online) (a) Cost function surface of the P0 as parameters k1 and k2 are varied; (b) displays the same data as (a) on the expanded scale; (c) corresponding contours near the nominal parameter value.
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Figure 3: Cost function surface and contours. (Color online) (a) Cost function surface of the P0 as parameters k1 and k2 are varied; (b) displays the same data as (a) on the expanded scale; (c) corresponding contours near the nominal parameter value.

Mentions: Here, we use the G1/S model to show the differences of the cost function surfaces between NLP-P0 and NLP-P3: in this case, the cost function of P0 is a highly irregular and complicated manifold with multiple local minima; the augmented cost function adopted in problem P3 is a much "smoother" function and hence it is easier for the NLP algorithm to converge to the solution. In order to simplify the analysis for exposition purpose, we only vary parameters k1 and k2 over the range k1 ϵ [0; 2 ] and k2 ϵ [0, 3.2 ] and fix all other parameters at their nominal values. Figure 3(a) displays the cost function surface of P0 , while Figure 3(b) exhibits the same data as Figure 3(a) on the expanded scale and Figure 3(c) is the corresponding contour plots. Figure 3(a) shows that the cost function surface of is a ridge, which drops suddenly from 109 to 0. However, Figure 3(b) reveals the cost function surface of P0 are actually banana-shaped valley around the nominal value of the fixed parameters, this unfavorable profile can slow down the convergence rate of the algorithm. Furthermore, there are many local minima in the banana-shaped valley. Some algorithms, such as simulated annealing, genetic algorithm, have been proposed to overcome these problem. However, these algorithms are all computationally demanding. In conclusion, these cost function features make the problem P0 a severe challenge to every optimization algorithm.


Parameter estimation in systems biology models using spline approximation.

Zhan C, Yeung LF - BMC Syst Biol (2011)

Cost function surface and contours. (Color online) (a) Cost function surface of the P0 as parameters k1 and k2 are varied; (b) displays the same data as (a) on the expanded scale; (c) corresponding contours near the nominal parameter value.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3750107&req=5

Figure 3: Cost function surface and contours. (Color online) (a) Cost function surface of the P0 as parameters k1 and k2 are varied; (b) displays the same data as (a) on the expanded scale; (c) corresponding contours near the nominal parameter value.
Mentions: Here, we use the G1/S model to show the differences of the cost function surfaces between NLP-P0 and NLP-P3: in this case, the cost function of P0 is a highly irregular and complicated manifold with multiple local minima; the augmented cost function adopted in problem P3 is a much "smoother" function and hence it is easier for the NLP algorithm to converge to the solution. In order to simplify the analysis for exposition purpose, we only vary parameters k1 and k2 over the range k1 ϵ [0; 2 ] and k2 ϵ [0, 3.2 ] and fix all other parameters at their nominal values. Figure 3(a) displays the cost function surface of P0 , while Figure 3(b) exhibits the same data as Figure 3(a) on the expanded scale and Figure 3(c) is the corresponding contour plots. Figure 3(a) shows that the cost function surface of is a ridge, which drops suddenly from 109 to 0. However, Figure 3(b) reveals the cost function surface of P0 are actually banana-shaped valley around the nominal value of the fixed parameters, this unfavorable profile can slow down the convergence rate of the algorithm. Furthermore, there are many local minima in the banana-shaped valley. Some algorithms, such as simulated annealing, genetic algorithm, have been proposed to overcome these problem. However, these algorithms are all computationally demanding. In conclusion, these cost function features make the problem P0 a severe challenge to every optimization algorithm.

Bottom Line: Moreover, the cores of our algorithms are LP and NLP based, which are flexible and consequently additional constraints can be embedded/removed easily.Eight system biology models are used for testing the proposed approaches.The proposed approaches have general application to identify unknown parameter values of a wide range of systems biology models.

View Article: PubMed Central - HTML - PubMed

Affiliation: Department of Electronic Engineering, City University of Hong Kong, PR China. zchoujun2@student.cityu.edu.hk

ABSTRACT

Background: Mathematical models for revealing the dynamics and interactions properties of biological systems play an important role in computational systems biology. The inference of model parameter values from time-course data can be considered as a "reverse engineering" process and is still one of the most challenging tasks. Many parameter estimation methods have been developed but none of these methods is effective for all cases and can overwhelm all other approaches. Instead, various methods have their advantages and disadvantages. It is worth to develop parameter estimation methods which are robust against noise, efficient in computation and flexible enough to meet different constraints.

Results: Two parameter estimation methods of combining spline theory with Linear Programming (LP) and Nonlinear Programming (NLP) are developed. These methods remove the need for ODE solvers during the identification process. Our analysis shows that the augmented cost function surfaces used in the two proposed methods are smoother; which can ease the optima searching process and hence enhance the robustness and speed of the search algorithm. Moreover, the cores of our algorithms are LP and NLP based, which are flexible and consequently additional constraints can be embedded/removed easily. Eight system biology models are used for testing the proposed approaches. Our results confirm that the proposed methods are both efficient and robust.

Conclusions: The proposed approaches have general application to identify unknown parameter values of a wide range of systems biology models.

Show MeSH