Reach Us
+44-1522-440391

**Seongho Kim ^{*}**

Biostatistics Core, Karmanos Cancer Institute, Department of Oncology, School of Medicine, Wayne State University, USA

- Corresponding Author:
- Seongho Kim

Biostatistics Core, Karmanos Cancer Institute

Department of Oncology, School of Medicine

Wayne State University, Detroit, MI 48201, USA

**Tel:**+1-313-576-8653

**E-mail:**[email protected]

**Received Date:** May 24, 2016; **Accepted Date:** May 24, 2016; **Published Date:** May 30, 2016

**Citation: **Seongho Kim (2016) Parameter Estimation Using Divide-and-Conquer Methods for Differential Equation Models. J Biom Biostat 7:305. doi:10.4172/2155-6180.1000305

**Copyright:** © 2016 Kim S. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Biometrics & Biostatistics

In systems **biology**, a key topic is the elucidation of the dynamic behavior of biological processes that are made up of complex biochemical networks. Statistical modeling is an important to capture the dynamics of biochemical networks such as metabolic networks, signal transduction pathways, and gene regulatory networks. These biochemical models have a set of parameters that represent the physical properties of the systems, such as kinetic constants and reaction rates. In general, the development of these models requires two steps: model structure construction and parameter estimation. The models are often constructed with time derivative expressions, such as ordinary differential equations (ODEs), to describe the change of certain quantities of interest over time [1,2]. The model parameters are then estimated by simulating the actual processes obtained from experimental analyses [3-5]. However, because the differential equation model has many uncertain parameters and limited measurement data, parameter estimation is a major bottleneck in the development of useful biochemical models [6,7].

Optimization algorithms cannot deal with the high dimensionality of search space due to calculation complexity. One way to circumvent this difficulty is to simplify complicated systems biology models using model order reduction methods. Model order reduction **methods** reduce the number of states and parameters of dynamical systems that are defined by ODEs [8]. Lumping is one model order reduction method in which the original states of the model are lumped or merged to a reduced number of pseudo-states, resulting in a fewer equations and parameters but with effectively the same or similar input-output behavior. Proper lumping is a special case of lumping where each of the original states contributes to only one of the pseudostates of the reduced system thereby forming groups that retain a clear physical interpretation. With these methods, the reduced systems include less information, but are supposed to retain the basic features or properties of the original models. Although computational expense is saved, it is highly likely that the simplification loses critical information, especially if there is excessive simplification. Another strategy is to use divide-and-conquer methods, which decompose a large network of interest into smaller sub-networks [9,10]. For example, Voit and Almeida [11] developed an approach to transforming the problem into several sets of decoupled algebraic equations, being processed efficiently in parallel or sequentially, in large genetic network models. Kimura et al. [12] employed a cooperative co-evolutionary algorithm with a decomposition strategy to handle large S-system models with noisy time-series data. When there are no closed loops, Koh et al. [13] decomposed the network into small, independent sub-networks and estimated the **parameter**s for each sub-network separately under the assumption that signals or mass flow in one direction. van Riel and Sontag [14] proposed a different approach to utilizing the modular structure of biochemical networks, providing the time courses of the intra-modular components that interact with neighboring modules. Those divide-and-conquer strategies, however, are not suitable for complex networks consisting of multiple closed or feedback loops, because dividing closed loops can change their intrinsic regulatory structures, greatly altering their dynamic features and the sensitivity of search parameters. Recently, to handle this difficulty, Maeda et al. [5] employed flux module decomposition that separates a complex, large-scale dynamic model into multiple flux modules without destroying its basic control architectures. However, it assumes that all parameters are necessary without accounting for differences in uncertainty of parameters.

To circumvent the aforementioned issues, we propose a divide-andconquer approach to avoiding unnecessary information loss while estimating high-dimensional parameters efficiently. To do this, we first divide a large complete system into sub-systems so that each subsystem has a smaller, manageable number of differential **equation**s. Then we estimate parameters for each sub-system, followed by refinement of the estimates through communication among subsystems. The success of the proposed algorithm depends on how the complete system is divided into small sub-systems.

We illustrate our proposed approaches with a simple threecompartment model. Its system of ordinary differential equations (ODEs) is as follows:

Where (Ka,Kb,Kc) are the parameters to estimate (i.e., Ka,Kb,Kc are the absorption rate, the distribution rate, the elimination rate constants, respectively); and . Its graphical representation is shown in **Figure 1a**. Using this model, we investigated the performance of the proposed approach in a simulation study. We generated 100 simulations and estimated the parameters using 1) a conventional approach (ONE) and 2) a divide-and-conquer approach (DAQ). The brief schematic representation of DAQ can be seen in **Figure 1b**.

As for DAQ, the parameters (Ka, Kb) are first estimated given Kc and then the parameters (Kb, Kc) are estimated given Ka. This procedure was repeated until convergence. **Table 1** displays the results of 100 simulation studies with mean squared errors (MSEs) and estimates’ bias by three levels of measurement errors. The performance of DAQ is comparable to that of ONE, and, in some cases, the biases of DAQ are smaller than these of ONE in **Table 1**.

s=0.1 | s=0.5 | s=1.0 | |||||
---|---|---|---|---|---|---|---|

TRUE | DAQ | ONE | DAQ | ONE | DAQ | ONE | |

Mean | Mean | Mean | Mean | Mean | Mean | ||

(SD) | (SD) | (SD) | (SD) | (SD) | (SD) | ||

MSE | 0.009 | 0.009 | 0.216 | 0.212 | 0.875 | 0.859 | |

-0.004 | -0.004 | -0.098 | -0.096 | -0.365 | -0.358 | ||

Bias | Bias | Bias | Bias | Bias | Bias | ||

(SD) | (SD) | (SD) | (SD) | (SD) | (SD) | ||

Ka | log(0.8) | 0.003 | 0.001 | 0.025 | 0.012 | 0.048 | 0.022 |

-0.006 | -0.007 | -0.043 | -0.045 | -0.084 | -0.09 | ||

Kb | log(10) | 0.005 | 0.001 | 0.045 | 0.034 | 0.104 | 0.073 |

-0.011 | -0.021 | -0.073 | -0.067 | -0.134 | -0.218 | ||

Kc | log(8) | -0.002 | -0.001 | -0.021 | -0.026 | -0.038 | -0.047 |

-0.006 | -0.014 | -0.044 | -0.044 | -0.086 | -0.123 |

**Table 1:** Results of 100 simulations of ONE and DAQ.

It is worth noting that, as the whole model is divided into smaller models, the computation expense decreases, but the information loss increases. For this reason, it is important to ensure that the decomposition is optimal, and future work will further need to find out the relationship between the decomposition and the information loss. Overall, as shown in the limited simulation study, the proposed approach preserves important properties of the original model and thereby increases the quality of the biochemical networks due to the property that the proposed approach does not depend on simplification. Furthermore, the proposed parameter **estimation** approach can be easily applied to other high-dimensional data such as genomics, transcriptomics, proteomics, and metabolomics. Therefore, the proposed work will benefit for many types of high-dimensional studies.

This work has been partially supported by NSF grant DMS-1312603. The Biostatistics Core is supported in part by NIH Cancer Center Support Grant P30 CA022453 to the Karmanos Cancer Institute at Wayne State University.

- Abdullah A, Deris S, Anwar S, Arjunan SN (2013) An evolutionary firefly algorithm for the estimation of nonlinear biological model parameters. PLoS One 8: e56310.
- Lillacci G, Khammash M (2010) Parameter estimation and model selection in computational biology. PLoS Comput Biol 6: e100069.
- Chou IC1, Voit EO (2009) Recent developments in parameter estimation and structure identification of biochemical and genomic systems. Math Biosci 219: 57-83.
- Tashkova K, Korosec P, Silc J, Todorovski L, Dzeroski S (2011) Parameter estimation with bio-inspired meta-heuristic optimization: modelling the dynamics of endocytosis. BMC Syst Biol 5: 159–185.
- Maeda K, Minamida H, Yoshida K, Kurata H (2013) Flus module decomposition for parameter estimation in a multiple-feedback loop model of biochemical networks. Bioprocess and Biosystems Engineering 36: 333-344.
- Kotte O, Heinemann M (2009) A divide-and-conquer approach to analyze underdetermined biochemical models. Bioinformatics 25: 519-525.
- van Rie NA (2006) Dynamic modelling and analysis of biochemical networks: mechanism-based models and model-based experiments. Brief Bioinform 7: 364-374.
- Okino MS, Mavrovouniotis ML (1998) Simplification of Mathematical Models of Chemical Reaction Systems. Chem Rev 98: 391-408.
- Kim S, Kim S (2006) A divide-and-conquer approach in applying EM for large recursive models with incomplete categorical data. Computational Statistics & Data Analysis 50: 611-641.
- Kim S, Kim S (2006) A note on collapsibility in recursive graphical models of contingency tables. Scandinavian Journal of Statistics 33: 575-590.
- Voit EO, Almeida J (2004) Decoupling dynamical systems for pathway identification from metabolic profiles. Bioinformatics 20: 1670-1681.
- Kimura S1, Ide K, Kashihara A, Kano M, Hatakeyama M, et al. (2005) Inference of S-system models of genetic networks using a cooperative coevolutionary algorithm. Bioinformatics 21: 1154-1163.
- Koh G, Teong HF, Clément MV, Hsu D, Thiagarajan PS (2006) A decompositional approach to parameter estimation in pathway modeling: a case study of the Akt and MAPK pathways and their crosstalk. Bioinformatics 22: e271-280.
- van Rie NA, Sontag ED (2006) Parameter estimation in models combining signal transduction and metabolic pathways: the dependent input approach. Syst Biol (Stevenage) 153: 263-274.

Select your language of interest to view the total content in your interested language

- Adomian Decomposition Method
- Algebra
- Algebraic Geometry
- Algorithm
- Analytical Geometry
- Applied Mathematics
- Artificial Intelligence Studies
- Axioms
- Balance Law
- Behaviometrics
- Big Data Analytics
- Big data
- Binary and Non-normal Continuous Data
- Binomial Regression
- Bioinformatics Modeling
- Biometrics
- Biostatistics methods
- Biostatistics: Current Trends
- Clinical Trail
- Cloud Computation
- Combinatorics
- Complex Analysis
- Computational Model
- Computational Sciences
- Computer Science
- Computer-aided design (CAD)
- Convection Diffusion Equations
- Cross-Covariance and Cross-Correlation
- Data Mining Current Research
- Deformations Theory
- Differential Equations
- Differential Transform Method
- Findings on Machine Learning
- Fourier Analysis
- Fuzzy Boundary Value
- Fuzzy Environments
- Fuzzy Quasi-Metric Space
- Genetic Linkage
- Geometry
- Hamilton Mechanics
- Harmonic Analysis
- Homological Algebra
- Homotopical Algebra
- Hypothesis Testing
- Integrated Analysis
- Integration
- Large-scale Survey Data
- Latin Squares
- Lie Algebra
- Lie Superalgebra
- Lie Theory
- Lie Triple Systems
- Loop Algebra
- Mathematical Modeling
- Matrix
- Microarray Studies
- Mixed Initial-boundary Value
- Molecular Modelling
- Multivariate-Normal Model
- Neural Network
- Noether's theorem
- Non rigid Image Registration
- Nonlinear Differential Equations
- Number Theory
- Numerical Solutions
- Operad Theory
- Physical Mathematics
- Quantum Group
- Quantum Mechanics
- Quantum electrodynamics
- Quasi-Group
- Quasilinear Hyperbolic Systems
- Regressions
- Relativity
- Representation theory
- Riemannian Geometry
- Robotics Research
- Robust Method
- Semi Analytical-Solution
- Sensitivity Analysis
- Smooth Complexities
- Soft Computing
- Soft biometrics
- Spatial Gaussian Markov Random Fields
- Statistical Methods
- Studies on Computational Biology
- Super Algebras
- Symmetric Spaces
- Systems Biology
- Theoretical Physics
- Theory of Mathematical Modeling
- Three Dimensional Steady State
- Topologies
- Topology
- mirror symmetry
- vector bundle

- Total views:
**8620** - [From(publication date):

June-2016 - Sep 21, 2019] - Breakdown by view type
- HTML page views :
**8491** - PDF downloads :
**129**

**Make the best use of Scientific Research and information from our 700 + peer reviewed, Open Access Journals**

International Conferences 2019-20