Some novel techniques of parameter estimation for dynamical models in biological systems

• Published in: IMA journal of applied mathematics Vol. 78; no. 2; pp. 235 - 260
• Main Authors: Liu, F., Burrage, K., Hamilton, N. A.
• Format: Journal Article
• Language: English
• Published: 01-04-2013
• Subjects: Bacteria; Biological; Dynamical systems; Mathematical analysis; Mathematical models; Nonlinear dynamics; Nonlinearity; Parameter estimation
• ISSN: 0272-4960; 1464-3634
• DOI: 10.1093/imamat/hxr046

Inverse problems based on using experimental data to estimate unknown parameters of a system often arise in biological and chaotic systems. In this paper, we consider parameter estimation in systems biology involving linear and non-linear complex dynamical models, including the Michaelis-Menten enzyme kinetic system, a dynamical model of competence induction in Bacillus subtilis bacteria and a model of feedback bypass in B. subtilis bacteria. We propose some novel techniques for inverse problems. Firstly, we establish an approximation of a non-linear differential algebraic equation that corresponds to the given biological systems. Secondly, we use the Picard contraction mapping, collage methods and numerical integration techniques to convert the parameter estimation into a minimization problem of the parameters. We propose two optimization techniques: a grid approximation method and a modified hybrid Nelder- Mead simplex search and particle swarm optimization (MH-NMSS-PSO) for non-linear parameter estimation. The two techniques are used for parameter estimation in a model of competence induction in B. subtilis bacteria with noisy data. The MH-NMSS-PSO scheme is applied to a dynamical model of competence induction in B. subtilis bacteria based on experimental data and the model for feedback bypass. Numerical results demonstrate the effectiveness of our approach.