Identification of Parameters in Systems Biology

Abstract

Systems Biology is an actively emerging interdisciplinary area between biology and applied mathematics, based on the idea of treating biological systems as a whole entity which is more than the sum of its interrelated components. One of the major goals of systems biology is to reveal, understand, and predict such properties through the development of mathematical models based on experimental data. In many cases, predictive models of systems biology are described by large systems of nonlinear differential equations. Quantitative identification of such systems requires the solution of inverse problems on the identification of parameters of the system. This dissertation explores the inverse problem for the identification of the finite dimensional set of parameters for systems of nonlinear ordinary differential equations (ODEs) arising in systems biology. Two numerical methods are implemented. The first method combines the ideas of Pontryagin optimization or Bellman’s quasilinearization with sensitivity analysis and Tikhonov’s regularization. The method is applied to various biological models such as the classical LotkaVolterra system, bistable switch model in genetic regulatory networks, gene regulation and repressilator models from synthetic biology. The numerical results and application to real data demonstrate the superlinear convergence with convergence rate close to quadratic. The method proved to be extremely effective in moderate scale models of systems biology. The results are published in a recent paper Mathematical Biosciences, 305(2018), 133-145. To address adaptation and scalability of the method for large-scale models of systems biology the modification of the method is developed by embedding a method of staggered corrector for sensitivity analysis and by enhancing multi-objective optimization which enables application of the new method to large-scale models with practically non-identifiable parameters based on multiple data sets, possibly with partial and noisy measurements. The new method is applied to benchmark model of three-step pathway modelled by 8 nonlinear ODEs with 36 unknown parameters and two control input parameters. The numerical results demonstrate the superlinear convergence with minimum five data sets and with minimum measurements per data set. The method is extremely robust with respect to partial and noisy measurements, and in terms of required number of measurements for each component of the system. Optimal choice of the Tikhonov regularization parameter significantly improves convergence rate, precision and convergence range of the algorithm. Software package qlopt is developed for both methods and posted in GitHub. MATLAB package AMIGO2 is used to demonstrate advantage of qlopt over most popular methods/software such as lsqnonlin, fmincon and nl2sol in an equivalent setting.