Parameter Synthesis in Nonlinear Dynamical Systems: Application to Systems Biology (original) (raw)

Parameter Reconstruction for Biochemical Networks Using Interval Analysis

Reliable Computing, 2006

In recent years, the modeling and simulation of biochemical networks has attracted increasing attention. Such networks are commonly modeled by systems of ordinary differential equations, a special class of which are known as S-systems. These systems are specifically designed to mimic kinetic reactions, and are sufficiently general to model genetic networks, metabolic networks, and signal transduction cascades. The parameters of an S-system correspond to various kinetic rates of the underlying reactions, and one of the main challenges is to determine approximate values of these parameters, given measured (or simulated) time traces of the involved reactants.

Estimation of parametric nonlinear ODEs for biological networks identification

2008

Ordinary Differential Equations (ODEs) provide a theoretical framework for a mechanistic description of biological networks (e.g. signalling pathway, gene regulatory network, metabolic pathway) as continuous time dynamical systems. Relevant ODEs are often nonlinear because they are derived from biochemical kinetics and based on law of mass action and its generalizations or Hill kinetics. We present two approaches devoted to the identification of parameters from time-series of the state variables in nonlinear ODEs. The first approach is based on a nonparametric estimation of the trajectory of the variables involved in the ODE. The parameters are learned in a second step by minimizing a distance between two estimates of the derivatives. In the second approach, dedicated to Bayesian estimation, we build a nonlinear state-space model from the ODEs and we estimate both parameters and hidden variables by approximate nonlinear filtering and smoothing (performed by the unscented transform).The two approaches are illustrated on numerical examples and discussed.

Global Sensitivity Analysis Challenges in Biological Systems Modeling

Industrial & Engineering Chemistry Research, 2009

Mammalian cell culture systems produce high-value biologics, such as monoclonal antibodies, which are increasingly being used clinically. A complete framework that interlinks model-based design of experiments (DOE) and model-based control and optimization to the actual industrial bioprocess could assist experimentation, hence reducing costs. However, high fidelity models have the inherent characteristic of containing a large number of parameters, which is further complicated by limitations in the current analytical techniques, thus resulting in the experimental validation of merely a small number of parameters. Sensitivity analysis techniques can provide valuable insight into model characteristics. Traditionally, the application of sensitivity analysis on models of biological systems has been treated more or less as a black box operation. In the present work, we elucidate the aspects of sensitivity analysis and identify, with reasoning, the most suitable group of sensitivity analysis methods for application to highly nonlinear dynamic models in the context of biological systems. Specifically, we perform computational experiments on antibody-producing mammalian cell culture models of different complexities and identify, as well as address, problems associated with such "real life" models. In conclusion, a novel global screening method (derivative based global sensitivity measures, DGSM) is proven to be the most time-efficient and robust alternative to the established variance-based Monte Carlo methods.

Bayesian parameter estimation for nonlinear modelling of biological pathways

2011

Background: The availability of temporal measurements on biological experiments has significantly promoted research areas in systems biology. To gain insight into the interaction and regulation of biological systems, mathematical frameworks such as ordinary differential equations have been widely applied to model biological pathways and interpret the temporal data. Hill equations are the preferred formats to represent the reaction rate in differential equation frameworks, due to their simple structures and their capabilities for easy fitting to saturated experimental measurements. However, Hill equations are highly nonlinearly parameterized functions, and parameters in these functions cannot be measured easily. Additionally, because of its high nonlinearity, adaptive parameter estimation algorithms developed for linear parameterized differential equations cannot be applied. Therefore, parameter estimation in nonlinearly parameterized differential equation models for biological pathways is both challenging and rewarding. In this study, we propose a Bayesian parameter estimation algorithm to estimate parameters in nonlinear mathematical models for biological pathways using time series data. Results: We used the Runge-Kutta method to transform differential equations to difference equations assuming a known structure of the differential equations. This transformation allowed us to generate predictions dependent on previous states and to apply a Bayesian approach, namely, the Markov chain Monte Carlo (MCMC) method. We applied this approach to the biological pathways involved in the left ventricle (LV) response to myocardial infarction (MI) and verified our algorithm by estimating two parameters in a Hill equation embedded in the nonlinear model. We further evaluated our estimation performance with different parameter settings and signal to noise ratios. Our results demonstrated the effectiveness of the algorithm for both linearly and nonlinearly parameterized dynamic systems.

Pitfalls in Finding Sloppy and Stiff Parameters in Biochemical Models

2015

Finding sloppy and stiff parameters have became one of the standard approaches to sensitivity analysis of models of biochemical systems. However, results obtained with any sensitivity analysis may be difficult to interpret and verify, due to the high complexity of biological models. This work shows possible traps in sensitivity analysis performed with standard methods. They are illustrated with a simple second order system and subsequently checked with a toy model of p53/Mdm2 regulatory module. For easier understanding of obtained results we proposed a new method to create parameters ranking. Additionally, to validate the method we compared obtained results with a procedure based on the study of changes in frequency of the model response. INTRODUCTION Dynamic computational models play an increasingly important role in systems biology. Such models are powerful tools that allow us to develop and test several hypotheses about complex biological systems [1], [2]. In the literature there...

Parameter sensitivity analysis for biochemical reaction networks

Mathematical Biosciences and Engineering

Biochemical reaction networks describe the chemical interactions occurring between molecular populations inside the living cell. These networks can be very noisy and complex and they often involve many variables and even more parameters. Parameter sensitivity analysis that studies the effects of parameter changes to the behaviour of biochemical networks can be a powerful tool in unravelling their key parameters and interactions. It can also be very useful in designing experiments that study these networks and in addressing parameter identifiability issues. This article develops a general methodology for analysing the sensitivity of probability distributions of stochastic processes describing the time-evolution of biochemical reaction networks to changes in their parameter values. We derive the coefficients that efficiently summarise the sensitivity of the probability distribution of the network to each parameter and discuss their properties. The methodology is scalable to large and complex stochastic reaction networks involving many parameters and can be applied to oscillatory networks. We use the two-dimensional Brusselator system as an illustrative example and apply our approach to the analysis of the Drosophila circadian clock. We investigate the impact of using stochastic over deterministic models and provide an analysis that can support key decisions for experimental design, such as the choice of variables and time-points to be observed.

Parameter subset selection techniques for problems in mathematical biology

Biological cybernetics, 2018

Patient-specific models for diagnostics and treatment planning require reliable parameter estimation and model predictions. Mathematical models of physiological systems are often formulated as systems of nonlinear ordinary differential equations with many parameters and few options for measuring all state variables. Consequently, it can be difficult to determine which parameters can reliably be estimated from available data. This investigation highlights pitfalls associated with practical parameter identifiability and subset selection. The latter refer to the process associated with selecting a subset of parameters that can be identified uniquely by parameter estimation protocols. The methods will be demonstrated using five examples of increasing complexity, as well as with patient-specific model predicting arterial blood pressure. This study demonstrates that methods based on local sensitivities are preferable in terms of computational cost and model fit when good initial parameter...

Novel metaheuristic for parameter estimation in nonlinear dynamic biological systems

BMC bioinformatics, 2006

We consider the problem of parameter estimation (model calibration) in nonlinear dynamic models of biological systems. Due to the frequent ill-conditioning and multi-modality of many of these problems, traditional local methods usually fail (unless initialized with very good guesses of the parameter vector). In order to surmount these difficulties, global optimization (GO) methods have been suggested as robust alternatives. Currently, deterministic GO methods can not solve problems of realistic size within this class in reasonable computation times. In contrast, certain types of stochastic GO methods have shown promising results, although the computational cost remains large. Rodriguez-Fernandez and coworkers have presented hybrid stochastic-deterministic GO methods which could reduce computation time by one order of magnitude while guaranteeing robustness. Our goal here was to further reduce the computational effort without loosing robustness.