Parameter uncertainty in biochemical models described by ordinary differential equations.

Improved mechanistic understanding of biochemical networks is one of the driving ambitions of Systems Biology. Computational modeling allows the integration of various sources of experimental data in order to put this conceptual understanding to the test in a quantitative manner. The aim of computational modeling is to obtain both predictive as well as explanatory models for complex phenomena, hereby providing useful approximations of reality with varying levels of detail. As the complexity required to describe different system increases, so does the need for determining how well such predictions can be made. Despite efforts to make tools for uncertainty analysis available to the field, these methods have not yet found widespread use in the field of Systems Biology. Additionally, the suitability of the different methods strongly depends on the problem and system under investigation. This review provides an introduction to some of the techniques available as well as gives an overview of the state-of-the-art methods for parameter uncertainty analysis.

[1]  M. Girolami,et al.  Inferring Signaling Pathway Topologies from Multiple Perturbation Measurements of Specific Biochemical Species , 2010, Science Signaling.

[2]  Peter A. J. Hilbers,et al.  Parameter adaptations during phenotype transitions in progressive diseases , 2011, BMC Systems Biology.

[3]  Robert Tibshirani,et al.  Bootstrap confidence intervals and bootstrap approximations , 1987 .

[4]  Peter A. J. Hilbers,et al.  A Bayesian approach to targeted experiment design , 2012, Bioinform..

[5]  Nicole Radde,et al.  Trajectory-oriented Bayesian experiment design versus Fisher A-optimal design: an in depth comparison study , 2012, Bioinform..

[6]  Jens Timmer,et al.  Dynamical modeling and multi-experiment fitting with PottersWheel , 2008, Bioinform..

[7]  Jacob Roll,et al.  Systems biology: model based evaluation and comparison of potential explanations for given biological data , 2009, The FEBS journal.

[8]  Ernst Dieter Gilles,et al.  Mathematical modeling and analysis of insulin clearance in vivo , 2008, BMC Systems Biology.

[9]  M. Girolami,et al.  Riemann manifold Langevin and Hamiltonian Monte Carlo methods , 2011, Journal of the Royal Statistical Society: Series B (Statistical Methodology).

[10]  S Waldherr,et al.  Parameter identification, experimental design and model falsification for biological network models using semidefinite programming. , 2010, IET systems biology.

[11]  P. Hartman Ordinary Differential Equations , 1965 .

[12]  Peter A. J. Hilbers,et al.  An integrated strategy for prediction uncertainty analysis , 2012, Bioinform..

[13]  Neil D. Lawrence,et al.  Accelerating Bayesian Inference over Nonlinear Differential Equations with Gaussian Processes , 2008, NIPS.

[14]  W. Stahel,et al.  Log-normal Distributions across the Sciences: Keys and Clues , 2001 .

[15]  Carmen G. Moles,et al.  Parameter estimation in biochemical pathways: a comparison of global optimization methods. , 2003, Genome research.

[16]  U Klingmüller,et al.  Predictive mathematical models of cancer signalling pathways , 2012, Journal of internal medicine.

[17]  Christopher R. Myers,et al.  Universally Sloppy Parameter Sensitivities in Systems Biology Models , 2007, PLoS Comput. Biol..

[18]  Ursula Klingmüller,et al.  Theoretical and experimental analysis links isoform- specific ERK signalling to cell fate decisions , 2009, Molecular systems biology.

[19]  J. Varah A Spline Least Squares Method for Numerical Parameter Estimation in Differential Equations , 1982 .

[20]  E. Gilles,et al.  Computational modeling of the dynamics of the MAP kinase cascade activated by surface and internalized EGF receptors , 2002, Nature Biotechnology.

[21]  Paul D. W. Kirk,et al.  Gaussian process regression bootstrapping: exploring the effects of uncertainty in time course data , 2009, Bioinform..

[22]  Gunnar Cedersund,et al.  A Hierarchical Whole-body Modeling Approach Elucidates the Link between in Vitro Insulin Signaling and in Vivo Glucose Homeostasis* , 2011, The Journal of Biological Chemistry.

[23]  David Welch,et al.  Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems , 2009, Journal of The Royal Society Interface.

[24]  Joseph P. Romano,et al.  A Review of Bootstrap Confidence Intervals , 1988 .

[25]  Jens Timmer,et al.  Systems-level interactions between insulin–EGF networks amplify mitogenic signaling , 2009, Molecular systems biology.

[26]  Sean R. Anderson,et al.  Repelled from the wound, or randomly dispersed? Reverse migration behaviour of neutrophils characterized by dynamic modelling , 2012, Journal of The Royal Society Interface.

[27]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[28]  Erika Cule,et al.  ABC-SysBio—approximate Bayesian computation in Python with GPU support , 2010, Bioinform..

[29]  Mark Girolami,et al.  Statistical analysis of nonlinear dynamical systems using differential geometric sampling methods , 2011, Interface Focus.

[30]  Carl E. Rasmussen,et al.  Gaussian processes for machine learning , 2005, Adaptive computation and machine learning.

[31]  Heinz Koeppl,et al.  ‘Glocal’ Robustness Analysis and Model Discrimination for Circadian Oscillators , 2009, PLoS Comput. Biol..

[32]  Barbara M. Bakker,et al.  Can yeast glycolysis be understood in terms of in vitro kinetics of the constituent enzymes? Testing biochemistry. , 2000, European journal of biochemistry.

[33]  Ajay Jasra,et al.  On population-based simulation for static inference , 2007, Stat. Comput..

[34]  S. Quake,et al.  A Systems Approach to Measuring the Binding Energy Landscapes of Transcription Factors , 2007, Science.

[35]  R. Eils,et al.  Mathematical modeling reveals threshold mechanism in CD95-induced apoptosis , 2004, The Journal of cell biology.

[36]  Fabian J Theis,et al.  High-dimensional Bayesian parameter estimation: case study for a model of JAK2/STAT5 signaling. , 2013, Mathematical biosciences.

[37]  Mark A. Girolami,et al.  BioBayes: A software package for Bayesian inference in systems biology , 2008, Bioinform..

[38]  Ana Rute Neves,et al.  The intricate side of systems biology. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[39]  K. S. Brown,et al.  Optimal experimental design in an epidermal growth factor receptor signalling and down-regulation model. , 2007, IET systems biology.

[40]  B. Palsson,et al.  The evolution of molecular biology into systems biology , 2004, Nature Biotechnology.

[41]  Bradley P. Carlin,et al.  Markov Chain Monte Carlo conver-gence diagnostics: a comparative review , 1996 .

[42]  Stacey D. Finley,et al.  Inferring relevant control mechanisms for interleukin‐12 signaling in naïve CD4+ T cells , 2011, Immunology and cell biology.

[43]  Jonas S. Almeida,et al.  Decoupling dynamical systems for pathway identification from metabolic profiles , 2004, Bioinform..

[44]  Klaas Nicolay,et al.  Silencing of glycolysis in muscle: experimental observation and numerical analysis , 2010, Experimental Physiology.

[45]  B. Efron,et al.  Bootstrap confidence intervals , 1996 .

[46]  Charles J. Geyer,et al.  Practical Markov Chain Monte Carlo , 1992 .

[47]  J. Banga,et al.  Structural Identifiability of Systems Biology Models: A Critical Comparison of Methods , 2011, PloS one.

[48]  Natal A. W. van Riel,et al.  Dynamic modelling and analysis of biochemical networks: mechanism-based models and model-based experiments , 2006, Briefings Bioinform..

[49]  A. Gelman,et al.  Physiological Pharmacokinetic Analysis Using Population Modeling and Informative Prior Distributions , 1996 .

[50]  Kwang-Hyun Cho,et al.  In silico identification of the key components and steps in IFN‐γ induced JAK‐STAT signaling pathway , 2005, FEBS letters.

[51]  A. Rundell,et al.  Comparative study of parameter sensitivity analyses of the TCR-activated Erk-MAPK signalling pathway. , 2006, Systems biology.

[52]  Hervé Delingette,et al.  Efficient probabilistic model personalization integrating uncertainty on data and parameters: Application to eikonal-diffusion models in cardiac electrophysiology. , 2011, Progress in biophysics and molecular biology.

[53]  Tina Toni,et al.  Designing attractive models via automated identification of chaotic and oscillatory dynamical regimes , 2011, Nature communications.

[54]  F. Bruggeman,et al.  The nature of systems biology. , 2007, Trends in microbiology.

[55]  John M. Hancock,et al.  A kinetic core model of the glucose-stimulated insulin secretion network of pancreatic β cells , 2007, Mammalian Genome.

[56]  Carol S. Woodward,et al.  Enabling New Flexibility in the SUNDIALS Suite of Nonlinear and Differential/Algebraic Equation Solvers , 2020, ACM Trans. Math. Softw..

[57]  Sandhya Dwarkadas,et al.  Parallel Metropolis coupled Markov chain Monte Carlo for Bayesian phylogenetic inference , 2002, Bioinform..

[58]  Jiguo Cao,et al.  Parameter estimation for differential equations: a generalized smoothing approach , 2007 .

[59]  A Kremling,et al.  Exploiting the bootstrap method for quantifying parameter confidence intervals in dynamical systems. , 2006, Metabolic engineering.

[60]  Mark A. Girolami,et al.  Bayesian ranking of biochemical system models , 2008, Bioinform..

[61]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and Linear Algebra , 1974 .

[62]  N A W van Riel,et al.  Modeling glucose and water dynamics in human skin. , 2008, Diabetes technology & therapeutics.

[63]  A. Barabasi,et al.  Network biology: understanding the cell's functional organization , 2004, Nature Reviews Genetics.

[64]  Andreas N. Philippou,et al.  Asymptotic normality of the maximum likelihood estimate in the independent not identically distributed case , 1975 .

[65]  Jens Timmer,et al.  Joining forces of Bayesian and frequentist methodology: a study for inference in the presence of non-identifiability , 2012, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[66]  Jens Timmer,et al.  An error model for protein quantification , 2007, Bioinform..

[67]  Gunnar Cedersund,et al.  Mass and Information Feedbacks through Receptor Endocytosis Govern Insulin Signaling as Revealed Using a Parameter-free Modeling Framework* , 2010, The Journal of Biological Chemistry.

[68]  Thomas Thorne,et al.  Calibrating spatio-temporal models of leukocyte dynamics against in vivo live-imaging data using approximate Bayesian computation. , 2012, Integrative biology : quantitative biosciences from nano to macro.

[69]  B. Efron The jackknife, the bootstrap, and other resampling plans , 1987 .

[70]  I. Chou,et al.  Recent developments in parameter estimation and structure identification of biochemical and genomic systems. , 2009, Mathematical biosciences.

[71]  H V Westerhoff,et al.  A metabolic control analysis of kinetic controls in ATP free energy metabolism in contracting skeletal muscle. , 2000, American journal of physiology. Cell physiology.

[72]  G. Cedersund,et al.  Conservation laws and unidentifiability of rate expressions in biochemical models. , 2007, IET systems biology.

[73]  E. Klipp,et al.  Biochemical networks with uncertain parameters. , 2005, Systems biology.

[74]  B. Efron Nonparametric standard errors and confidence intervals , 1981 .

[75]  J. Timmer,et al.  Identification of nucleocytoplasmic cycling as a remote sensor in cellular signaling by databased modeling , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[76]  Albert Compte,et al.  Workflow for generating competing hypothesis from models with parameter uncertainty , 2011, Interface Focus.

[77]  Fabio Rigat,et al.  Parallel hierarchical sampling: A general-purpose interacting Markov chains Monte Carlo algorithm , 2012, Comput. Stat. Data Anal..

[78]  Ursula Klingmüller,et al.  Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood , 2009, Bioinform..

[79]  Radford M. Neal Sampling from multimodal distributions using tempered transitions , 1996, Stat. Comput..

[80]  Jens Timmer,et al.  Likelihood based observability analysis and confidence intervals for predictions of dynamic models , 2011, BMC Systems Biology.

[81]  Peng Qiu,et al.  Optimal experiment selection for parameter estimation in biological differential equation models , 2012, BMC Bioinformatics.

[82]  H. Jeffreys An invariant form for the prior probability in estimation problems , 1946, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[83]  Ursula Klingmüller,et al.  Tests for cycling in a signalling pathway , 2004 .

[84]  Maksat Ashyraliyev,et al.  Systems biology: parameter estimation for biochemical models , 2009, The FEBS journal.

[85]  Mustafa Khammash,et al.  Parameter Estimation and Model Selection in Computational Biology , 2010, PLoS Comput. Biol..

[86]  Daniel A Beard,et al.  Oxidative ATP synthesis in skeletal muscle is controlled by substrate feedback. , 2007, American journal of physiology. Cell physiology.

[87]  David J. Klinke,et al.  An empirical Bayesian approach for model-based inference of cellular signaling networks , 2009, BMC Bioinformatics.

[88]  Mark A. Girolami,et al.  Estimating Bayes factors via thermodynamic integration and population MCMC , 2009, Comput. Stat. Data Anal..

[89]  J. Schaber,et al.  Model-based inference of biochemical parameters and dynamic properties of microbial signal transduction networks. , 2011, Current opinion in biotechnology.

[90]  T. Maiwald,et al.  Materials and Methods SOM Text Figs. S1 to S16 References Materials and Methods , 2022 .

[91]  Jacob Roll,et al.  Model-Based Hypothesis Testing of Key Mechanisms in Initial Phase of Insulin Signaling , 2008, PLoS Comput. Biol..

[92]  Kwang-Hyun Cho,et al.  Experimental Design in Systems Biology, Based on Parameter Sensitivity Analysis Using a Monte Carlo Method: A Case Study for the TNFα-Mediated NF-κ B Signal Transduction Pathway , 2003, Simul..