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Malizia F, Corso A, Gambuzza LV, Russo G, Latora V, Frasca M. Reconstructing higher-order interactions in coupled dynamical systems. Nat Commun 2024; 15:5184. [PMID: 38890277 PMCID: PMC11189584 DOI: 10.1038/s41467-024-49278-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/12/2023] [Accepted: 05/30/2024] [Indexed: 06/20/2024] Open
Abstract
Higher-order interactions play a key role for the operation and function of a complex system. However, how to identify them is still an open problem. Here, we propose a method to fully reconstruct the structural connectivity of a system of coupled dynamical units, identifying both pairwise and higher-order interactions from the system time evolution. Our method works for any dynamics, and allows the reconstruction of both hypergraphs and simplicial complexes, either undirected or directed, unweighted or weighted. With two concrete applications, we show how the method can help understanding the complexity of bacterial systems, or the microscopic mechanisms of interaction underlying coupled chaotic oscillators.
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Affiliation(s)
- Federico Malizia
- Dipartimento di Fisica ed Astronomia, Università di Catania, Catania, Italy
- Network Science Institute, Northeastern University London, London, E1W 1LP, UK
| | - Alessandra Corso
- Department of Electrical Electronic and Computer Science Engineering, University of Catania, Catania, Italy
| | - Lucia Valentina Gambuzza
- Department of Electrical Electronic and Computer Science Engineering, University of Catania, Catania, Italy
| | - Giovanni Russo
- Department of Mathematics and Computer Science, University of Catania, Catania, Italy
| | - Vito Latora
- Dipartimento di Fisica ed Astronomia, Università di Catania, Catania, Italy
- School of Mathematical Sciences, Queen Mary University of London, London, E1 4NS, UK
- INFN, Catania, Italy
- Complexity Science Hub, Josefstäadter Strasse 39, A 1080, Vienna, Austria
| | - Mattia Frasca
- Department of Electrical Electronic and Computer Science Engineering, University of Catania, Catania, Italy.
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2
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Gallo L, Frasca M, Latora V, Russo G. Lack of practical identifiability may hamper reliable predictions in COVID-19 epidemic models. SCIENCE ADVANCES 2022; 8:eabg5234. [PMID: 35044820 PMCID: PMC8769547 DOI: 10.1126/sciadv.abg5234] [Citation(s) in RCA: 9] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/12/2023]
Abstract
Compartmental models are widely adopted to describe and predict the spreading of infectious diseases. The unknown parameters of these models need to be estimated from the data. Furthermore, when some of the model variables are not empirically accessible, as in the case of asymptomatic carriers of coronavirus disease 2019 (COVID-19), they have to be obtained as an outcome of the model. Here, we introduce a framework to quantify how the uncertainty in the data affects the determination of the parameters and the evolution of the unmeasured variables of a given model. We illustrate how the method is able to characterize different regimes of identifiability, even in models with few compartments. Last, we discuss how the lack of identifiability in a realistic model for COVID-19 may prevent reliable predictions of the epidemic dynamics.
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Affiliation(s)
- Luca Gallo
- Department of Physics and Astronomy, University of Catania, Catania 95125, Italy
- INFN Sezione di Catania, Via S. Sofia, 64, Catania 95125, Italy
| | - Mattia Frasca
- Department of Electrical, Electronics and Computer Science Engineering, University of Catania, Catania 95125, Italy
- Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti,” Consiglio Nazionale delle Ricerche (IASI-CNR), 00185 Roma 00185, Italy
- Corresponding author.
| | - Vito Latora
- Department of Physics and Astronomy, University of Catania, Catania 95125, Italy
- INFN Sezione di Catania, Via S. Sofia, 64, Catania 95125, Italy
- School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK
- Complexity Science Hub Vienna, A-1080 Vienna, Austria
| | - Giovanni Russo
- Department of Mathematics and Computer Science, University of Catania, Catania 95125, Italy
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3
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Kumbhari A, Rose D, Lee PP, Kim PS. A minimal model of T cell avidity may identify subtherapeutic vaccine schedules. Math Biosci 2021; 334:108556. [PMID: 33539903 DOI: 10.1016/j.mbs.2021.108556] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/08/2020] [Revised: 01/24/2021] [Accepted: 01/25/2021] [Indexed: 11/17/2022]
Abstract
T cells protect the body from cancer by recognising tumour-associated antigens. Recognising these antigens depends on multiple factors, one of which is T cell avidity, i.e., the total interaction strength between a T cell and a cancer cell. While both high- and low-avidity T cells can kill cancer cells, durable anti-cancer immune responses require the selection of high-avidity T cells. Previous experimentation with anti-cancer vaccines, however, has shown that most vaccines elicit low-avidity T cells. Optimising vaccine schedules may remedy this by preferentially selecting high-avidity T cells. Here, we use mathematical modelling to develop a simple, phenomenological model of avidity selection that may identify vaccine schedules that disproportionately favour low-avidity T cells. We calibrate our model to our prior, more complex model, and then validate it against several experimental data sets. We find that the sensitivity of the model's parameters change with vaccine dosage, which allows us to use a patient's data and clinical history to screen for suitable vaccine strategies.
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Affiliation(s)
- Adarsh Kumbhari
- School of Mathematics and Statistics, University of Sydney, Sydney, NSW, Australia
| | - Danya Rose
- School of Mathematics and Statistics, University of Sydney, Sydney, NSW, Australia
| | - Peter P Lee
- Department of Immuno-Oncology, City of Hope and Beckman Research Institute, Duarte, CA, USA
| | - Peter S Kim
- School of Mathematics and Statistics, University of Sydney, Sydney, NSW, Australia.
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Dale R, Oswald S, Jalihal A, LaPorte MF, Fletcher DM, Hubbard A, Shiu SH, Nelson ADL, Bucksch A. Overcoming the Challenges to Enhancing Experimental Plant Biology With Computational Modeling. FRONTIERS IN PLANT SCIENCE 2021; 12:687652. [PMID: 34354723 PMCID: PMC8329482 DOI: 10.3389/fpls.2021.687652] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/29/2021] [Accepted: 06/01/2021] [Indexed: 05/10/2023]
Abstract
The study of complex biological systems necessitates computational modeling approaches that are currently underutilized in plant biology. Many plant biologists have trouble identifying or adopting modeling methods to their research, particularly mechanistic mathematical modeling. Here we address challenges that limit the use of computational modeling methods, particularly mechanistic mathematical modeling. We divide computational modeling techniques into either pattern models (e.g., bioinformatics, machine learning, or morphology) or mechanistic mathematical models (e.g., biochemical reactions, biophysics, or population models), which both contribute to plant biology research at different scales to answer different research questions. We present arguments and recommendations for the increased adoption of modeling by plant biologists interested in incorporating more modeling into their research programs. As some researchers find math and quantitative methods to be an obstacle to modeling, we provide suggestions for easy-to-use tools for non-specialists and for collaboration with specialists. This may especially be the case for mechanistic mathematical modeling, and we spend some extra time discussing this. Through a more thorough appreciation and awareness of the power of different kinds of modeling in plant biology, we hope to facilitate interdisciplinary, transformative research.
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Affiliation(s)
- Renee Dale
- Donald Danforth Plant Science Center, St. Louis, MO, United States
- *Correspondence: Renee Dale
| | - Scott Oswald
- Warnell School of Forestry and Natural Resources and Institute of Bioinformatics, University of Georgia, Athens, GA, United States
| | - Amogh Jalihal
- Department of Systems Biology, Harvard Medical School, Boston, MA, United States
| | - Mary-Francis LaPorte
- Department of Plant Sciences, University of California, Davis, Davis, CA, United States
| | - Daniel M. Fletcher
- Bioengineering Sciences Research Group, Department of Mechanical Engineering, School of Engineering, Faculty of Engineering and Physical Sciences, University of Southampton, Southampton, United Kingdom
| | - Allen Hubbard
- Donald Danforth Plant Science Center, St. Louis, MO, United States
| | - Shin-Han Shiu
- Department of Plant Biology and Department of Computational Mathematics, Science, and Engineering, Michigan State University, East Lansing, MI, United States
| | | | - Alexander Bucksch
- Warnell School of Forestry and Natural Resources and Institute of Bioinformatics, University of Georgia, Athens, GA, United States
- Department of Plant Biology, University of Georgia, Athens, GA, United States
- Institute of Bioinformatics, University of Georgia, Athens, GA, United States
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5
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Tsiantis N, Banga JR. Using optimal control to understand complex metabolic pathways. BMC Bioinformatics 2020; 21:472. [PMID: 33087041 PMCID: PMC7579911 DOI: 10.1186/s12859-020-03808-8] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/12/2020] [Accepted: 10/13/2020] [Indexed: 12/16/2022] Open
Abstract
BACKGROUND Optimality principles have been used to explain the structure and behavior of living matter at different levels of organization, from basic phenomena at the molecular level, up to complex dynamics in whole populations. Most of these studies have assumed a single-criteria approach. Such optimality principles have been justified from an evolutionary perspective. In the context of the cell, previous studies have shown how dynamics of gene expression in small metabolic models can be explained assuming that cells have developed optimal adaptation strategies. Most of these works have considered rather simplified representations, such as small linear pathways, or reduced networks with a single branching point, and a single objective for the optimality criteria. RESULTS Here we consider the extension of this approach to more realistic scenarios, i.e. biochemical pathways of arbitrary size and structure. We first show that exploiting optimality principles for these networks poses great challenges due to the complexity of the associated optimal control problems. Second, in order to surmount such challenges, we present a computational framework which has been designed with scalability and efficiency in mind, including mechanisms to avoid the most common pitfalls. Third, we illustrate its performance with several case studies considering the central carbon metabolism of S. cerevisiae and B. subtilis. In particular, we consider metabolic dynamics during nutrient shift experiments. CONCLUSIONS We show how multi-objective optimal control can be used to predict temporal profiles of enzyme activation and metabolite concentrations in complex metabolic pathways. Further, we also show how to consider general cost/benefit trade-offs. In this study we have considered metabolic pathways, but this computational framework can also be applied to analyze the dynamics of other complex pathways, such as signal transduction or gene regulatory networks.
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Affiliation(s)
- Nikolaos Tsiantis
- Bioprocess Engineering Group, Spanish National Research Council, IIM-CSIC, C/Eduardo Cabello 6, 36208 Vigo, Spain
- Department of Chemical Engineering, University of Vigo, 36310 Vigo, Spain
| | - Julio R. Banga
- Bioprocess Engineering Group, Spanish National Research Council, IIM-CSIC, C/Eduardo Cabello 6, 36208 Vigo, Spain
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Rybiński M, Möller S, Sunnåker M, Lormeau C, Stelling J. TopoFilter: a MATLAB package for mechanistic model identification in systems biology. BMC Bioinformatics 2020; 21:34. [PMID: 31996136 PMCID: PMC6990465 DOI: 10.1186/s12859-020-3343-y] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/29/2019] [Accepted: 01/08/2020] [Indexed: 12/27/2022] Open
Abstract
Background To develop mechanistic dynamic models in systems biology, one often needs to identify all (or minimal) representations of the biological processes that are consistent with experimental data, out of a potentially large set of hypothetical mechanisms. However, a simple enumeration of all alternatives becomes quickly intractable when the number of model parameters grows. Selecting appropriate dynamic models out of a large ensemble of models, taking the uncertainty in our biological knowledge and in the experimental data into account, is therefore a key current problem in systems biology. Results The TopoFilter package addresses this problem in a heuristic and automated fashion by implementing the previously described topological filtering method for Bayesian model selection. It includes a core heuristic for searching the space of submodels of a parametrized model, coupled with a sampling-based exploration of the parameter space. Recent developments of the method allow to balance exhaustiveness and speed of the model space search, to efficiently re-sample parameters, to parallelize the search, and to use custom scoring functions. We use a theoretical example to motivate these features and then demonstrate TopoFilter’s applicability for a yeast signaling network with more than 250’000 possible model structures. Conclusions TopoFilter is a flexible software framework that makes Bayesian model selection and reduction efficient and scalable to network models of a complexity that represents contemporary problems in, for example, cell signaling. TopoFilter is open-source, available under the GPL-3.0 license at https://gitlab.com/csb.ethz/TopoFilter. It includes installation instructions, a quickstart guide, a description of all package options, and multiple examples.
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Affiliation(s)
- Mikołaj Rybiński
- Department of Biosystems Science and Engineering and SIB Swiss Institute of Bioinformatics, ETH Zurich, Mattenstr. 26, Basel, 4058, Switzerland.,ID Scientific IT Services, ETH Zurich, Zurich, 8092, Switzerland
| | - Simon Möller
- Department of Biosystems Science and Engineering and SIB Swiss Institute of Bioinformatics, ETH Zurich, Mattenstr. 26, Basel, 4058, Switzerland
| | - Mikael Sunnåker
- Department of Biosystems Science and Engineering and SIB Swiss Institute of Bioinformatics, ETH Zurich, Mattenstr. 26, Basel, 4058, Switzerland
| | - Claude Lormeau
- Department of Biosystems Science and Engineering and SIB Swiss Institute of Bioinformatics, ETH Zurich, Mattenstr. 26, Basel, 4058, Switzerland.,Life Science Zurich Ph.D. program "Systems Biology", Zurich, 8092, Switzerland
| | - Jörg Stelling
- Department of Biosystems Science and Engineering and SIB Swiss Institute of Bioinformatics, ETH Zurich, Mattenstr. 26, Basel, 4058, Switzerland.
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Pitt JA, Banga JR. Parameter estimation in models of biological oscillators: an automated regularised estimation approach. BMC Bioinformatics 2019; 20:82. [PMID: 30770736 PMCID: PMC6377730 DOI: 10.1186/s12859-019-2630-y] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/18/2018] [Accepted: 01/14/2019] [Indexed: 12/24/2022] Open
Abstract
BACKGROUND Dynamic modelling is a core element in the systems biology approach to understanding complex biosystems. Here, we consider the problem of parameter estimation in models of biological oscillators described by deterministic nonlinear differential equations. These problems can be extremely challenging due to several common pitfalls: (i) a lack of prior knowledge about parameters (i.e. massive search spaces), (ii) convergence to local optima (due to multimodality of the cost function), (iii) overfitting (fitting the noise instead of the signal) and (iv) a lack of identifiability. As a consequence, the use of standard estimation methods (such as gradient-based local ones) will often result in wrong solutions. Overfitting can be particularly problematic, since it produces very good calibrations, giving the impression of an excellent result. However, overfitted models exhibit poor predictive power. Here, we present a novel automated approach to overcome these pitfalls. Its workflow makes use of two sequential optimisation steps incorporating three key algorithms: (1) sampling strategies to systematically tighten the parameter bounds reducing the search space, (2) efficient global optimisation to avoid convergence to local solutions, (3) an advanced regularisation technique to fight overfitting. In addition, this workflow incorporates tests for structural and practical identifiability. RESULTS We successfully evaluate this novel approach considering four difficult case studies regarding the calibration of well-known biological oscillators (Goodwin, FitzHugh-Nagumo, Repressilator and a metabolic oscillator). In contrast, we show how local gradient-based approaches, even if used in multi-start fashion, are unable to avoid the above-mentioned pitfalls. CONCLUSIONS Our approach results in more efficient estimations (thanks to the bounding strategy) which are able to escape convergence to local optima (thanks to the global optimisation approach). Further, the use of regularisation allows us to avoid overfitting, resulting in more generalisable calibrated models (i.e. models with greater predictive power).
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Affiliation(s)
- Jake Alan Pitt
- (Bio)Process Engineering Group, IIM-CSIC, Eduardo Cabello 6, Vigo, 36208 Spain
- RWTH Aachen University, Faculty of Medicine, Joint Research Centre for Computational Biomedicine (JRC-COMBINE), Aachen, Germany
| | - Julio R. Banga
- (Bio)Process Engineering Group, IIM-CSIC, Eduardo Cabello 6, Vigo, 36208 Spain
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Yvinec R, Crépieux P, Reiter E, Poupon A, Clément F. Advances in computational modeling approaches of pituitary gonadotropin signaling. Expert Opin Drug Discov 2018; 13:799-813. [DOI: 10.1080/17460441.2018.1501025] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/23/2023]
Affiliation(s)
- Romain Yvinec
- PRC, INRA, CNRS, IFCE, Université de Tours, Nouzilly, France
| | | | - Eric Reiter
- PRC, INRA, CNRS, IFCE, Université de Tours, Nouzilly, France
| | - Anne Poupon
- PRC, INRA, CNRS, IFCE, Université de Tours, Nouzilly, France
| | - Frédérique Clément
- Inria, Université Paris-Saclay, Palaiseau, France
- LMS, Ecole Polytechnique, CNRS, Université Paris-Saclay, Palaiseau, France
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9
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Widmer LA, Stelling J. Bridging intracellular scales by mechanistic computational models. Curr Opin Biotechnol 2018; 52:17-24. [PMID: 29486391 DOI: 10.1016/j.copbio.2018.02.005] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2017] [Accepted: 02/11/2018] [Indexed: 12/31/2022]
Abstract
The impact of intracellular spatial organization beyond classical compartments on processes such as cell signaling is increasingly recognized. A quantitative, mechanistic understanding of cellular systems therefore needs to account for different scales in at least three coordinates: time, molecular abundances, and space. Mechanistic mathematical models may span all these scales, but corresponding multi-scale models need to resolve mechanistic details on small scales while maintaining computational tractability for larger ones. This typically results in models that combine different levels of description: from a microscopic representation of chemical reactions up to continuum dynamics in space and time. We highlight recent progress in bridging these model classes and outline current challenges in multi-scale models such as active transport and dynamic geometries.
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Affiliation(s)
- Lukas Andreas Widmer
- Department of Biosystems Science and Engineering and Swiss Institute of Bioinformatics, ETH Zürich, Basel, Switzerland; Systems Biology PhD Program, Life Science Zurich Graduate School, Zurich, Switzerland
| | - Jörg Stelling
- Department of Biosystems Science and Engineering and Swiss Institute of Bioinformatics, ETH Zürich, Basel, Switzerland.
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Loiben AM, Soueid-Baumgarten S, Kopyto RF, Bhattacharya D, Kim JC, Cosgrove BD. Data-Modeling Identifies Conflicting Signaling Axes Governing Myoblast Proliferation and Differentiation Responses to Diverse Ligand Stimuli. Cell Mol Bioeng 2017; 10:433-450. [PMID: 31719871 DOI: 10.1007/s12195-017-0508-5] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/22/2017] [Accepted: 08/27/2017] [Indexed: 01/03/2023] Open
Abstract
Introduction Skeletal muscle tissue development and regeneration relies on the proliferation, maturation and fusion of muscle progenitor cells (myoblasts), which arise transiently from muscle stem cells (satellite cells). Following muscle damage, myoblasts proliferate and differentiate in response to temporally-varying inflammatory cytokines, growth factors, and extracellular matrix cues, which stimulate a shared network of intracellular signaling pathways. Here we present an integrated data-modeling approach to elucidate synergies and antagonisms among proliferation and differentiation signaling axes in myoblasts stimulated by regeneration-associated ligands. Methods We treated mouse primary myoblasts in culture with combinations of eight regeneration-associated growth factors and cytokines in mixtures that induced additive, synergistic, and antagonistic effects on myoblast proliferation and differentiation responses. For these combinatorial stimuli, we measured the activation dynamics of seven signal transduction pathways using multiplexed phosphoprotein assays and scored proliferation and differentiation responses based on expression of myogenic commitment factors to assemble a cue-signaling-response data compendium. We interrogated the relationship between these signals and responses by partial least-squares (PLS) regression modeling. Results Partial least-squares data-modeling accurately predicted response outcomes in cross-validation on the training compendium (cumulative R 2 = 0.96). The PLS model highlighted signaling axes that distinctly govern myoblast proliferation (MEK-ERK, Stat3) and differentiation (JNK) in response to these combinatorial cues, and we confirmed these signal-response associations with small molecule perturbations. Unexpectedly, we observed that a negative feedback circuit involving the phosphatase DUSP6/MKP-3 auto-regulates MEK-ERK signaling in myoblasts. Conclusion This data-modeling approach identified conflicting signaling axes that underlie muscle progenitor cell proliferation and differentiation.
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Affiliation(s)
- Alexander M Loiben
- 1Meinig School of Biomedical Engineering, Cornell University, Ithaca, NY 14853 USA
| | | | - Ruth F Kopyto
- 2Biological Sciences, College of Agriculture and Life Sciences, Cornell University, Ithaca, NY 14853 USA
| | - Debadrita Bhattacharya
- 3Graduate Field of Biochemistry, Molecular and Cell Biology, Department of Molecular Biology and Genetics, Cornell University, Ithaca, NY 14853 USA
| | - Joseph C Kim
- 1Meinig School of Biomedical Engineering, Cornell University, Ithaca, NY 14853 USA
| | - Benjamin D Cosgrove
- 1Meinig School of Biomedical Engineering, Cornell University, Ithaca, NY 14853 USA
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Villaverde AF, Barreiro A, Papachristodoulou A. Structural Identifiability of Dynamic Systems Biology Models. PLoS Comput Biol 2016; 12:e1005153. [PMID: 27792726 PMCID: PMC5085250 DOI: 10.1371/journal.pcbi.1005153] [Citation(s) in RCA: 91] [Impact Index Per Article: 11.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/06/2016] [Accepted: 09/19/2016] [Indexed: 12/20/2022] Open
Abstract
A powerful way of gaining insight into biological systems is by creating a nonlinear differential equation model, which usually contains many unknown parameters. Such a model is called structurally identifiable if it is possible to determine the values of its parameters from measurements of the model outputs. Structural identifiability is a prerequisite for parameter estimation, and should be assessed before exploiting a model. However, this analysis is seldom performed due to the high computational cost involved in the necessary symbolic calculations, which quickly becomes prohibitive as the problem size increases. In this paper we show how to analyse the structural identifiability of a very general class of nonlinear models by extending methods originally developed for studying observability. We present results about models whose identifiability had not been previously determined, report unidentifiabilities that had not been found before, and show how to modify those unidentifiable models to make them identifiable. This method helps prevent problems caused by lack of identifiability analysis, which can compromise the success of tasks such as experiment design, parameter estimation, and model-based optimization. The procedure is called STRIKE-GOLDD (STRuctural Identifiability taKen as Extended-Generalized Observability with Lie Derivatives and Decomposition), and it is implemented in a MATLAB toolbox which is available as open source software. The broad applicability of this approach facilitates the analysis of the increasingly complex models used in systems biology and other areas.
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Affiliation(s)
- Alejandro F. Villaverde
- Department of Engineering Science, University of Oxford, Oxford, United Kingdom
- Department of Systems & Control Engineering, University of Vigo, Vigo, Spain
- * E-mail:
| | - Antonio Barreiro
- Department of Systems & Control Engineering, University of Vigo, Vigo, Spain
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