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Padoan A, Forni F, Sepulchre R. Balanced truncation for model reduction of biological oscillators. BIOLOGICAL CYBERNETICS 2021; 115:383-395. [PMID: 34382116 PMCID: PMC8382660 DOI: 10.1007/s00422-021-00888-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 10/28/2020] [Accepted: 07/14/2021] [Indexed: 06/13/2023]
Abstract
Model reduction is a central problem in mathematical biology. Reduced order models enable modeling of a biological system at different levels of complexity and the quantitative analysis of its properties, like sensitivity to parameter variations and resilience to exogenous perturbations. However, available model reduction methods often fail to capture a diverse range of nonlinear behaviors observed in biology, such as multistability and limit cycle oscillations. The paper addresses this need using differential analysis. This approach leads to a nonlinear enhancement of classical balanced truncation for biological systems whose behavior is not restricted to the stability of a single equilibrium. Numerical results suggest that the proposed framework may be relevant to the approximation of classical models of biological systems.
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Affiliation(s)
- Alberto Padoan
- Department of Engineering, University of Cambridge, Cambridge, UK
| | - Fulvio Forni
- Department of Engineering, University of Cambridge, Cambridge, UK
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Przedborski M, Sharon D, Chan S, Kohandel M. A mean-field approach for modeling the propagation of perturbations in biochemical reaction networks. Eur J Pharm Sci 2021; 165:105919. [PMID: 34175448 DOI: 10.1016/j.ejps.2021.105919] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/26/2021] [Revised: 05/17/2021] [Accepted: 06/20/2021] [Indexed: 12/12/2022]
Abstract
Often, the time evolution of a biochemical reaction network is crucial for determining the effects of combining multiple pharmaceuticals. Here we illustrate a mathematical framework for modeling the dominant temporal behaviour of a complicated molecular pathway or biochemical reaction network in response to an arbitrary perturbation, such as resulting from the administration of a therapeutic agent. The method enables the determination of the temporal evolution of a target protein as the perturbation propagates through its regulatory network. The mathematical approach is particularly useful when the experimental data that is available for characterizing or parameterizing the regulatory network is limited or incomplete. To illustrate the method, we consider the examples of the regulatory networks for the target proteins c-Myc and Chop, which play an important role in venetoclax resistance in acute myeloid leukemia. First we show how the networks that regulate each target protein can be reduced to a mean-field model by identifying the distinct effects that groups of proteins in the regulatory network have on the target protein. Then we show how limited protein-level data can be used to further simplify the mean-field model to pinpoint the dominant effects of the network perturbation on the target protein. This enables a further reduction in the number of parameters in the model. The result is an ordinary differential equation model that captures the temporal evolution of the expression of a target protein when one or more proteins in its regulatory network have been perturbed. Finally, we show how the dominant effects predicted by the mathematical model agree with RNA sequencing data for the regulatory proteins comprising the molecular network, despite the model not having a priori knowledge of this data. Thus, while the approach gives a simplified model for the expression of the target protein, it allows for the interpretation of the effects of the perturbation on the regulatory network itself. This method can be easily extended to sets of target proteins to model components of a larger systems biology model, and provides an approach for partially integrating RNA sequencing data and protein expression data. Moreover, it is a general approach that can be used to study drug effects on specific protein(s) in any disease or condition.
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Affiliation(s)
- Michelle Przedborski
- Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada.
| | - David Sharon
- Princess Margaret Cancer Centre, Toronto, ON, Canada
| | - Steven Chan
- Princess Margaret Cancer Centre, Toronto, ON, Canada
| | - Mohammad Kohandel
- Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada
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Vijayakumar S, Conway M, Lió P, Angione C. Seeing the wood for the trees: a forest of methods for optimization and omic-network integration in metabolic modelling. Brief Bioinform 2019; 19:1218-1235. [PMID: 28575143 DOI: 10.1093/bib/bbx053] [Citation(s) in RCA: 20] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/13/2017] [Indexed: 11/13/2022] Open
Abstract
Metabolic modelling has entered a mature phase with dozens of methods and software implementations available to the practitioner and the theoretician. It is not easy for a modeller to be able to see the wood (or the forest) for the trees. Driven by this analogy, we here present a 'forest' of principal methods used for constraint-based modelling in systems biology. This provides a tree-based view of methods available to prospective modellers, also available in interactive version at http://modellingmetabolism.net, where it will be kept updated with new methods after the publication of the present manuscript. Our updated classification of existing methods and tools highlights the most promising in the different branches, with the aim to develop a vision of how existing methods could hybridize and become more complex. We then provide the first hands-on tutorial for multi-objective optimization of metabolic models in R. We finally discuss the implementation of multi-view machine learning approaches in poly-omic integration. Throughout this work, we demonstrate the optimization of trade-offs between multiple metabolic objectives, with a focus on omic data integration through machine learning. We anticipate that the combination of a survey, a perspective on multi-view machine learning and a step-by-step R tutorial should be of interest for both the beginner and the advanced user.
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Affiliation(s)
| | - Max Conway
- Computer Laboratory, University of Cambridge, UK
| | - Pietro Lió
- Computer Laboratory, University of Cambridge, UK
| | - Claudio Angione
- Department of Computer Science and Information Systems, Teesside University, UK
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Snowden TJ, van der Graaf PH, Tindall MJ. Methods of Model Reduction for Large-Scale Biological Systems: A Survey of Current Methods and Trends. Bull Math Biol 2017; 79:1449-1486. [PMID: 28656491 PMCID: PMC5498684 DOI: 10.1007/s11538-017-0277-2] [Citation(s) in RCA: 56] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/10/2016] [Accepted: 03/30/2017] [Indexed: 01/31/2023]
Abstract
Complex models of biochemical reaction systems have become increasingly common in the systems biology literature. The complexity of such models can present a number of obstacles for their practical use, often making problems difficult to intuit or computationally intractable. Methods of model reduction can be employed to alleviate the issue of complexity by seeking to eliminate those portions of a reaction network that have little or no effect upon the outcomes of interest, hence yielding simplified systems that retain an accurate predictive capacity. This review paper seeks to provide a brief overview of a range of such methods and their application in the context of biochemical reaction network models. To achieve this, we provide a brief mathematical account of the main methods including timescale exploitation approaches, reduction via sensitivity analysis, optimisation methods, lumping, and singular value decomposition-based approaches. Methods are reviewed in the context of large-scale systems biology type models, and future areas of research are briefly discussed.
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Affiliation(s)
- Thomas J Snowden
- Department of Mathematics and Statistics, University of Reading, Reading, RG6 6AX, UK.,Certara QSP, University of Kent Innovation Centre, Canterbury, CT2 7FG, UK
| | - Piet H van der Graaf
- Certara QSP, University of Kent Innovation Centre, Canterbury, CT2 7FG, UK.,Leiden Academic Centre for Drug Research, Universiteit Leiden, Leiden, 2333 CC, Netherlands
| | - Marcus J Tindall
- Department of Mathematics and Statistics, University of Reading, Reading, RG6 6AX, UK. .,The Institute for Cardiovascular and Metabolic Research (ICMR), University of Reading, Reading, RG6 6AX, UK.
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Carapezza G, Umeton R, Costanza J, Angione C, Stracquadanio G, Papini A, Lió P, Nicosia G. Efficient behavior of photosynthetic organelles via Pareto optimality, identifiability, and sensitivity analysis. ACS Synth Biol 2013; 2:274-88. [PMID: 23654280 DOI: 10.1021/sb300102k] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
In this work, we develop methodologies for analyzing and cross comparing metabolic models. We investigate three important metabolic networks to discuss the complexity of biological organization of organisms, modeling, and system properties. In particular, we analyze these metabolic networks because of their biotechnological and basic science importance: the photosynthetic carbon metabolism in a general leaf, the Rhodobacter spheroides bacterium, and the Chlamydomonas reinhardtii alga. We adopt single- and multi-objective optimization algorithms to maximize the CO 2 uptake rate and the production of metabolites of industrial interest or for ecological purposes. We focus both on the level of genes (e.g., finding genetic manipulations to increase the production of one or more metabolites) and on finding concentration enzymes for improving the CO 2 consumption. We find that R. spheroides is able to absorb an amount of CO 2 until 57.452 mmol h (-1) gDW (-1) , while C. reinhardtii obtains a maximum of 6.7331. We report that the Pareto front analysis proves extremely useful to compare different organisms, as well as providing the possibility to investigate them with the same framework. By using the sensitivity and robustness analysis, our framework identifies the most sensitive and fragile components of the biological systems we take into account, allowing us to compare their models. We adopt the identifiability analysis to detect functional relations among enzymes; we observe that RuBisCO, GAPDH, and FBPase belong to the same functional group, as suggested also by the sensitivity analysis.
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Affiliation(s)
- Giovanni Carapezza
- Department of Mathematics and
Computer Science, University of Catania, Italy
| | - Renato Umeton
- University of
Rome “La
Sapienza”, S. Andrea Hospital, and Department of Biological
Engineering, Massachussets Institute of Technology, United States
| | - Jole Costanza
- Department of Mathematics and
Computer Science, University of Catania, Italy
| | | | | | - Alessio Papini
- Department of Evolutionary Biology, University of Florence, Italy
| | - Pietro Lió
- Computer
Laboratory, University of Cambridge, U.K
| | - Giuseppe Nicosia
- Department of Mathematics and
Computer Science, University of Catania, Italy
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Lampre I, Marignier JL, Mirdamadi-Esfahani M, Pernot P, Archirel P, Mostafavi M. Oxidation of Bromide Ions by Hydroxyl Radicals: Spectral Characterization of the Intermediate BrOH•–. J Phys Chem A 2013; 117:877-87. [DOI: 10.1021/jp310759u] [Citation(s) in RCA: 31] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/07/2023]
Affiliation(s)
- Isabelle Lampre
- Laboratoire
de Chimie Physique, UMR 8000 CNRS/Université Paris-Sud, Faculté des
Sciences d’Orsay, Bât. 349, 91405 Orsay Cedex, France
| | - Jean-Louis Marignier
- Laboratoire
de Chimie Physique, UMR 8000 CNRS/Université Paris-Sud, Faculté des
Sciences d’Orsay, Bât. 349, 91405 Orsay Cedex, France
| | - Malaknaz Mirdamadi-Esfahani
- Laboratoire
de Chimie Physique, UMR 8000 CNRS/Université Paris-Sud, Faculté des
Sciences d’Orsay, Bât. 349, 91405 Orsay Cedex, France
| | - Pascal Pernot
- Laboratoire
de Chimie Physique, UMR 8000 CNRS/Université Paris-Sud, Faculté des
Sciences d’Orsay, Bât. 349, 91405 Orsay Cedex, France
| | - Pierre Archirel
- Laboratoire
de Chimie Physique, UMR 8000 CNRS/Université Paris-Sud, Faculté des
Sciences d’Orsay, Bât. 349, 91405 Orsay Cedex, France
| | - Mehran Mostafavi
- Laboratoire
de Chimie Physique, UMR 8000 CNRS/Université Paris-Sud, Faculté des
Sciences d’Orsay, Bât. 349, 91405 Orsay Cedex, France
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Costanza J, Carapezza G, Angione C, Lió P, Nicosia G. Robust design of microbial strains. Bioinformatics 2012; 28:3097-104. [PMID: 23044547 DOI: 10.1093/bioinformatics/bts590] [Citation(s) in RCA: 47] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/22/2023] Open
Abstract
MOTIVATION Metabolic engineering algorithms provide means to optimize a biological process leading to the improvement of a biotechnological interesting molecule. Therefore, it is important to understand how to act in a metabolic pathway in order to have the best results in terms of productions. In this work, we present a computational framework that searches for optimal and robust microbial strains that are able to produce target molecules. Our framework performs three tasks: it evaluates the parameter sensitivity of the microbial model, searches for the optimal genetic or fluxes design and finally calculates the robustness of the microbial strains. We are capable to combine the exploration of species, reactions, pathways and knockout parameter spaces with the Pareto-optimality principle. RESULTS Our framework provides also theoretical and practical guidelines for design automation. The statistical cross comparison of our new optimization procedures, performed with respect to currently widely used algorithms for bacteria (e.g. Escherichia coli) over different multiple functions, reveals good performances over a variety of biotechnological products. AVAILABILITY http://www.dmi.unict.it/nicosia/pathDesign.html. CONTACT nicosia@dmi.unict.it or pl219@cam.ac.uk SUPPLEMENTARY INFORMATION Supplementary data are available at Bioinformatics online.
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Affiliation(s)
- Jole Costanza
- Department of Mathematics and Computer Science, University of Catania, Viale A. Doria 6, 95125 Catania, Italy
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Multi-objective Optimisation, Sensitivity and Robustness Analysis in FBA Modelling. COMPUTATIONAL METHODS IN SYSTEMS BIOLOGY 2012. [DOI: 10.1007/978-3-642-33636-2_9] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/19/2023]
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Zhang HX, Goutsias J. Reducing experimental variability in variance-based sensitivity analysis of biochemical reaction systems. J Chem Phys 2011; 134:114105. [PMID: 21428605 DOI: 10.1063/1.3563539] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Sensitivity analysis is a valuable task for assessing the effects of biological variability on cellular behavior. Available techniques require knowledge of nominal parameter values, which cannot be determined accurately due to experimental uncertainty typical to problems of systems biology. As a consequence, the practical use of existing sensitivity analysis techniques may be seriously hampered by the effects of unpredictable experimental variability. To address this problem, we propose here a probabilistic approach to sensitivity analysis of biochemical reaction systems that explicitly models experimental variability and effectively reduces the impact of this type of uncertainty on the results. The proposed approach employs a recently introduced variance-based method to sensitivity analysis of biochemical reaction systems [Zhang et al., J. Chem. Phys. 134, 094101 (2009)] and leads to a technique that can be effectively used to accommodate appreciable levels of experimental variability. We discuss three numerical techniques for evaluating the sensitivity indices associated with the new method, which include Monte Carlo estimation, derivative approximation, and dimensionality reduction based on orthonormal Hermite approximation. By employing a computational model of the epidermal growth factor receptor signaling pathway, we demonstrate that the proposed technique can greatly reduce the effect of experimental variability on variance-based sensitivity analysis results. We expect that, in cases of appreciable experimental variability, the new method can lead to substantial improvements over existing sensitivity analysis techniques.
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Affiliation(s)
- Hong-Xuan Zhang
- Procter & Gamble Co., Miami Valley Innovation Center, Cincinnati, Ohio 45253, USA
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