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Kravtsova N, Chamberlin HM, Dawes AT. Efficient parameter generation for constrained models using MCMC. Sci Rep 2023; 13:16285. [PMID: 37770498 PMCID: PMC10539337 DOI: 10.1038/s41598-023-43433-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/12/2023] [Accepted: 09/23/2023] [Indexed: 09/30/2023] Open
Abstract
Mathematical models of complex systems rely on parameter values to produce a desired behavior. As mathematical and computational models increase in complexity, it becomes correspondingly difficult to find parameter values that satisfy system constraints. We propose a Markov Chain Monte Carlo (MCMC) approach for the problem of constrained model parameter generation by designing a Markov chain that efficiently explores a model's parameter space. We demonstrate the use of our proposed methodology to analyze responses of a newly constructed bistability-constrained model of protein phosphorylation to perturbations in the underlying protein network. Our results suggest that parameter generation for constrained models using MCMC provides powerful tools for modeling-aided analysis of complex natural processes.
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Affiliation(s)
- Natalia Kravtsova
- Department of Mathematics, The Ohio State University, Columbus, OH, USA
| | - Helen M Chamberlin
- Department of Molecular Genetics, The Ohio State University, Columbus, OH, USA
| | - Adriana T Dawes
- Department of Mathematics, The Ohio State University, Columbus, OH, USA.
- Department of Molecular Genetics, The Ohio State University, Columbus, OH, USA.
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2
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Linden NJ, Kramer B, Rangamani P. Bayesian parameter estimation for dynamical models in systems biology. PLoS Comput Biol 2022; 18:e1010651. [PMID: 36269772 PMCID: PMC9629650 DOI: 10.1371/journal.pcbi.1010651] [Citation(s) in RCA: 13] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/19/2022] [Revised: 11/02/2022] [Accepted: 10/12/2022] [Indexed: 11/07/2022] Open
Abstract
Dynamical systems modeling, particularly via systems of ordinary differential equations, has been used to effectively capture the temporal behavior of different biochemical components in signal transduction networks. Despite the recent advances in experimental measurements, including sensor development and '-omics' studies that have helped populate protein-protein interaction networks in great detail, modeling in systems biology lacks systematic methods to estimate kinetic parameters and quantify associated uncertainties. This is because of multiple reasons, including sparse and noisy experimental measurements, lack of detailed molecular mechanisms underlying the reactions, and missing biochemical interactions. Additionally, the inherent nonlinearities with respect to the states and parameters associated with the system of differential equations further compound the challenges of parameter estimation. In this study, we propose a comprehensive framework for Bayesian parameter estimation and complete quantification of the effects of uncertainties in the data and models. We apply these methods to a series of signaling models of increasing mathematical complexity. Systematic analysis of these dynamical systems showed that parameter estimation depends on data sparsity, noise level, and model structure, including the existence of multiple steady states. These results highlight how focused uncertainty quantification can enrich systems biology modeling and enable additional quantitative analyses for parameter estimation.
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Affiliation(s)
- Nathaniel J. Linden
- Department of Mechanical and Aerospace Engineering, University of California San Diego, San Diego, California, United States of America
| | - Boris Kramer
- Department of Mechanical and Aerospace Engineering, University of California San Diego, San Diego, California, United States of America
- * E-mail: (BK); (PR)
| | - Padmini Rangamani
- Department of Mechanical and Aerospace Engineering, University of California San Diego, San Diego, California, United States of America
- * E-mail: (BK); (PR)
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3
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Renardy M, Joslyn LR, Millar JA, Kirschner DE. To Sobol or not to Sobol? The effects of sampling schemes in systems biology applications. Math Biosci 2021; 337:108593. [PMID: 33865847 PMCID: PMC8184610 DOI: 10.1016/j.mbs.2021.108593] [Citation(s) in RCA: 13] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/03/2021] [Revised: 03/21/2021] [Accepted: 03/22/2021] [Indexed: 12/13/2022]
Abstract
Computational and mathematical models in biology rely heavily on the parameters that characterize them. However, robust estimates for their values are typically elusive and thus a large parameter space becomes necessary for model study, particularly to make translationally impactful predictions. Sampling schemes exploring parameter spaces for models are used for a variety of purposes in systems biology, including model calibration and sensitivity analysis. Typically, random sampling is used; however, when models have a high number of unknown parameters or the models are highly complex, computational cost becomes an important factor. This issue can be reduced through the use of efficient sampling schemes such as Latin hypercube sampling (LHS) and Sobol sequences. In this work, we compare and contrast three sampling schemes - random sampling, LHS, and Sobol sequences - for the purposes of performing both parameter sensitivity analysis and model calibration. In addition, we apply these analyses to different types of computational and mathematical models of varying complexity: a simple ODE model, a complex ODE model, and an agent-based model. In general, the sampling scheme had little effect when used for calibration efforts, but when applied to sensitivity analyses, Sobol sequences exhibited faster convergence. While the observed benefit to convergence is relatively small, Sobol sequences are computationally less expensive to compute than LHS samples and also have the benefit of being deterministic, which allows for better reproducibility of results.
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Affiliation(s)
- Marissa Renardy
- Department of Microbiology and Immunology, University of Michigan Medical School, Ann Arbor, MI, United States
| | - Louis R Joslyn
- Department of Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor, MI, United States
| | - Jess A Millar
- Department of Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor, MI, United States
| | - Denise E Kirschner
- Department of Microbiology and Immunology, University of Michigan Medical School, Ann Arbor, MI, United States; Department of Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor, MI, United States.
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4
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Jacob B, Drawert B, Yi TM, Petzold L. An arbitrary Lagrangian Eulerian smoothed particle hydrodynamics (ALE-SPH) method with a boundary volume fraction formulation for fluid-structure interaction. ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2021; 128:274-289. [PMID: 34040286 PMCID: PMC8143034 DOI: 10.1016/j.enganabound.2021.04.006] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
We present a new weakly-compressible smoothed particle hydrodynamics (SPH) method capable of modeling non-slip fixed and moving wall boundary conditions. The formulation combines a boundary volume fraction (BVF) wall approach with the transport-velocity SPH method. The resulting method, named SPH-BVF, offers detection of arbitrarily shaped solid walls on-the-fly, with small computational overhead due to its local formulation. This simple framework is capable of solving problems that are difficult or infeasible for standard SPH, namely flows subject to large shear stresses or at moderate Reynolds numbers, and mass transfer in deformable boundaries. In addition, the method extends the transport-velocity formulation to reaction-diffusion transport of mass in Newtonian fluids and linear elastic solids, which is common in biological structures. Taken together, the SPH-BVF method provides a good balance of simplicity and versatility, while avoiding some of the standard obstacles associated with SPH: particle penetration at the boundaries, tension instabilities and anisotropic particle alignments, that hamper SPH from being applied to complex problems such as fluid-structure interaction in a biological system.
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Affiliation(s)
- Bruno Jacob
- Department of Mechanical Engineering, University of California-Santa Barbara, Santa Barbara, California, 93106, USA
| | - Brian Drawert
- Department of Computer Science, University of North Carolina at Asheville, Asheville, North Carolina, 28804, USA
| | - Tau-Mu Yi
- Department of Molecular, Cellular, and Developmental Biology, University of California-Santa Barbara, Santa Barbara, California 93106, USA
| | - Linda Petzold
- Department of Mechanical Engineering, University of California-Santa Barbara, Santa Barbara, California, 93106, USA
- Department of Computer Science, University of California-Santa Barbara, Santa Barbara, California, 93106, USA
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5
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Paun LM, Husmeier D. Markov chain Monte Carlo with Gaussian processes for fast parameter estimation and uncertainty quantification in a 1D fluid-dynamics model of the pulmonary circulation. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING 2021; 37:e3421. [PMID: 33249755 PMCID: PMC7901000 DOI: 10.1002/cnm.3421] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 03/19/2020] [Revised: 11/07/2020] [Accepted: 11/18/2020] [Indexed: 06/12/2023]
Abstract
The past few decades have witnessed an explosive synergy between physics and the life sciences. In particular, physical modelling in medicine and physiology is a topical research area. The present work focuses on parameter inference and uncertainty quantification in a 1D fluid-dynamics model for quantitative physiology: the pulmonary blood circulation. The practical challenge is the estimation of the patient-specific biophysical model parameters, which cannot be measured directly. In principle this can be achieved based on a comparison between measured and predicted data. However, predicting data requires solving a system of partial differential equations (PDEs), which usually have no closed-form solution, and repeated numerical integrations as part of an adaptive estimation procedure are computationally expensive. In the present article, we demonstrate how fast parameter estimation combined with sound uncertainty quantification can be achieved by a combination of statistical emulation and Markov chain Monte Carlo (MCMC) sampling. We compare a range of state-of-the-art MCMC algorithms and emulation strategies, and assess their performance in terms of their accuracy and computational efficiency. The long-term goal is to develop a method for reliable disease prognostication in real time, and our work is an important step towards an automatic clinical decision support system.
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Affiliation(s)
- L. Mihaela Paun
- School of Mathematics and StatisticsUniversity of GlasgowGlasgowUK
| | - Dirk Husmeier
- School of Mathematics and StatisticsUniversity of GlasgowGlasgowUK
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6
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Valdez AR, Rocha BM, da Fonseca Façanha JM, de Souza AVO, Pérez-Gramatges A, Chapiro G, Santos RWD. Foam-Assisted Water–Gas Flow Parameters: From Core-Flood Experiment to Uncertainty Quantification and Sensitivity Analysis. Transp Porous Media 2021. [DOI: 10.1007/s11242-021-01550-0] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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7
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Zhang Y, Young LS. DNN-assisted statistical analysis of a model of local cortical circuits. Sci Rep 2020; 10:20139. [PMID: 33208805 PMCID: PMC7674455 DOI: 10.1038/s41598-020-76770-3] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/13/2020] [Accepted: 10/20/2020] [Indexed: 01/27/2023] Open
Abstract
In neuroscience, computational modeling is an effective way to gain insight into cortical mechanisms, yet the construction and analysis of large-scale network models—not to mention the extraction of underlying principles—are themselves challenging tasks, due to the absence of suitable analytical tools and the prohibitive costs of systematic numerical exploration of high-dimensional parameter spaces. In this paper, we propose a data-driven approach assisted by deep neural networks (DNN). The idea is to first discover certain input-output relations, and then to leverage this information and the superior computation speeds of the well-trained DNN to guide parameter searches and to deduce theoretical understanding. To illustrate this novel approach, we used as a test case a medium-size network of integrate-and-fire neurons intended to model local cortical circuits. With the help of an accurate yet extremely efficient DNN surrogate, we revealed the statistics of model responses, providing a detailed picture of model behavior. The information obtained is both general and of a fundamental nature, with direct application to neuroscience. Our results suggest that the methodology proposed can be scaled up to larger and more complex biological networks when used in conjunction with other techniques of biological modeling.
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Affiliation(s)
- Yaoyu Zhang
- School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSC and Qing Yuan Research Institute, Shanghai Jiao Tong University, Shanghai, 200240, China.
| | - Lai-Sang Young
- School of Mathematics and School of Natural Sciences, Institute for Advanced Study, Princeton, NJ, 08540, USA. .,Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA.
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8
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Makrygiorgos G, Maggioni GM, Mesbah A. Surrogate modeling for fast uncertainty quantification: Application to 2D population balance models. Comput Chem Eng 2020. [DOI: 10.1016/j.compchemeng.2020.106814] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/01/2022]
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9
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Hazarika S, Li H, Wang KC, Shen HW, Chou CS. NNVA: Neural Network Assisted Visual Analysis of Yeast Cell Polarization Simulation. IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS 2020; 26:34-44. [PMID: 31425114 DOI: 10.1109/tvcg.2019.2934591] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
Complex computational models are often designed to simulate real-world physical phenomena in many scientific disciplines. However, these simulation models tend to be computationally very expensive and involve a large number of simulation input parameters, which need to be analyzed and properly calibrated before the models can be applied for real scientific studies. We propose a visual analysis system to facilitate interactive exploratory analysis of high-dimensional input parameter space for a complex yeast cell polarization simulation. The proposed system can assist the computational biologists, who designed the simulation model, to visually calibrate the input parameters by modifying the parameter values and immediately visualizing the predicted simulation outcome without having the need to run the original expensive simulation for every instance. Our proposed visual analysis system is driven by a trained neural network-based surrogate model as the backend analysis framework. In this work, we demonstrate the advantage of using neural networks as surrogate models for visual analysis by incorporating some of the recent advances in the field of uncertainty quantification, interpretability and explainability of neural network-based models. We utilize the trained network to perform interactive parameter sensitivity analysis of the original simulation as well as recommend optimal parameter configurations using the activation maximization framework of neural networks. We also facilitate detail analysis of the trained network to extract useful insights about the simulation model, learned by the network, during the training process. We performed two case studies, and discovered multiple new parameter configurations, which can trigger high cell polarization results in the original simulation model. We evaluated our results by comparing with the original simulation model outcomes as well as the findings from previous parameter analysis performed by our experts.
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Renardy M, Hult C, Evans S, Linderman JJ, Kirschner DE. Global sensitivity analysis of biological multi-scale models. CURRENT OPINION IN BIOMEDICAL ENGINEERING 2019; 11:109-116. [PMID: 32864523 DOI: 10.1016/j.cobme.2019.09.012] [Citation(s) in RCA: 19] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/27/2022]
Abstract
Mathematical models of biological systems need to both reflect and manage the inherent complexities of biological phenomena. Through their versatility and ability to capture behavior at multiple scales, multi-scale models offer a valuable approach. Due to the typically nonlinear and stochastic nature of multi-scale models as well as unknown parameter values, various types of uncertainty are present; thus, effective assessment and quantification of such uncertainty through sensitivity analysis is important. In this review, we discuss global sensitivity analysis in the context of multi-scale and multi-compartment models and highlight its value in model development and analysis. We present an overview of sensitivity analysis methods, approaches for extending such methods to a multi-scale setting, and examples of how sensitivity analysis can inform model reduction. Through schematics and references to past work, we aim to emphasize the advantages and usefulness of such techniques.
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Affiliation(s)
- Marissa Renardy
- University of Michigan Medical School, Department of Microbiology and Immunology
| | - Caitlin Hult
- University of Michigan Medical School, Department of Microbiology and Immunology
- University of Michigan, Department of Chemical Engineering
| | - Stephanie Evans
- University of Michigan Medical School, Department of Microbiology and Immunology
| | | | - Denise E Kirschner
- University of Michigan Medical School, Department of Microbiology and Immunology
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11
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Cosi FG, Giese W, Neubert W, Luther S, Chamakuri N, Parlitz U, Falcke M. Multiscale Modeling of Dyadic Structure-Function Relation in Ventricular Cardiac Myocytes. Biophys J 2019; 117:2409-2419. [PMID: 31635789 PMCID: PMC6990380 DOI: 10.1016/j.bpj.2019.09.023] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/31/2019] [Revised: 09/11/2019] [Accepted: 09/16/2019] [Indexed: 01/20/2023] Open
Abstract
Cardiovascular disease is often related to defects of subcellular components in cardiac myocytes, specifically in the dyadic cleft, which include changes in cleft geometry and channel placement. Modeling of these pathological changes requires both spatially resolved cleft as well as whole cell level descriptions. We use a multiscale model to create dyadic structure-function relationships to explore the impact of molecular changes on whole cell electrophysiology and calcium cycling. This multiscale model incorporates stochastic simulation of individual L-type calcium channels and ryanodine receptor channels, spatially detailed concentration dynamics in dyadic clefts, rabbit membrane potential dynamics, and a system of partial differential equations for myoplasmic and lumenal free Ca2+ and Ca2+-binding molecules in the bulk of the cell. We found action potential duration, systolic, and diastolic [Ca2+] to respond most sensitively to changes in L-type calcium channel current. The ryanodine receptor channel cluster structure inside dyadic clefts was found to affect all biomarkers investigated. The shape of clusters observed in experiments by Jayasinghe et al. and channel density within the cluster (characterized by mean occupancy) showed the strongest correlation to the effects on biomarkers.
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Affiliation(s)
- Filippo G Cosi
- Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany; Georg-August-Universität Göttingen, Institute for the Dynamics of Complex Systems, Göttingen, Germany; DZHK (German Center for Cardiovascular Research), Partner Site Göttingen, Göttingen, Germany
| | - Wolfgang Giese
- Max Delbrück Center for Molecular Medicine in the Helmholtz Association, Berlin, Germany
| | - Wilhelm Neubert
- Max Delbrück Center for Molecular Medicine in the Helmholtz Association, Berlin, Germany
| | - Stefan Luther
- Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany; Georg-August-Universität Göttingen, Institute for the Dynamics of Complex Systems, Göttingen, Germany; DZHK (German Center for Cardiovascular Research), Partner Site Göttingen, Göttingen, Germany
| | - Nagaiah Chamakuri
- Institute of Applied Mathematics, University of Hohenheim, Stuttgart, Germany
| | - Ulrich Parlitz
- Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany; Georg-August-Universität Göttingen, Institute for the Dynamics of Complex Systems, Göttingen, Germany; DZHK (German Center for Cardiovascular Research), Partner Site Göttingen, Göttingen, Germany
| | - Martin Falcke
- Max Delbrück Center for Molecular Medicine in the Helmholtz Association, Berlin, Germany; DZHK (German Center for Cardiovascular Research), Partner Site Berlin, Berlin, Germany; Department of Physics, Humboldt University Berlin, Germany.
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12
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Paulson JA, Martin-Casas M, Mesbah A. Fast uncertainty quantification for dynamic flux balance analysis using non-smooth polynomial chaos expansions. PLoS Comput Biol 2019; 15:e1007308. [PMID: 31469832 PMCID: PMC6742419 DOI: 10.1371/journal.pcbi.1007308] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/14/2019] [Revised: 09/12/2019] [Accepted: 07/31/2019] [Indexed: 01/05/2023] Open
Abstract
We present a novel surrogate modeling method that can be used to accelerate the solution of uncertainty quantification (UQ) problems arising in nonlinear and non-smooth models of biological systems. In particular, we focus on dynamic flux balance analysis (DFBA) models that couple intracellular fluxes, found from the solution of a constrained metabolic network model of the cellular metabolism, to the time-varying nature of the extracellular substrate and product concentrations. DFBA models are generally computationally expensive and present unique challenges to UQ, as they entail dynamic simulations with discrete events that correspond to switches in the active set of the solution of the constrained intracellular model. The proposed non-smooth polynomial chaos expansion (nsPCE) method is an extension of traditional PCE that can effectively capture singularities in the DFBA model response due to the occurrence of these discrete events. The key idea in nsPCE is to use a model of the singularity time to partition the parameter space into two elements on which the model response behaves smoothly. Separate PCE models are then fit in both elements using a basis-adaptive sparse regression approach that is known to scale well with respect to the number of uncertain parameters. We demonstrate the effectiveness of nsPCE on a DFBA model of an E. coli monoculture that consists of 1075 reactions and 761 metabolites. We first illustrate how traditional PCE is unable to handle problems of this level of complexity. We demonstrate that over 800-fold savings in computational cost of uncertainty propagation and Bayesian estimation of parameters in the substrate uptake kinetics can be achieved by using the nsPCE surrogates in place of the full DFBA model simulations. We then investigate the scalability of the nsPCE method by utilizing it for global sensitivity analysis and maximum a posteriori estimation in a synthetic metabolic network problem with a larger number of parameters related to both intracellular and extracellular quantities.
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Affiliation(s)
- Joel A. Paulson
- Department of Chemical and Biomolecular Engineering, University of California, Berkeley, Berkeley, California, United States of America
| | - Marc Martin-Casas
- Department of Chemical and Biomolecular Engineering, University of California, Berkeley, Berkeley, California, United States of America
| | - Ali Mesbah
- Department of Chemical and Biomolecular Engineering, University of California, Berkeley, Berkeley, California, United States of America
- * E-mail:
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