1
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Xiang X, Zhou J, Deng Y, Yang X. Identifying the generator matrix of a stationary Markov chain using partially observable data. CHAOS (WOODBURY, N.Y.) 2024; 34:023132. [PMID: 38386908 DOI: 10.1063/5.0156458] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/29/2023] [Accepted: 01/23/2024] [Indexed: 02/24/2024]
Abstract
Given that most states in real-world systems are inaccessible, it is critical to study the inverse problem of an irreversibly stationary Markov chain regarding how a generator matrix can be identified using minimal observations. The hitting-time distribution of an irreversibly stationary Markov chain is first generalized from a reversible case. The hitting-time distribution is then decoded via the taboo rate, and the results show remarkably that under mild conditions, the generator matrix of a reversible Markov chain or a specific case of irreversibly stationary ones can be identified by utilizing observations from all leaves and two adjacent states in each cycle. Several algorithms are proposed for calculating the generator matrix accurately, and numerical examples are presented to confirm their validity and efficiency. An application to neurophysiology is provided to demonstrate the applicability of such statistics to real-world data. This means that partially observable data can be used to identify the generator matrix of a stationary Markov chain.
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Affiliation(s)
- Xuyan Xiang
- School of Mathematics and Physics Science, Hunan University of Arts and Science, Changde 415000, China
- College of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China
| | - Jieming Zhou
- College of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China
| | - Yingchun Deng
- College of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China
| | - Xiangqun Yang
- College of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China
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2
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Felice JI, Milesi V, Fabricius G. Exploration of the Parameter Space of an Ion Channel Kinetic Model by a Markov-Chain-Based Methodology. J Chem Inf Model 2024; 64:555-562. [PMID: 38159289 DOI: 10.1021/acs.jcim.3c01811] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/03/2024]
Abstract
In this work, we propose a methodology based on Monte Carlo Markov chains to explore the parameter space of kinetic models for ion channels. The methodology allows the detection of potential parameter sets of a model that are compatible with experimentally obtained whole-cell currents, which could remain hidden when methods focus on obtaining the parameters that provide the best fit. To show its implementation and utility, we considered a four-state kinetic model proposed in the literature to describe the activation of the voltage-gated proton channel (Hv1), Biophysical Journal, 2014, 107, 1564. In that work, a set of values for the rate transitions that describe the channel kinetics at different intracellular H+ concentration (pHi) were obtained by the Simplex method. With our approach, we find that, in fact, there is more than one parameter set for each pHi, which renders the same open probability temporal course within the experimental error margin for all of the considered voltages. The large differences that we obtained for the values of some rate constants among the different solutions show that there is more than one possible interpretation of this channel behavior as a function of pHi. We also simulated a proposed new experimental condition where it is possible to observe that different sets of parameters yield different results. Our study highlights the importance of a comprehensive analysis of parameter space in kinetic models and the utility of the proposed methodology for finding potential solutions.
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Affiliation(s)
- Juan I Felice
- UNLP, CONICET, Asociado CIC PBA, Facultad de Ciencias Exactas, Departamento de Ciencias Biológicas, Instituto de Estudios Inmunológicos y Fisiopatológicos (IIFP), La Plata 1900, Argentina
- CCT CONICET La Plata, Consejo Nacional de Investigaciones Científicas y Técnicas, La Plata 1900, Argentina
| | - Verónica Milesi
- UNLP, CONICET, Asociado CIC PBA, Facultad de Ciencias Exactas, Departamento de Ciencias Biológicas, Instituto de Estudios Inmunológicos y Fisiopatológicos (IIFP), La Plata 1900, Argentina
- CCT CONICET La Plata, Consejo Nacional de Investigaciones Científicas y Técnicas, La Plata 1900, Argentina
| | - Gabriel Fabricius
- CONICET and Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas (INIFTA), CC 16, Suc. 4, La Plata 1900, Argentina
- CCT CONICET La Plata, Consejo Nacional de Investigaciones Científicas y Técnicas, La Plata 1900, Argentina
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3
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Araujo R, Brumley D, Cursons J, Day K, Faria M, Flegg JA, Germano D, Hunt H, Hunter P, Jenner A, Johnston S, McCaw JM, Maini P, Miller C, Muskovic W, Osborne J, Pan M, Rajagopal V, Shahidi N, Siekmann I, Stumpf M, Zanca A. Frontiers of Mathematical Biology: A workshop honouring Professor Edmund Crampin. Math Biosci 2023; 359:109007. [PMID: 37062447 DOI: 10.1016/j.mbs.2023.109007] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/23/2023] [Revised: 03/28/2023] [Accepted: 04/02/2023] [Indexed: 04/18/2023]
Affiliation(s)
- Robyn Araujo
- School of Mathematical Sciences, Queensland University of Technology, Australia
| | - Douglas Brumley
- School of Mathematics and Statistics, The University of Melbourne, Australia
| | | | - Karen Day
- Bio21 Institute, The University of Melbourne, Australia
| | - Matthew Faria
- Department of Biomedical Engineering, The University of Melbourne, Australia
| | - Jennifer A Flegg
- School of Mathematics and Statistics, The University of Melbourne, Australia
| | - Domenic Germano
- School of Mathematics and Statistics, The University of Melbourne, Australia
| | - Hilary Hunt
- Department of Biology, University of Oxford, United Kingdom
| | - Peter Hunter
- Auckland Bioengineering Institute, University of Auckland, New Zealand
| | - Adrianne Jenner
- School of Mathematical Sciences, Queensland University of Technology, Australia
| | - Stuart Johnston
- School of Mathematics and Statistics, The University of Melbourne, Australia
| | - James M McCaw
- School of Mathematics and Statistics, The University of Melbourne, Australia; Melbourne School of Population and Global Health, The University of Melbourne, Australia.
| | - Philip Maini
- Mathematical Institute, University of Oxford, United Kingdom
| | - Claire Miller
- Auckland Bioengineering Institute, University of Auckland, New Zealand
| | | | - James Osborne
- School of Mathematics and Statistics, The University of Melbourne, Australia
| | - Michael Pan
- School of Mathematics and Statistics, The University of Melbourne, Australia
| | - Vijay Rajagopal
- Department of Biomedical Engineering, The University of Melbourne, Australia
| | - Niloofar Shahidi
- Auckland Bioengineering Institute, University of Auckland, New Zealand
| | - Ivo Siekmann
- School of Computer Science and Mathematics, Liverpool John Moores University, United Kingdom
| | - Michael Stumpf
- School of Mathematics and Statistics, The University of Melbourne, Australia; Melbourne Integrative Genomics, The University of Melbourne, Australia
| | - Adriana Zanca
- School of Mathematics and Statistics, The University of Melbourne, Australia
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4
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Grosu GF, Hopp AV, Moca VV, Bârzan H, Ciuparu A, Ercsey-Ravasz M, Winkel M, Linde H, Mureșan RC. The fractal brain: scale-invariance in structure and dynamics. Cereb Cortex 2023; 33:4574-4605. [PMID: 36156074 PMCID: PMC10110456 DOI: 10.1093/cercor/bhac363] [Citation(s) in RCA: 5] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/02/2022] [Revised: 08/09/2022] [Accepted: 08/10/2022] [Indexed: 11/12/2022] Open
Abstract
The past 40 years have witnessed extensive research on fractal structure and scale-free dynamics in the brain. Although considerable progress has been made, a comprehensive picture has yet to emerge, and needs further linking to a mechanistic account of brain function. Here, we review these concepts, connecting observations across different levels of organization, from both a structural and functional perspective. We argue that, paradoxically, the level of cortical circuits is the least understood from a structural point of view and perhaps the best studied from a dynamical one. We further link observations about scale-freeness and fractality with evidence that the environment provides constraints that may explain the usefulness of fractal structure and scale-free dynamics in the brain. Moreover, we discuss evidence that behavior exhibits scale-free properties, likely emerging from similarly organized brain dynamics, enabling an organism to thrive in an environment that shares the same organizational principles. Finally, we review the sparse evidence for and try to speculate on the functional consequences of fractality and scale-freeness for brain computation. These properties may endow the brain with computational capabilities that transcend current models of neural computation and could hold the key to unraveling how the brain constructs percepts and generates behavior.
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Affiliation(s)
- George F Grosu
- Department of Experimental and Theoretical Neuroscience, Transylvanian Institute of Neuroscience, Str. Ploiesti 33, 400157 Cluj-Napoca, Romania
- Faculty of Electronics, Telecommunications and Information Technology, Technical University of Cluj-Napoca, Str. Memorandumului 28, 400114 Cluj-Napoca, Romania
| | | | - Vasile V Moca
- Department of Experimental and Theoretical Neuroscience, Transylvanian Institute of Neuroscience, Str. Ploiesti 33, 400157 Cluj-Napoca, Romania
| | - Harald Bârzan
- Department of Experimental and Theoretical Neuroscience, Transylvanian Institute of Neuroscience, Str. Ploiesti 33, 400157 Cluj-Napoca, Romania
- Faculty of Electronics, Telecommunications and Information Technology, Technical University of Cluj-Napoca, Str. Memorandumului 28, 400114 Cluj-Napoca, Romania
| | - Andrei Ciuparu
- Department of Experimental and Theoretical Neuroscience, Transylvanian Institute of Neuroscience, Str. Ploiesti 33, 400157 Cluj-Napoca, Romania
- Faculty of Electronics, Telecommunications and Information Technology, Technical University of Cluj-Napoca, Str. Memorandumului 28, 400114 Cluj-Napoca, Romania
| | - Maria Ercsey-Ravasz
- Department of Experimental and Theoretical Neuroscience, Transylvanian Institute of Neuroscience, Str. Ploiesti 33, 400157 Cluj-Napoca, Romania
- Faculty of Physics, Babes-Bolyai University, Str. Mihail Kogalniceanu 1, 400084 Cluj-Napoca, Romania
| | - Mathias Winkel
- Merck KGaA, Frankfurter Straße 250, 64293 Darmstadt, Germany
| | - Helmut Linde
- Department of Experimental and Theoretical Neuroscience, Transylvanian Institute of Neuroscience, Str. Ploiesti 33, 400157 Cluj-Napoca, Romania
- Merck KGaA, Frankfurter Straße 250, 64293 Darmstadt, Germany
| | - Raul C Mureșan
- Department of Experimental and Theoretical Neuroscience, Transylvanian Institute of Neuroscience, Str. Ploiesti 33, 400157 Cluj-Napoca, Romania
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5
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Nieto Ramos A, Fenton FH, Cherry EM. Bayesian inference for fitting cardiac models to experiments: estimating parameter distributions using Hamiltonian Monte Carlo and approximate Bayesian computation. Med Biol Eng Comput 2023; 61:75-95. [PMID: 36322242 DOI: 10.1007/s11517-022-02685-y] [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: 04/15/2022] [Accepted: 10/02/2022] [Indexed: 01/07/2023]
Abstract
Customization of cardiac action potential models has become increasingly important with the recognition of patient-specific models and virtual patient cohorts as valuable predictive tools. Nevertheless, developing customized models by fitting parameters to data poses technical and methodological challenges: despite noise and variability associated with real-world datasets, traditional optimization methods produce a single "best-fit" set of parameter values. Bayesian estimation methods seek distributions of parameter values given the data by obtaining samples from the target distribution, but in practice widely known Bayesian algorithms like Markov chain Monte Carlo tend to be computationally inefficient and scale poorly with the dimensionality of parameter space. In this paper, we consider two computationally efficient Bayesian approaches: the Hamiltonian Monte Carlo (HMC) algorithm and the approximate Bayesian computation sequential Monte Carlo (ABC-SMC) algorithm. We find that both methods successfully identify distributions of model parameters for two cardiac action potential models using model-derived synthetic data and an experimental dataset from a zebrafish heart. Although both methods appear to converge to the same distribution family and are computationally efficient, HMC generally finds narrower marginal distributions, while ABC-SMC is less sensitive to the algorithmic settings including the prior distribution.
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Affiliation(s)
- Alejandro Nieto Ramos
- School of Mathematical Sciences, Rochester Institute of Technology, 1 Lomb Memorial Drive, 14623, Rochester, NY, USA.,Epilepsy Center, Neurological Institute, Cleveland Clinic, 9500 Euclid Avenue, Cleveland, OH, 44195, USA
| | - Flavio H Fenton
- School of Physics, Georgia Institute of Technology, 837 State Street NW, 30332, Atlanta, GA, USA
| | - Elizabeth M Cherry
- School of Computational Science and Engineering, Georgia Institute of Technology, 756 West Peachtree Street, 30308, Atlanta, GA, USA.
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6
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Moffett AS, Cui G, Thomas PJ, Hunt WD, McCarty NA, Westafer RS, Eckford AW. Permissive and nonpermissive channel closings in CFTR revealed by a factor graph inference algorithm. BIOPHYSICAL REPORTS 2022; 2:100083. [PMID: 36425670 PMCID: PMC9680790 DOI: 10.1016/j.bpr.2022.100083] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 05/14/2022] [Accepted: 10/13/2022] [Indexed: 06/16/2023]
Abstract
The closing of the gated ion channel in the cystic fibrosis transmembrane conductance regulator can be categorized as nonpermissive to reopening, which involves the unbinding of ADP or ATP, or permissive, which does not. Identifying the type of closing is of interest as interactions with nucleotides can be affected in mutants or by introducing agonists. However, all closings are electrically silent and difficult to differentiate. For single-channel patch-clamp traces, we show that the type of the closing can be accurately determined by an inference algorithm implemented on a factor graph, which we demonstrate using both simulated and lab-obtained patch-clamp traces.
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Affiliation(s)
- Alexander S. Moffett
- Department of Electrical Engineering and Computer Science, York University, Toronto, ON, Canada
| | - Guiying Cui
- Emory + Children’s Center for Cystic Fibrosis and Airways Disease Research, Emory University School of Medicine and Children’s Healthcare of Atlanta, Atlanta, Georgia
| | - Peter J. Thomas
- Department of Mathematics, Applied Mathematics, and Statistics, Case Western Reserve University, Cleveland, Ohio
| | - William D. Hunt
- School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia
| | - Nael A. McCarty
- Emory + Children’s Center for Cystic Fibrosis and Airways Disease Research, Emory University School of Medicine and Children’s Healthcare of Atlanta, Atlanta, Georgia
| | | | - Andrew W. Eckford
- Department of Electrical Engineering and Computer Science, York University, Toronto, ON, Canada
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7
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Münch JL, Paul F, Schmauder R, Benndorf K. Bayesian inference of kinetic schemes for ion channels by Kalman filtering. eLife 2022; 11:e62714. [PMID: 35506659 PMCID: PMC9342998 DOI: 10.7554/elife.62714] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/02/2020] [Accepted: 04/22/2022] [Indexed: 11/16/2022] Open
Abstract
Inferring adequate kinetic schemes for ion channel gating from ensemble currents is a daunting task due to limited information in the data. We address this problem by using a parallelized Bayesian filter to specify hidden Markov models for current and fluorescence data. We demonstrate the flexibility of this algorithm by including different noise distributions. Our generalized Kalman filter outperforms both a classical Kalman filter and a rate equation approach when applied to patch-clamp data exhibiting realistic open-channel noise. The derived generalization also enables inclusion of orthogonal fluorescence data, making unidentifiable parameters identifiable and increasing the accuracy of the parameter estimates by an order of magnitude. By using Bayesian highest credibility volumes, we found that our approach, in contrast to the rate equation approach, yields a realistic uncertainty quantification. Furthermore, the Bayesian filter delivers negligibly biased estimates for a wider range of data quality. For some data sets, it identifies more parameters than the rate equation approach. These results also demonstrate the power of assessing the validity of algorithms by Bayesian credibility volumes in general. Finally, we show that our Bayesian filter is more robust against errors induced by either analog filtering before analog-to-digital conversion or by limited time resolution of fluorescence data than a rate equation approach.
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Affiliation(s)
- Jan L Münch
- Institut für Physiologie II, Universitätsklinikum Jena, Friedrich Schiller University JenaJenaGermany
| | - Fabian Paul
- Department of Biochemistry and Molecular Biology, University of ChicagoChicagoUnited States
| | - Ralf Schmauder
- Institut für Physiologie II, Universitätsklinikum Jena, Friedrich Schiller University JenaJenaGermany
| | - Klaus Benndorf
- Institut für Physiologie II, Universitätsklinikum Jena, Friedrich Schiller University JenaJenaGermany
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8
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Jæger KH, Edwards AG, Giles WR, Tveito A. A computational method for identifying an optimal combination of existing drugs to repair the action potentials of SQT1 ventricular myocytes. PLoS Comput Biol 2021; 17:e1009233. [PMID: 34383746 PMCID: PMC8360568 DOI: 10.1371/journal.pcbi.1009233] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/16/2021] [Accepted: 07/01/2021] [Indexed: 01/26/2023] Open
Abstract
Mutations are known to cause perturbations in essential functional features of integral membrane proteins, including ion channels. Even restricted or point mutations can result in substantially changed properties of ion currents. The additive effect of these alterations for a specific ion channel can result in significantly changed properties of the action potential (AP). Both AP shortening and AP prolongation can result from known mutations, and the consequences can be life-threatening. Here, we present a computational method for identifying new drugs utilizing combinations of existing drugs. Based on the knowledge of theoretical effects of existing drugs on individual ion currents, our aim is to compute optimal combinations that can ‘repair’ the mutant AP waveforms so that the baseline AP-properties are restored. More specifically, we compute optimal, combined, drug concentrations such that the waveforms of the transmembrane potential and the cytosolic calcium concentration of the mutant cardiomyocytes (CMs) becomes as similar as possible to their wild type counterparts after the drug has been applied. In order to demonstrate the utility of this method, we address the question of computing an optimal drug for the short QT syndrome type 1 (SQT1). For the SQT1 mutation N588K, there are available data sets that describe the effect of various drugs on the mutated K+ channel. These published findings are the basis for our computational analysis which can identify optimal compounds in the sense that the AP of the mutant CMs resembles essential biomarkers of the wild type CMs. Using recently developed insights regarding electrophysiological properties among myocytes from different species, we compute optimal drug combinations for hiPSC-CMs, rabbit ventricular CMs and adult human ventricular CMs with the SQT1 mutation. Since the ‘composition’ of ion channels that form the AP is different for the three types of myocytes under consideration, so is the composition of the optimal drug. Poly-pharmacology (using multiple drugs to treat disease) has been proposed for improving cardiac anti-arrhythmic therapy for at least two decades. However, the specific arrhythmia contexts in which polytherapy is likely to be both safe and effective have remained elusive. Type 1 short QT syndrome (SQT1) is a rare form of cardiac arrhythmia that results from mutations to the human Ether-á-go-go Related Gene (hERG) potassium channel. Functionally, these mutations are remarkably consistent in that they permit the channel to open earlier during each heart beat. While hundreds of compounds are known to inhibit hERG channels, the specific effect of SQT1 mutations that allows for early channel opening also limits the ability of most of those compounds to correct SQT1 dysfunction. Here, we have applied a suite of ventricular cardiomyocyte computational models to ask whether polytherapy may offer a more effective therapeutic strategy in SQT1, and if so, what the likely characteristics of that strategy are. Our analyses suggest that simultaneous induction of late sodium current and partial hERG blockade offers a promising strategy. While no activators of late sodium current have been clinically approved, several experimental compounds are available and may provide a basis for interrogating this strategy. The method presented here can be used to compute optimal drug combinations provided that the effect of each drug on every relevant ion channel is known.
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MESH Headings
- Action Potentials/drug effects
- Amino Acid Substitution
- Animals
- Anti-Arrhythmia Agents/administration & dosage
- Arrhythmias, Cardiac/drug therapy
- Arrhythmias, Cardiac/genetics
- Arrhythmias, Cardiac/physiopathology
- Computational Biology
- Drug Combinations
- Drug Design
- Drug Therapy, Combination/methods
- ERG1 Potassium Channel/drug effects
- ERG1 Potassium Channel/genetics
- ERG1 Potassium Channel/physiology
- Heart Conduction System/abnormalities
- Heart Conduction System/physiopathology
- Heart Defects, Congenital/drug therapy
- Heart Defects, Congenital/genetics
- Heart Defects, Congenital/physiopathology
- Humans
- Induced Pluripotent Stem Cells/drug effects
- Induced Pluripotent Stem Cells/physiology
- Models, Cardiovascular
- Mutation, Missense
- Myocytes, Cardiac/drug effects
- Myocytes, Cardiac/physiology
- Rabbits
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Affiliation(s)
| | - Andrew G. Edwards
- Simula Research Laboratory, Oslo, Norway
- Department of Pharmacology, University of California, Davis, California United States of America
| | - Wayne R. Giles
- Simula Research Laboratory, Oslo, Norway
- Department of Physiology and Pharmacology, Faculty of Medicine, University of Calgary, Calgary, Canada
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Statistical Approach to Incorporating Experimental Variability into a Mathematical Model of the Voltage-Gated Na + Channel and Human Atrial Action Potential. Cells 2021; 10:cells10061516. [PMID: 34208565 PMCID: PMC8234464 DOI: 10.3390/cells10061516] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/08/2021] [Revised: 06/01/2021] [Accepted: 06/12/2021] [Indexed: 11/29/2022] Open
Abstract
The voltage-gated Na+ channel Nav1.5 is critical for normal cardiac myocyte excitability. Mathematical models have been widely used to study Nav1.5 function and link to a range of cardiac arrhythmias. There is growing appreciation for the importance of incorporating physiological heterogeneity observed even in a healthy population into mathematical models of the cardiac action potential. Here, we apply methods from Bayesian statistics to capture the variability in experimental measurements on human atrial Nav1.5 across experimental protocols and labs. This variability was used to define a physiological distribution for model parameters in a novel model formulation of Nav1.5, which was then incorporated into an existing human atrial action potential model. Model validation was performed by comparing the simulated distribution of action potential upstroke velocity measurements to experimental measurements from several different sources. Going forward, we hope to apply this approach to other major atrial ion channels to create a comprehensive model of the human atrial AP. We anticipate that such a model will be useful for understanding excitability at the population level, including variable drug response and penetrance of variants linked to inherited cardiac arrhythmia syndromes.
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10
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Pein F, Eltzner B, Munk A. Analysis of patchclamp recordings: model-free multiscale methods and software. EUROPEAN BIOPHYSICS JOURNAL : EBJ 2021; 50:187-209. [PMID: 33837454 PMCID: PMC8071803 DOI: 10.1007/s00249-021-01506-8] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 08/14/2020] [Revised: 01/02/2021] [Accepted: 01/25/2021] [Indexed: 12/27/2022]
Abstract
Analysis of patchclamp recordings is often a challenging issue. We give practical guidance how such recordings can be analyzed using the model-free multiscale idealization methodology JSMURF, JULES, and HILDE. We provide an operational manual how to use the accompanying software available as an R-package and as a graphical user interface. This includes selection of the right approach and tuning of parameters. We also discuss advantages and disadvantages of model-free approaches in comparison to hidden Markov model approaches and explain how they complement each other.
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Affiliation(s)
- Florian Pein
- Statistical Laboratory, DPMMS, University of Cambridge, Cambridge, UK.
| | - Benjamin Eltzner
- Institute for Mathematical Stochastics, Georg-August-University of Goettingen, Göttingen, Germany
| | - Axel Munk
- Institute for Mathematical Stochastics, Georg-August-University of Goettingen, Göttingen, Germany
- Max Planck Institute for Biophysical Chemistry, Göttingen, Germany
- Felix Bernstein Institute for Mathematical Statistics in the Biosciences, Göttingen, Germany
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11
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Pein F, Bartsch A, Steinem C, Munk A. Heterogeneous Idealization of Ion Channel Recordings - Open Channel Noise. IEEE Trans Nanobioscience 2020; 20:57-78. [PMID: 33052850 DOI: 10.1109/tnb.2020.3031202] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Abstract
We propose a new model-free segmentation method for idealizing ion channel recordings. This method is designed to deal with heterogeneity of measurement errors. This in particular applies to open channel noise which, in general, is particularly difficult to cope with for model-free approaches. Our methodology is able to deal with lowpass filtered data which provides a further computational challenge. To this end we propose a multiresolution testing approach, combined with local deconvolution to resolve the lowpass filter. Simulations and statistical theory confirm that the proposed idealization recovers the underlying signal very accurately at presence of heterogeneous noise, even when events are shorter than the filter length. The method is compared to existing approaches in computer experiments and on real data. We find that it is the only one which allows to identify openings of the PorB porine at two different temporal scales. An implementation is available as an R package.
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12
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13
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Hibbah EH, El Maroufy H, Fuchs C, Ziad T. An MCMC computational approach for a continuous time state-dependent regime switching diffusion process. J Appl Stat 2019; 47:1354-1374. [PMID: 35706700 PMCID: PMC9041985 DOI: 10.1080/02664763.2019.1677573] [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: 12/22/2018] [Accepted: 10/02/2019] [Indexed: 10/25/2022]
Abstract
State-dependent regime switching diffusion processes or hybrid switching diffusion (HSD) processes are hard to simulate with classical methods which leads us to adopt a Markov chain Monte Carlo (MCMC) Bayesian approach very convenient to estimate complicated models such as the HSD one. In the HSD, the diffusion component is dependent on the switching discrete hidden regimes and the transition rates of the regime switching are dependent on the diffusion observations. Since in reality phenomena are only observed in discrete times, data imputation is called for to create more observations so as to have good approximations for the density of the diffusion process. Three categories of entities will be computed in a Bayesian context: The latent imputed observations, the regime switching states, and the parameters of the models. The latent imputed data is updated at random time intervals in block using a Metropolis Hastings algorithm. The switching states are computed by an adaptation of a forward filtering backward smoothing algorithm to the HSD model. The parameters are estimated after prior specifications and conditional posterior densities formulation using Gibbs sampler or Metropolis Hastings algorithm.
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Affiliation(s)
- El Houcine Hibbah
- Department of Applied Mathematics, Faculty of Sciences and Technics, Sultan Mouly Slimane University, Morocco
| | - Hamid El Maroufy
- Department of Applied Mathematics, Faculty of Sciences and Technics, Sultan Mouly Slimane University, Morocco
| | - Christiane Fuchs
- Faculty of Business Administration and Economics, Bielefeld University, Bielefield, Germany
- Helmholtz Zentrum Munchen, German Research Center for Environmental Health GmbH, Institute of Computational Biology, Neuherberg, Germany
- Chair of Mathematical Modeling of Biology Systems, Technisché Universität München, Garching, Germany
| | - Taib Ziad
- Faculty of Mathematics, Chalmers University of Technology, Göteborg, Sweden
- Statistical Science Director at Astrazeneca, Sweden
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14
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Tavakoli M, Tsekouras K, Day R, Dunn KW, Pressé S. Quantitative Kinetic Models from Intravital Microscopy: A Case Study Using Hepatic Transport. J Phys Chem B 2019; 123:7302-7312. [PMID: 31298856 PMCID: PMC6857640 DOI: 10.1021/acs.jpcb.9b04729] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
The liver performs critical physiological functions, including metabolizing and removing substances, such as toxins and drugs, from the bloodstream. Hepatotoxicity itself is intimately linked to abnormal hepatic transport, and hepatotoxicity remains the primary reason drugs in development fail and approved drugs are withdrawn from the market. For this reason, we propose to analyze, across liver compartments, the transport kinetics of fluorescein-a fluorescent marker used as a proxy for drug molecules-using intravital microscopy data. To resolve the transport kinetics quantitatively from fluorescence data, we account for the effect that different liver compartments (with different chemical properties) have on fluorescein's emission rate. To do so, we develop ordinary differential equation transport models from the data where the kinetics is related to the observable fluorescence levels by "measurement parameters" that vary across different liver compartments. On account of the steep non-linearities in the kinetics and stochasticity inherent to the model, we infer kinetic and measurement parameters by generalizing the method of parameter cascades. For this application, the method of parameter cascades ensures fast and precise parameter estimates from noisy time traces.
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Affiliation(s)
- Meysam Tavakoli
- Department of Physics, Indiana University-Purdue University, Indianapolis, Indiana 46202, United States
| | | | - Richard Day
- Department of Cellular and Integrative Physiology, Indiana University School of Medicine, Indianapolis, Indiana 46202, United States
| | - Kenneth W. Dunn
- Department of Medicine and Biochemistry, Indiana University School of Medicine, Indianapolis, Indiana 46202, United States
| | - Steve Pressé
- Center for Biological Physics, Arizona State University, Tempe, Arizona 85287, United States
- School of Molecular Sciences, Arizona State University, Tempe, Arizona 85287, United States
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15
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Diehn M, Munk A, Rudolf D. Maximum likelihood estimation in hidden Markov models with inhomogeneous noise. ESAIM-PROBAB STAT 2019. [DOI: 10.1051/ps/2018017] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022]
Abstract
We consider parameter estimation in finite hidden state space Markov models with time-dependent inhomogeneous noise, where the inhomogeneity vanishes sufficiently fast. Based on the concept of asymptotic mean stationary processes we prove that the maximum likelihood and a quasi-maximum likelihood estimator (QMLE) are strongly consistent. The computation of the QMLE ignores the inhomogeneity, hence, is much simpler and robust. The theory is motivated by an example from biophysics and applied to a Poisson- and linear Gaussian model.
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16
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Tveito A, Maleckar MM, Lines GT. Computing Optimal Properties of Drugs Using Mathematical Models of Single Channel Dynamics. COMPUTATIONAL AND MATHEMATICAL BIOPHYSICS 2018. [DOI: 10.1515/cmb-2018-0004] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
AbstractSingle channel dynamics can be modeled using stochastic differential equations, and the dynamics of the state of the channel (e.g. open, closed, inactivated) can be represented using Markov models. Such models can also be used to represent the effect of mutations as well as the effect of drugs used to alleviate deleterious effects of mutations. Based on the Markov model and the stochastic models of the single channel, it is possible to derive deterministic partial differential equations (PDEs) giving the probability density functions (PDFs) of the states of the Markov model. In this study, we have analyzed PDEs modeling wild type (WT) channels, mutant channels (MT) and mutant channels for which a drug has been applied (MTD). Our aim is to show that it is possible to optimize the parameters of a given drug such that the solution of theMTD model is very close to that of the WT: the mutation’s effect is, theoretically, reduced significantly.We will present the mathematical framework underpinning this methodology and apply it to several examples. In particular, we will show that it is possible to use the method to, theoretically, improve the properties of some well-known existing drugs.
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Affiliation(s)
- Aslak Tveito
- 1Simula Research Laboratory, Norway and Department of Informatics, The University of Oslo,Oslo, Norway
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17
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Pein F, Tecuapetla-Gomez I, Schutte OM, Steinem C, Munk A. Fully Automatic Multiresolution Idealization for Filtered Ion Channel Recordings: Flickering Event Detection. IEEE Trans Nanobioscience 2018; 17:300-320. [PMID: 29994220 DOI: 10.1109/tnb.2018.2845126] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Abstract
We propose a new model-free segmentation method, JULES, which combines recent statistical multiresolution techniques with local deconvolution for idealization of ion channel recordings. The multiresolution criterion takes into account scales down to the sampling rate enabling the detection of flickering events, i.e., events on small temporal scales, even below the filter frequency. For such small scales the deconvolution step allows for a precise determination of dwell times and, in particular, of amplitude levels, a task which is not possible with common thresholding methods. This is confirmed theoretically and in a comprehensive simulation study. In addition, JULES can be applied as a preprocessing method for a refined hidden Markov analysis. Our new methodology allows us to show that gramicidin A flickering events have the same amplitude as the slow gating events. JULES is available as an R function jules in the package clampSeg.
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18
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Beattie KA, Hill AP, Bardenet R, Cui Y, Vandenberg JI, Gavaghan DJ, de Boer TP, Mirams GR. Sinusoidal voltage protocols for rapid characterisation of ion channel kinetics. J Physiol 2018; 596:1813-1828. [PMID: 29573276 PMCID: PMC5978315 DOI: 10.1113/jp275733] [Citation(s) in RCA: 43] [Impact Index Per Article: 7.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/19/2017] [Accepted: 02/19/2018] [Indexed: 12/21/2022] Open
Abstract
Key points Ion current kinetics are commonly represented by current–voltage relationships, time constant–voltage relationships and subsequently mathematical models fitted to these. These experiments take substantial time, which means they are rarely performed in the same cell. Rather than traditional square‐wave voltage clamps, we fitted a model to the current evoked by a novel sum‐of‐sinusoids voltage clamp that was only 8 s long. Short protocols that can be performed multiple times within a single cell will offer many new opportunities to measure how ion current kinetics are affected by changing conditions. The new model predicts the current under traditional square‐wave protocols well, with better predictions of underlying currents than literature models. The current under a novel physiologically relevant series of action potential clamps is predicted extremely well. The short sinusoidal protocols allow a model to be fully fitted to individual cells, allowing us to examine cell–cell variability in current kinetics for the first time.
Abstract Understanding the roles of ion currents is crucial to predict the action of pharmaceuticals and mutations in different scenarios, and thereby to guide clinical interventions in the heart, brain and other electrophysiological systems. Our ability to predict how ion currents contribute to cellular electrophysiology is in turn critically dependent on our characterisation of ion channel kinetics – the voltage‐dependent rates of transition between open, closed and inactivated channel states. We present a new method for rapidly exploring and characterising ion channel kinetics, applying it to the hERG potassium channel as an example, with the aim of generating a quantitatively predictive representation of the ion current. We fitted a mathematical model to currents evoked by a novel 8 second sinusoidal voltage clamp in CHO cells overexpressing hERG1a. The model was then used to predict over 5 minutes of recordings in the same cell in response to further protocols: a series of traditional square step voltage clamps, and also a novel voltage clamp comprising a collection of physiologically relevant action potentials. We demonstrate that we can make predictive cell‐specific models that outperform the use of averaged data from a number of different cells, and thereby examine which changes in gating are responsible for cell–cell variability in current kinetics. Our technique allows rapid collection of consistent and high quality data, from single cells, and produces more predictive mathematical ion channel models than traditional approaches. Ion current kinetics are commonly represented by current–voltage relationships, time constant–voltage relationships and subsequently mathematical models fitted to these. These experiments take substantial time, which means they are rarely performed in the same cell. Rather than traditional square‐wave voltage clamps, we fitted a model to the current evoked by a novel sum‐of‐sinusoids voltage clamp that was only 8 s long. Short protocols that can be performed multiple times within a single cell will offer many new opportunities to measure how ion current kinetics are affected by changing conditions. The new model predicts the current under traditional square‐wave protocols well, with better predictions of underlying currents than literature models. The current under a novel physiologically relevant series of action potential clamps is predicted extremely well. The short sinusoidal protocols allow a model to be fully fitted to individual cells, allowing us to examine cell–cell variability in current kinetics for the first time.
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Affiliation(s)
- Kylie A Beattie
- Computational Biology, Department of Computer Science, University of Oxford, Oxford, OX1 3QD, UK.,Division of Applied Regulatory Science, Office of Clinical Pharmacology, Office of Translational Sciences, Center for Drug Evaluation and Research, Food and Drug Administration, Silver Spring, MD, USA
| | - Adam P Hill
- Department of Molecular Cardiology and Biophysics, Victor Chang Cardiac Research Institute, Sydney, NSW, 2010, Australia.,St Vincent's Clinical School, UNSW Sydney, Darlinghurst, NSW, 2010, Australia
| | - Rémi Bardenet
- CNRS & CRIStAL, Université de Lille, 59651 Villeneuve d'Ascq, Lille, France
| | - Yi Cui
- Safety Evaluation and Risk Management, Global Clinical Safety and Pharmacovigilance, GlaxoSmithKline, Uxbridge, UB11 1BS, UK
| | - Jamie I Vandenberg
- Department of Molecular Cardiology and Biophysics, Victor Chang Cardiac Research Institute, Sydney, NSW, 2010, Australia.,St Vincent's Clinical School, UNSW Sydney, Darlinghurst, NSW, 2010, Australia
| | - David J Gavaghan
- Computational Biology, Department of Computer Science, University of Oxford, Oxford, OX1 3QD, UK
| | - Teun P de Boer
- Department of Medical Physiology, Division of Heart & Lungs, University Medical Center Utrecht, Utrecht, The Netherlands
| | - Gary R Mirams
- Centre for Mathematical Medicine & Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK
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Adams C, Stroberg W, DeFazio RA, Schnell S, Moenter SM. Gonadotropin-Releasing Hormone (GnRH) Neuron Excitability Is Regulated by Estradiol Feedback and Kisspeptin. J Neurosci 2018; 38:1249-1263. [PMID: 29263236 PMCID: PMC5792479 DOI: 10.1523/jneurosci.2988-17.2017] [Citation(s) in RCA: 28] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/18/2017] [Revised: 11/21/2017] [Accepted: 12/09/2017] [Indexed: 01/03/2023] Open
Abstract
Gonadotropin-releasing hormone (GnRH) neurons produce the central output controlling fertility and are regulated by steroid feedback. A switch from estradiol negative to positive feedback initiates the GnRH surge, ultimately triggering ovulation. This occurs on a daily basis in ovariectomized, estradiol-treated (OVX+E) mice; GnRH neurons are suppressed in the morning and activated in the afternoon. To test the hypotheses that estradiol and time of day signals alter GnRH neuron responsiveness to stimuli, GFP-identified GnRH neurons in brain slices from OVX+E or OVX female mice were recorded during the morning or afternoon. No differences were observed in baseline membrane potential. Current-clamp revealed GnRH neurons fired more action potentials in response to current injection during positive feedback relative to all other groups, which were not different from each other despite reports of differing ionic conductances. Kisspeptin increased GnRH neuron response in cells from OVX and OVX+E mice in the morning but not afternoon. Paradoxically, excitability in kisspeptin knock-out mice was similar to the maximum observed in control mice but was unchanged by time of day or estradiol. A mathematical model applying a Markov Chain Monte Carlo method to estimate probability distributions for estradiol- and time of day-dependent parameters was used to predict intrinsic properties underlying excitability changes. A single identifiable distribution of solutions accounted for similar GnRH neuron excitability in all groups other than positive feedback despite different underlying conductance properties; this was attributable to interdependence of voltage-gated potassium channel properties. In contrast, redundant solutions may explain positive feedback, perhaps indicative of the importance of this state for species survival.SIGNIFICANCE STATEMENT Infertility affects 15%-20% of couples; failure to ovulate is a common cause. Understanding how the brain controls ovulation is critical for new developments in both infertility treatment and contraception. Gonadotropin-releasing hormone (GnRH) neurons are the final common pathway for central neural control of ovulation. We studied how estradiol feedback regulates GnRH excitability, a key determinant of neural firing rate using laboratory and computational approaches. GnRH excitability is upregulated during positive feedback, perhaps driving increased neural firing rate at this time. Kisspeptin increased GnRH excitability and was essential for estradiol regulation of excitability. Modeling predicts that multiple combinations of changes to GnRH intrinsic conductances can produce the firing response in positive feedback, suggesting the brain has many ways to induce ovulation.
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Affiliation(s)
| | | | | | - Santiago Schnell
- Departments of Molecular and Integrative Physiology
- Computational Medicine and Bioinformatics
| | - Suzanne M Moenter
- Departments of Molecular and Integrative Physiology,
- Obstetrics and Gynecology, and
- Internal Medicine, University of Michigan, Ann Arbor, Michigan, 48109
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20
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Epstein M, Calderhead B, Girolami MA, Sivilotti LG. Bayesian Statistical Inference in Ion-Channel Models with Exact Missed Event Correction. Biophys J 2017; 111:333-348. [PMID: 27463136 DOI: 10.1016/j.bpj.2016.04.053] [Citation(s) in RCA: 23] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/22/2015] [Revised: 03/15/2016] [Accepted: 04/08/2016] [Indexed: 12/29/2022] Open
Abstract
The stochastic behavior of single ion channels is most often described as an aggregated continuous-time Markov process with discrete states. For ligand-gated channels each state can represent a different conformation of the channel protein or a different number of bound ligands. Single-channel recordings show only whether the channel is open or shut: states of equal conductance are aggregated, so transitions between them have to be inferred indirectly. The requirement to filter noise from the raw signal further complicates the modeling process, as it limits the time resolution of the data. The consequence of the reduced bandwidth is that openings or shuttings that are shorter than the resolution cannot be observed; these are known as missed events. Postulated models fitted using filtered data must therefore explicitly account for missed events to avoid bias in the estimation of rate parameters and therefore assess parameter identifiability accurately. In this article, we present the first, to our knowledge, Bayesian modeling of ion-channels with exact missed events correction. Bayesian analysis represents uncertain knowledge of the true value of model parameters by considering these parameters as random variables. This allows us to gain a full appreciation of parameter identifiability and uncertainty when estimating values for model parameters. However, Bayesian inference is particularly challenging in this context as the correction for missed events increases the computational complexity of the model likelihood. Nonetheless, we successfully implemented a two-step Markov chain Monte Carlo method that we called "BICME", which performs Bayesian inference in models of realistic complexity. The method is demonstrated on synthetic and real single-channel data from muscle nicotinic acetylcholine channels. We show that parameter uncertainty can be characterized more accurately than with maximum-likelihood methods. Our code for performing inference in these ion channel models is publicly available.
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Affiliation(s)
- Michael Epstein
- Department of Mathematics, Imperial College London, London, UK; CoMPLEX, University College London, London, UK
| | - Ben Calderhead
- Department of Mathematics, Imperial College London, London, UK.
| | - Mark A Girolami
- Department of Statistics, University of Warwick, Coventry, UK
| | - Lucia G Sivilotti
- Department of Neuroscience, Physiology & Pharmacology, University College London, London, UK
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21
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Hwang W, Lee IB, Hong SC, Hyeon C. Decoding Single Molecule Time Traces with Dynamic Disorder. PLoS Comput Biol 2016; 12:e1005286. [PMID: 28027304 PMCID: PMC5226833 DOI: 10.1371/journal.pcbi.1005286] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/11/2016] [Revised: 01/11/2017] [Accepted: 12/07/2016] [Indexed: 12/11/2022] Open
Abstract
Single molecule time trajectories of biomolecules provide glimpses into complex folding landscapes that are difficult to visualize using conventional ensemble measurements. Recent experiments and theoretical analyses have highlighted dynamic disorder in certain classes of biomolecules, whose dynamic pattern of conformational transitions is affected by slower transition dynamics of internal state hidden in a low dimensional projection. A systematic means to analyze such data is, however, currently not well developed. Here we report a new algorithm—Variational Bayes-double chain Markov model (VB-DCMM)—to analyze single molecule time trajectories that display dynamic disorder. The proposed analysis employing VB-DCMM allows us to detect the presence of dynamic disorder, if any, in each trajectory, identify the number of internal states, and estimate transition rates between the internal states as well as the rates of conformational transition within each internal state. Applying VB-DCMM algorithm to single molecule FRET data of H-DNA in 100 mM-Na+ solution, followed by data clustering, we show that at least 6 kinetic paths linking 4 distinct internal states are required to correctly interpret the duplex-triplex transitions of H-DNA. We have developed a new algorithm to better decode single molecule data with dynamic disorder. Our new algorithm, which represents a substantial improvement over other methodologies, can detect the presence of dynamic disorder in each trajectory and quantify the kinetic characteristics of underlying energy landscape. As a model system, we applied our algorithm to the single molecule FRET time traces of H-DNA. While duplex-triplex transitions of H-DNA are conventionally interpreted in terms of two-state kinetics, slowly varying dynamic patterns corresponding to hidden internal states can also be identified from the individual time traces. Our algorithm reveals that at least 4 distinct internal states are required to correctly interpret the data.
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Affiliation(s)
- Wonseok Hwang
- Korea Institute for Advanced Study, Seoul, Republic of Korea
| | - Il-Buem Lee
- Department of Physics, Korea University, Seoul, Republic of Korea
| | - Seok-Cheol Hong
- Korea Institute for Advanced Study, Seoul, Republic of Korea
- Department of Physics, Korea University, Seoul, Republic of Korea
| | - Changbong Hyeon
- Korea Institute for Advanced Study, Seoul, Republic of Korea
- * E-mail:
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22
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Boie S, Chen J, Sanderson MJ, Sneyd J. The relative contributions of store-operated and voltage-gated Ca 2+ channels to the control of Ca 2+ oscillations in airway smooth muscle. J Physiol 2016; 595:3129-3141. [PMID: 27502470 DOI: 10.1113/jp272996] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/24/2016] [Accepted: 08/03/2016] [Indexed: 02/02/2023] Open
Abstract
KEY POINTS Agonist-dependent oscillations in the concentration of free cytosolic calcium are a vital mechanism for the control of airway smooth muscle contraction and thus are a critical factor in airway hyper-responsiveness. Using a mathematical model, closely tied to experimental work, we show that the oscillations in membrane potential accompanying the calcium oscillations have no significant effect on the properties of the calcium oscillations. In addition, the model shows that calcium entry through store-operated calcium channels is critical for calcium oscillations, but calcium entry through voltage-gated channels has much less effect. The model predicts that voltage-gated channels are less important than store-operated channels in the control of airway smooth muscle tone. ABSTRACT Airway smooth muscle contraction is typically the key mechanism underlying airway hyper-responsiveness, and the strength of muscle contraction is determined by the frequency of oscillations of intracellular calcium (Ca2+ ) concentration. In airway smooth muscle cells, these Ca2+ oscillations are caused by cyclic Ca2+ release from the sarcoplasmic reticulum, although Ca2+ influx via plasma membrane channels is also necessary to sustain the oscillations over longer times. To assess the relative contributions of store-operated and voltage-gated Ca2+ channels to this Ca2+ influx, we generated a comprehensive mathematical model, based on experimental Ca2+ measurements in mouse precision-cut lung slices, to simulate Ca2+ oscillations and changes in membrane potential. Agonist-induced Ca2+ oscillations are accompanied by oscillations in membrane potential, although the membrane potential oscillations are too small to generate large Ca2+ currents through voltage-gated Ca2+ channels, and thus have little effect on the Ca2+ oscillations. Ca2+ entry through voltage-gated channels only becomes important when the cell is depolarized (e.g. by a high external K+ concentration). As a result, agonist-induced Ca2+ oscillations are critically dependent on Ca2+ entry through store-operated channels but do not depend strongly on Ca2+ entry though voltage-gated channels.
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Affiliation(s)
- Sebastian Boie
- Department of Mathematics, The University of Auckland, Auckland, New Zealand
| | - Jun Chen
- Department of Microbiology and Physiological Systems, University of Massachusetts Medical School, Worcester, MA, USA
| | - Michael J Sanderson
- Department of Microbiology and Physiological Systems, University of Massachusetts Medical School, Worcester, MA, USA
| | - James Sneyd
- Department of Mathematics, The University of Auckland, Auckland, New Zealand
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23
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Siekmann I, Fackrell M, Crampin EJ, Taylor P. Modelling modal gating of ion channels with hierarchical Markov models. Proc Math Phys Eng Sci 2016; 472:20160122. [PMID: 27616917 PMCID: PMC5014102 DOI: 10.1098/rspa.2016.0122] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/18/2016] [Accepted: 07/22/2016] [Indexed: 11/12/2022] Open
Abstract
Many ion channels spontaneously switch between different levels of activity. Although this behaviour known as modal gating has been observed for a long time it is currently not well understood. Despite the fact that appropriately representing activity changes is essential for accurately capturing time course data from ion channels, systematic approaches for modelling modal gating are currently not available. In this paper, we develop a modular approach for building such a model in an iterative process. First, stochastic switching between modes and stochastic opening and closing within modes are represented in separate aggregated Markov models. Second, the continuous-time hierarchical Markov model, a new modelling framework proposed here, then enables us to combine these components so that in the integrated model both mode switching as well as the kinetics within modes are appropriately represented. A mathematical analysis reveals that the behaviour of the hierarchical Markov model naturally depends on the properties of its components. We also demonstrate how a hierarchical Markov model can be parametrized using experimental data and show that it provides a better representation than a previous model of the same dataset. Because evidence is increasing that modal gating reflects underlying molecular properties of the channel protein, it is likely that biophysical processes are better captured by our new approach than in earlier models.
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Affiliation(s)
- Ivo Siekmann
- Systems Biology Laboratory, Melbourne School of Engineering, University of Melbourne, Melbourne, Australia
- Centre for Systems Genomics, University of Melbourne, Melbourne, Australia
| | - Mark Fackrell
- School of Mathematics and Statistics, University of Melbourne, Melbourne, Australia
| | - Edmund J. Crampin
- Systems Biology Laboratory, Melbourne School of Engineering, University of Melbourne, Melbourne, Australia
- Centre for Systems Genomics, University of Melbourne, Melbourne, Australia
- School of Mathematics and Statistics, University of Melbourne, Melbourne, Australia
- School of Medicine, University of Melbourne, Melbourne, Australia
- Australian Research Council Centre of Excellence in Convergent Bio-Nano Science and Technology, Melbourne, Australia
| | - Peter Taylor
- School of Mathematics and Statistics, University of Melbourne, Melbourne, Australia
- Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers, Melbourne, Australia
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24
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Tveito A, Lines GT, Edwards AG, McCulloch A. Computing rates of Markov models of voltage-gated ion channels by inverting partial differential equations governing the probability density functions of the conducting and non-conducting states. Math Biosci 2016; 277:126-35. [PMID: 27154008 PMCID: PMC4894014 DOI: 10.1016/j.mbs.2016.04.011] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/14/2016] [Revised: 03/12/2016] [Accepted: 04/24/2016] [Indexed: 11/18/2022]
Abstract
Markov models are ubiquitously used to represent the function of single ion channels. However, solving the inverse problem to construct a Markov model of single channel dynamics from bilayer or patch-clamp recordings remains challenging, particularly for channels involving complex gating processes. Methods for solving the inverse problem are generally based on data from voltage clamp measurements. Here, we describe an alternative approach to this problem based on measurements of voltage traces. The voltage traces define probability density functions of the functional states of an ion channel. These probability density functions can also be computed by solving a deterministic system of partial differential equations. The inversion is based on tuning the rates of the Markov models used in the deterministic system of partial differential equations such that the solution mimics the properties of the probability density function gathered from (pseudo) experimental data as well as possible. The optimization is done by defining a cost function to measure the difference between the deterministic solution and the solution based on experimental data. By evoking the properties of this function, it is possible to infer whether the rates of the Markov model are identifiable by our method. We present applications to Markov model well-known from the literature.
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Affiliation(s)
- Aslak Tveito
- Simula Research Laboratory, Center for Biomedical Computing, P.O. Box 134, Lysaker 1325, Norway.
| | - Glenn T Lines
- Simula Research Laboratory, Center for Biomedical Computing, P.O. Box 134, Lysaker 1325, Norway
| | - Andrew G Edwards
- Simula Research Laboratory, Center for Biomedical Computing, P.O. Box 134, Lysaker 1325, Norway; Department of Biosciences, University of Oslo, Oslo, Norway
| | - Andrew McCulloch
- Department of Bioengineering, University of California, San Diego, USA
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25
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Mirams GR, Pathmanathan P, Gray RA, Challenor P, Clayton RH. Uncertainty and variability in computational and mathematical models of cardiac physiology. J Physiol 2016; 594:6833-6847. [PMID: 26990229 PMCID: PMC5134370 DOI: 10.1113/jp271671] [Citation(s) in RCA: 93] [Impact Index Per Article: 11.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/30/2015] [Accepted: 02/28/2016] [Indexed: 12/22/2022] Open
Abstract
KEY POINTS Mathematical and computational models of cardiac physiology have been an integral component of cardiac electrophysiology since its inception, and are collectively known as the Cardiac Physiome. We identify and classify the numerous sources of variability and uncertainty in model formulation, parameters and other inputs that arise from both natural variation in experimental data and lack of knowledge. The impact of uncertainty on the outputs of Cardiac Physiome models is not well understood, and this limits their utility as clinical tools. We argue that incorporating variability and uncertainty should be a high priority for the future of the Cardiac Physiome. We suggest investigating the adoption of approaches developed in other areas of science and engineering while recognising unique challenges for the Cardiac Physiome; it is likely that novel methods will be necessary that require engagement with the mathematics and statistics community. ABSTRACT The Cardiac Physiome effort is one of the most mature and successful applications of mathematical and computational modelling for describing and advancing the understanding of physiology. After five decades of development, physiological cardiac models are poised to realise the promise of translational research via clinical applications such as drug development and patient-specific approaches as well as ablation, cardiac resynchronisation and contractility modulation therapies. For models to be included as a vital component of the decision process in safety-critical applications, rigorous assessment of model credibility will be required. This White Paper describes one aspect of this process by identifying and classifying sources of variability and uncertainty in models as well as their implications for the application and development of cardiac models. We stress the need to understand and quantify the sources of variability and uncertainty in model inputs, and the impact of model structure and complexity and their consequences for predictive model outputs. We propose that the future of the Cardiac Physiome should include a probabilistic approach to quantify the relationship of variability and uncertainty of model inputs and outputs.
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Affiliation(s)
- Gary R Mirams
- Computational Biology, Department of Computer Science, University of Oxford, Oxford, OX1 3QD, UK
| | - Pras Pathmanathan
- US Food and Drug Administration, 10903 New Hampshire Avenue, Silver Spring, MD, 20993, USA
| | - Richard A Gray
- US Food and Drug Administration, 10903 New Hampshire Avenue, Silver Spring, MD, 20993, USA
| | - Peter Challenor
- College of Engineering, Mathematics and Physical Science, University of Exeter, Exeter, EX4 4QF, UK
| | - Richard H Clayton
- Insigneo institute for in-silico medicine and Department of Computer Science, University of Sheffield, Regent Court, Sheffield, S1 4DP, UK
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26
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Hines KE. A primer on Bayesian inference for biophysical systems. Biophys J 2016; 108:2103-13. [PMID: 25954869 DOI: 10.1016/j.bpj.2015.03.042] [Citation(s) in RCA: 38] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/18/2014] [Revised: 03/15/2015] [Accepted: 03/18/2015] [Indexed: 12/20/2022] Open
Abstract
Bayesian inference is a powerful statistical paradigm that has gained popularity in many fields of science, but adoption has been somewhat slower in biophysics. Here, I provide an accessible tutorial on the use of Bayesian methods by focusing on example applications that will be familiar to biophysicists. I first discuss the goals of Bayesian inference and show simple examples of posterior inference using conjugate priors. I then describe Markov chain Monte Carlo sampling and, in particular, discuss Gibbs sampling and Metropolis random walk algorithms with reference to detailed examples. These Bayesian methods (with the aid of Markov chain Monte Carlo sampling) provide a generalizable way of rigorously addressing parameter inference and identifiability for arbitrarily complicated models.
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Affiliation(s)
- Keegan E Hines
- Department of Neuroscience, University of Texas at Austin, Austin, Texas.
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Johnstone RH, Chang ETY, Bardenet R, de Boer TP, Gavaghan DJ, Pathmanathan P, Clayton RH, Mirams GR. Uncertainty and variability in models of the cardiac action potential: Can we build trustworthy models? J Mol Cell Cardiol 2015; 96:49-62. [PMID: 26611884 PMCID: PMC4915860 DOI: 10.1016/j.yjmcc.2015.11.018] [Citation(s) in RCA: 87] [Impact Index Per Article: 9.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 05/31/2015] [Revised: 10/13/2015] [Accepted: 11/17/2015] [Indexed: 01/07/2023]
Abstract
Cardiac electrophysiology models have been developed for over 50 years, and now include detailed descriptions of individual ion currents and sub-cellular calcium handling. It is commonly accepted that there are many uncertainties in these systems, with quantities such as ion channel kinetics or expression levels being difficult to measure or variable between samples. Until recently, the original approach of describing model parameters using single values has been retained, and consequently the majority of mathematical models in use today provide point predictions, with no associated uncertainty. In recent years, statistical techniques have been developed and applied in many scientific areas to capture uncertainties in the quantities that determine model behaviour, and to provide a distribution of predictions which accounts for this uncertainty. In this paper we discuss this concept, which is termed uncertainty quantification, and consider how it might be applied to cardiac electrophysiology models. We present two case studies in which probability distributions, instead of individual numbers, are inferred from data to describe quantities such as maximal current densities. Then we show how these probabilistic representations of model parameters enable probabilities to be placed on predicted behaviours. We demonstrate how changes in these probability distributions across data sets offer insight into which currents cause beat-to-beat variability in canine APs. We conclude with a discussion of the challenges that this approach entails, and how it provides opportunities to improve our understanding of electrophysiology. Uncertainty and variability in action potential models can be quantified. A probabilistic method for inferring maximal current densities is developed and applied. We use this to infer the currents responsible for canine beat-to-beat variability. Emulation of mathematical models provides rich information at low computational cost. The importance of considering uncertainty and variability in future is discussed.
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Affiliation(s)
- Ross H Johnstone
- Computational Biology, Dept. of Computer Science, University of Oxford, Oxford OX1 3QD, UK
| | - Eugene T Y Chang
- Insigneo Institute for in-silico Medicine and Department of Computer Science, University of Sheffield, Sheffield S1 4DP, UK
| | - Rémi Bardenet
- CNRS & CRIStAL, Université de Lille, 59651 Villeneuve d'Ascq, France
| | - Teun P de Boer
- Division of Heart & Lungs, Department of Medical Physiology, University Medical Center Utrecht, Utrecht, The Netherlands
| | - David J Gavaghan
- Computational Biology, Dept. of Computer Science, University of Oxford, Oxford OX1 3QD, UK
| | - Pras Pathmanathan
- U.S. Food and Drug Administration, 10903 New Hampshire Avenue, Silver Spring, MD 20993, USA.
| | - Richard H Clayton
- Insigneo Institute for in-silico Medicine and Department of Computer Science, University of Sheffield, Sheffield S1 4DP, UK.
| | - Gary R Mirams
- Computational Biology, Dept. of Computer Science, University of Oxford, Oxford OX1 3QD, UK.
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28
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Hines KE, Bankston JR, Aldrich RW. Analyzing single-molecule time series via nonparametric Bayesian inference. Biophys J 2015; 108:540-56. [PMID: 25650922 DOI: 10.1016/j.bpj.2014.12.016] [Citation(s) in RCA: 35] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/22/2014] [Revised: 12/03/2014] [Accepted: 12/08/2014] [Indexed: 12/22/2022] Open
Abstract
The ability to measure the properties of proteins at the single-molecule level offers an unparalleled glimpse into biological systems at the molecular scale. The interpretation of single-molecule time series has often been rooted in statistical mechanics and the theory of Markov processes. While existing analysis methods have been useful, they are not without significant limitations including problems of model selection and parameter nonidentifiability. To address these challenges, we introduce the use of nonparametric Bayesian inference for the analysis of single-molecule time series. These methods provide a flexible way to extract structure from data instead of assuming models beforehand. We demonstrate these methods with applications to several diverse settings in single-molecule biophysics. This approach provides a well-constrained and rigorously grounded method for determining the number of biophysical states underlying single-molecule data.
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Affiliation(s)
- Keegan E Hines
- Center for Learning and Memory and Department of Neuroscience, University of Texas at Austin, Austin, Texas
| | - John R Bankston
- Department of Physiology and Biophysics, University of Washington School of Medicine, Seattle, Washington
| | - Richard W Aldrich
- Center for Learning and Memory and Department of Neuroscience, University of Texas at Austin, Austin, Texas.
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29
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Tewari SG, Zhou Y, Otto BJ, Dash RK, Kwok WM, Beard DA. Markov chain Monte Carlo based analysis of post-translationally modified VDAC gating kinetics. Front Physiol 2015; 5:513. [PMID: 25628567 PMCID: PMC4292549 DOI: 10.3389/fphys.2014.00513] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/03/2014] [Accepted: 12/09/2014] [Indexed: 12/17/2022] Open
Abstract
The voltage-dependent anion channel (VDAC) is the main conduit for permeation of solutes (including nucleotides and metabolites) of up to 5 kDa across the mitochondrial outer membrane (MOM). Recent studies suggest that VDAC activity is regulated via post-translational modifications (PTMs). Yet the nature and effect of these modifications is not understood. Herein, single channel currents of wild-type, nitrosated, and phosphorylated VDAC are analyzed using a generalized continuous-time Markov chain Monte Carlo (MCMC) method. This developed method describes three distinct conducting states (open, half-open, and closed) of VDAC activity. Lipid bilayer experiments are also performed to record single VDAC activity under un-phosphorylated and phosphorylated conditions, and are analyzed using the developed stochastic search method. Experimental data show significant alteration in VDAC gating kinetics and conductance as a result of PTMs. The effect of PTMs on VDAC kinetics is captured in the parameters associated with the identified Markov model. Stationary distributions of the Markov model suggest that nitrosation of VDAC not only decreased its conductance but also significantly locked VDAC in a closed state. On the other hand, stationary distributions of the model associated with un-phosphorylated and phosphorylated VDAC suggest a reversal in channel conformation from relatively closed state to an open state. Model analyses of the nitrosated data suggest that faster reaction of nitric oxide with Cys-127 thiol group might be responsible for the biphasic effect of nitric oxide on basal VDAC conductance.
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Affiliation(s)
- Shivendra G Tewari
- Department of Molecular and Integrative Physiology, University of Michigan Ann Arbor, MI, USA
| | - Yifan Zhou
- HD Biosciences Corporation Shanghai, China
| | - Bradley J Otto
- Department of Anesthesiology, Medical College of Wisconsin Milwaukee, WI, USA
| | - Ranjan K Dash
- Department of Physiology, Medical College of Wisconsin Milwaukee, WI, USA ; Biotechnology and Bioengineering Center, Medical College of Wisconsin Milwaukee, WI, USA
| | - Wai-Meng Kwok
- Department of Anesthesiology, Medical College of Wisconsin Milwaukee, WI, USA ; Department of Pharmacology and Toxicology, Medical College of Wisconsin Milwaukee, WI, USA
| | - Daniel A Beard
- Department of Molecular and Integrative Physiology, University of Michigan Ann Arbor, MI, USA
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30
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Hotz T, Schütte OM, Sieling H, Polupanow T, Diederichsen U, Steinem C, Munk A. Idealizing ion channel recordings by a jump segmentation multiresolution filter. IEEE Trans Nanobioscience 2014; 12:376-86. [PMID: 24235310 DOI: 10.1109/tnb.2013.2284063] [Citation(s) in RCA: 28] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Abstract
Based on a combination of jump segmentation and statistical multiresolution analysis for dependent data, a new approach called J-SMURF to idealize ion channel recordings has been developed. It is model-free in the sense that no a-priori assumptions about the channel’s characteristics have to be made; it thus complements existing methods which assume a model for the channel's dynamics, like hidden Markov models. The method accounts for the effect of an analog filter being applied before the data analysis, which results in colored noise, by adapting existing muliresolution statistics to this situation. J-SMURF’s ability to denoise the signal without missing events even when the signal-to-noise ratio is low is demonstrated on simulations as well as on ion current traces obtained from gramicidin A channels reconstituted into solvent-free planar membranes. When analyzing a newly synthesized acylated system of a fatty acid modified gramicidin channel, we are able to give statistical evidence for unknown gating characteristics such as subgating.
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31
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Siekmann I, Sneyd J, Crampin EJ. Statistical analysis of modal gating in ion channels. Proc Math Phys Eng Sci 2014. [DOI: 10.1098/rspa.2014.0030] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Ion channels regulate the concentrations of ions within cells. By stochastically opening and closing its pore, they enable or prevent ions from crossing the cell membrane. However, rather than opening with a constant probability, many ion channels switch between several different levels of activity even if the experimental conditions are unchanged. This phenomenon is known as modal gating: instead of directly adapting its activity, the channel seems to mix sojourns in active and inactive modes in order to exhibit intermediate open probabilities. Evidence is accumulating that modal gating rather than modulation of opening and closing at a faster time scale is the primary regulatory mechanism of ion channels. However, currently, no method is available for reliably calculating sojourns in different modes. In order to address this challenge, we develop a statistical framework for segmenting single-channel datasets into segments that are characteristic for particular modes. The algorithm finds the number of mode changes, detects their locations and infers the open probabilities of the modes. We apply our approach to data from the inositol-trisphosphate receptor. Based upon these results, we propose that mode changes originate from alternative conformational states of the channel protein that determine a certain level of channel activity.
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Affiliation(s)
- Ivo Siekmann
- National ICT Australia, Victorian Research Laboratory, Melbourne, Australia
- Systems Biology Laboratory, Melbourne School of Engineering, The University of Melbourne, Melbourne, Australia
| | - James Sneyd
- Department of Mathematics, The University of Auckland, Auckland, New Zealand
| | - Edmund J. Crampin
- National ICT Australia, Victorian Research Laboratory, Melbourne, Australia
- Systems Biology Laboratory, Melbourne School of Engineering, The University of Melbourne, Melbourne, Australia
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32
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Calderhead B, Epstein M, Sivilotti L, Girolami M. Bayesian approaches for mechanistic ion channel modeling. Methods Mol Biol 2013; 1021:247-72. [PMID: 23715989 DOI: 10.1007/978-1-62703-450-0_13] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/16/2023]
Abstract
We consider the Bayesian analysis of mechanistic models describing the dynamic behavior of ligand-gated ion channels. The opening of the transmembrane pore in an ion channel is brought about by conformational changes in the protein, which results in a flow of ions through the pore. Remarkably, given the diameter of the pore, the flow of ions from a small number of channels or indeed from a single ion channel molecule can be recorded experimentally. This produces a large time-series of high-resolution experimental data, which can be used to investigate the gating process of these channels. We give a brief overview of the achievements and limitations of alternative maximum-likelihood approaches to this type of modeling, before investigating the statistical issues associated with analyzing stochastic model reaction mechanisms from a Bayesian perspective. Finally, we compare a number of Markov chain Monte Carlo algorithms that may be used to tackle this challenging inference problem.
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Affiliation(s)
- Ben Calderhead
- Department of Computing Science, University of Glasgow, Glasgow, Scotland
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33
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Cao P, Donovan G, Falcke M, Sneyd J. A stochastic model of calcium puffs based on single-channel data. Biophys J 2013; 105:1133-42. [PMID: 24010656 PMCID: PMC3852038 DOI: 10.1016/j.bpj.2013.07.034] [Citation(s) in RCA: 46] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/03/2013] [Revised: 07/03/2013] [Accepted: 07/24/2013] [Indexed: 11/17/2022] Open
Abstract
Calcium puffs are local transient Ca(2+) releases from internal Ca(2+) stores such as the endoplasmic reticulum or the sarcoplasmic reticulum. Such release occurs through a cluster of inositol 1,4,5-trisphosphate receptors (IP3Rs). Based on the IP3R model (which is determined by fitting to stationary single-channel data) and nonstationary single-channel data, we construct a new IP3R model that includes time-dependent rates of mode switches. A point-source model of Ca(2+) puffs is then constructed based on the new IP3R model and is solved by a hybrid Gillespie method with adaptive timing. Model results show that a relatively slow recovery of an IP3R from Ca(2+) inhibition is necessary to reproduce most of the experimental outcomes, especially the nonexponential interpuff interval distributions. The number of receptors in a cluster could be severely underestimated when the recovery is sufficiently slow. Furthermore, we find that, as the number of IP3Rs increases, the average duration of puffs initially increases but then becomes saturated, whereas the average decay time keeps increasing linearly. This gives rise to the observed asymmetric puff shape.
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Affiliation(s)
- Pengxing Cao
- Department of Mathematics, The University of Auckland, Auckland, New Zealand
| | - Graham Donovan
- Department of Mathematics, The University of Auckland, Auckland, New Zealand
| | - Martin Falcke
- Mathematical Cell Physiology, Max Delbrück Center for Molecular Medicine, Berlin, Germany
| | - James Sneyd
- Department of Mathematics, The University of Auckland, Auckland, New Zealand
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34
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Siekmann I, Sneyd J, Crampin EJ. MCMC can detect nonidentifiable models. Biophys J 2013; 103:2275-86. [PMID: 23283226 DOI: 10.1016/j.bpj.2012.10.024] [Citation(s) in RCA: 54] [Impact Index Per Article: 4.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/19/2012] [Revised: 06/14/2012] [Accepted: 10/18/2012] [Indexed: 11/26/2022] Open
Abstract
Continuous-time Markov models have been considered the best representation for the stochastic dynamics of ion channels for more than thirty years. For most single-channel data sets, several open and closed states are required for accurately representing the dynamics. However, each data point only shows if the channel is open or closed but not in which state it is. Consequently, some model structures are inherently overparameterized and therefore, in principle, unsuitable for representing any data--those models are called "nonidentifiable". As of this writing, it seems to be poorly understood which continuous-time Markov models are identifiable and which are not, therefore the unconscious use of a nonidentifiable model is a considerable concern. To address this problem, an improved variant of a recently published Markov-chain Monte Carlo method is presented. The algorithm is tested using test data as well as experimental data. We demonstrate that, opposed to a widely used maximum-likelihood estimator, it gives clear warning signs when a nonidentifiable model is used for fitting. Furthermore, for test data that was generated from a nonidentifiable model, the Markov-chain Monte Carlo results recover much more information from the data than maximum-likelihood estimation.
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Affiliation(s)
- Ivo Siekmann
- Auckland Bioengineering Institute, The University of Auckland, Auckland, New Zealand.
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35
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Comparison of models for IP3 receptor kinetics using stochastic simulations. PLoS One 2013; 8:e59618. [PMID: 23630568 PMCID: PMC3629942 DOI: 10.1371/journal.pone.0059618] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/14/2012] [Accepted: 02/15/2013] [Indexed: 12/07/2022] Open
Abstract
Inositol 1,4,5-trisphosphate receptor (IP3R) is a ubiquitous intracellular calcium (Ca2+) channel which has a major role in controlling Ca2+ levels in neurons. A variety of computational models have been developed to describe the kinetic function of IP3R under different conditions. In the field of computational neuroscience, it is of great interest to apply the existing models of IP3R when modeling local Ca2+ transients in dendrites or overall Ca2+ dynamics in large neuronal models. The goal of this study was to evaluate existing IP3R models, based on electrophysiological data. This was done in order to be able to suggest suitable models for neuronal modeling. Altogether four models (Othmer and Tang, 1993; Dawson etal., 2003; Fraiman and Dawson, 2004; Doi etal., 2005) were selected for a more detailed comparison. The selection was based on the computational efficiency of the models and the type of experimental data that was used in developing the model. The kinetics of all four models were simulated by stochastic means, using the simulation software STEPS, which implements the Gillespie stochastic simulation algorithm. The results show major differences in the statistical properties of model functionality. Of the four compared models, the one by Fraiman and Dawson (2004) proved most satisfactory in producing the specific features of experimental findings reported in literature. To our knowledge, the present study is the first detailed evaluation of IP3R models using stochastic simulation methods, thus providing an important setting for constructing a new, realistic model of IP3R channel kinetics for compartmental modeling of neuronal functions. We conclude that the kinetics of IP3R with different concentrations of Ca2+ and IP3 should be more carefully addressed when new models for IP3R are developed.
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36
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Siekmann I, Wagner LE, Yule D, Crampin EJ, Sneyd J. A kinetic model for type I and II IP3R accounting for mode changes. Biophys J 2012; 103:658-68. [PMID: 22947927 PMCID: PMC3443778 DOI: 10.1016/j.bpj.2012.07.016] [Citation(s) in RCA: 39] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/18/2012] [Revised: 07/11/2012] [Accepted: 07/13/2012] [Indexed: 10/28/2022] Open
Abstract
Based upon an extensive single-channel data set, a Markov model for types I and II inositol trisphosphate receptors (IP(3)R) is developed. The model aims to represent accurately the kinetics of both receptor types of IP(3)R depending on the concentrations of inositol trisphosphate (IP(3)), adenosine trisphosphate (ATP), and intracellular calcium (Ca(2+)). In particular, the model takes into account that for some combinations of ligands the IP(3)R switches between extended periods of inactivity alternating with intervals of bursting activity (mode changes). In a first step, the inactive and active modes are modeled separately. It is found that, within modes, both receptor types are ligand-independent. In a second step, the submodels are connected by transition rates. Ligand-dependent regulation of the channel activity is achieved by modulating these transitions between active and inactive modes. As a result, a compact representation of the IP(3)R is obtained that accurately captures stochastic single-channel dynamics including mode changes in a model with six states and 10 rate constants, only two of which are ligand-dependent.
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Affiliation(s)
- Ivo Siekmann
- Auckland Bioengineering Institute, The University of Auckland, Auckland, New Zealand.
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37
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Lauzon AM, Bates JHT, Donovan G, Tawhai M, Sneyd J, Sanderson MJ. A multi-scale approach to airway hyperresponsiveness: from molecule to organ. Front Physiol 2012; 3:191. [PMID: 22701430 PMCID: PMC3371674 DOI: 10.3389/fphys.2012.00191] [Citation(s) in RCA: 36] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/28/2012] [Accepted: 05/21/2012] [Indexed: 12/13/2022] Open
Abstract
Airway hyperresponsiveness (AHR), a characteristic of asthma that involves an excessive reduction in airway caliber, is a complex mechanism reflecting multiple processes that manifest over a large range of length and time scales. At one extreme, molecular interactions determine the force generated by airway smooth muscle (ASM). At the other, the spatially distributed constriction of the branching airways leads to breathing difficulties. Similarly, asthma therapies act at the molecular scale while clinical outcomes are determined by lung function. These extremes are linked by events operating over intermediate scales of length and time. Thus, AHR is an emergent phenomenon that limits our understanding of asthma and confounds the interpretation of studies that address physiological mechanisms over a limited range of scales. A solution is a modular computational model that integrates experimental and mathematical data from multiple scales. This includes, at the molecular scale, kinetics, and force production of actin-myosin contractile proteins during cross-bridge and latch-state cycling; at the cellular scale, Ca2+ signaling mechanisms that regulate ASM force production; at the tissue scale, forces acting between contracting ASM and opposing viscoelastic tissue that determine airway narrowing; at the organ scale, the topographic distribution of ASM contraction dynamics that determine mechanical impedance of the lung. At each scale, models are constructed with iterations between theory and experimentation to identify the parameters that link adjacent scales. This modular model establishes algorithms for modeling over a wide range of scales and provides a framework for the inclusion of other responses such as inflammation or therapeutic regimes. The goal is to develop this lung model so that it can make predictions about bronchoconstriction and identify the pathophysiologic mechanisms having the greatest impact on AHR and its therapy.
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Affiliation(s)
- Anne-Marie Lauzon
- Meakins-Christie Laboratories, Department of Medicine, McGill University Montreal, QC, Canada
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38
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A model for Ca2+ oscillations stimulated by the type 5 metabotropic glutamate receptor: An unusual mechanism based on repetitive, reversible phosphorylation of the receptor. Biochimie 2011; 93:2132-8. [DOI: 10.1016/j.biochi.2011.09.010] [Citation(s) in RCA: 33] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/08/2011] [Accepted: 09/11/2011] [Indexed: 11/17/2022]
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39
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Piasta KN, Theobald DL, Miller C. Potassium-selective block of barium permeation through single KcsA channels. ACTA ACUST UNITED AC 2011; 138:421-36. [PMID: 21911483 PMCID: PMC3182450 DOI: 10.1085/jgp.201110684] [Citation(s) in RCA: 47] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
Abstract
Ba2+, a doubly charged analogue of K+, specifically blocks K+ channels by virtue of electrostatic stabilization in the permeation pathway. Ba2+ block is used here as a tool to determine the equilibrium binding affinity for various monovalent cations at specific sites in the selectivity filter of a noninactivating mutant of KcsA. At high concentrations of external K+, the block-time distribution is double exponential, marking at least two Ba2+ sites in the selectivity filter, in accord with a Ba2+-containing crystal structure of KcsA. By analyzing block as a function of extracellular K+, we determined the equilibrium dissociation constant of K+ and of other monovalent cations at an extracellular site, presumably S1, to arrive at a selectivity sequence for binding at this site: Rb+ (3 µM) > Cs+ (23 µM) > K+ (29 µM) > NH4+ (440 µM) >> Na+ and Li+ (>1 M). This represents an unusually high selectivity for K+ over Na+, with |ΔΔG0| of at least 7 kcal mol−1. These results fit well with other kinetic measurements of selectivity as well as with the many crystal structures of KcsA in various ionic conditions.
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Affiliation(s)
- Kene N Piasta
- Department of Biochemistry, Howard Hughes Medical Institute, Brandeis University, Waltham, MA 02454, USA
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