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Veron G, Maltsev VA, Stern MD, Maltsev AV. Elementary intracellular Ca signals approximated as a transition of release channel system from a metastable state. JOURNAL OF APPLIED PHYSICS 2023; 134:124701. [PMID: 37744735 PMCID: PMC10517864 DOI: 10.1063/5.0151255] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/20/2023] [Accepted: 09/04/2023] [Indexed: 09/26/2023]
Abstract
Cardiac muscle contraction is initiated by an elementary Ca signal (called Ca spark) which is achieved by collective action of Ca release channels in a cluster. The mechanism of this synchronization remains uncertain. We approached Ca spark activation as an emergent phenomenon of an interactive system of release channels. We constructed a weakly lumped Markov chain that applies an Ising model formalism to such release channel clusters and probable open channel configurations and demonstrated that spark activation is described as a system transition from a metastable to an absorbing state, analogous to the pressure required to overcome surface tension in bubble formation. This yielded quantitative estimates of the spark generation probability as a function of various system parameters. We performed numerical simulations to find spark probabilities as a function of sarcoplasmic reticulum Ca concentration, obtaining similar values for spark activation threshold as our analytic model, as well as those reported in experimental studies. Our parametric sensitivity analyses also showed that the spark activation threshold decreased as Ca sensitivity of RyR activation and RyR cluster size increased.
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Affiliation(s)
- Guillermo Veron
- Cellular Biophysics Section, Laboratory of Cardiovascular Science, National Institute on Aging, NIH, Baltimore, Maryland 21224, USA
| | - Victor A. Maltsev
- Cellular Biophysics Section, Laboratory of Cardiovascular Science, National Institute on Aging, NIH, Baltimore, Maryland 21224, USA
| | - Michael D. Stern
- Cellular Biophysics Section, Laboratory of Cardiovascular Science, National Institute on Aging, NIH, Baltimore, Maryland 21224, USA
| | - Anna V. Maltsev
- School of Mathematical Sciences, Queen Mary University of London, London E14NS, United Kingdom
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Qu Z, Yan D, Song Z. Modeling Calcium Cycling in the Heart: Progress, Pitfalls, and Challenges. Biomolecules 2022; 12:1686. [PMID: 36421700 PMCID: PMC9687412 DOI: 10.3390/biom12111686] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/10/2022] [Revised: 11/08/2022] [Accepted: 11/11/2022] [Indexed: 11/16/2022] Open
Abstract
Intracellular calcium (Ca) cycling in the heart plays key roles in excitation-contraction coupling and arrhythmogenesis. In cardiac myocytes, the Ca release channels, i.e., the ryanodine receptors (RyRs), are clustered in the sarcoplasmic reticulum membrane, forming Ca release units (CRUs). The RyRs in a CRU act collectively to give rise to discrete Ca release events, called Ca sparks. A cell contains hundreds to thousands of CRUs, diffusively coupled via Ca to form a CRU network. A rich spectrum of spatiotemporal Ca dynamics is observed in cardiac myocytes, including Ca sparks, spark clusters, mini-waves, persistent whole-cell waves, and oscillations. Models of different temporal and spatial scales have been developed to investigate these dynamics. Due to the complexities of the CRU network and the spatiotemporal Ca dynamics, it is challenging to model the Ca cycling dynamics in the cardiac system, particularly at the tissue sales. In this article, we review the progress of modeling of Ca cycling in cardiac systems from single RyRs to the tissue scale, the pros and cons of the current models and different modeling approaches, and the challenges to be tackled in the future.
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Affiliation(s)
- Zhilin Qu
- Department of Medicine, David Geffen School of Medicine, University of California, A2-237 CHS, 650 Charles E. Young Drive South, Los Angeles, CA 90095, USA
- Department of Computational Medicine, David Geffen School of Medicine, University of California, Los Angeles, CA 90095, USA
| | - Dasen Yan
- Peng Cheng Laboratory, Shenzhen 518066, China
| | - Zhen Song
- Peng Cheng Laboratory, Shenzhen 518066, China
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Maltsev AV, Stern MD, Maltsev VA. Disorder in Ca2+ release unit locations confers robustness but cuts flexibility of heart pacemaking. J Gen Physiol 2022; 154:e202113061. [PMID: 35943725 PMCID: PMC9366202 DOI: 10.1085/jgp.202113061] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/22/2021] [Revised: 05/04/2022] [Accepted: 06/21/2022] [Indexed: 12/12/2022] Open
Abstract
Excitation-contraction coupling kinetics is dictated by the action potential rate of sinoatrial-nodal cells. These cells generate local Ca releases (LCRs) that activate Na/Ca exchanger current, which accelerates diastolic depolarization and determines the pace. LCRs are generated by clusters of ryanodine receptors, Ca release units (CRUs), residing in the sarcoplasmic reticulum. While CRU distribution exhibits substantial heterogeneity, its functional importance remains unknown. Using numerical modeling, here we show that with a square lattice distribution of CRUs, Ca-induced-Ca-release propagation during diastolic depolarization is insufficient for pacemaking within a broad range of realistic ICaL densities. Allowing each CRU to deviate randomly from its lattice position allows sparks to propagate, as observed experimentally. As disorder increases, the CRU distribution exhibits larger empty spaces and simultaneously CRU clusters, as in Poisson clumping. Propagating within the clusters, Ca release becomes synchronized, increasing action potential rate and reviving pacemaker function of dormant/nonfiring cells. However, cells with fully disordered CRU positions could not reach low firing rates and their β-adrenergic-receptor stimulation effect was substantially decreased. Inclusion of Cav1.3, a low-voltage activation L-type Ca channel isoform into ICaL, strongly increases recruitment of CRUs to fire during diastolic depolarization, increasing robustness of pacemaking and complementing effects of CRU distribution. Thus, order/disorder in CRU locations along with Cav1.3 expression regulates pacemaker function via synchronization of CRU firing. Excessive CRU disorder and/or overexpression of Cav1.3 boosts pacemaker function in the basal state, but limits the rate range, which may contribute to heart rate range decline with age and disease.
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Affiliation(s)
- Anna V. Maltsev
- School of Mathematics, Queen Mary University of London, London, UK
| | - Michael D. Stern
- Laboratory of Cardiovascular Science, Intramural Research Program, National Institute on Aging, National Institutes of Health, Baltimore, MD
| | - Victor A. Maltsev
- Laboratory of Cardiovascular Science, Intramural Research Program, National Institute on Aging, National Institutes of Health, Baltimore, MD
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Maltsev AV, Stern MD, Lakatta EG, Maltsev VA. Functional Heterogeneity of Cell Populations Increases Robustness of Pacemaker Function in a Numerical Model of the Sinoatrial Node Tissue. Front Physiol 2022; 13:845634. [PMID: 35574456 PMCID: PMC9091312 DOI: 10.3389/fphys.2022.845634] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/30/2021] [Accepted: 03/15/2022] [Indexed: 11/19/2022] Open
Abstract
Each heartbeat is initiated by specialized pacemaker cells operating within the sinoatrial node (SAN). While individual cells within SAN tissue exhibit substantial heterogeneity of their electrophysiological parameters and Ca cycling, the role of this heterogeneity for cardiac pacemaker function remains mainly unknown. Here we investigated the problem numerically in a 25 × 25 square grid of connected coupled-clock Maltsev-Lakatta cell models. The tissue models were populated by cells with different degree of heterogeneity of the two key model parameters, maximum L-type Ca current conductance (gCaL) and sarcoplasmic reticulum Ca pumping rate (Pup). Our simulations showed that in the areas of Pup-gCaL parametric space at the edge of the system stability, where action potential (AP) firing is absent or dysrhythmic in SAN tissue models populated with identical cells, rhythmic AP firing can be rescued by populating the tissues with heterogeneous cells. This robust SAN function is synergistic with respect to heterogeneity in gCaL and Pup and can be further strengthened by clustering of cells with similar properties. The effect of cell heterogeneity is not due to a simple summation of activity of intrinsically firing cells naturally present in heterogeneous SAN; rather AP firing cells locally and critically interact with non-firing/dormant cells. When firing cells prevail, they recruit many dormant cells to fire, strongly enhancing overall SAN function; and vice versa, prevailing dormant cells suppress AP firing in cells with intrinsic automaticity and halt SAN function. The transitions between firing and non-firing states of the system are sharp, resembling phase transitions in statistical physics. Furthermore, robust function of heterogeneous SAN tissue requires weak cell coupling, a known property of the central area of SAN where cardiac impulse emerges; stronger cell coupling reduces AP firing rate and ultimately halts SAN automaticity at the edge of stability.
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Shrivastava R, Ghosh S. Collective Dynamics of Ion Channels on Bilayer Lipid Membranes. ACS OMEGA 2021; 6:7544-7557. [PMID: 33778266 PMCID: PMC7992176 DOI: 10.1021/acsomega.0c06061] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/12/2020] [Accepted: 02/02/2021] [Indexed: 05/06/2023]
Abstract
Ion channels self-organize on cellular and organelle membranes as clusters and mutually modulate their gating behavior. It has been reported that the efficient information transfer is achieved by cooperative clustering of ion channels. To address the origin and nature of collective dynamics in ion channel clusters, a statistical mechanical model, namely, the Zimm-Bragg-type model in two dimensions with unequal weight distribution in channel-channel interactions, has been proposed. Nearest neighbor interaction along with next-nearest neighbor interaction has been considered, assuming symmetric spatial organization. The multichannel bilayer electrophysiology recordings of the voltage-dependent anion channel (VDAC) from rat brain mitochondria have been analyzed in order to test and further extend the model. The model successfully describes the multichannel gating behavior and self-organization of the VDAC cluster.
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Fill M, Gillespie D. Simulating cardiac Ca 2+ release units: effects of RyR cluster size and Ca 2+ buffers on diastolic Ca 2+ leak. Pflugers Arch 2021; 473:435-446. [PMID: 33608799 DOI: 10.1007/s00424-021-02539-w] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/03/2020] [Revised: 01/27/2021] [Accepted: 02/05/2021] [Indexed: 10/22/2022]
Abstract
Leak of Ca2+ out of the cardiac sarcoplasmic reticulum (SR) via ryanodine receptors (RyRs) during diastole is vital to regulate SR Ca2+ levels. This leak can become deleterious when large spontaneous RyR-mediated Ca2+ release events evoke proarrhythmic Ca2+ waves that can lead to delayed after-depolarizations. Here, we model diastolic SR Ca2+ leak at individual SR Ca2+ release sites using computer simulations of RyR arrays like those in the dyadic cleft. The results show that RyR arrays size has a significant effect on SR Ca2+ leak, with bigger arrays producing larger and more frequent Ca2+ release events. Moreover, big RyR arrays are more susceptible to small changes in the levels of dyadic Ca2+ buffers. Such changes in buffering shift Ca2+ leak from small Ca2+ release events (involving few open RyRs) to larger events (with many open RyRs). Moreover, by analyzing a large parameter space of possible buffering and SR Ca2+ loads, we find further evidence for the hypothesis that SR Ca2+ leak by RyR arrays can undergo a sudden phase transition.
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Affiliation(s)
- Michael Fill
- Department of Physiology and Biophysics, Rush University Medical Center, Chicago, IL, USA
| | - Dirk Gillespie
- Department of Physiology and Biophysics, Rush University Medical Center, Chicago, IL, USA.
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Maltsev AV, Stern MD. Social heterogeneity drives complex patterns of the COVID-19 pandemic: insights from a novel Stochastic Heterogeneous Epidemic Model (SHEM). MEDRXIV : THE PREPRINT SERVER FOR HEALTH SCIENCES 2020. [PMID: 32676619 DOI: 10.1101/2020.07.10.20150813] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/24/2022]
Abstract
In todays absence of a vaccine and impactful treatments, the most effective way to combat the virus is to find and implement mitigation strategies. An invaluable resource in this task is numerical modeling that can reveal key factors in COVID-19 pandemic development. On the other hand, it has become evident that regional infection curves of COVID-19 exhibit complex patterns which often differ from curves predicted by forecasting models. The wide variations in attack rate observed among different social strata suggest that this may be due to social heterogeneity not accounted for by regional models. We investigated this hypothesis by developing and using a new Stochastic Heterogeneous Epidemic Model (SHEM) that focuses on subpopulations that are vulnerable in the sense of having an increased likelihood of spreading infection among themselves. We found that the isolation or embedding of vulnerable sub-clusters in a major population hub generated complex stochastic infection patterns which included multiple peaks and growth periods, an extended plateau, a prolonged tail, or a delayed second wave of infection. Embedded vulnerable groups became hotspots that drove infection despite efforts of the main population to socially distance, while isolated groups suffered delayed but intense infection. Amplification of infection by these hotspots facilitated transmission from one urban area to another, causing the epidemic to hopscotch in a stochastic manner to places it would not otherwise reach, resembling a microcosm of the situation worldwide as of September 2020. Our results suggest that social heterogeneity is a key factor in the formation of complex infection propagation patterns. Thus, the mitigation of vulnerable groups is essential to control the COVID-19 pandemic worldwide. The design of our new model allows it to be applied in future studies of real-world scenarios on any scale, limited only by computing memory and the ability to determine the underlying topology and parameters.
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Gillespie D. Recruiting RyRs to Open in a Ca 2+ Release Unit: Single-RyR Gating Properties Make RyR Group Dynamics. Biophys J 2019; 118:232-242. [PMID: 31839264 DOI: 10.1016/j.bpj.2019.11.021] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/18/2019] [Revised: 11/01/2019] [Accepted: 11/19/2019] [Indexed: 01/01/2023] Open
Abstract
In cardiac myocytes, clusters of type-2 ryanodine receptors (RyR2s) release Ca2+ from the sarcoplasmic reticulum (SR) via a positive feedback mechanism in which fluxed Ca2+ activates nearby RyRs. Although the general principles of this are understood, less is known about how single-RyR gating properties define the RyR group dynamics in an array of many channels. Here, we examine this using simulations with three models of RyR gating that have identical open probabilities: the commonly used two-state Markov gating model, one that utilizes multiple exponentials to fit single-channel open time (OT) and closed time (CT) distributions, and an extension of this multiexponential model that also includes experimentally measured correlations between single-channel OTs and CTs. The simulations of RyR clusters that utilize the multiexponential gating model produce infrequent Ca2+ release events with relatively few open RyRs. Ca2+ release events become even smaller when OT/CT correlations are included. This occurs because the correlations produce a small but consistent bias against recruiting more RyRs to open during the middle of a Ca2+ release event, between the initiation and termination phases (which are unaltered compared to the uncorrelated simulations). In comparison, the two-state model produces frequent, large, and long Ca2+ release events because it had a recruitment bias in favor of opening more RyRs. This difference stems from the two-state model's single-RyR OT and CT distributions being qualitatively different from the experimental ones. Thus, the details of single-RyR gating can profoundly affect SR Ca2+ release even if open probability and mean OTs and CTs are identical. We also show that Ca2+ release events can terminate spontaneously without any reduction in SR [Ca2+], luminal regulation, Ca2+-dependent inactivation, or physical coupling between RyRs when Ca2+ flux is below a threshold value. This supports and extends the pernicious attrition/induction decay hypothesis that SR Ca2+ release events terminate below a threshold Ca2+ flux.
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Affiliation(s)
- Dirk Gillespie
- Department of Physiology and Biophysics, Rush University Medical Center, Chicago, Illinois.
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Mechanisms of Calcium Leak from Cardiac Sarcoplasmic Reticulum Revealed by Statistical Mechanics. Biophys J 2019; 116:2212-2223. [PMID: 31103231 DOI: 10.1016/j.bpj.2019.04.031] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/14/2018] [Revised: 04/17/2019] [Accepted: 04/26/2019] [Indexed: 11/21/2022] Open
Abstract
Heart muscle contraction is normally activated by a synchronized Ca release from sarcoplasmic reticulum (SR), a major intracellular Ca store. However, under abnormal conditions, Ca leaks from the SR, decreasing heart contraction amplitude and increasing risk of life-threatening arrhythmia. The mechanisms and regimes of SR operation generating the abnormal Ca leak remain unclear. Here, we employed both numerical and analytical modeling to get mechanistic insights into the emergent Ca leak phenomenon. Our numerical simulations using a detailed realistic model of the Ca release unit reveal sharp transitions resulting in Ca leak. The emergence of leak is closely mapped mathematically to the Ising model from statistical mechanics. The system steady-state behavior is determined by two aggregate parameters: the analogs of magnetic field (h) and the inverse temperature (β) in the Ising model, for which we have explicit formulas in terms of SR [Ca] and release channel opening and closing rates. The classification of leak regimes takes the shape of a phase β-h diagram, with the regime boundaries occurring at h = 0 and a critical value of β (β∗) that we estimate using a classical Ising model and mean field theory. Our theory predicts that a synchronized Ca leak will occur when h > 0 and β >β∗, and a disordered leak occurs when β <β∗ and h is not too negative. The disorder leak is distinguished from synchronized leak (in long-lasting sparks) by larger Peierls contour lengths, an output parameter reflecting degree of disorder. Thus, in addition to our detailed numerical model approach, we also offer an instantaneous computational tool using analytical formulas of the Ising model for respective ryanodine receptor parameters and SR Ca load that describe and classify phase transitions and leak emergence.
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Du P, Calder S, Angeli TR, Sathar S, Paskaranandavadivel N, O'Grady G, Cheng LK. Progress in Mathematical Modeling of Gastrointestinal Slow Wave Abnormalities. Front Physiol 2018; 8:1136. [PMID: 29379448 PMCID: PMC5775268 DOI: 10.3389/fphys.2017.01136] [Citation(s) in RCA: 23] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/03/2017] [Accepted: 12/22/2017] [Indexed: 12/19/2022] Open
Abstract
Gastrointestinal (GI) motility is regulated in part by electrophysiological events called slow waves, which are generated by the interstitial cells of Cajal (ICC). Slow waves propagate by a process of "entrainment," which occurs over a decreasing gradient of intrinsic frequencies in the antegrade direction across much of the GI tract. Abnormal initiation and conduction of slow waves have been demonstrated in, and linked to, a number of GI motility disorders. A range of mathematical models have been developed to study abnormal slow waves and applied to propose novel methods for non-invasive detection and therapy. This review provides a general outline of GI slow wave abnormalities and their recent classification using multi-electrode (high-resolution) mapping methods, with a particular emphasis on the spatial patterns of these abnormal activities. The recently-developed mathematical models are introduced in order of their biophysical scale from cellular to whole-organ levels. The modeling techniques, main findings from the simulations, and potential future directions arising from notable studies are discussed.
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Affiliation(s)
- Peng Du
- Auckland Bioengineering Institute, University of Auckland, Auckland, New Zealand
| | - Stefan Calder
- Auckland Bioengineering Institute, University of Auckland, Auckland, New Zealand
| | - Timothy R. Angeli
- Auckland Bioengineering Institute, University of Auckland, Auckland, New Zealand
| | - Shameer Sathar
- Auckland Bioengineering Institute, University of Auckland, Auckland, New Zealand
| | | | - Gregory O'Grady
- Auckland Bioengineering Institute, University of Auckland, Auckland, New Zealand
- Department of Surgery, University of Auckland, Auckland, New Zealand
| | - Leo K. Cheng
- Auckland Bioengineering Institute, University of Auckland, Auckland, New Zealand
- Department of Surgery, Vanderbilt University, Nashville, TN, United States
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Gambardella J, Trimarco B, Iaccarino G, Santulli G. New Insights in Cardiac Calcium Handling and Excitation-Contraction Coupling. ADVANCES IN EXPERIMENTAL MEDICINE AND BIOLOGY 2017; 1067:373-385. [PMID: 28956314 DOI: 10.1007/5584_2017_106] [Citation(s) in RCA: 52] [Impact Index Per Article: 7.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/05/2023]
Abstract
Excitation-contraction (EC) coupling denotes the conversion of electric stimulus in mechanic output in contractile cells. Several studies have demonstrated that calcium (Ca2+) plays a pivotal role in this process. Here we present a comprehensive and updated description of the main systems involved in cardiac Ca2+ handling that ensure a functional EC coupling and their pathological alterations, mainly related to heart failure.
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Affiliation(s)
- Jessica Gambardella
- Department of Advanced Biomedical Sciences, "Federico II" University, Naples, Italy.,Department of Medicine, Surgery and Dentistry, Scuola Medica Salernitana, University of Salerno, Fisciano, Italy
| | - Bruno Trimarco
- Department of Advanced Biomedical Sciences, "Federico II" University, Naples, Italy
| | - Guido Iaccarino
- Department of Medicine, Surgery and Dentistry, Scuola Medica Salernitana, University of Salerno, Fisciano, Italy
| | - Gaetano Santulli
- Department of Advanced Biomedical Sciences, "Federico II" University, Naples, Italy. .,Department of Medicine, Albert Einstein College of Medicine, 1300 Morris Park Ave, Forch 525, 10461, New York, NY, USA.
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