Decoding myocardial Ca²⁺ signals across multiple spatial scales: a role for sensitivity analysis.

Numerous studies have employed mathematical modeling to quantitatively understand release of Ca(2+) from the sarcoplasmic reticulum (SR) in the heart. Models have been used to investigate physiologically important phenomena such as triggering of SR Ca(2+) release by Ca(2+) entry across the cell membrane and spontaneous leak of Ca(2+) from the SR in quiescent heart cells. In this review we summarize studies that have modeled myocardial Ca(2+) at different spatial scales: the sub-cellular level, the cellular level, and the multicellular level. We discuss each category of models from the standpoint of parameter sensitivity analysis, a common simulation procedure that can generate quantitative, comprehensive predictions about how changes in conditions influence model output. We propose that this is a useful perspective for conceptualizing models, in part because a sensitivity analysis requires the investigator to define the relevant parameters and model outputs. This procedure therefore helps to illustrate the capabilities and limitations of each model. We further suggest that in future studies, sensitivity analyses will aid in simplifying complex models and in suggesting experiments to differentiate between competing models built with different assumptions. We conclude with a discussion of unresolved questions that are likely to be addressed over the next several years.

[1]  J J Rice,et al.  Modeling gain and gradedness of Ca2+ release in the functional unit of the cardiac diadic space. , 1999, Biophysical journal.

[2]  Gil Bub,et al.  Dynamical Mechanism for Subcellular Alternans in Cardiac Myocytes , 2009, Circulation research.

[3]  A. Zima,et al.  Alteration of sarcoplasmic reticulum Ca2+ release termination by ryanodine receptor sensitization and in heart failure , 2009, The Journal of physiology.

[4]  Alan Garfinkel,et al.  Spark-Induced Sparks As a Mechanism of Intracellular Calcium Alternans in Cardiac Myocytes , 2010, Circulation research.

[5]  Yoram Rudy,et al.  Local control of β-adrenergic stimulation: Effects on ventricular myocyte electrophysiology and Ca(2+)-transient. , 2011, Journal of molecular and cellular cardiology.

[6]  Henggui Zhang,et al.  Alternans of cardiac calcium cycling in a cluster of ryanodine receptors: a simulation study. , 2008, American journal of physiology. Heart and circulatory physiology.

[7]  Zeyun Yu,et al.  Three-dimensional geometric modeling of membrane-bound organelles in ventricular myocytes: bridging the gap between microscopic imaging and mathematical simulation. , 2008, Journal of structural biology.

[8]  Blanca Rodríguez,et al.  Impact of ionic current variability on human ventricular cellular electrophysiology. , 2009, American journal of physiology. Heart and circulatory physiology.

[9]  B. Knollmann,et al.  Calsequestrin 2 deletion shortens the refractoriness of Ca²⁺ release and reduces rate-dependent Ca²⁺-alternans in intact mouse hearts. , 2012, Journal of molecular and cellular cardiology.

[10]  Takumi Washio,et al.  A three-dimensional simulation model of cardiomyocyte integrating excitation-contraction coupling and metabolism. , 2011, Biophysical journal.

[11]  Michael J. Holst,et al.  Numerical Analysis of Ca2+ Signaling in Rat Ventricular Myocytes with Realistic Transverse-Axial Tubular Geometry and Inhibited Sarcoplasmic Reticulum , 2010, PLoS Comput. Biol..

[12]  G. Langer,et al.  Calcium concentration and movement in the diadic cleft space of the cardiac ventricular cell. , 1996, Biophysical journal.

[13]  A. Di Maio,et al.  Di Maio, A , 2012 .

[14]  Michael D. Stern,et al.  Local Control Models of Cardiac Excitation–Contraction Coupling , 1999, The Journal of general physiology.

[15]  A Garfinkel,et al.  Model of intracellular calcium cycling in ventricular myocytes. , 2003, Biophysical journal.

[16]  S. Huke,et al.  Ablation of triadin causes loss of cardiac Ca2+ release units, impaired excitation–contraction coupling, and cardiac arrhythmias , 2009, Proceedings of the National Academy of Sciences.

[17]  G. Smith,et al.  Ryanodine receptor allosteric coupling and the dynamics of calcium sparks. , 2008, Biophysical journal.

[18]  J. Restrepo,et al.  A rabbit ventricular action potential model replicating cardiac dynamics at rapid heart rates. , 2007, Biophysical journal.

[19]  E. Sobie Parameter sensitivity analysis in electrophysiological models using multivariable regression. , 2009, Biophysical journal.

[20]  Aristide C. Chikando,et al.  Mitochondria in cardiomyocyte Ca2+ signaling. , 2009, The international journal of biochemistry & cell biology.

[21]  Zhilin Qu,et al.  Criticality in intracellular calcium signaling in cardiac myocytes. , 2012, Biophysical journal.

[22]  V. Bondarenko,et al.  Transmural heterogeneity of repolarization and Ca2+ handling in a model of mouse ventricular tissue. , 2010, American journal of physiology. Heart and circulatory physiology.

[23]  E. Entcheva,et al.  Cardiac cellular coupling and the spread of early instabilities in intracellular Ca2+. , 2012, Biophysical journal.

[24]  M. Stern,et al.  Theory of excitation-contraction coupling in cardiac muscle. , 1992, Biophysical journal.

[25]  Susan L. Wearne,et al.  Neuronal Firing Sensitivity to Morphologic and Active Membrane Parameters , 2007, PLoS Comput. Biol..

[26]  J. Shadid,et al.  Interplay of ryanodine receptor distribution and calcium dynamics. , 2006, Biophysical journal.

[27]  Wei Chen,et al.  The statistics of calcium-mediated focal excitations on a one-dimensional cable. , 2012, Biophysical Journal.

[28]  Christian Soeller,et al.  Estimation of the sarcoplasmic reticulum Ca2+ release flux underlying Ca2+ sparks. , 2002, Biophysical journal.

[29]  John Jeremy Rice,et al.  Integrative modeling of the cardiac ventricular myocyte , 2011, Wiley interdisciplinary reviews. Systems biology and medicine.

[30]  Jeffrey J Saucerman,et al.  Synergy between CaMKII substrates and β-adrenergic signaling in regulation of cardiac myocyte Ca(2+) handling. , 2010, Biophysical journal.

[31]  Jun Hu,et al.  Dynamic interreceptor coupling contributes to the consistent open duration of ryanodine receptors. , 2009, Biophysical journal.

[32]  Heping Cheng,et al.  Calcium sparks. , 2008, Physiological reviews.

[33]  Donald M Bers,et al.  Dynamic Calcium Movement Inside Cardiac Sarcoplasmic Reticulum During Release , 2011, Circulation research.

[34]  G. Smith,et al.  Spontaneous Calcium Sparks and Calcium Homeostasis in a Minimal Model of Permeabilized Ventricular Myocytes , 2010 .

[35]  R. Winslow,et al.  An integrative model of the cardiac ventricular myocyte incorporating local control of Ca2+ release. , 2002, Biophysical journal.

[36]  Joseph L Greenstein,et al.  Mechanisms of excitation-contraction coupling in an integrative model of the cardiac ventricular myocyte. , 2006, Biophysical journal.

[37]  L. Izu,et al.  Large currents generate cardiac Ca2+ sparks. , 2001, Biophysical journal.

[38]  H Zhang,et al.  Models of cardiac tissue electrophysiology: progress, challenges and open questions. , 2011, Progress in biophysics and molecular biology.

[39]  Zhilin Qu,et al.  Calcium alternans in cardiac myocytes: order from disorder. , 2013, Journal of molecular and cellular cardiology.

[40]  Eric A Sobie,et al.  Predicting local SR Ca(2+) dynamics during Ca(2+) wave propagation in ventricular myocytes. , 2010, Biophysical journal.

[41]  J. Clark,et al.  Mathematical model of an adult human atrial cell: the role of K+ currents in repolarization. , 1998, Circulation research.

[42]  Dmitry Terentyev,et al.  Intra‐sarcoplasmic reticulum Ca2+ oscillations are driven by dynamic regulation of ryanodine receptor function by luminal Ca2+ in cardiomyocytes , 2009, The Journal of physiology.

[43]  Petter Laake,et al.  T‐tubule disorganization and reduced synchrony of Ca2+ release in murine cardiomyocytes following myocardial infarction , 2006, The Journal of physiology.

[44]  C. Soeller,et al.  Numerical simulation of local calcium movements during L-type calcium channel gating in the cardiac diad. , 1997, Biophysical journal.

[45]  Eric A Sobie,et al.  Quantification of repolarization reserve to understand interpatient variability in the response to proarrhythmic drugs: a computational analysis. , 2011, Heart rhythm.

[46]  Donald M Bers,et al.  Regulation of cardiac sarcoplasmic reticulum Ca release by luminal [Ca] and altered gating assessed with a mathematical model. , 2005, Biophysical journal.

[47]  Eric A Sobie,et al.  Termination of cardiac Ca(2+) sparks: an investigative mathematical model of calcium-induced calcium release. , 2002, Biophysical journal.

[48]  Wei Chen,et al.  A mathematical model of spontaneous calcium release in cardiac myocytes. , 2011, American journal of physiology. Heart and circulatory physiology.

[49]  Alan Garfinkel,et al.  So little source, so much sink: requirements for afterdepolarizations to propagate in tissue. , 2010, Biophysical journal.

[50]  Zeyun Yu,et al.  Modelling cardiac calcium sparks in a three‐dimensional reconstruction of a calcium release unit , 2012, The Journal of physiology.

[51]  R. Winslow,et al.  Mechanisms of altered excitation-contraction coupling in canine tachycardia-induced heart failure, II: model studies. , 1999, Circulation research.

[52]  M. S. Jafri,et al.  Effect of Ca2+ on cardiac mitochondrial energy production is modulated by Na+ and H+ dynamics. , 2007, American journal of physiology. Cell physiology.

[53]  M. Huertas,et al.  The dynamics of luminal depletion and the stochastic gating of Ca2+-activated Ca2+ channels and release sites. , 2007, Journal of theoretical biology.

[54]  Andrew D McCulloch,et al.  Modeling beta-adrenergic control of cardiac myocyte contractility in silico. , 2003, The Journal of biological chemistry.

[55]  W. Lederer,et al.  The control of calcium release in heart muscle. , 1995, Science.

[56]  R. Winslow,et al.  An integrated model of cardiac mitochondrial energy metabolism and calcium dynamics. , 2003, Biophysical journal.

[57]  Eric A. Sobie,et al.  Dynamics of calcium sparks and calcium leak in the heart. , 2011, Biophysical journal.

[58]  Yoram Rudy,et al.  Role of activated CaMKII in abnormal calcium homeostasis and I(Na) remodeling after myocardial infarction: insights from mathematical modeling. , 2008, Journal of molecular and cellular cardiology.

[59]  N. Macquaide,et al.  Dyssynchrony of Ca2+ release from the sarcoplasmic reticulum as subcellular mechanism of cardiac contractile dysfunction. , 2011, Journal of molecular and cellular cardiology.

[60]  Raimond L Winslow,et al.  Control and regulation of mitochondrial energetics in an integrated model of cardiomyocyte function. , 2009, Biophysical journal.

[61]  Henggui Zhang,et al.  Cardiac cell modelling: observations from the heart of the cardiac physiome project. , 2011, Progress in biophysics and molecular biology.

[62]  Ye Chen-Izu,et al.  Ca²⁺ waves in the heart. , 2013, Journal of molecular and cellular cardiology.

[63]  C W Balke,et al.  Local control of excitation‐contraction coupling in rat heart cells. , 1994, The Journal of physiology.

[64]  Eric A Sobie,et al.  A probability density approach to modeling local control of calcium-induced calcium release in cardiac myocytes. , 2007, Biophysical journal.

[65]  Yoram Rudy,et al.  Multiscale modeling of calcium cycling in cardiac ventricular myocyte: macroscopic consequences of microscopic dyadic function. , 2011, Biophysical journal.

[66]  Eric A. Sobie,et al.  Parameter sensitivity analysis of stochastic models provides insights into cardiac calcium sparks. , 2013, Biophysical journal.

[67]  Ona Z Liu,et al.  Does the Goldilocks Principle apply to calcium release restitution in heart cells? , 2012, Journal of molecular and cellular cardiology.

[68]  Eric A Sobie,et al.  Orphaned ryanodine receptors in the failing heart. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[69]  Eric A Sobie,et al.  Moment closure for local control models of calcium-induced calcium release in cardiac myocytes. , 2008, Biophysical journal.

[70]  Amrita X. Sarkar,et al.  Exploiting mathematical models to illuminate electrophysiological variability between individuals , 2012, The Journal of physiology.

[71]  Eric A Sobie,et al.  Dynamic local changes in sarcoplasmic reticulum calcium: physiological and pathophysiological roles. , 2012, Journal of molecular and cellular cardiology.

[72]  A. Tanskanen,et al.  A simplified local control model of calcium-induced calcium release in cardiac ventricular myocytes. , 2004, Biophysical journal.

[73]  D. Noble A modification of the Hodgkin—Huxley equations applicable to Purkinje fibre action and pacemaker potentials , 1962, The Journal of physiology.

[74]  R. Winslow,et al.  Cardiac Ca2+ dynamics: the roles of ryanodine receptor adaptation and sarcoplasmic reticulum load. , 1998, Biophysical journal.

[75]  B. O’Rourke,et al.  Mitochondrial Ca2+ uptake: tortoise or hare? , 2009, Journal of molecular and cellular cardiology.

[76]  Joseph L Greenstein,et al.  Role of CaMKII in RyR leak, EC coupling and action potential duration: a computational model. , 2010, Journal of molecular and cellular cardiology.

[77]  Alan Garfinkel,et al.  Spatially Discordant Alternans in Cardiac Tissue: Role of Calcium Cycling , 2006 .

[78]  Eric A Sobie,et al.  Models of cardiac excitation-contraction coupling in ventricular myocytes. , 2010, Mathematical biosciences.

[79]  Eduardo Ríos,et al.  The quantal nature of Ca2+ sparks and in situ operation of the ryanodine receptor array in cardiac cells. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[80]  George S. B. Williams,et al.  Ca2+ dynamics in the mitochondria - state of the art. , 2011, Journal of molecular and cellular cardiology.

[81]  Donald M Bers,et al.  A mathematical treatment of integrated Ca dynamics within the ventricular myocyte. , 2004, Biophysical journal.

[82]  M. Diaz,et al.  Integrative Analysis of Calcium Cycling in Cardiac Muscle , 2000, Circulation research.

[83]  Donald M Bers,et al.  How does stochastic ryanodine receptor-mediated Ca leak fail to initiate a Ca spark? , 2011, Biophysical journal.

[84]  J. Keizer,et al.  A simple numerical model of calcium spark formation and detection in cardiac myocytes. , 1998, Biophysical journal.

[85]  Denis Noble,et al.  How the Hodgkin–Huxley equations inspired the Cardiac Physiome Project , 2012, The Journal of physiology.

[86]  A. Zahradníková,et al.  Spatial and temporal Ca2+, Mg2+, and ATP2- dynamics in cardiac dyads during calcium release. , 2007, Biochimica et biophysica acta.

[87]  E. Marder,et al.  How Multiple Conductances Determine Electrophysiological Properties in a Multicompartment Model , 2009, The Journal of Neuroscience.

[88]  Eric A Sobie,et al.  Recovery of cardiac calcium release is controlled by sarcoplasmic reticulum refilling and ryanodine receptor sensitivity. , 2011, Cardiovascular research.

[89]  Clara Franzini-Armstrong,et al.  Ca2+ blinks: rapid nanoscopic store calcium signaling. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[90]  N. Trayanova Whole-heart modeling: applications to cardiac electrophysiology and electromechanics. , 2011, Circulation research.

[91]  Eric A. Sobie,et al.  Regression Analysis for Constraining Free Parameters in Electrophysiological Models of Cardiac Cells , 2009, PLoS Comput. Biol..

[92]  Donald M Bers,et al.  Calmodulin mediates differential sensitivity of CaMKII and calcineurin to local Ca2+ in cardiac myocytes. , 2008, Biophysical journal.

[93]  Joseph L Greenstein,et al.  Integrative Systems Models of Cardiac Excitation–Contraction Coupling , 2011, Circulation research.

[94]  Alan Garfinkel,et al.  Multi-scale modeling in biology: how to bridge the gaps between scales? , 2011, Progress in biophysics and molecular biology.

[95]  Gregory D Smith,et al.  Ca2+ alternans in a cardiac myocyte model that uses moment equations to represent heterogeneous junctional SR Ca2+. , 2010, Biophysical journal.