Langevin, population density and moment-based modeling of local and global aspects of intercellular calcium signaling

Markov chain models of the coupled gating of intracellular calcium (Ca2+) channels are often used to study the stochastic dynamic of local Ca2+ release events and whole cell Ca2+ homeostasis. However, the runtime of the Markov chain description of Ca2+ channel gating is exponential in the number of Ca2+ channel states and may thus result in a combinatorial state space explosion when the number of channel states is large. This dissertation presents several novel stochastic modeling approaches th a t capture im portant aspects of Ca2+ signaling while improving com putational efficiency. This dissertation presents several novel stochastic modeling approaches th a t capture im portant aspects of calcium Ca2+ signaling. First, we present a Ca2+ release site modeling approach based on a Langevin description of stochastic Ca2+ release. This Langevin model facilitates our investigation of correlations between successive puff/spark amplitudes, durations and inter-spark intervals, and how such puff/spark statistics depend on the number of channels per release site and the kinetics of Ca2+-m ediated inactivation of open channels. Second, we show th a t when the Ca2+ channel model is minimal, Langevin equations in a whole cell model involving a large number of release sites may be replaced by a single Fokker-Planck equation. This yields an extremely compact and efficient local/global whole cell model th a t reproduces and helps interpret recent experiments investigating Ca2+ homeostasis in permeabilized ventricular myocytes. Last but not least, we present a population density and moment-based approach to modeling L-type Ca2+ channels. Our approaches account for the effect of heterogeneity of local Ca2+ signals on whole cell Ca currents. Moreover, they facilitate the study of domain Ca-mediated inactivation of L-type Ca channels. CONTENTS Acknowledgments. iv D edication . v Chapter 1 In tro d u c tio n . 1 1.1 Calcium h o m e o s ta s is . 2 1.2 Calcium c h a n n e ls . 4 1.3 Local and global calcium signaling . 6 1.4 M athem atical model of calcium s ig n a lin g . 7 1.4.1 Markov chain model of calcium channel g a t in g . 7 1.4.2 M athematical model of calcium h o m e o s ta s is . 9 2 Calcium-activation kinetics modulate successive puff/spark amplitude, duration and inter-event-interval correlations in a Langevin model of stochastic calcium release 11 2.1 In tro d u c tio n . 11 2.2 Model formulation . 14 2.2.1 Markov chain model of a calcium release s i t e . 14 2 . 2.2 The Langevin description of a calcium release s i t e . 15 2.3 R e s u lts . 16 2.3.1 Representative Langevin simulations . 17 2.3.2 Validation of Langevin ap p ro ach . 20 2.3.3 Analysis of spark s ta t is t ic s . 21 2.4 D iscu ssio n . 23 2.4.1 Comparison to e x p e r im e n t. 24 2.4.2 Comparison to prior theoretical w o r k . 26 2.4. 3 Advantages and limitations of the Langevin a p p ro a c h . 27 2.5 Appendix . 28 2.5. 1 Derivation of the Langevin CaRU m o d e l . 28 2.5. 2 Langevin equation boundary conditions . 29 2.5. 3 Numerical scheme of the Langevin m o d e l . 30 3 Calcium homeostasis in a local/global whole cell model of permeabilized ventric­ ular m y o c y te s . 32 3.1 In tro d u c tio n . 32 3.2 Model formulation . 35 3.2.1 Markov chain description of a calcium release s i t e . 35 3.2.2 Langevin calcium release site m o d e l . 36 3.2.3 Equivalence of Markov chain and Langevin formulations . 38 3.2.4 Full local/global whole cell m o d e l . 40 3.2.5 Reduced local/global whole cell model . 43 3.2. 6 Fokker-Planck local/global whole cell m o d e l . 44 3.3 R e s u lts . 46 3.3.1 Calcium homeostasis in the local/global whole cell m o d e l 46 3.3.2 Sparkand non-spark-mediated SR calcium release . 51 3.4 D iscu ssio n . 55 3.4.1 Summary of main f in d in g s . 55 3.4.2 Physiological significance. 57 3.4. 3 Comparison to other whole cell m o d e ls . 58 3.4. 4 Limitations and extensions of the model . 60 3.5 Appendix . 61 3.5. 1 Comparison of Markov chain and Langevin CaRU m o d e ls . 61 3.5. 2 Calcium release flux and CaRU s i z e . 63 4 A population density and moment-based approach to modeling domain calciumm ediated inactivation of L-type calcium c h a n n e ls . 65 4.1 In tro d u c tio n . 65 4.2 D iscu ssio n . 84