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Stochastic modeling of calcium-regulated calcium influx and discrete calcium ions

Stochastic modeling of calcium-regulated calcium influx and discrete calcium ions. Seth Weinberg Acknowledgements: Xiao Wang, Yan Hao , Gregory Smith. Motivation. Calcium plays a key role in regulating cell signaling processes, such as myocyte contraction and synaptic transmission

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Stochastic modeling of calcium-regulated calcium influx and discrete calcium ions

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  1. Stochastic modeling of calcium-regulated calcium influx and discrete calcium ions Seth Weinberg Acknowledgements: Xiao Wang, Yan Hao, Gregory Smith

  2. Motivation • Calcium plays a key role in regulating cell signaling processes, such as myocyte contraction and synaptic transmission • Due to the small number of channels in a release site (~20 – 100), stochastic fluctuations can influence overall dynamics • Resting concentrations 100 nM and subspace volumes on the order of 10-17– 10-16 L • ~0.6 – 6 calcium ions • Hypothesis: Fluctuations due to small number of ions can also influence dynamics, perhaps induce sparks

  3. Model formulation • Markov chain model of a calcium-regulated calcium channel • Calcium modeled by a continuous differential equation

  4. Complications using Markov chains • For N channels and M states per channel • b(20, 2) = 21 • b(20, 3) = 231 • b(20, 4) = 1771 • b(20, 12) = 5.7e8

  5. Chemical Langevin Equation • General equation for M reactions • Two-state channel fraction of open channels

  6. Including discrete calcium ions • Elementary reactions • Calcium-binding to the closed channel opens the channel • Calcium fluxes into and out of volume

  7. Langevin formulation • Stochastic differential equations:

  8. Integration techniques • Not so simple to integrate stochastic differential equations! • Ito vsStratonovich calculus – different assumptions regarding Riemman integrals, leads to different integration techniques, not equivalent • Euler method is simple • Other methods, complex to implement

  9. Sample problem • Analytical solution

  10. Matlab simulation • Euler, Milstein, stochastic RK4

  11. Sample simulation N = 20 channels, Wds = 10-17 L

  12. Calcium spark scores Parameter space for one set of parameters

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