Stochastic modeling of calcium-regulated calcium influx and discrete calcium ions Seth Weinberg...
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![Page 1: Stochastic modeling of calcium-regulated calcium influx and discrete calcium ions Seth Weinberg Acknowledgements: Xiao Wang, Yan Hao, Gregory Smith.](https://reader030.fdocuments.in/reader030/viewer/2022032600/56649db45503460f94aa55bd/html5/thumbnails/1.jpg)
Stochastic modeling of calcium-regulated calcium influx and
discrete calcium ionsSeth Weinberg
Acknowledgements: Xiao Wang, Yan Hao, Gregory Smith
![Page 2: Stochastic modeling of calcium-regulated calcium influx and discrete calcium ions Seth Weinberg Acknowledgements: Xiao Wang, Yan Hao, Gregory Smith.](https://reader030.fdocuments.in/reader030/viewer/2022032600/56649db45503460f94aa55bd/html5/thumbnails/2.jpg)
MotivationCalcium 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
![Page 3: Stochastic modeling of calcium-regulated calcium influx and discrete calcium ions Seth Weinberg Acknowledgements: Xiao Wang, Yan Hao, Gregory Smith.](https://reader030.fdocuments.in/reader030/viewer/2022032600/56649db45503460f94aa55bd/html5/thumbnails/3.jpg)
Model formulationMarkov chain model of a calcium-regulated
calcium channel
Calcium modeled by a continuous differential equation
![Page 4: Stochastic modeling of calcium-regulated calcium influx and discrete calcium ions Seth Weinberg Acknowledgements: Xiao Wang, Yan Hao, Gregory Smith.](https://reader030.fdocuments.in/reader030/viewer/2022032600/56649db45503460f94aa55bd/html5/thumbnails/4.jpg)
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
![Page 5: Stochastic modeling of calcium-regulated calcium influx and discrete calcium ions Seth Weinberg Acknowledgements: Xiao Wang, Yan Hao, Gregory Smith.](https://reader030.fdocuments.in/reader030/viewer/2022032600/56649db45503460f94aa55bd/html5/thumbnails/5.jpg)
Chemical Langevin Equation
General equation for M reactions
Two-state channel fraction of open channels
![Page 6: Stochastic modeling of calcium-regulated calcium influx and discrete calcium ions Seth Weinberg Acknowledgements: Xiao Wang, Yan Hao, Gregory Smith.](https://reader030.fdocuments.in/reader030/viewer/2022032600/56649db45503460f94aa55bd/html5/thumbnails/6.jpg)
Including discrete calcium ions
Elementary reactionsCalcium-binding to the closed channel opens the
channel
Calcium fluxes into and out of volume
![Page 7: Stochastic modeling of calcium-regulated calcium influx and discrete calcium ions Seth Weinberg Acknowledgements: Xiao Wang, Yan Hao, Gregory Smith.](https://reader030.fdocuments.in/reader030/viewer/2022032600/56649db45503460f94aa55bd/html5/thumbnails/7.jpg)
Langevin formulationStochastic differential equations:
![Page 8: Stochastic modeling of calcium-regulated calcium influx and discrete calcium ions Seth Weinberg Acknowledgements: Xiao Wang, Yan Hao, Gregory Smith.](https://reader030.fdocuments.in/reader030/viewer/2022032600/56649db45503460f94aa55bd/html5/thumbnails/8.jpg)
Integration techniquesNot so simple to integrate stochastic differential
equations! Ito vs Stratonovich calculus – different
assumptions regarding Riemman integrals, leads to different integration techniques, not equivalent
Euler method is simple
Other methods, complex to implement
![Page 9: Stochastic modeling of calcium-regulated calcium influx and discrete calcium ions Seth Weinberg Acknowledgements: Xiao Wang, Yan Hao, Gregory Smith.](https://reader030.fdocuments.in/reader030/viewer/2022032600/56649db45503460f94aa55bd/html5/thumbnails/9.jpg)
Sample problem
Analytical solution
![Page 10: Stochastic modeling of calcium-regulated calcium influx and discrete calcium ions Seth Weinberg Acknowledgements: Xiao Wang, Yan Hao, Gregory Smith.](https://reader030.fdocuments.in/reader030/viewer/2022032600/56649db45503460f94aa55bd/html5/thumbnails/10.jpg)
Matlab simulationEuler, Milstein, stochastic RK4
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Sample simulation
N = 20 channels, Wds = 10-17 L
![Page 12: Stochastic modeling of calcium-regulated calcium influx and discrete calcium ions Seth Weinberg Acknowledgements: Xiao Wang, Yan Hao, Gregory Smith.](https://reader030.fdocuments.in/reader030/viewer/2022032600/56649db45503460f94aa55bd/html5/thumbnails/12.jpg)
Calcium spark scoresParameter space for one set of parameters