1. Seth Weinberg Acknowledgements: Xiao Wang, Yan Hao, Gregory Smith
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 – 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 vs Stratonovich 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 = L

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