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

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

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

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

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

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

Langevin formulation Stochastic differential equations:

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

Sample problem Analytical solution

Matlab simulation Euler, Milstein, stochastic RK4

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

Calcium spark scores Parameter space for one set of parameters