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

Stochastic modeling of calcium-regulated calcium influx and

discrete calcium ionsSeth Weinberg

Acknowledgements: Xiao Wang, Yan Hao, Gregory Smith

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

Model formulationMarkov 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 reactionsCalcium-binding to the closed channel opens the

channel

Calcium fluxes into and out of volume

Langevin formulationStochastic differential equations:

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

Sample problem

Analytical solution

Matlab simulationEuler, Milstein, stochastic RK4

Sample simulation

N = 20 channels, Wds = 10-17 L

Calcium spark scoresParameter space for one set of parameters