Post on 23-Dec-2015
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