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

Including discrete calcium ions

Elementary reactionsCalcium-binding to the closed channel opens the

channel

Calcium fluxes into and out of volume

Langevin formulationDifferential equations:

Integration and evaluationEuler-Maruyama method, fixed time step

Spark score statistics

Deterministic systemSpontaneous activity

Not a triggered responseNo sparks in the deterministic system

Stochastic systemvrel = 0.1 ms-1 Score

Stochastic systemvrel = 0.2 ms-1 Score

Stochastic System vrel = 0.1 ms-1 0.15 ms-1

0.2 ms-1 0.3 ms-1

Implications/conclusionsCalcium fluctuations

Can “induce” calcium sparks under conditions of small calcium release

Can “suppress” calcium sparks under conditions of greater calcium release

Necessary to include the effects of the discrete calcium ions Continuous description of calcium inaccurate

Future goalsIncorporate more biophysically detailed channel

modelsCalcium-activation, -inactivation; calsequestrin,

calmodulin regulationSpatial coupling between small volumes (10,000

dyadic subspaces)Multiscale modeling – brings a new set of

challenges

Moving towards multiscale modeling

Langevin formulationComputationally fast,

compared with SSA, tau-leaping

Slow, compared with deterministic

Appropriate algorithm depends on time-step, propensity functionLangevin

inappropriate in some cases, unnecessary in others

Gillespie, 2007

Adaptive modelingPropensity functions vary throughout the simulation

Recall Ca2+ concentration ranges from 0.1 – 100 uM Ideally, algorithm could vary depending current state

Further, could make time-step as large as possible

Future work in multiscale modeling can utilizing these efficient algorithms

Partition-leaping (Harris, 2006)

Thank you!