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Modeling 3D Calcium Waves from StochasticCalcium sparks in a Sarcomere Using COMSOL
Multiphysics c©
Zana Coulibaly1, Leighton Izu2, and Bradford Peercy1
1 Department of Mathematics and StatisticsUniversity of Maryland, Baltimore County
2 Department of PharmacologyUniversity of California Davis
Biological Background and Motivation Mathematical Model Implementation Results References
Cardiac Myocyte Schematic
(a) Heart muscle. (b) Heart tissues.
(c) Sarcomere schematics. (d) Calcium dynamics. [Bers, 2002]
Figure 1 : Cardiac cell schematic.
Biological Background and Motivation Mathematical Model Implementation Results References
Goal
Our goal is to model calcium dynamics in a sarcomere. Thisinvolves:
modeling the random release of calcium from calciumrelease units (CRU),
modeling the diffusion and interaction of calcium withother chemical species,
changing the distribution of CRUs on a z-disc.
Biological Background and Motivation Mathematical Model Implementation Results References
Diffusion, release and buffer equations
Main equations [Izu 2001, Gobbert 2008]
∂c
∂t= ∇ · (Dc∇c)− Jpump + Jleak + Jrelease +
∑i
Ri(c, bi, Bi),
∂bi∂t
= ∇ · (Dbi∇bi) +Ri(c, bi, Bi),
∂Bi
∂t= ∇ · (DBi∇Bi)−Ri(c, bi, Bi).
Buffer reaction
c+ bi
k+i
⇀↽k−i
Bi.
Stochastic Release Mechanism
Jrelease(c,x) =∑j
σ(t, Tmj )S(t;Topen)δ(x− x̂j),
S(t, Topen) =
{1 if α ≤ Jprob(c),0 if α > Jprob(c),
(1)
Jprob(c) = Pmaxcm
Kmprob + cm
, α ∼ U [0, 1].
⇒ A CRU is allowed to open every ms. After opening, a CRUundergoes a refractory period of about 100 ms.
Biological Background and Motivation Mathematical Model Implementation Results References
Comsol-Matlab Implementation
(a) CRU spacing [Izu 2006]. (b) CRU spacing in simulations. (c) Release current [Soeller 2012].
z-disc 2
z-disc 1
(d) The sarcomere domain. (e) Programming logic.
Biological Background and Motivation Mathematical Model Implementation Results References
Stochastic sparks organizing into a wave
Biological Background and Motivation Mathematical Model Implementation Results References
Forced sparks organizing into a wave
Biological Background and Motivation Mathematical Model Implementation Results References
References
Izu et al., Evolution of Cardiac Calcium Waves fromStochastic Calcium Sparks , Biophysical Journal Volume80, 2001.
Bers, Cardiac Excitation-contraction Coupling, NatureVolume 45, 2002.
Izu et al., Three-Dimensional Distribution of RyanodineReceptor Clusters in Cardiac Myocytes, BiophysicalJournal Volume 91, 2006.
Gobbert, Long-time Simulations on High ResolutionMeshes to Model Calcium Waves in a Heart Cell, SIAMJournal of Scientific Computing, 2008.