modeling 3d calcium waves from stochastic calcium sparks ... · after opening, a cru undergoes a...

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

czana1@umbc.edu, ltizu@ucdavis.edu, bpeercy@umbc.edu

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.