GPU-enabled stochastic spatiotemporal model of rat ventricular myocyte calcium dynamicsHoang-Trong Minh Tuan1, George S.B. Williams1,2, W. Jonathan Lederer2, and M. Saleet Jafri1,2

1George Mason University, Virginia, USA ; 2University of Maryland - Baltimore, Maryland, USA

Introduction

Dysfunction of normal calcium dynamics is a major factor in certain types of cardiac arrhythmias. Thesecardiac arrhythmias are thought to result from Ca2+waves which occur when Ca2+release propagates fromone calcium release unit (CRU) to another outside of the normal time during systole resulting in depolarizationof the cell’s outer membrane. Experimental evidences suggest that the local elevations of [Ca2+] at thoseCRUs - Ca2+sparks - and their recruitment results in a global [Ca2+]i transient. However, the studying ofsuch calcium dynamics at a detail whole-cell level is computational prohibitive.Such challenges of whole-cell modeling is tackled by a novel computational method with Markov-chainMonte-carlo (MCMC) simulation. This exact stochastic algorithm greatly reduces computation time by usingan adaptive time step approach and a compact-form representation of the Markov-chain state space. TheUltra-Fast MCMC method has proven its efficiency in memory usage and computational performance, par-ticularly when utilizing next generation graphics processing units (GPU) computing architecture - codenameFermi from nVidia. The authors presented an ongoing effort to study the calcium dynamics at the whole-celllevel, for the first time, that incorporates detail structure of rat ventricle myocytes, i.e. spatial organization of20,000 release sites and microsecond level resolution of clusters of ryanodine receptor type 2 (RyR2).

3D Whole-Cell Model

I Rat ventricular myocyte:I Cell dimension: 100× 20× 18(µm3) (Vcell = 36pL)I grid size: 0.2µm→ Nmesh = 4, 500, 000 mesh pointsI V T

nsr = 3.5%Vcell, V Tmyo = 50%Vcell

I 20,000 CRUs/cellI inter-CRU distances:

I 2µm along longitudinal direction (X)I 0.8µm along transversal direction (Y,Z)

CRU Model

CRU structureI A subspace: Vds = 300× 300× 15(nm3) = 1.35× 10−6pLI A junctional SR: Vjsr = 300× 300× 50(nm3) = 4.5× 10−6pLI A part of network SR: Vnsr = V T

nsr/Nmesh

I A part of cytoplasm: Vmyo = V Tmyo/Nmesh

RyR2I 45 RyR2s

I 2-state minimal modelI gating in a stochastic mannerI assumption of allosteric coupling a∗ = 7.14286× 10−2.

Fluxes & ratesI Jefflux: from subspaceI Jrefill: from local NSRI Jryr: from JSR

DiffusionI Dmyo = 300µm2.s−1 (all directions)I Dnsr = 100µm2.s−1 (all directions)

3D Spatial Monte Carlo Formulation

1 ≤ i ≤ 500, 1 ≤ j ≤ 100, 1 ≤ k ≤ 90, 1 ≤ n ≤ 20, 000

∂[Ca2+]myo

∂t= βmyo(Jefflux

V Tmyo

Vmyo− Jserca) +Dmyo∇2[Ca2+]myo (1)

∂[Ca2+]nsr

∂t= βnsr

λnsr(−Jrefill

VnsrTVnsr

+ Jserca) +Dnsr∇2[Ca2+]nsr (2)

∂[Ca2+](n)ds

∂t= βds

λds(Jryr − Jefflux) (3)

∂[Ca2+](n)jsr

∂t=βjsr

λjsr(Jrefill − Jryr) (4)

with λjsr, λds, λnsr are volume fractions, and buffering

βmyo =(

1 + BTmyoKmyom

(Kmyom +Ca2+

i )2

)−1

βds = 0.1βjsr =

(1 + BTjsrK

jsrm

(K jsrm +Ca2+

i )2

)−1

βnsr = 1The systems are solved using explicit Euler method (1st order in time, 2nd order in space).

Modeling Channels

Markov-chain modelingI Markov-chain model of a single channel is represented in the form of a state transition matrixI Based on the dependent variables, the matrix is decomposed into a number of components matrices whose

elements are rate constants

Modeling Channels

Markov-chain modeling

Figure: Markov-chain of a 2-state channel and its transition rate matrices

Transition rate matrixI Build the transition rate matrix (in compact form) for homogeneous/heterogeneous cluster

Figure: Matrices of transition rates are formulated in compact form

Benchmark Results

Benchmark system

I Intel Quad-core Nehalem dual-CPU (8MB cache,2.53GHz)

I nVidia Fermi C2050 GPU cardI 6x4GB DDR3 RAMI SATA2 (32MB buffer)

I 1sec simulation, Vm-clamp 100msI adaptive timestep: 1 µs - 10 nsI I/O: every 10 time-stepsI Data format: HDF5, Silo

Runtime

Table: Non-spatial simulation (V-clamp)

Type No I/O I/OCPU 69min 70min52secCPU-GPU 3min40sec 4m26sec

Table: Spatial simulation (Resting)

Type No I/O I/OCPU-GPU 3h9min 3h50min

Simulation results

Dynamics of dyastolic calcium sparks:

1 A closed-view of [Ca2+]ds fluctuations in thespatial models

2 Dynamics of local cytosolic calcium in themesh points containing calcium release sites.

1 Dynamics of diastolic calcium spark triggeringand termination

2 Dynamics of calcium at junctional SRs

Acknowledgement

I NSF DMS-0443843, NIH P01 HL67849, R01 HL36974, and S10 RR023028

Simulation results (cont.)

Spontaneous calcium:

3D view of dyastolic calcium dynamics in a single whole-cell slice:

Sparks propagation under calcium overload condition

Discussion

I a novel, fast computational method that can serve as a framework for any Markov-chain Monte-carlowhole-cell simulation of excitable cells was proposed

I simulation results show a similar dyastolic behavior to our whole-cell non-spatial model, with givencomponents; yet microscopic level is somehow different with many fluctuations. This can be explained by thevariation in cytosolic concentration between very small regions of microdomain levels.

I Ca2+waves are observed under calcium overloaded condition.I the full excitation-coupling whole-cell model will be incorporated for Ca2+-entrained arrhythmias study under

different pathological conditions.

http://binf.gmu.edu email:thoangtr@gmu.edu