Calcium Dynamics

Intracellular Ca²⁺ signals connect molecular events at ion channels to cell-scale physiological responses. This past project studied the mathematics and computation of Ca²⁺ diffusion, buffering, stochastic release, and whole-cell signaling in excitable and non-excitable cells.

Rapid Buffering and Domains

Early work focused on local Ca²⁺ domains near open channels. An analytical steady-state solution of the rapid buffering approximation gave an upper bound on local Ca²⁺ elevations when the approximation is valid, and numerical and asymptotic analyses clarified the relationship between rapid buffering and excess-buffer approximations.

This line of work contributed to broader modeling approaches for intracellular Ca²⁺ waves, sparks, local domains, and reaction-diffusion systems.

Puffs, Sparks, and Waves

In collaboration with experimental groups studying cardiac myocytes, the project developed numerical models of Ca²⁺ spark formation and detection, including the mobility of fluorescent indicator dyes. Related work studied the spark-to-wave transition in Ca²⁺-overloaded cardiac cells and showed how saltatory wave propagation differs from classical traveling waves in excitable media.

The lab also modeled stochastic Ca²⁺ puffs and sparks using Markov chain models of coupled intracellular Ca²⁺ channels. These studies examined release-site termination mechanisms including stochastic attrition, Ca²⁺-dependent inactivation, allosteric interactions, and release-site ultrastructure.

Multiscale Models

Later work developed and benchmarked computational methods for large Markov models of Ca²⁺-regulated channels, including approaches that exploit Kronecker structure. Whole-cell modeling included minimal and realistic models of Ca²⁺ responses, IP₃ receptor dynamics, and local-control models of cardiac excitation-contraction coupling.

A major theme was the use of probability-density and moment-closure methods to represent heterogeneous local Ca²⁺ signals efficiently while retaining the stochastic features needed for biologically realistic models.

Selected Publications

• Smith GD. Analytical steady-state solution to the rapid buffering approximation near an open Ca²⁺ channel. Biophys. J. 71(6):3064-3072, 1996. [doi:10.1016/S0006-3495(96)79500-0] [PMID:8968577]
• Smith GD, Wagner J, and Keizer J. Validity of the rapid buffering approximation near a point source of Ca²⁺ ions. Biophys. J. 70(6):2527-2539, 1996. [doi:10.1016/S0006-3495(96)79824-7] [PMID:8744292]
• Smith GD, Keizer J, Stern M, Lederer WJ, and Cheng H. A simple numerical model of Ca²⁺ spark formation and detection in cardiac myocytes. Biophys. J. 75(7):15-32, 1998. [doi:10.1016/S0006-3495(98)77491-0] [PMID:9649364]
• Smith GD, Dai L, Miura R, Sherman A. Asymptotic analysis of equations for the buffered diffusion of intracellular Ca²⁺. SIAM. J. Appl. Math. 61(5):1816-1838, 2001. [JSTOR]
• Williams GSB, Huertas MA, Sobie EA, Jafri MS, and Smith GD. A probability density approach to modeling local control of Ca²⁺-induced Ca²⁺ release in cardiac myocytes. Biophys. J. 92(7):2311-28, 2007. [doi:10.1529/biophysj.106.099861] [PMID:17237200]
• Thul R, Smith GD, Coombes S. A bidomain threshold model of propagating calcium waves. J. Mathematical Biology 56(4):435-63, 2008. [doi:10.1007/s00285-007-0123-5]
• Groff JR, DeRemigio H, and Smith GD. Markov chain models of ion channels and Ca²⁺ release sites. In: Stochastic Methods in Neuroscience. Laing C and Gabriel L, eds. Pages 29-64. Oxford University Press. 2009. [doi:10.1093/acprof:oso/9780199235070.003.0002]