In most cell types the endoplasmic reticulum (ER) has integrative and regenerative properties analogous to the excitable membranes of neurons.  Considerable insight into ER "Ca²⁺ excitability" has been obtained through the development of whole cell models of Ca²⁺ handling that are based on molecular mechanisms, e.g., single channel kinetics of the inositol 1,4,5-trisphosphate (IP₃) receptor Ca²⁺ channel.  However, a limitation of whole cell models to date is the assumption that IP₃ receptors (IP₃Rs) are globally coupled by the bulk (or average) cytosolic [Ca²⁺].  In fact, IP₃Rs are coupled locally via buffered Ca²⁺ diffusion at IP₃-sensitive Ca²⁺ release sites---clusters of IP₃Rs located in close opposition that activate and inactivate in a concerted fashion, giving rise to global Ca²⁺ responses as well as localized Ca²⁺ elevations known as "Ca²⁺ puffs."

Computational Biology research in the Smith lab focuses on the development and testing of computational models of the dynamics of Ca²⁺ handling in myocytes (muscle cells), rat basophilic leukemia cells (an experimental model for mucosal mast cells), and other cell types.  We are interested in Ca²⁺ regulation at IP₃-sensitive Ca²⁺ release sites, kinetic models of the IP₃R consistent with advancing knowledge of distinct Ca²⁺ regulation and single channel kinetics of various IP₃R subtypes, and mathematical descriptions of the buffered diffusion of intracellular Ca²⁺ that serve as the theoretical foundation for our understanding of Ca²⁺ microdomains near open Ca²⁺ channels.

(see CV for reprints)

Numerical and analytical results related to localized Ca²⁺ elevations (Ca²⁺ ``puffs'' and ``sparks'') viewed as a reaction-diffusion problem. 

Punctate releases of Ca²⁺, called Ca²⁺ sparks, originate at the regular array of t-tubules in cardiac myocytes.  In collaboration with the Jon Lederer (Medical Biotechnology Center, UMBI; School of Medicine, UMAB) and co-workers, I have developed a simple numerical model of Ca²⁺ spark formation and detection in this cell type. The model involves the numerical solution to a set of reaction-diffusion equations that incorporate Ca²⁺ release, cytosolic diffusion, and resequestration by SR Ca²⁺-ATPases, as well as the association and dissociation of Ca²⁺ with endogenous Ca²⁺-binding sites and the diffusible indicator dye fluo-3. A realistic feature of these calculations is that the time-dependent spatial profile of fluorescence is convoluted with a point spread function to simulate the contribution of out-of-focus signal [with JE Keizer, MD Stern, WJ Lederer, and H Cheng].

Models of RyR-mediated Ca²⁺-induced Ca²⁺ release and the spark-to-wave transition in cardiac myocytes.

During Ca²⁺ overload, Ca²⁺ sparks serve as sites for the initiation of Ca²⁺ waves in myocytes. In collaboration with JE Keizer, S Ponce-Dawson, and J Pearson, I have carried out computer simulations of spark-induced waves to explore the influence of the regular array of release sites on their propagation. These computer simulations with a linear array of Ca²⁺ release sites give a wave speed proportional to the Ca²⁺ diffusion constant rather than its square root, as is true for classical traveling waves in an excitable medium. Future work includes extending this model to three dimensions and developing a stochastic description of saltatory Ca²⁺ wave propagation.

Singular and regular perturbation analysis of reaction-diffusion equations associated with the buffered diffusion of Ca²⁺.

The `domain' Ca<sup>2+</sup> concentration near an open Ca<sup>2+</sup> channel can be estimated by obtaining hemispherically symmetric steady-state solutions to a reaction-diffusion formulation for the buffered diffusion of intracellular Ca<sup>2+</sup>. After nondimensionalizing these equations and scaling space so that both reaction terms and the source amplitude are order one, we identify two dimensionless parameters (eps_c and eps_b) that correspond to the scaled mobility of dimensionless Ca<sup>2+</sup> and buffer, respectively.    Using perturbation methods, we derive second order approximations for the Ca<sup>2+</sup> and buffer profiles in three asymptotic limits. 1) An `excess buffer approximation' (EBA), where the mobility of buffer exceeds that of Ca²⁺, and the fast diffusion of buffer toward the Ca²⁺ channel prevents buffer saturation (cf. Neher, 1986).  2) A `rapid buffer approximation' (RBA), where the diffusive time scale for Ca<sup>2+</sup> and buffer are comparable, but slow compared to reaction, resulting in saturation of buffer near the Ca<sup>2+</sup> channel (cf. Wagner and Keizer, 1994; Smith, 1996). 3) A recently identified (nearly) `immobile buffer approximation' (IBA), where the diffusion of Ca²⁺ is fast compared to buffer [with L Dai, R Miura, and A Sherman].

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