Date Awarded


Document Type


Degree Name

Doctor of Philosophy (Ph.D.)


Applied Science


Gregory D Smith


Markov chain models have played an important role in understanding the relationship between single channel gating of intracellular calcium (Ca2+) channels, specifically 1,4,5-trisphosphate receptors (IP3Rs) and ryanodine receptors (RyRs), and the stochastic dynamics of Ca2+ release events, known as Ca2+ puffs and sparks. Mechanistic Ca2+ release site models are defined by the composition of single channel models whose transition probabilities depend on the local calcium concentration and thus the state of the other channels. Unfortunately, the large state space of such compositional models impedes simulation and computational analysis of the whole cell Ca2+ signaling in which the stochastic dynamics of localized Ca2+ release events play an important role. This dissertation introduces, implements and validates the application of several automated model reduction techniques that significantly reduce the computational cost of mechanistic compositionally defined Ca2+ release site models.;A common feature of Ca2+ channel models is the separation of time scales. For example, the well-known bell-shaped equilibrium open probability of IP3Rs can be reproduced by Markov Chain models that include transitions mediated by fast Ca2+ activation and slower Ca2+ inactivation. Chapter 2 introduces an automated model reduction technique that is based on fast/slow analysis that leverages these time scale differences. Rate constants in the single channel model are categorized as either fast or slow, groups of release site states that are connected by fast transitions are identified and lumped, and transition rates between reduced states are chosen consistent with the conditional probability distributions among states within each group. The fast/slow reduction approach is validated by the fact that puff/spark statistics can be efficiently computed from reduced Ca2+ release site models with small and transient error.;For Markov chain Ca2+ release site models without time-scale separation, the manner in which the full model states should be aggregated for optimal reduction is difficult to determine a priori. In Chapter 3, a genetic algorithm based approach that mimics the inheritance, mutation and selection processes of natural evolution is implemented to reduce these models. Given a full model of interest and target reduced model size, this genetic algorithm searches for set partitions, each corresponding to a potential scheme for state aggregation, that lead to reduced models that well-approximate the full model. A whole cell model with coupled local and global Ca2+ signaling is simplified by replacing a compositionally defined full Ca2+ release site model with a reduced model obtained through the genetic algorithm.;In Chapter 4, a Langevin formulation of Ca2+ release sites is introduced as an alternative model reduction technique that is applicable when the number of channels per Ca2+ release site is too large for the previously discussed reduction methods, but not so large that the stochasticity of Ca2+ release is negligible. The Langevin formulation for coupled intracellular Ca2+ channels results in stochastic differntial equations that well-approximate the corresponding Markov chain models when release sites possess as few as 20 channels, and the agreement improves as the number of channels per release site increases. Importantly, the computational time required by the Langevin approach does not increase with the size of Ca2+ release site.



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