Mathematicalmodelsofcalciumrelease sites derived fromMarkovchainmodelsof intracellularcalciumchannels exhibit collective gating reminiscent of the experimentally observed phenomenon ofcalciumpuffs andsparks.Suchmodelsoften take the form of stochastic automata networks in which the transition probabilities of each channel depend on the localcalciumconcentration and thus the state of the other channels. In order to overcome the state-spaceexplosionthat occurs in such compositionally definedcalciumrelease sitemodels,we have implemented several automated procedures formodelreduction using fast/slow analysis. After categorizing rate constants in the single channelmodelas either fast or slow, groups of states in the expanded release sitemodelthat are connected by fast transitions are lumped, and transition rates between reduced states are chosen consistent with the conditionalprobability distributionamong states within each group. For small problems these conditionalprobability distributionscan be numerically calculated from the fullmodelwithout approximation. For large problems the conditionalprobability distributionscan be approximated without the construction of the fullmodelby assuming rapid mixing of states connected by fast transitions. Alternatively, iterative aggregation/disaggregation may be employed to obtain reducedcalciumrelease sitemodelsin a memory-efficient fashion. Benchmarking of several different iterative aggregation/disaggregation-based fast/slow reduction schemes establishes the effectiveness of automatedcalciumrelease site reduction utilizing the Koury–McAllister–Stewart method. Mathematical modeling has played an important role in understanding the relationship between single channel gating of intracellular calcium(Ca2+)'>(Ca 2+ ) (Ca2+) channels and the stochastic dynamics of Ca2+'>Ca 2+ Ca2+ release events known as Ca2+'>Ca 2+ Ca2+ puffs and sparks. Ca2+'>Ca 2+ 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. Because the large state space of such models impedes computational analysis of the dynamics of Ca2+'>Ca 2+ Ca2+ release sites, we implement and validate the application of several automated model reduction techniques that leverage separation of time scales, a common feature of single channel models of inositol 1,4,5-trisphosphate receptors (IP3Rs)'>(IP 3 Rs) (IP3Rs) and ryanodine receptors (RyRs). The authors show for the first time that memory-efficient iterative aggregation/disaggregation (IAD)-based numerical schemes are effective for fast/slow reduction in compositionally defined Ca2+'>Ca 2+ Ca2+ release sitemodels.
Dynamics—A Change of Modeling Paradigm?
Hao, Yan; Kemper, Peter; and Smith, Gregory D., Reduction of calcium release site models via fast/slow analysis and iterative aggregation/disaggregation (2009). Dynamics—A Change of Modeling Paradigm?, 19(037107).