Reduction of calcium release site models via optimized state aggregation

Abstract

Background: Markov chain models of calcium release sites in living cells exhibit stochastic dynamics reminiscent of the experimentally observed phenomenon of calcium puffs and sparks. Such models often take the form of stochastic automata networks in which the transition probabilities for each of a large number of intercellular channel models depend on the local calcium concentration and thus the state of nearby channels. The state-space size in such compositionally defined calcium release site models increases exponentially as the number of channels increases, which is referred to as "state-space explosion". Methods: In order to overcome the state-space explosion problem, we utilized the idea of "coarse graining" and implemented an automated procedure that reduces the state space by aggregating and lumping states of the full release site model. For a given state aggregation scheme, the transition rates between reduced states are chosen consistent with the conditional probability distribution among states within each group. A genetic algorithm-based approach is then applied to select the state aggregation schemes that lead to reduced models that approximate the observable behaviors of the full model. Results: The genetic algorithm-based approach is implemented in Matlab® and applied to two different release site models. The approach found reduced models that approximate the full model in the number of open channels, spark statistics, and the jump probability matrix as a function of time. Conclusions: A novel automated genetic algorithm-based searching technique is implemented to find reduced calcium release site models that approximate observable behaviors of the full Markov chain models that possess intractable state-spaces. As compared to the full model, the reduced models produce quantitatively similar results using significantly less computational resources.

The Physics behind Systems Biology

“…The notion of minimal models is nicely illustrated in the methodological comparison of modeling approaches of varying levels of detail in [30]. Another perspective on modeling is given in [139], where coarse-graining of the state space is used to simplify Markov-chain models of calcium dynamics.…”

“…In fact, many models in Physics, such as Newton's equations for multi-body dynamics, have suffered from expensive computations, and the Physics community has thus introduced several concepts for handling this problem, such as the well-known mean field theory for a simplified description of many interacting particles (see [138] for a review of such approaches with an emphasis on applications in Biology). One article in this topical issue introduces model reduction techniques based on optimized state aggregation for an efficient reduction of stochastic calcium release site models [139].…”