Abstract
A stochastic reaction network model of Ca2+ dynamics in synapses (Pepke et al PLoS Comput. Biol. 6 e1000675) is expressed and simulated using rule-based reaction modeling notation in dynamical grammars and in MCell. The model tracks the response of calmodulin and CaMKII to calcium influx in synapses. Data from numerically intensive simulations is used to train a reduced model that, out of sample, correctly predicts the evolution of interaction parameters characterizing the instantaneous probability distribution over molecular states in the much larger fine-scale models. The novel model reduction method, ‘graph-constrained correlation dynamics’, requires a graph of plausible state variables and interactions as input. It parametrically optimizes a set of constant coefficients appearing in differential equations governing the time-varying interaction parameters that determine all correlations between variables in the reduced model at any time slice.
Highlights
Given a stochastic reaction network, even one specified by high-level “parameterized reactions” or “rule-based” notation [2,3,4,5,6], there is a corresponding Chemical Master Equation (CME) for the evolution of probability distributions over all possible molecular states of the system
The degree of model reduction obtained by GraphConstrained Correlation Dynamics” (GCCD) in the calmodulin-dependent protein kinase II (CaMKII) example is large
In the MCell simulations we may conservatively count it as the integer-valued populations of the following classes of molecular species: free Ca2+, free CaM (3 × 3 + 1 = 10 species), monomeric CaMKII subunit which can bind CaM in any of its states and can be phosphorylated (3 × 3 × 2 = 18 species), and dimerize if at most one subunit is phosphorylated (9 × 9 + 9 × 10/2 = 126 species; phosphorylated dimers dissociate before they can doubly phosphorylate) for a total of 155 species each of which has a dynamical random variable, and a total reduction from 155 to just 8 dynamical variables, which is very large
Summary
Given a stochastic reaction network, even one specified by high-level “parameterized reactions” or “rule-based” notation [2,3,4,5,6], there is a corresponding Chemical Master Equation (CME) for the evolution of probability distributions over all possible molecular states of the system. These states are described in terms of discrete-valued random variables. The sampling approach requires a lot of computing power to sample enough trajectories, and poses substantial obstacles for analysis
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