Abstract
Hybrid deterministic-stochastic methods provide an efficient alternative to a fully stochastic treatment of models which include components with disparate levels of stochasticity. However, general-purpose hybrid solvers for spatially resolved simulations of reaction-diffusion systems are not widely available. Here we describe fundamentals of a general-purpose spatial hybrid method. The method generates realizations of a spatially inhomogeneous hybrid system by appropriately integrating capabilities of a deterministic partial differential equation solver with a popular particle-based stochastic simulator, Smoldyn. Rigorous validation of the algorithm is detailed, using a simple model of calcium ‘sparks’ as a testbed. The solver is then applied to a deterministic-stochastic model of spontaneous emergence of cell polarity. The approach is general enough to be implemented within biologist-friendly software frameworks such as Virtual Cell.
Highlights
It is not uncommon for a cell-biological model to include some components that might be stochastic in nature, whereas others, if uncoupled, would behave deterministically
Calcium ‘sparks’ in cardiomyocytes is one such example, in which dynamics of calcium ions, which are usually present in large numbers, can be described deterministically, whereas the channels open and close stochastically
The calcium influx through the channels renders the entire system stochastic, but a fully stochastic treatment accounting for each calcium ion is computationally expensive
Summary
It is not uncommon for a cell-biological model to include some components that might be stochastic in nature (small copy numbers, rare events), whereas others, if uncoupled, would behave deterministically (large copy numbers, fast reactions). Through their interaction, fluctuations in a stochastic subsystem may induce significant random perturbations in the ‘deterministic’ one, rendering the entire system stochastic. Stochastic openings of voltage-sensitive ion channels depend on the ‘deterministic’ membrane potential, which, in turn, is affected by stochastic electric currents passing through the channels [2]. Stochasticity in otherwise deterministic cellular subsystems may be brought about by their coupling to dynamics of cytoskeletal filaments, translation events, and other processes involving macromolecules and small organelles present in small numbers
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