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Numerical Approach to Spatial Deterministic-Stochastic Models Arising in Cell Biology

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  • James C Schaff
  • Fei Gao
  • Ye Li
  • Igor L Novak
  • Boris M Slepchenko

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.Author Summary: Mechanisms of some cellular phenomena involve interactions of molecular systems of which one can be described deterministically, while the other is inherently stochastic. 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. Fortunately, such systems can be efficiently solved by hybrid methods in which deterministic and stochastic algorithms are appropriately integrated. Here we describe fundamentals of a general-purpose deterministic-stochastic method for simulating spatially resolved systems. The internal workings of the method are explained and illustrated by applications to very different phenomena such as calcium ‘sparks’, stochastically gated reactions, and spontaneous cell polarization.

Suggested Citation

  • James C Schaff & Fei Gao & Ye Li & Igor L Novak & Boris M Slepchenko, 2016. "Numerical Approach to Spatial Deterministic-Stochastic Models Arising in Cell Biology," PLOS Computational Biology, Public Library of Science, vol. 12(12), pages 1-23, December.
    • Handle: RePEc:plo:pcbi00:1005236
    DOI: 10.1371/journal.pcbi.1005236
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    References listed on IDEAS

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    1. Steven J. Altschuler & Sigurd B. Angenent & Yanqin Wang & Lani F. Wu, 2008. "On the spontaneous emergence of cell polarity," Nature, Nature, vol. 454(7206), pages 886-889, August.
    2. Steven S Andrews & Nathan J Addy & Roger Brent & Adam P Arkin, 2010. "Detailed Simulations of Cell Biology with Smoldyn 2.1," PLOS Computational Biology, Public Library of Science, vol. 6(3), pages 1-10, March.
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    Cited by:

    1. Sabir, Zulqurnain & Wahab, Hafiz Abdul & Umar, Muhammad & Sakar, Mehmet Giyas & Raja, Muhammad Asif Zahoor, 2020. "Novel design of Morlet wavelet neural network for solving second order Lane–Emden equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 172(C), pages 1-14.

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