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. 2016 Dec 13;12(12):e1005236.
doi: 10.1371/journal.pcbi.1005236. eCollection 2016 Dec.

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

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Free PMC article

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

James C Schaff et al. PLoS Comput Biol. .
Free PMC article

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.

Conflict of interest statement

The authors have declared that no competing interests exist.

Figures

Fig 1
Fig 1. Test with a separable stochastic subsystem.
(A) Channel arrangement (upper panel) and a snapshot of simulation results for U(r,t) at t = 1 s (lower panel). (B) The multi-trial mean, <|Ωcell|1ΩcellU(r,t)dr>, converges to an exact expectation value with the increasing number of trials and decreasing time step Δt (data points for Δt = 0.02 s, 0.006 s, and 0.002 s are shown as triangles, squares, and circles, respectively). Inset: convergence with respect to Δt (shown for N = 104).
Fig 2
Fig 2. Time-dependent solutions of a fully coupled problem with large D.
Results from VCell hybrid (dots) are validated against a reference solution of the corresponding well-mixed system obtained by Gibson-Bruck nonspatial solver (solid line). Probability density functions of U and probability distributions of the numbers of open channels in insets (black columns for VCell hybrid, white columns for Gibson-Bruck) are based on 10,000 realizations by each solver and shown for three time points, t = 1 s, 2 s, 3 s.
Fig 3
Fig 3. Steady-state solution of a fully coupled system in the limit of large D.
As in Fig 2, the near steady-state solution of a fully coupled system was obtained by VCell hybrid for the fast-diffusion limit and validated against Gibson-Bruck stochastic nonspatial solver. Results from each solver are based on 20,000 realizations and shown for t = 30 s, the time long enough for the system to become sufficiently close to its steady state.
Fig 4
Fig 4. Spatial hybrid vs. Fokker-Planck formulation in the limit of large D.
Probability density function of dimensionless continuous variable ρ is shown for t = 30 s, sufficient for accurate approximation of the system’s steady state. Results from VCell hybrid (dots) are based on 10,000 realizations, and Fokker-Planck equations (Eq (4)) were solved by VCell fully implicit advection-diffusion solver with mesh size Δρ/ρmax = 0.004 (solid curve). The L2-norm of the difference between the two solutions is 0.0137.
Fig 5
Fig 5. Validation of spatially inhomogeneous solutions (finite D), imax = 2.
Solutions for probability density function p(ρ) are shown for τ = 1. Results from VCell hybrid (triangles for i = 0 and circles for i = 1) are based on 12,500 realizations. Solutions of the corresponding functional Fokker-Planck equation (solid curves) were obtained with Δρ = 1.25e-3. The L2-norms of the differences of the two solutions are ≈ 1.9% (i = 0) and 3.3% (i = 1) of the respective maximum values.
Fig 6
Fig 6. Validation of spatially inhomogeneous solutions (finite diffusion), imax = 3.
Solutions for probability density function p(ρ) are shown for τ = 1. Results from VCell hybrid for positions near the channel (i = 0), squares), away from the channel (i = 2, circles) and in between (i = 1, triangles) are based on 12,500 realizations. The curves are extrapolations to Δρ = 0 of numerical solutions of the corresponding Fokker-Planck equation, computed with Δρs ∝ (1.5)-s 0, s = 0,1,…,5. The L2-norms of the differences of the two solutions are ≈ 1.3% (i = 0 and i = 1) and 1.5% (i = 2) of the respective maximum values.
Fig 7
Fig 7. Hybrid solution of a system of stochastically-gated reactions, Eqs (6 and 7).
For t >> τD, the probability that a macro-molecule remains unbound (dots with error bars) deviates from an exponential and approaches the power-law predicted in [53] (dashed curve). Inset: initial exponential decay of the relaxation function obtained with h = 0.2 μm (solid curve) and h = 0.034 μm (dashed curve).
Fig 8
Fig 8. Cell polarization: coalescence of a multi-cluster system into a single cluster.
Distributions of proteins recruited to the membrane from the interior (top) and active receptors (bottom), obtained by solving the model of Eqs (8–11) with VCell spatial hybrid for parameters specified in the text. Local surface densities increase in large clusters, as the number of clusters diminishes (see color scales).

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References

    1. Rüdiger S (2014) Stochastic models of intracellular calcium signals. Phys Rep 534: 39–47.
    1. Schneidman E, Freedman B, Segev I (1998) Ion channel stochasticity may be critical in determining the reliability and precision of spike timing. Neural Comput 10: 1679–1703. - PubMed
    1. Pahle J (2009) Biochemical simulations: stochastic, approximate stochastic and hybrid approaches. Brief Bioinform 10: 53–64. - PMC - PubMed
    1. Choi T, Maurya MR, Tartakovsky DM, Subramaniam S (2010) Stochastic hybrid modeling of intracellular calcium dynamics. J Chem Phys 133: 165101. - PMC - PubMed
    1. Gardiner C (2004) Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Third Edition: Springer-Verlag.

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