Table 1.
Computational approaches for simulating spatial and temporal stochasticity in biochemical reaction networks.
Additional approaches not listed here are referenced via reviews in the text.
Fig 1.
Computational modeling schematics.
(A) Molecules at the cell membrane and a thin slice of adjacent cytoplasm are simulated explicitly. Both compartments are modeled as 2D surface. (B) In the quasi-3D simulations, a well-stirred compartment representing bulk cytoplasm is added to approximate 3D effects. (C) Reaction scheme for a Turing-type model of Cdc42-dependent yeast polarity establishment from Goryachev and Polkhilko [4].
Fig 2.
Illustration of the different reaction regimes (reaction-limited, diffusion-influenced, and diffusion-limited regimes) and the range of validity of the 2D theory.
The left panel shows estimated rate constants (yellow diamonds) for the 2D second order reaction A+B → C obtained by fitting chemical rate equations (black curves, right panels) to results from particle-based simulations (yellow curves, right panels). The reaction limit, , is indicated by the black dot-dashed line and the estimate for the diffusion limit
is represented by the red curve. The results from the 2D
theory are shown as the green-dashed line. Parameters chosen are DA = DB = ½D (x-axis of left panel), λ = 2.5554 s-1, rmax = 2.5 μm,
= 0.05 μm. Simulations were conducted on a L = 5 μm domain.
Fig 3.
Empirical estimates for macroscopic rate constants in the yeast polarity model for the two different parameter sets.
Results from particle-based simulations that include membrane exchange are shown as yellow curves. Fits to the simulation results using appropriate rate equations are shown as black curves. Top row, parameters used for purely 2D simulation. Bottom row, parameters used for quasi-3D simulation. The rate constants are reported in Table 2.
Table 2.
Microscopic parameters and effective macroscopic rate constants for reversible/irreversible bimolecular reactions of the form A + B ↔ C.
Fig 4.
Simulations of polarity establishment within the Turing unstable regime.
Snapshots of total Cdc42-GTP (both Cdc42-GTP and Bem1-GEF-Cdc42-GTP). Top: Particle-based simulations. Red dots represent individual molecules. Bottom: Reaction-diffusion partial differential equation simulations. (A) Individual molecules in particle-based simulations, and individual pixels in 100x100 grid RDE simulations. Scale bar, 0.5 μm. (B) To compare the polarity patches, 2D histograms of the final polarized states were computed, where both distributions were binned on coarsened 20x20 grids.
Fig 5.
Variability in 2D polarization from microscopic fluctuations.
(A) Measurements of H(r = 0.5 μm) at 10 second intervals across n = 5 particle-based simulation realizations. (B) Measurements of H(r = 0.5 μm) across the corresponding RDE simulations. (C) The pairwise distance distribution P(r) and our polarity metric H(r) for polarized (red) and uniform (black) particle distributions. (D) Snapshots of total Cdc42-GTP for each particle-based realization at t = 100 and t = 200 seconds. In some cases, particle coordinates were re-centered after simulation to keep polarity patches from visually wrapping around to the other side of the periodic domain. Corresponding RDE snapshots are in Fig H in S1 Text. Scale bars 0.5 μm.
Fig 6.
Stochasticity facilitates polarization.
Bifurcation plots showing polarization, measured by H(r = 0.5 μm), versus parameters influencing the total particle numbers in the simulation. (A) Varying Cdc42 concentration on a fixed domain. (B) Varying GEF concentration on a fixed domain. (C) Varying the simulation area and particle numbers at constant concentrations. Bifurcations were found via linear stability analysis of the deterministic RDEs.
Fig 7.
Reservoir approach schematics and validation.
(A) Molecules can diffuse in and out of the reservoir. Although distinct molecules are shown for illustration, the reservoir is perfectly mixed. (B) Particles at a depth z must diffuse a distance of either zimpl—z to enter, or z–zimpl to exit, the explicit simulation domain. The integrals are solved numerically over discrete slices with thickness Δz. (C) Time courses of the number of molecules in the explicit domain, comparing our approach and a non-reactive Brownian dynamics simulation. The shaded regions represent the mean±1 S.D. over 5 realizations. (D) Time-averaged comparisons, mean±1 S.D. of fluctuations, over 500s, 1 realization.
Fig 8.
Quasi-3D particle-based simulations of the polarity establishment model.
Shown are snapshots of total Cdc42-GTP. Scale bar, 1.0 μm. Corresponding 2D histograms of the local number of molecules are shown.
Fig 9.
Quantitative comparisons of polarization in quasi-3D particle-based simulations and corresponding RDEs.
Top: time courses of H(r = 2 μm). Results across multiple realizations of [Cdc42] = 0.150 μM are shown. Bottom: Plots of H(r) at final time points. By 1800s, the q3D-RDEs did not fully polarize, so the H(r) starting from a pre-polarized distribution is shown instead.
Fig 10.
The effect of Cdc42 concentration on polarization for quasi-3D particle-based simulations.
A bifurcation diagram comparing polarity, measured via H(r = 2 μm), in the particle-based and reaction-diffusion simulations as a function of Cdc42 concentration. Simulations with pre-polarized RDEs were used to identify an estimated range for the bifurcation point. All other points are given by the mean±1s.d. (n = 5 realizations, except for t = 600s particle-based simulations at [Cdc42] = 0.150 μM, n = 3, and 0.155 μM, n = 4).
Table 3.
Parameters used to perform simulations described in the main text.