Fig 1.
Simulation of a bimolecular reaction using a particle-based method and the spatial Gillespie approach.
In our particle-based simulations (A) molecules undergo a random walk in continuous space and discrete time intervals Δt. If a pair of reacting molecules are within a distance ρ, they can react with probability rate λ. In the spatial Gillespie approach (B), the domain is discretized using a grid, here with square elements of size h. Molecule jumps to adjacent grid elements and reactions take place within grid elements at random times. The propensity of a reaction within a grid element is proportional to the number of molecules (nA, nB) and a mesoscopic rate constant kmeso. (C) The scale-dependent mesoscopic rate kh ensures that the mean association time of two molecules in the reaction-diffusion master equation (τc) matches an analogous microscopic representation (τR). (D) A concentration-dependent mesoscopic rate constant is defined as kc = Ac/τRc where Ac is the mean free area between molecules of the most abundant reactant in the grid element, estimated as Ac = h2/max(nA, nB), and τRc is the mean association time calculated from a microscopic model of two molecules that react in a circular domain with area πRc2 = Ac. See Methods section for further details.
Fig 2.
Comparison of spatial Gillespie simulations using the mesoscopic rates kh and kc with particle-based simulations.
(A-D) Results for the mean number of species A from spatial Gillespie simulations using the mesoscopic rate kh with initial low abundance of reactants in (A, B) (total A = total B = 5, total C = 0 at t = 0) and initial high abundance (total A = total B = 5000, total C = 0 at t = 0) in (C-D). (E-H) show corresponding simulations to (A-D) but using the mesoscopic rate kc. In all the simulations, the degree of diffusion control is λπρ2/Dtot = 50, with Dtot = 2D and D = 0.0025μm2/s, ρ = 0.005 μm and λ = 3183.1/s. The size of the domain is L = 1μm. For the reversible reaction A+B ↔ C, the microscopic dissociation rate constant kdmicro is 1/s in (B, F), and 10/s in (D, H).
Fig 3.
Cdc42 polarization simulated with particle-based simulations and spatial Gillespie simulations using kh and kc.
(A) Reactions in a model for polarization in budding yeast. Species that are not bound to the membrane (brown), dwell in the cytosol. Membrane and cytosol are represented in the simulations as juxtaposed 2D squared domains. (B) Time series of a particle-based simulation of the polarization model in (A). In each snapshot, red dots show the positions on the membrane of all active Cdc42 molecules (Cdc42T and Cdc42T-GEF). The lower panels in (B) show a quantification of active Cdc42 clustering using the H(r) function (see Methods for details). (C, D) Spatial Gillespie simulations using kh (C) and kc (D) with the same model parameters as the particle-based simulation in (A) and grid element size h = 5ρ. Pseudo-coordinates for each molecule are randomly sampled from the containing grid element and displayed as a red dot to facilitate comparison with particle-based simulations. Model parameters are presented in Table 1.
Table 1.
For reactions, the values listed are for microscopic rate constants. For second-order reactions, the corresponding λ values can be calculated from Eqs 2 or 26, depending on the type of reaction. The parameters kh, khd, kc and kcd used in spatial Gillespie simulations are computed from Eqs 8, 13, 16 and 19, respectively. The parameters for the 3D model correspond with the 2D parameters, except when they have been scaled to account for dimensionality (Methods). Both 2D and 3D particle-based simulations were performed using Smoldyn [22,23] specifying reaction probabilities for second-order interactions. Such probabilities are computed from the rate constants presented here as described in the Methods.
Fig 4.
kh and kc are benchmarked against particle-based simulations by looking at polarization dynamics and equilibrium.
(A) Time evolution of the clustering of total active Cdc42 from simulations of the polarity model in Fig 3A with parameters in Table 1. We contrast results from particle-based simulations and the spatial Gillespie approach using either kh or kc and a grid element size h = 5ρ. Clustering at a particular timepoint is quantified as the mean of H(r = 1.1μm) over 30 simulations. Uncertainty intervals are computed as mean ± standard deviation and presented as a shaded region enveloping the mean. (B) Clustering at equilibrium from simulations in (A) for different values of the total amount of GEF. The clustering at equilibrium is computed as the mean of H(r = 1.1μm) between 250s and 300s. The box plots were generated with the clustering at equilibrium from 30 simulations. (C) and (D) are corresponding figures to (A) and (B) respectively, the only difference is that the spatial Gillespie simulations are run with h = 2.5ρ and using only kc as kh cannot be computed for such h in this model (see Methods).
Fig 5.
Direct recruitment of polarity factors to the patch enables highly mobile clusters.
(A) Snapshots of the distribution of total active Cdc42 over a 1hr simulation. The black dot in each frame is the centroid of the polarity cluster, and the green dot is the centroid when the polarity cluster first formed. (B) Diffusivity of the centroid of the polarity patch (Dpatch) as a function of the total amount of GEF molecules in the simulations. Dpatch was obtained from the mean squared displacement (MSD) of the patch centroid by fitting the equation MSD(Δti) = 4Dpatch Δtiβ to the data, where Δti is a particular time interval, and β reflects the degree of anomalous diffusion. (C) Patch diffusivities as a function of total GEF for simulations where k2b has been decreased by a factor of 1/16 and k5a has been increased by a factor of 10 relative to the parameters in Table 1. For these simulations β ≈ 1. (D) Snapshots from a representative simulation in (C) as indicated by the red arrow. (E) Patch diffusivities as a function of total GEF for simulations where k6 has been increased to either 1 μm2/s or 50 μm2/s. With k6 = 1 μm2/s polarization is lost when the number of GEFs is below 200. With k6 = 50 μm2/s the simulations show polarization at even lower GEF amounts, in this case, β varied between 0.85 and 1. (F) Snapshots from a representative simulation in (E) as indicated by the red arrow. (G) Patch diffusivities as a function of total GEF after adding Reaction 7 to the model. β values were ≈ 0.85 for the two data points with highest mobilities, and close to 1 for the other points. (H) Snapshots from a representative simulation in (G) as indicated by the red arrow. Error bars for patch centroid diffusivities are standard errors from the least-squares fit use to compute Dpatch.
Fig 6.
Cluster mobility correlates with fast distribution fluctuations and short dwell times of Cdc42T.
Comparison of a “Low mobility” model (from Fig 5C) and a “High mobility” model (from Fig 5G). (A) Patch centroid diffusivities, (B) dwell time of Cdc42T at the cluster, (C) Snapshots of the lateral profile of the concentration of total Cdc42T molecules for the High mobility model and (D) Low mobility model. (E) Coefficient of variation of the distribution of total Cdc42T molecules, CVpatch (see Methods), as a function of the number of total GEF molecules. (F) Patch centroid diffusivity as a function CVpatch for the Low mobility and High mobility models. Error bars for patch centroid diffusivities are standard errors from the least-squared fit used to compute Dpatch. For all other quantities, the error bars are the standard deviation from estimations in 5 independent simulations.
Fig 7.
Association reactions at the membrane stabilize clusters by trapping polarity factors at the patch.
Patch centroid diffusivity (Dpatch) as a function of the total number of GEF molecules and k2a (A), k3 (B), k4a (C). The size of the dots reflects the magnitude of Dpatch as indicated in the legend at the right. For each case, the value of the other two rate constants is shown above the panel. (D-F) The amounts of Cdc42 and GEF at the patch are quantified as the total Cdc42T (black) and Cdc42T-GEF (blue) respectively for the corresponding points enclosed by dashed boxes in (A-C). (G-I) Dwell times at the patch of Cdc42T (black) and GEF (blue) for the corresponding points enclosed by dashed boxes in (A-C) (see Methods for details). Error bars are the standard deviation from estimations in 10 independent simulations.
Fig 8.
Increasing the rate constant of Cdc42T-GEFm association (k4a) induces an abrupt change in patch mobility.
(A) Patch centroid diffusivity as a function of the number of GEF molecules for the model that includes Reaction 7 (Updated model, black) and the same model but setting k4a = 0 (red). (B) Patch diffusivity as a function of k4a for simulations with GEF = 100. Error bars are standard errors from the least-squared fit used to compute Dpatch.
Fig 9.
3D particle-based simulations recapitulate 2D spatial Gillespie simulations results from Fig 8.
(A) Snapshots of 3D particle-based simulations for different values of k4a. Cdc42T molecules are shown as red dots on a spherical surface representing the cell membrane. Rate constants are estimated from the ones used in Fig 8B as described in the Methods. Parameters are presented in Table 1. (B) Patch centroid diffusivity as a function of k4a for 3D particle-based simulations (red) with GEF = 100. For comparison we also show results from 2D spatial Gillespie simulations in Fig 8. Error bars are standard errors from the least-squared fit used to compute Dpatch.