A. An improved approach for stochastic reaction-diffusion simulations

A.1. Background.

At the microscopic level, second-order reactions can be thought of as occurring in two steps. First, the two reactants must diffuse to close enough proximity for the reaction to occur. Second, after the reactants encounter one another, there may still be enthalpic and entropic barriers to overcome before the reaction can proceed. Second-order reactions are diffusion-limited when, the “search” time for two reactants to encounter one another is substantially longer than the time for the reaction to proceed once the reactants are close enough to interact. Conversely, the reactions are reaction-limited when the diffusional search time is short in comparison to the reaction time.

Particle-based simulations that track the position of each molecule in the system provide a microscopic representation of chemically reacting systems (Fig 1A). Typically, in these simulations it is assumed that once the reactants meet, the reaction occurs instantaneously [22,23] or requires a single kinetic step to proceed [24–26] (see Methods for a discussion of microscopic models for second-order reactions). For the particle-based simulations presented here, we followed an implementation of the latter approach in which molecules react with a rate λ when their separation is within an interaction radius ρ [26]. Therefore, the microscopic parameters required for simulating second-order reactions are the diffusion coefficients for the reacting molecular species, ρ and λ. Simulating first order reactions, such as the dissociation of molecular complexes, requires specifying a single rate constant for the reaction. For particular cases, mathematical expressions that relate the microscopic parameters to experimentally measured macroscopic rate constants (i.e., rate constants that appear in mass action kinetics) have been derived [21,24,26]. Such mathematical expressions can be used to parameterize particle-based simulations that are consistent with experimental measurements.

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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 (n_A, n_B) and a mesoscopic rate constant k_meso. (C) The scale-dependent mesoscopic rate k_h ensures that the mean association time of two molecules in the reaction-diffusion master equation (τ_c) matches an analogous microscopic representation (τ_R). (D) A concentration-dependent mesoscopic rate constant is defined as k_c = A_c/τ_Rc where A_c is the mean free area between molecules of the most abundant reactant in the grid element, estimated as A_c = h²/max(n_A, n_B), and τ_Rc is the mean association time calculated from a microscopic model of two molecules that react in a circular domain with area πR_c² = A_c. See Methods section for further details.

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Microscopically-detailed, particle-based simulations are typically computationally expensive. Therefore, to reduce simulation times it is common to use an approximate description of the system based on the reaction-diffusion master equation (RDME). In this approach, space is discretized into a grid of volume elements and molecules are assumed to be “well-mixed” within the volume elements (Fig 1B). This is a mesoscopic description in the sense that the size of the grid elements is normally larger than the size of a molecule (microscopic scale), but still significantly smaller than the whole system (macroscopic scale). Molecules can jump to adjacent grid elements with a propensity proportional to the diffusion coefficient and reactions occur with a propensity proportional to the reactants’ abundances within the grid element and a mesoscopic rate constant k_meso. As the exact solution of the RDME can be difficult to obtain, realizations of the system are typically simulated with an efficient spatial version of the Gillespie algorithm [17].

A challenge with RDME simulations is determining parameters that produce results consistent with the underlying microscopic dynamics. This can be especially problematic when second-order reactions are diffusion-limited because simulation results can strongly depend on the grid spacing h [27]. This issue was addressed recently through the derivation of mesoscopic rate constants in terms of microscopic parameters that ensure compliance with the microscopic kinetics in the limit of low molecular abundances [18–20]. Specifically, this approach considered a domain containing two diffusing molecules that undergo an association reaction. A mesoscopic rate constant is then derived using the requirement that the mean association time for the mesoscopic description is equivalent to that of the microscopic representation (Fig 1C, Methods). Because this mesoscopic rate constant depends on h, it has been referred to as a “scale-dependent” rate constant. If the association reaction is reversible, a mesoscopic dissociation rate constant is computed by ensuring that the equilibrium behavior of the mesoscopic description is identical to that of the microscopic representation (Methods).

In Section A.2. we evaluate the scale-dependent mesoscopic rate k_h derived in [19,20] by comparing simulations of simple reaction schemes using k_h with results from particle-based simulations. We note that although k_h yields accurate results at low concentrations, it shows significant deviations from particle-based simulations in scenarios with high molecular abundances. To address this issue, in Section A.3. we propose a concentration-dependent mesoscopic rate k_c that produces accurate results over a broad range of concentrations. In Section A.4 we evaluate both k_h and k_c in a biochemical model for polarity establishment in yeast.

A.2. Evaluation of the scale-dependent mesoscopic rate k_h.

We evaluated the scale-dependent mesoscopic rate k_h [19,20] by considering two prototypical reactions: irreversible association A+B → C and reversible association A+B ↔ C. A measure of the degree to which the reaction is controlled by diffusion is the dimensionless ratio λπρ²/(D_A+D_B) where D_A and D_B are the diffusion coefficients of the A and B molecules respectively. For the results presented in this section, we set the value of this ratio to 50 to ensure the reactions are strongly controlled by diffusion. Because our goal is to model reaction and diffusion at the cell membrane, simulations are performed in a 2D domain with periodic boundary conditions.

To test the accuracy of k_h, we performed simulations in which either the size of computational domain was fixed and the molecular abundances varied (Fig 2A–2D) or the molecular abundances fixed and the size of the computational domain varied (S1 Fig). For each case, we present time series for the mean number of molecules of one of the reactants. At low reactant concentrations, simulations using k_h are consistent with the microscopic dynamics as expected (Figs 2A and 2B and S1G–S1J) [19,20]. We observed small differences for the reversible reaction that may be attributed to differences in the assumptions used to derive k_h and the method we used to perform particle-based simulations. At higher concentrations, however, the early kinetics of both the reversible and irreversible reactions showed significant deviations from the particle-based results (Figs 2C and 2D and S1A–S1F). At later times the deviations are reduced as the number of molecules decreases. We note that it is not possible to achieve higher accuracies by reducing the size of the grid elements, because for this parameter regime, it is not possible to calculate k_h for h smaller than ~5ρ (Methods) [19,20].

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Fig 2. Comparison of spatial Gillespie simulations using the mesoscopic rates k_h and k_c with particle-based simulations.

(A-D) Results for the mean number of species A from spatial Gillespie simulations using the mesoscopic rate k_h 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 k_c. In all the simulations, the degree of diffusion control is λπρ²/D_tot = 50, with D_tot = 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 k^d_micro is 1/s in (B, F), and 10/s in (D, H).

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We also computed timeseries of the standard deviation to quantify the fluctuations around the mean for each case in Fig 2A–2D (S2A–S2D Fig). This metric also showed agreement between spatial Gillespie simulations using k_h and particle-based simulations at low concentrations, and increased deviations at high concentrations at early times. Overall, the mesoscopic approach using k_h provides a good approximation to the microscale dynamics for irreversible and reversible diffusion-controlled reactions only if the density of reactants is low.

To understand why k_h showed accurate results for systems with low abundances but loses accuracy at high concentrations, remember that k_h was derived to reproduce the association time in a two-molecule system (Fig 1C, Methods). In a dilute system containing n pairs of reacting molecules, the expected time for an association event is similar to that in a set of n independent 2-molecule systems, and simulations using k_h provide a good approximation of the kinetics. At high concentrations, however, molecules are more likely to associate with partners in their vicinity before diffusing over a significant portion of the spatial domain. Because the derivation of k_h does not consider such multi-molecule effects, simulations lose accuracy at higher concentrations.

A.4. Evaluation of mesoscale simulations in a model of the yeast polarity circuit.

It is known that the RDME approach can generate anomalous behavior due to numerical artifacts from using a finite grid [28]. Therefore, we compared results from the mesoscale rates k_h and k_c, and particle-based simulations, in a reaction-diffusion model for polarity establishment in budding yeast adapted from [29] (Fig 3A). Central to this biochemical network is the Rho-GTPase Cdc42. Cdc42 can exist in an inactive (GDP bound) form Cdc42D that shuttles between the membrane and the cytosol, and an active (GTP bound) form Cdc42T that localizes to the cell membrane. Deactivation occurs as Cdc42 hydrolyzes GTP into GDP, a process accelerated by GTPase activating proteins (GAPs). GAP activity is considered implicitly using a pseudo-first order deactivation rate constant. The activation of Cdc42 is catalyzed by the guanine nucleotide exchange factor Cdc24 (GEF) that binds Cdc42D and facilitates the exchange of GDP for GTP. GEF molecules can shuttle between membrane and cytosol, and at the membrane they activate Cdc42. Once GEF binds Cdc42D, it activates the Rho-GTPase and dissociates from the resulting Cdc42T in a single step. There is positive feedback in the levels of active Cdc42 because Cdc42T can bind a GEF molecule forming a complex Cdc42T-GEF that can activate neighboring Cdc42D molecules. In this model Cdc42T can recruit both membrane-bound and cytosolic GEF molecules. In the following simulations, membrane and cytosol are represented as coincident 2D square domains. What distinguishes the membrane and cytosol are the diffusion coefficients for the molecular species in each domain. The parameters were adapted from a 3D macroscopic model [13,29] into a 2D microscopic representation (Methods) and are presented in Table 1.

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Fig 3. Cdc42 polarization simulated with particle-based simulations and spatial Gillespie simulations using k_h and k_c.

(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 k_h (C) and k_c (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.

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Table 1. Model parameters.

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 k_h, k_h^d, k_c and k_c^d 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.

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Particle-based simulations initialized with all Cdc42 and GEF molecules randomly located in the cytosol evolved into a polarized distribution of total Cdc42T (Cdc42T and Cdc42T-GEF) at the membrane (Fig 3B) [16]. At early times, small fluctuations in the levels of Cdc42T are amplified by positive feedback reactions resulting in growing clusters of the active Rho-GTPase (Fig 3B 10 s). Several clusters can form that compete for polarity factors in the cytosol until just one cluster remains (Fig 3B, 60 s—120 s). The remaining cluster grows, depleting polarity factors in the cytosol, until a steady state is reached when the inward and outward fluxes of molecules to the cluster balance (Fig 3B 120 s– 300 s).

In Fig 3C and 3D we present snapshots of representative mesoscopic simulations using the scale-dependent (k_h) and concentration-dependent (k_c) rates, respectively. To facilitate comparison with particle-based simulations, spatial pseudo-coordinates for each molecule were obtained by randomly sampling within the grid element containing the molecule. Mesoscopic simulations using both k_h and k_c reached a steady-state polarity cluster (Fig 3C and 3D 300 s) with a size comparable to that of the particle-based simulation in Fig 3A. There were differences, however, in the time evolution of polarization between the different methods. The simulation using k_h took longer to evolve into a single polarity cluster, while the simulation using k_c polarized faster compared to the particle-based approach. We note that these simulations were run using a grid element h = 5ρ which corresponds to the finest grid possible using k_h.

To quantify the dynamics of polarization we measured clustering of total Cdc42T with the function H(r) (Fig 3B lower panels) [16,30,31] (Methods). H(r) has the desired properties that H(r) = 0 for a random distribution of molecules, and for a clustered distribution, H(r) shows a maximum at a value r = r_max which provides a measure of the cluster size. H(r) showed a maximum at r = 1.1 μm that increased over time (Fig 3B lower panels). We therefore chose H(r = 1.1 μm) as a metric for polarization. We note that quantifying clustering with max H(r) does not qualitatively change the results. In Fig 4A we show the time evolution of the mean of H(r = 1.1 μm) over different realizations for the different simulation methods. The shading indicates standard deviation to illustrate the variability in polarization dynamics. At early times, simulations using k_h matched particle-based simulations but showed deviations at later times, taking longer to polarize. On the other hand, simulations using k_c displayed overall faster polarization than the other methods. By defining the polarization time as the moment when the standard deviation stabilizes, it is apparent that simulations using k_h have a longer time of polarization (240 s) compared to particle-based simulations (180 s), while using k_c results in faster polarization (130 s).

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Fig 4. k_h and k_c 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 k_h or k_c 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 k_c as k_h cannot be computed for such h in this model (see Methods).

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We further characterized the simulation approaches by looking at the steady state behavior for different amounts of available GEF molecules (Fig 4B). While particle-based simulations lost polarity at GEF amounts of around 300 molecules, mesoscopic simulations using k_h polarized for GEF abundances of around 100, showing polarization in a regime where the microscopic simulations do not spontaneously polarize. On the other hand, mesoscopic simulations using k_c showed polarization for GEF amounts greater than 400, failing to polarize at a value of 400 GEF where the microscopic approach shows polarization.

As the mesoscopic simulations with both k_h and k_c presented in Fig 4A and 4B showed discrepancies with respect to particle-based simulations, we sought to obtain more accurate results by reducing the grid element size, which so far was set to h = 5ρ. We were able to do this only for simulations with the concentration-dependent rate k_c, because for the parameters of the model, k_h cannot be computed for grid elements smaller than 5ρ (see Methods). For h = 2.5ρ we observed a significant improvement in accuracy using k_c both in the dynamics of polarization (Fig 4C) and in the equilibrium behavior for different values of the number of GEF molecules (Fig 4D).

The increase in accuracy using h = 2.5ρ came with a cost of ≈ 5X increase in computation time with respect to simulations using h = 5ρ. Particle based simulations, on the other hand, were ≈ 10X more computationally expensive with respect to mesoscopic simulations using h = 5ρ. We note that particle-based simulations were performed with the highly optimized simulation platform Smoldyn [22,23], whereas mesoscopic simulations were run using our own custom written C code, which has not been optimized for computational performance.

In summary, we found that the mesoscopic rate k_c provided more accurate results than k_h in simulations of elementary reactions at high concentrations. During simulations of the polarity model, both high and low concentrations of reactants can exist simultaneously. Therefore, it is harder to definitively establish which mesoscopic rate provides a better description of the system. We favor the concentration-dependent rate k_c for several reasons: 1) it shows more accurate results at the higher concentrations found in the polarity site, 2) it does not artificially generate polarization in parameter regimes where particle-based simulations do not polarize (although it fails to show polarization for some parameters where the particle-based simulations do polarize), and 3) k_c allows the use of smaller grid elements producing increased accuracy.

B. Mechanisms regulating mobility of the polarity site

B.1. Requirements for high patch mobility.

When yeast cells are presented with pheromone from a potential mating partner, Cdc42 forms dynamic clusters that explore the membrane for 10–100 min before stabilizing in a region close to the pheromone source [5,6]. Therefore, we sought to determine if intrinsic fluctuations are sufficient to explain patch mobility by performing spatial Gillespie simulations using our new concentration-dependent mesoscopic rate constants for bimolecular reactions.

To assess patch mobility, we tracked the centroid of the distribution for active Cdc42 over time (Fig 5A). However, with our initial parameterization (Table 1), the patch did not move significantly over a 60 min time interval. It has been suggested that the amount of GEF in the cytosol strongly regulates cluster dynamics [5]. In previous modeling studies, GEF abundances ranged from 500–2000 molecules [13,29,32,33]. These studies primarily focused on situations where the patch was stable. However, immediately following exposure to pheromone, when the polarity clusters are highly mobile, it is possible that the amount of available GEF is significantly smaller. Therefore, we ran simulations with GEF levels of 700 molecules and fewer. To quantify patch movement, we computed the mean squared displacement (MSD) of the patch centroid over time and used these values to estimate an effective diffusion coefficient D_patch (Fig 5B). As anticipated, decreasing GEF abundance increased patch mobility, but the system lost polarity at moderate GEF levels (≈ 450 molecules) before patch movement increased substantially. Therefore, we investigated if varying any of the rate constants would allow the system to polarize at lower GEF abundances. After testing all the reactions in the model (S3 Fig), we observed that the following modifications allowed the system to polarize with low GEF abundances (100 or fewer molecules): 1) decreasing the rate constant k_2b for Cdc42T deactivation, 2) decreasing the rate constant of dissociation of the Cdc42T-GEF complex k_4b, 3) increasing the rate constant k_5a for membrane binding of cytosolic Cdc42D and 4) increasing the rate constant k₆ for association of Cdc42T with cytosolic GEF to form the Cdc42T-GEF complex. We computed patch mobility in simulations where both k_2b and k_5a where modified, (Fig 5C and 5D) and where k₆ was increased (Fig 5E and 5F), because these changes resulted in polarization down to 15 GEF molecules. Interestingly, as the total GEF abundance was decreased, only increasing k₆ generated highly mobile patches (compare Fig 5D and 5F which are representative realizations from the points indicated by the red arrows in Fig 5C and 5E).

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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 (D_patch) as a function of the total amount of GEF molecules in the simulations. D_patch was obtained from the mean squared displacement (MSD) of the patch centroid by fitting the equation MSD(Δt_i) = 4D_patch Δt_i^β to the data, where Δt_i 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 k_2b has been decreased by a factor of 1/16 and k_5a 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 D_patch.

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The rate constant k₆ governs the direct recruitment of cytosolic GEF to the patch through complex formation with Cdc42T. This reaction can occur rapidly, because diffusion in the cytosol is fast compared to diffusion in the membrane. Another potentially fast reaction not considered in the model is the recruitment of cytosolic Cdc42D directly to the patch. In cells, most inactive Cdc42 molecules are found in the cytosol bound to GDI proteins that hold them in their inactive state. In our original model, for a cytosolic Cdc42D molecule to be activated it must first be inserted in the membrane, a step implicitly representing dissociation from the GDI protein. Once at the membrane Cdc42D has to laterally diffuse to react with a GEF. However, prior work has suggested that GEFs may displace Rho-GTPases from their GDI proteins [34,35]. Based on this observation, we updated our model to include a reaction where Cdc42T-GEF recruits and activates cytosolic Cdc42D (Fig 5G). An analogous reaction was included in a model for polarization by Klunder et al. (2013) [33]. Using a rate constant comparable to the one employed by Klunder et al. and maintaining the original values of the other rate constants, we observed a highly mobile patch at low GEF abundances (Fig 5G and 5H). Our results suggest that direct recruitment of fast diffusing cytosolic polarity factors to the patch promotes mobility of the polarity cluster.

To gain further insight into mechanisms that generate a dynamic patch, we investigated several different patch properties. We first observed that the models shown in Fig 5C and 5G, which have substantially different patch mobility at low GEF abundance (replotted in Fig 6A on a log-log scale as “High mobility” and “Low mobility” models), have nearly identical amounts of active Cdc42 and Cdc42T-GEF complex, and the fluctuations in the abundance of these species were also similar between the two models (S4 Fig). However, the dwell time of Cdc42T was shorter in the high mobility model, indicating a faster cycling of Cdc42 between the cluster and the cytosol as compared to the low mobility model (Fig 6B).

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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, CV_patch (see Methods), as a function of the number of total GEF molecules. (F) Patch centroid diffusivity as a function CV_patch 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 D_patch. For all other quantities, the error bars are the standard deviation from estimations in 5 independent simulations.

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We also observed that patches with high mobility appeared to show larger spatial fluctuations in the distribution of active Cdc42 (Fig 6C), as compared to the low mobility case (Fig 6D). We quantified spatial fluctuations in the Cdc42T distribution by computing the coefficient of variation CV_patch (see Methods) and observed that this quantity strongly correlates with D_patch (Fig 6E).

These results suggest that patch mobility is correlated with spatial fluctuations in the Cdc42T distribution, rather than fluctuations in the abundance of Cdc42T or its GEF. Furthermore, low GEF abundance and rapid Cdc42 cycling between the cluster and the cytosol are required for high patch mobility.

B.2. Potential mechanisms for regulating polarity cluster dynamics.

During mating, the polarity site transitions from being highly mobile to a more stable state. Therefore, we investigated mechanisms that would allow regulation of patch movement. We note that increasing k₆ or adding Reaction 7 in our original model enabled highly mobile clusters at low GEF abundances, but reversing such modifications eliminates polarization instead of stabilizing the patch (see for example Fig 5E, k₆ = 1 μm²/s). We therefore systematically evaluated how all other reactions affect patch mobility in a model that included Reaction 7, which we refer to as the updated model (Figs 7A–7C, S5 and S6). We found that the following reactions robustly modulate cluster mobility: activation of Cdc42D_m by GEF_m (k_2a), activation of Cdc42D_m by Cdc42T-GEF (k₃) and association of Cdc42T with GEF_m to form the Cdc42T-GEF complex (k_4a). We note that these reactions all involve the association between two membrane-bound molecules.

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Fig 7. Association reactions at the membrane stabilize clusters by trapping polarity factors at the patch.

Patch centroid diffusivity (D_patch) as a function of the total number of GEF molecules and k_2a (A), k₃ (B), k_4a (C). The size of the dots reflects the magnitude of D_patch 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.

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To investigate the role of k_2a, k₃ and k_4a in stabilizing the polarity site, we systematically varied the GEF abundance and one of these rates, while setting the other two equal to zero in the cases of k₃ and k_4a or a very small value in the case of k_2a (a minimal value is required to start Cdc42 activation) (Fig 7A–7C). We found that increasing k_4a was most effective at reducing patch mobility as it stabilizes the patch at lower values of the rate constant and a wider range of GEF abundances compared to k_2a and k₃. To understand why varying k_4a produced a more dramatic effect on patch mobility, we quantified the amounts and dwell times of Cdc42T and GEF at the patch as a function of k_4a, k_2a and k₃ (Fig 7D–7I). Increasing any of the three rates produced a modest increase in Cdc42, but only increasing k_4a produced a substantial increase in GEF at the patch (Fig 7D–7F). On the other hand, increasing k_2a and k₃ lengthened the dwell time only of Cdc42, while increasing k_4a extended the dwell time only of GEF (Fig 7G–7I). These results suggest that k_2a and k₃ stabilize the patch mainly by prolonging the residence time of Cdc42 at the patch, while k_4a reduces cluster mobility by trapping GEF at the patch and increasing its abundance.

Because k_4a had the largest effect on patch mobility, we studied the effects of varying this parameter using the original set of parameter values in the updated model. Setting k_4a equal to zero resulted in an increase of more than an order of magnitude in patch mobility for low GEF abundances (Fig 8A). Increasing k_4a with GEF = 100 resulted in an abrupt change in cluster mobility around a value of 0.003 μm²/s (Fig 8B), in agreement with the results seen using the reduced values of k_2a and k₃ in Fig 7C. After this sharp decrease, the mobility of the patch gradually increased with increasing k_4a and appeared to plateau as the reaction becomes diffusion-limited.

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Fig 8. Increasing the rate constant of Cdc42T-GEF_m association (k_4a) 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 k_4a = 0 (red). (B) Patch diffusivity as a function of k_4a for simulations with GEF = 100. Error bars are standard errors from the least-squared fit used to compute D_patch.

https://doi.org/10.1371/journal.pcbi.1008525.g008

C. Validation through 3d particle-based simulations

So far, all our results have been obtained using 2D simulations with periodic boundary conditions and assumed the only difference between the membrane and cytosol is the rate at which molecules diffuse. However, real yeast cells are 3D objects with distinct membrane and cytosolic compartments. Therefore, to ensure our results remained valid in 3D, we used Smoldyn [22,23] to perform particle-based simulations on a sphere, translating 2D parameters to 3D as described in the Methods. Because varying k_4a had the largest effect on patch mobility, we chose to validate these results. In agreement with our 2D simulations, low values of k_4a produced a mobile polarity patch and patch mobility decreased rapidly with increasing values of this rate constant (Fig 9A). Quantifying patch movement as a function of k_4a in the 3D simulations produced similar trends as results from the 2D spatial Gillespie simulations (Fig 9B). We note, however, that in the 3D particle-based simulations, the transition from a mobile to static patch appears to take place at a slightly higher values of k_4a (between 0.005 and 0.01 μm²/s), and for each value of k_4a, D_patch is higher, in comparison with the 2D spatial Gillespie simulations. The good agreement between the full 3D and approximate 2D simulation results validates the use of the more computationally efficient 2D simulations to investigate dynamics of the polarity patch. These results also provide further support for a mechanism for stabilizing the polarity patch by increasing the rate at which membrane-bound GEF and Cdc42T associate.

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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 k_4a. 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 k_4a 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 D_patch.

https://doi.org/10.1371/journal.pcbi.1008525.g009

Efficient and accurate stochastic simulations

To investigate the effect of molecular-level fluctuations on cluster mobility, we performed stochastic simulations of the biochemical network of Cdc42 polarization in yeast. Stochastic effects are most accurately captured using particle-based (microscopic) simulations [16,22,23,42–44] or a convergent reaction-diffusion master equation [45]. However, the long-time scales associated with the movement of the polarity site make the use of such simulations computationally prohibitive to perform extensive investigations. We therefore used less accurate, but more computationally efficient simulations based on the spatial Gillespie method. Despite improvements on the accuracy of spatial Gillespie simulations [18–20], we found that simulations lost accuracy in the diffusion-limited regime at high molecular concentrations. To increase the accuracy of such simulations, we implemented concentration-dependent mesoscopic rate constants building on ideas from Yogurtcu et al [21]. An alternative approach that could also provide accurate and computationally efficient results is using hybrid microscopic-mesoscopic simulations [46–51], for example coupling a microscopic formulation of the cell membrane with a mesoscopic representation of the cytosol. Such methods, of course, imply higher implementation complexity.

Highly dynamic clusters

Relocation of polarity clusters has commonly been explained via negative feedback mechanisms [52]. Negative feedback reactions can destabilize positive-feedback driven clusters resulting in travelling waves [36], or oscillations where the clusters disappear and reappear at different locations of the membrane [41,53]. Directed vesicle delivery is also a form of negative feedback that dilutes the polarity cluster and induces wandering motion [7,13,14]. Here, we document how biochemical noise can induce relocation of polarity clusters without an explicit negative feedback mechanism. Essential for noise-driven cluster motion are low abundances of limiting polarity factors and fast cycling of polarity factors between the cluster and the cytosol. Fast cluster-cytosol cycling can be promoted by reactions where polarity factors are directly recruited to the cluster.

Regulation of cluster dynamics

During yeast mating, stabilization of highly mobile clusters has been attributed to increased cytosolic levels of the polarity factor Cdc24, the Cdc42 GEF, as pheromone induced MAPK activity triggers its nuclear export [5]. Besides increased GEF abundance, our study suggests that accelerating the kinetics of second-order reactions at the membrane involved in the activation of Cdc42 can stabilize highly dynamic clusters. Increasing the rate constants of such reactions seem to stabilize a mobile cluster by increasing the abundance and trapping polarity factors at the patch. Interestingly, modulating the rate constant of one such reaction, association of membrane-bound Cdc42 and GEF, produced a switch-like transition from a mobile to a stable polarity patch. The Cdc42/GEF interaction, therefore, is a likely target of pheromone induced signaling during regulation of polarity cluster dynamics. Interestingly, this interaction, which in yeast cells is bridged by the scaffold protein Bem1, is thought to be regulated in another context involving cell-cycle control [54].

Further evidence highlights the importance of biochemical events taking place at the cell membrane, and properties of the cell membrane itself in the regulation of polarity cluster dynamics. In fission yeast cells, which also display mobile patches in the early stages of mating, cluster dynamics is known to be under the control of a GAP (Cdc42 inactivator molecule) that localizes at the cell membrane [55]. During spore germination of fission yeast, initial uniform growth is associated with highly dynamic Cdc42 clusters. Upon rupture of the outer spore wall, the clusters stabilize into a single cluster in the direction where rupture takes place, giving rise to directed growth [40]. Other studies have documented that membrane tension [56] and membrane curvature [57,58] can influence cluster stability. Additional mechanisms that may be used by cells to regulate biochemical events at the membrane and control cluster dynamics include crowding of signaling molecules [59,60], restricting the diffusion of molecules with cytoskeletal barriers [11,61,62] and confining molecules into high affinity subdomains [63].

In summary, our results demonstrate the power of using accurate and efficient mesoscopic simulations to inform more detailed, but computationally costly, particle-based simulations. Our studies also provided considerable insight into the mechanisms used by cells to harness random molecular behavior and regulate the dynamic properties of their polarity sites.

Spatial Gillespie simulations

In this coarse-grained approach, space is discretized into grid units, and the state of the system is given by the number of molecules of each species in each grid unit. The system evolves continuously in time according to the reaction-diffusion master equation, which is the spatial extension of the chemical master equation for well-mixed systems. We simulated individual realizations with the Next Subvolume method [17] which is an efficient implementation of the spatial version of the stochastic simulation algorithm [64]. We ran simulations on a square domain of size L discretized with a Cartesian mesh with grid element size h. In the spatial Gillespie algorithm, diffusion is treated as a reaction that results in a molecule transitioning from its current location to a neighboring grid unit. If there are n molecules of a given species in a particular grid unit, the propensity k_jump for one of those molecules to transition to a neighboring grid unit is n D/h², where D is the diffusion coefficient. In 2D, there are 4 neighbor cells and the total propensity of jump is 4 n D/h². The reaction propensities within a grid unit are estimated in the same way as the well-stirred Gillespie algorithm [64]. For example, the propensity of a second-order reaction for the association of the species A and B, is computed as k_meso n_A n_B/h² where k_meso is the mesoscopic rate constant and n_A and n_B are the numbers of molecules A and B in the particular grid unit. The difference in the versions of the spatial Gillespie methods we use here are the different ways in which k_meso is computed.

Microscopic models for reacting particles

There are two common and related models for describing bimolecular reactions at the microscopic level, we refer to them as the Smoluchowski model [24], and the λ-ρ model [25,26]. Both consider diffusing molecules as particles undergoing Brownian motion, but they differ in how reactions are described. The Smoluchowski model represents reacting molecules as solid spheres that can react when separated exactly by a distance σ. In this model a molecule A will react with probability rate k_micro[B]_σ, where [B]_σ is the concentration of its reactive partner B at the reactive distance σ. In the λ-ρ model, reacting molecules are considered point-particles that react with probability rate λ when they are separated by a distance equal or less than ρ.

Both formalisms provide expressions for the macroscopic rate constant k in terms of the microscopic parameters for 3D systems [24,26]. In the reaction-limit where 4πσD >> k_micro (Smoluchowski model) or D >> λ ρ² (λ-ρ model), where D is the sum of the diffusion coefficients of A and B, k = k_micro and k = 4πρ³λ/3 in the Smoluchowski and λ-ρ models respectively. In the opposite limit, when reactions take place immediately when the molecules meet (diffusion-limit), the corresponding equations are k = 4πσD and k = 4πρD.

In 2D there are no general closed form expressions relating macroscopic and microscopic rate constants. However, in the reaction-limit, which in 2D requires D >> k_micro in the Smoluchowski model or D >> λρ² in the λ-ρ model, macroscopic rate constants can be approximated as k = k_micro and k = πρ²λ, respectively. For diffusion-limited reactions macroscopic rates are not well defined and depend on local concentrations [21].

Because it is easier to implement, we followed the λ-ρ formalism and treated particle-based simulations based on this description as the ground truth. However, because the derivations of k_h and k_c make use of the Smoluchowski model, to compare particle-based and RDME simulations we take σ = ρ and used an approximate relation between λ and k_micro. To obtain this relation we formulated a microscopic rate equation for the probability rate that a molecule A reacts when there is a B molecule within a distance ρ: (1) and defining [B]_ρ = 1/(πρ²) in 2D we obtain: (2)

The scale-dependent mesoscopic rate constant k_h in a 2D system

To account for the fact that, in the RDME setting, the encounter time of two molecules, and therefore, the reaction time, diverges as the grid element size decreases, Hellander et al. [19] derived a mesoscopic rate constant k_h using the condition that the mean association time for two molecules diffusing in a specified domain in the RDME representation is equivalent to the exact result for a microscopic description following the Smoluchowski approach.

The mean association time τ_micro for two molecules diffusing on a domain with area A in the microscopic formulation can be estimated from [18]. Assuming that one molecule diffuses on a circular domain with reflective boundary conditions and radius R such that πR² = A, and the other is static at the center of the domain: (3) with (4) (5) where ρ is the reactive radius and the parameter α is defined as: (6) where D is the sum of the diffusion coefficients of the two molecules.

The mean association time in the mesoscopic formulation τ_meso is estimated for a square domain of length L with square grid units of length h [19] as: (7) k_h is the mesoscopic rate constant that ensures that τ_meso = τ_micro provided that L² = πR². This leads to the following expression for k_h: (8) where (9)

Note that k_h can be computed only if: (10)

This implies a lower bound on h: (11)

When h is below this bound, there is not k_h for which the equality τ_meso = τ_micro holds.

Mesoscopic dissociation rate k_h^d

The mesoscopic dissociation rate constant is derived so that the steady state concentrations of a reversible second-order reaction of a two-molecules system are the same in the mesoscopic and microscopic formulations [20]. The steady state is characterized by the ratio of the average unbound time to the total time, which can be computed as the ratio of the mean rebinding time to the sum of the mean rebinding time and the mean dissociation time. Therefore, the condition to be satisfied is: (12)

The mean rebinding time in the mesoscopic simulation τ_meso^rebind was shown to be L²/k_meso and a good approximation for the mean rebinding time in the microscopic formulation τ_micro^rebind in the 2D disk domain is L²/k_micro. The mean dissociation times in the mesoscopic (τ_meso^d) and microscopic (τ_micro^d) formulations are respectively 1/k_meso^d and 1/k_micro^d. After replacing k_meso and k_meso^d by k_h and k_h^d the equilibrium condition then reduces to: (13)

Mesoscopic concentration-dependent rate constant k_c

The mesoscopic rate constant k_c aims to provide accurate results in systems with high concentrations relative to the space discretization. In this case, several reacting molecules are often found together in a grid element. Therefore, reaction rates are dominated by the time it takes for molecules to react once they are in the same grid element and it is reasonable to ignore the time for molecules to diffuse into the same grid unit.

Let us consider a grid element containing a single A molecule and one or more of its reacting partner B at a concentration [B]_h. In this scenario we define k_c via the mesoscopic rate equation: (14) where τ_Rc is the mean reaction time. To account for the fact that at higher [B]_h the mean-free-path before reaction is shorter (and therefore, so is the reaction time), τ_Rc is approximated as the mean association time of two molecules diffusing on a domain with area A_c = h²/n_B where n_B is the number of B molecules in the grid element. The time τ_Rc is computed using Eq 3 with: (15)

Taking [B]_h = h²/n_B = 1/A_c and replacing in Eq 14 we get: (16) where the function F and the constant α are defined in Eqs 4 and 6.

For crowded situations where R_c ≤ ρ, we set k_c = k_micro. Although technically there is not a lower bound on h, for realistic simulations h should be greater than ρ.

In the more general case where there can be more than one A or B molecules in the grid element we get: (17) where max(n_A, n_B) is the number of the most abundant reactant within the grid, and k_c is computed from Eq 16 using: (18)

Mesoscopic dissociation rate k_c^d

We estimate the mesoscopic dissociation rate k_c^d in a similar way as described above for a two-molecule system (Eqs 12 and 13) under the assumption that the pair of molecules diffuse on a domain with area A_c: (19)

2D particle-based simulations

The simulations performed to benchmark our methods (Fig 2) were carried out using our own custom written software. All particle-based polarity simulations (2D and 3D) were performed using Smoldyn [22,23]. In all particle-based simulations used the λ-ρ approach for second-order reactions. While this is not the default scheme in Smoldyn, which typically approximates the original formulation of Smoluchowski model where reacting molecules always react upon contact, the software also admits λ-ρ parameters as described below.

In these particle-based simulations space is continuous and time is discretized in intervals Δt. Molecules are considered point particles and their Brownian motion is simulated with the Euler-Maruyama method: If x(t), y(t) are the position coordinates of a given particle at time t moving in a 2D domain, the position coordinates at time t + Δt are calculated as: (20) (21) where Z_i, and Z_j are independent random numbers drawn from a standard normal distribution, and D_m is the diffusion coefficient. Every time step, the new positions of all the particles are calculated.

Bimolecular reactions occur with probability P = 1-exp(-λ Δt) during a time interval Δt, when two reactants are within a distance ρ (the reaction radius). This probability is approximated as P ≈ λ Δt for small Δt. For simulations with Smoldyn we input to the software the probability P.

First-order reactions in our custom made simulations (Fig 2), occur during a time interval Δt with probability P_i = 1-exp(-k_i Δt), which can be approximated as P_i = k_i Δt for small Δt, where k_i is the reaction rate constant. For simulations performed with Smoldyn we only input the rate constant.

In our custom-made simulations (Fig 2), when a dissociation event for two molecules in a complex occurs, one molecule is set at the position previously occupied by the complex and the second is placed at distance ρ + ε apart from the first. ε is a small number just to ensure reactants will not be within a reactive distance the next simulation time step, and the orientation of the second molecule is chosen randomly from a uniform distribution. For simulations with Smoldyn, dissociation reactions are handled with the software default routines, but we specify the separation distance after dissociation as ρ + ε.

Simulations of simple reversible and irreversible reactions (Fig 2) were performed with Δt ≈ (0.1ρ)²/(4D_tot) with D_tot = 2D_m. This ensures a root mean squared displacement < 0.1ρ over a simulation time-step resulting in accurate results at the expense of computational resources. The parameters of such simulations are given in the captions of the corresponding figures.

In the polarity simulations we used Δt = (ρ)²/(4D_cyto) after observing similar results with preliminary simulations using smaller time steps. In this case, the root mean squared displacement over a simulation time-step for cytosolic components is ρ, and it is less than ρ for membrane components since D_cyto >> D_memb.

The parameters for the polarity establishment model are given in Table 1. We present the reaction parameters as microscopic rate constants k_micro. For second-order reactions, the input to Smoldyn is the reaction probability P = λ Δt, and λ is related to k_micro as λ = k_micro/πρ² according to Eq 2.

Reactions of the polarization model

GEF_c → GEF_m                                                 R-1a

GEF_m → GEF_c                                                 R-1b

GEF_m + Cdc42D_m → GEF_m + Cdc42T                       R-2a

Cdc42T → Cdc42D_m                                              R-2b

Cdc42T-GEF + Cdc42D_m → Cdc42T-GEF + Cdc42T          R-3

Cdc42T + GEF_m → Cdc42T-GEF                                    R-4a

Cdc42T-GEF → Cdc42T + GEF_m                                    R-4b

Cdc42D_c → Cdc42D_m                                                    R-5a

Cdc42D_m → Cdc42D_c                                                    R-5b

Cdc42T + GEF_c → Cdc42T-GEF                                     R-6

Cdc42T-GEF + Cdc42D_c → Cdc42T-GEF + Cdc42T             R-7

In the above reactions, GEF_c and GEF_m are cytosolic and membrane bound GEF, respectively. Cdc42D_c and Cdc42D_m are cytosolic and membrane-bound inactive Cdc42 (Cdc42-GDP), respectively. Cdc42T is membrane-bound active Cdc42 (Cdc42-GTP). Cdc42T-GEF is the membrane-bound complex of Cdc42T and GEF.

Rate constants for the 2D stochastic polarization model

The rate constants for the 2D model in Fig 3A, presented in Table 1, were adapted from [13] which is a modified version of the model in [29]. That model is based on deterministic reaction-diffusion equations and is therefore a macroscopic representation. On the other hand, our stochastic simulations are parameterized with microscopic rate constants. For first-order and second-order reaction-limited reactions, the macroscopic rate constants from the model in [13] can be used directly in our simulations. However, for 2D diffusion-influenced reactions the conversion is more complicated [16]. For the purposes of this work, we used the macroscopic rate constants in [13] as a first approximation for the microscopic parameters and explored the effects of varying the rate constants over several orders of magnitude.

While the cell cytosol is a 3D compartment, in our 2D simulations the cytosol and the membrane are juxtaposed two-dimensional domains. This is a computationally efficient representation that neglects cytosolic gradients perpendicular to the membrane since, diffusion at the cytosol is fast compared to the timescale of reactions. To obtain equivalent rate constants for this purely 2D system we scaled cytosolic concentrations in the rate equations in [13] as: (22) where [C_c]^3D is the molar concentration of the cytosolic component C_c in the original equations, [C_c]^2D is the concentration of C_c in the 2D cytosol, V_c is the volume of the cell cytosol and A_m is the membrane area.

In [13], concentrations at the membrane are expressed in molar units assuming that the membrane was a volumetric compartment with thickness Δz. We therefore also scaled concentrations of species at the membrane as: (23) where [C_m]^2D is the concentration of the membrane-bound species C_m in units of mass/area, and [C_m]^3D is the molar concentration of C_m. From the scaled reaction-diffusion equations we obtain the scaled 2D reaction rate constants k^2D in terms of the rate constants k^3D in [13]:

• Rate constants for first-order reactions that involve a transition from the cytosol to the membrane (R-1a and R-5a) are scaled as:

(24)

• Second-order rate constants for reactions taking place at the membrane (R-2a, R-3, R-4a) are scaled as:

(25)

• Second-order rate constants for reactions in which a cytosolic species reacts with a membrane-bound species (R-6, R-7) are scaled as:

(26)

• The rate constants for first-order reactions in which the reactant and the product are bound to the membrane (R-2b and R-4b) are unchanged.
• Reactions 1b and 5b are the reverse of reactions 1a and 5a respectively, and to obtain the 2D rate constants it is necessary to multiply the corresponding volumetric rate constants by a factor of 1/η. However, that factor cancels out with a factor of η present in the reaction-diffusion equations used in [13,29], which takes into account the difference between the cytosol volume and the effective volume of the membrane. Therefore, k_1b and k_5b are unchanged.

The resulting 2D rate constants are presented in Table 1 using molecules as the unit of mass and μm as the unit of length.

We note one significant difference between the model parameters used here, and those used by McClure et al. We used a smaller number of total Cdc42 molecules based on the work of Watson et al. [65]. For our particle-based simulations, we used a smaller reaction radius ρ than in our previous publication [16] to make this value closer to the size of the reacting proteins. This modification reduced the rate of Cdc42 activation, affecting polarization. To compensate for this effect, we increased the rate constants of Reactions 5a, 5b and 6 by a factor of 10.

3D particle-based simulations

3D simulations were performed using Smoldyn [22,23]. Membrane-bound species diffuse on the surface of a sphere with radius R and cytosolic components diffuse within the interior of the sphere. The parameter values used in the particle-based 3D simulations are given in Table 1. They correspond with the 2D parameters except when they have been converted to account for differences in the system dimensionality as described below.

The reaction rate constants were obtained from the 2D model. Rate constants for reactions that take place exclusively in the membrane do not need to be modified. These are Reactions 1b, 2a, 2b, 3, 4a, 4b and 5b.

The rate constants for reactions in which a cytosolic species binds the membrane or a membrane-bound molecule (Reactions 1a, 5a, 6, and 7) are estimated from the 2D model using the scaling introduced in the Methods subsection “Rate constants for the 2D stochastic polarization model” ignoring the factor Δz. With that scaling these 3D rate constants are obtained by multiplying the corresponding 2D rate constants by the factor V_c/A_m.

Although we report the microscopic rate constant for all reactions, we parameterized second-order reactions within Smoldyn, providing the probability of reaction P during a time-step Δt when the separation of reactants is within ρ, approximated as P = λ Δt, for small λ Δt. For reactions where both reactants are at the membrane, the relation between λ and k_micro is the same as in the 2D simulations (Eq 2).

For second-order reactions where a cytosolic molecule reacts with a membrane-bound molecule and the product is at the membrane (Reactions 6, 7) λ is expressed in terms of k_micro as: (27)

This follows from an expression equivalent to Eq 1 for the probability rate that a membrane bound molecule A^m reacts with a cytosolic molecule B^c located within a distance ρ: (28) where [B^c]_ρ = 3/(2πρ³) is the concentration of a molecule in a half sphere of radius ρ.

Quantification of clustering with H(r)

Before introducing the function H(r) let us consider some typical metrics to quantify molecules aggregation. We first introduce the pair-wise molecule density n(r): (29) where m_i(r) Δr is the number of molecules between distance r and r + Δr from molecule i and N is the total number of molecules. In a 2D domain with area A, if molecules are uniformly distributed, n(r) should converge to n_unif(r) = 2 π r (N-1) / A. The pair-wise correlation function g(r), commonly used in physics, can be computed as g(r) = n(r)/n_unif(r), so that g(r) = 1 for a uniform distribution of molecules. Alternatively a distribution of pair-wise distances P(r) can be obtained normalizing n(r) by the N -1 comparisons involved in the calculation of each m_i(r) [31], (30)

For a uniform distribution of molecules P_unif(r) = 2 π r / A.

The H(r) function quantifies the difference between the cumulative distribution function (sometimes referred to as the Ripley’s K function) from the system of interest and the case with uniformly distributed molecules. In 2D, H(r) is defined as: (31)

With this definition H(r) = 0 if the molecule distribution has no structure (uniformly distributed) since . Also, H(r) < 0 indicates dispersion and H(r) > 0 indicates clustering or aggregation. Furthermore, the value of r where H(r) is maximum provides an estimate of the cluster length-scale [30].

Mean squared displacement (MSD) and effective polarity cluster diffusivity (D_patch)

After a polarity cluster has formed, the distribution of active Cdc42 is translated to the center of the domain to reduce border effects, and the centroid of the distribution is recorded every 1min. The centroid of the patch is calculated accounting for the toroidal geometry of the domain resulting from the periodic boundary conditions. The mean squared displacement (MSD) for a particular time interval Δt_i is computed from all centroid trajectories over time intervals of length Δt_i from multiple simulations. We discarded centroid jumps over the domain boundary by breaking trajectories containing jumps larger than a maximum jump max_jump into sub-trajectories containing only jumps smaller than max_jump. We empirically set max_jump = 6μm from visual inspection of centroid trajectories. Each MSD curve is obtained from 50 simulations of 1hr each except for S6 Fig where we used 5 simulations. The effective diffusivity of the polarity cluster (D_patch) can be obtained by fitting the equation MSD(Δt_i) = 4D_patch Δt_i^β to the MSD data, where Δt_i is a particular time interval, and a reflects the degree of anomalous diffusion. In practice we took logarithms to the data and fit the equation log(MSD(Δt_i)) = log(4D_patch) + β log(Δt_i) using only the first data points that showed a linear behavior in a log-log plot. The MSD during the smallest interval computed (1min) reflects rapid variations in the position of the centroid within the polarity cluster and do not contribute to the long scale displacement of the distribution. We therefore subtracted MSD(1min) to all MSD data before estimating D_patch. MSDs in the 3D particle-based simulations were computed from geodesic displacements of the cluster on the spherical surface.

Coefficient of variation of the distribution of active Cdc42T: CV_patch

CV_patch was computed as a weighted average over space of the local coefficient of variation over time of the amount of Cdc42T. The average over space is weighted by the mean local abundance of Cdc42T: (32)

Here CV_j^T is the mean divided by the standard deviation over a time interval T of the amount of Cdc42T at location j. <Cdc42T_j>^T is the average amount of Cdc42T at location j over a period of time T. The space average is computed over the whole simulation domain (A) as Cdc42T is mainly located at the polarity site. The time averages are computed over a short period T = 1 min to ensure the mean distribution of Cdc42T does not relocate significantly. One estimation of CV_patch is obtained from a single simulation that has reached steady state with samples taken every second to compute time averages. In Fig 6E and 6F we plotted the mean and standard deviation (error bars) from 5 independent measurements of CV_patch.

Dwell times at the patch

To compute the dwell time of Cdc42 at the patch, we introduced in the model additional tagged Cdc42 species (Cdc42_m^tagged, Cdc42T^tagged, Cdc42T^tagged-GEF) that have the same behavior as the untagged versions, except that if Cdc42_m^tagged jumps from the membrane to the cytosol it converts into untagged Cdc42D_c. Simulations are initialized with no tagged species, and are run until the distribution of polarity factors reaches steady state. At this point, Cdc42_m, Cdc42T, Cdc42T-GEF are converted into the tagged versions in a region surrounding the polarity patch and we record the decay in the amount of the tagged molecules. The same idea is used to estimate the dwell time at the patch of GEF (introducing GEF_m^tagged and Cdc42T-GEF^tagged). For Cdc42, we ignored the initial rapid decay coming from membrane detachment of inactive Cdc42. The dwell time at the patch is obtained by fitting an exponential decay function to the data. We reported the mean and standard deviation (error bars) from 10 or 30 independent measurements of the dwell time.