Stochastic Model of Vesicular Sorting in Cellular Organelles

The proper sorting of membrane components by regulated exchange between cellular organelles is crucial to intracellular organization. This process relies on the budding and fusion of transport vesicles, and should be strongly influenced by stochastic fluctuations, considering the relatively small size of many organelles. We identify the perfect sorting of two membrane components initially mixed in a single compartment as a first passage process, and we show that the mean sorting time exhibits two distinct regimes as a function of the ratio of vesicle fusion to budding rates. Low ratio values lead to fast sorting but result in a broad size distribution of sorted compartments dominated by small entities. High ratio values result in two well-defined sorted compartments but sorting is exponentially slow. Our results suggest an optimal balance between vesicle budding and fusion for the rapid and efficient sorting of membrane components and highlight the importance of stochastic effects for the steady-state organization of intracellular compartments.

Figure 1

Sketch of the sorting mechanism. A “mother” compartment contains both $A$ and $B$ membrane components, which are individually exported by vesicle budding (rate $K$). Secreted vesicles with the same identity fuse with one another to create pure $A$ and $B$ compartments (rate $k_f$) and may fuse back with the mother compartment with a composition-dependent rate. Pure $A$ and $B$ compartments do not fuse together. Stochastic fluctuations lead to the permanent separation of $A$ and $B$ components into two independent distributions of pure compartments. A typical time trace for the mother compartment composition (with fraction of $B$ component $\varphi$) is shown. Full sorting occurs when $\varphi =0$ (red star).

Figure 2

Dimensionless mean first passage time $K\tau$ as a function of the ratio of fusion to budding rate $k_f/K$ for different initial compositions and sizes. The three groups of curves correspond to different initial compositions (from left to right: ${\varphi }_0=0.8$, 0.5, and 0.2). The simulation results are shown for two different initial compartment sizes in each case (red circles: $N_0=20$; red crosses: $N_0=100$). The black curves show the analytical results for $N_0\to \infty$, the dashed line is the full solution [Eq. (S.11) in the Supplemental Material 21], and solid lines are the asymptotic limits for $K\gg k_f$ ($t_0$) and $K\ll k_f$ [Eq. (5)]. The light blue curves show the results of the discrete model [Eq. (7)]. The crossover from fast to slow sorting [Eq. (9)] is shown with blue dots. Inset: Size dependence of the mean first passage time $K\tau$ (with ${\varphi }_0=0.5$ and $k_f/K=50$). The error bars are standard deviations over many independent simulations.

Figure 3

(a) Steady-state size distribution of one-component compartments undergoing fusion at a rate $k_f$ and shedding vesicles at a rate $K$, for different values of the fusion to budding-rate ratio $k_f/K$. The total membrane area is $N={10}^3$. The single largest compartment corresponds to the red part of the distribution, whose integral equals unity. It contains a fraction $\rho$ of the total area, given in parentheses, consistent with Eq. (11). The typical size of the small compartments, obtained in the Supplemental Material [21], is displayed on each graph (black dots). (b) Sorting phase diagram showing how the interplay between vesicular export and fusion influences both the sorting time and the distribution of sorted components. The color background represents the fraction $\rho$ of the total area contained in the largest compartment. The boundary between fast and slow sorting is indicated by the solid blue lines when only one component can be exported in budding vesicles [Eq. (9)], and by the red dotted line when both components can be exported.