Fig 1.
Left: Sketch of the recycling scheme: The minority phase (red) is brought to the membrane as monomers (a) and recycled by vesicles of a given size nv.
Clusters of size smaller than nv are recycled entirely in one vesicle (b), while larger clusters are fragmented and loose an area nv during a recycling event (c). Right: Stationary cluster size distribution in the limit of infinite systems, for different values of the size of a recycling vesicle nv. These results are obtained by numerically solving the master equation (Eqs (2, 3)). The “whole cluster recycling” limit (nv → ∞), corresponding to Eq 4, is shown in red. The “monomer recycling” limit (nv = 1) is shown in blue. Intermediate values of nv are the dashed lines.
Fig 2.
Bottom: Snapshot of the cluster size distribution at two different times for the monomer recycling scheme (left) and the whole cluster recycling scheme (right).
The black arrows illustrate the dynamics of the large cluster. Top: Evolution of the size of the largest cluster with time for the two schemes. The parameters are J = 2.2 ⋅ 10−4 and K = 2 ⋅ 10−3 (whole cluster recycling) or J = 1.1 ⋅ 10−4 and K = 1 ⋅ 10−3 (monomer recycling). This ensure in both cases that ϕ = 0.1 and κ = 100. The system size is Ns = 1000.
Fig 3.
Steady-state distributions of cluster size for small systems, obtained by simulations (averaged over a large number of independent simulations) compared to the infinite membrane result.
Simulations are shown for different system sizes for the whole cluster recycling (a) and monomer recycling (b) schemes. Comparison with the hybrid analytical model developed in the text for the whole cluster recycling (c) and monomer recycling (d) schemes show excellent agreement for strong confinement (with Ns = 100). All plots are for ϕ = 0.1 and κ = 180.
Fig 4.
Efficiency of a two-step enzymatic reaction taking place on the membrane of an organelle as a function of the organelle’s size.
(a): Sketch of the reaction described in the text, in which a substrate S is transformed into a product P via and intermediate I, each reactions being catalysed by different enzymes E1 and E2, both confined within clusters. (b1–b3) Reaction efficiency for a single membrane unit, defined as a circular cluster of size Rd in a circular membrane patch of size , for ϕ = 0.1 and for different value of the ratio of reaction to degradation rates μ (Rd is given in unit of
). (c) Average reaction efficiency for the full systems, consisting of clusters with fluctuating sizes. The average size distribution (sketched) depends on the level of confinement (the system’s size), as quantified in Fig 3. The average efficiency η* (Eq 13) is normalised by that of an homogeneous system η0 Eq 10. All parameters are given in the text.