Fig 1.
The structure-resolved reaction-diffusion model for clathrin recruitment and assembly on membranes captures structure, valency, and excluded volume.
Clathrin trimers are initialized in solution, but bind to adaptor proteins that are localized to the surface. Clathrin binds to adaptors only after the adaptors have localized to the surface, which is consistent with the behavior of the adaptor AP-2[16]. Adaptor proteins can bind specific lipids on the membrane surface when initialized in solution. Clathrin-clathrin sites and clathrin-adaptor sites exclude one another with a radius of σ = 5nm (blue circles) and σ = 1nm, respectively, but only when they are unbound; bound sites do not exclude volume. Therefore, at all times in the simulation, center-of-mass (COM) sites on trimers exclude other trimer COMs at σ = 10nm (purple circles), to ensure the lattices do not overlap with one another.
Fig 2.
Reaction network for the model captures energetics and kinetics of binding in 3D, 2D, and between 3D-2D.
For components clathrin (labeled C), adaptor (A), and lipid (P), only the specific reactions shown in black arrows occur between the connected interfaces. The legend indicates what parameters per reaction are specified, where a 2D reaction relates to the corresponding 3D reaction via a lengthscale h due to the change in dimensionality of the search. When a clathrin-adaptor complex forms (C.A), it binds other clathrin more strongly via ΔGcoop. The lower panel shows how the addition of a monomer can close a hexagonal loop. The free energy of forming both bonds is modeled with negative cooperativity, via an energy penalty ΔGstrain.
Table 1.
Each parameter varied in the RD model to reproduce in vitro experimental kinetics, along with the optimal values we found.
Other energies, geometries, and concentrations were fixed by experimental data.
Fig 3.
Structure-resolved reaction-diffusion simulations of clathrin assembly kinetics on membranes reproduce in vitro experiment.
(A) Simulations were set up to mimic experimental conditions from published work of Pucadyil et al [25], where fluorescence of clathrin accumulating on membrane tubules was measured with time. A constant (fluctuating) solution concentration of clathrin (at 80nM) was maintained through exchange with a large volume reservoir. (B) Fluorescence of clathrin on membranes is averaged over multiple tubules (black line). The experimental fluorescence data was reported in arbitrary units [25]. This means there is a free scale factor we use to match with our simulations, which measure clathrin accumulation in units of copy numbers per μm2. The model result is shown in blue (averaged from 4 simulation trajectories). Both the simulation data (gray dashed line) and experimental data can be fit to Eq 1A, producing timescales (growth rate k and lag time τ) that can be directly compared to one another. The initial growth is approximately linear, with a steepness given by kE, with E the maximal extent of clathrin on the membrane (Eq 1B). (C) Snapshots of one simulation trajectory at different time points, also see S1 Movie. (D) The macroscopic timescales of the lag time τ (black dots) and initial growth steepness kE (red dots) vary with changes to six model parameters, as shown in each sub-plot D1-D6 and listed in Table 1. We plot the initial growth steepness kE rather than the growth rate k, as it was more amenable to theoretical predictions. The dashed black and red lines are the theoretical fits from our phenomenological models to τ (Eq 2) and kE (Eq 3). For each subplot, the non-varied parameters are otherwise fixed to the optimal values of Table 1.
Fig 4.
Clathrin lattices at physiologic-like geometries fail to assemble until adaptor concentration reaches 0.6μM.
(A) Kinetics of the largest clathrin lattice assembly is faster and reaches larger sizes with increasing adaptor concentration. All simulations have 0.65μM clathrin. (B) Lag times and steepness of initial growth determined from simulation (black and red dots, respectively) follow the theoretical expressions of Eq 2 and Eq 3 relatively well. We excluded the low concentrations where the kinetics is not well-described by the lag and exponential growth model of Eq 1. (C) Lag times are typically slower in these geometrically physiologic-like simulations and rate-limited by localization of clathrin to a membrane (term 1 in Eq 2) with far fewer adaptor binding sites than in vitro. (D) The size of the largest lattices increases with increasing adaptor concentration, transitioning from small to large lattices at ~0.8μM (black squares). At this point, the stoichiometry of clathrin:adaptor has reached 1:1 (orange circles).
Fig 5.
Clathrin lattices start as monomers that face an initial barrier to growth, with a stable size reached only after significant growth.
(A) The probability of observing clathrin lattices of size n can be converted to an energy-like metric −ln (P(n)), which at equilibrium is a true free energy in units of kBT. An initial barrier (in n) to nucleation plateaus, followed by a relatively flat region, where structures have comparable probability. Across the full time-dependent trajectory shown in the upper inset, we analyze lattices sampled across the full trajectory (green dots), the growth phase (yellow dots), and equilibrium (orange dots). During growth, clathrin forms intermediate size lattices, while at equilibrium, only small and large lattices are visible. [AP2] = 1.6μM. (B) To quantify the end of the initial barrier to growth (n1), and the start to the stabilized growth (n2), we define a plateau at constant −ln (P(n)) that defines these intercepts for each trajectory (Text A in S1 Text). (C) With increasing adaptor concentration, larger lattices become stabilized. The shape of the curves are consistent with the data in Fig 4A; the free energy forms a trough at higher concentrations, consistent with the average size of lattices at equilibrium. (D) The critical size where the barrier plateaus is ~n1 = 25, independent of adaptor concentration. The noisy plateau region is followed by a well that starts at n2 and increases with increasing adaptor concentration. (Inset) With higher adaptor concentration, the time to cross the first barrier at n1 is faster, following the inverse of adaptors τobs∝1/[AP2], with a proportionality constant 1/[0.026μM-1s-1].
Fig 6.
Curved-cage assembly in solution requires added stabilization, which is comparable to membrane bending energies.
(A) Clathrin cages with adaptors present do not form in solution (black dots), using the same model as on the membrane (open circles), despite the same concentration of clathrin. With added stabilization for all clathrin-clathrin bonds (ΔG) and reduced strain (ΔGstrain), solutions cages start to form (red and blue), although growth is less cooperative without dimensional reduction following membrane localization. (B) Membrane bending energy per clathrin trimer as the sizes n of curved lattices are increased. More highly curved cages (α = 98o) cost more energy per trimer to bend the membrane. The membrane energy is proportional to the bending modulus κ, where we use κ = 20 kBT consistent with measurements on the plasma membrane [67].