A master equation approach to actin polymerization applied to endocytosis in yeast

Abstract

We present a Master Equation approach to calculating polymerization dynamics and force generation by branched actin networks at membranes. The method treats the time evolution of the F-actin distribution in three dimensions, with branching included as a directional spreading term. It is validated by comparison with stochastic simulations of force generation by actin polymerization at obstacles coated with actin "nucleation promoting factors" (NPFs). The method is then used to treat the dynamics of actin polymerization and force generation during endocytosis in yeast, using a model in which NPFs form a ring around the endocytic site, centered by a spot of molecules attaching the actin network strongly to the membrane. We find that a spontaneous actin filament nucleation mechanism is required for adequate forces to drive the process, that partial inhibition of branching and polymerization lead to different characteristic responses, and that a limited range of polymerization-rate values provide effective invagination and obtain correct predictions for the effects of mutations in the active regions of the NPFs.

Fig 1. Model schematic.

The membrane deformation is determined by a balance of forces between turgor pressure, the membrane bending energy, and actin pulling and pushing forces.

Fig 2. Schematic of discretized version of Eq 1 for the case of filaments having length four subunits (l¯max=4).

The subunits in solid circles can generate a branch of length $\overline{l}(\overline{y},t)$ that reaches the open circle $(\overline{x},\overline{y})$. $\overline{x}$ and $\overline{y}$ are dimensionless coordinates normalized by $a/\sqrt{2}$, where a is the step size per added subunit. See Eq. S30 for definition of $\overline{l}(\overline{y},t)$.

Fig 3. ME validation.

We simulate (A) pushing of an obstacle by an actin network, using a stochastic method and the ME approach. We calculate (B) F-actin count and (C) velocity as functions of constant external force.

Fig 4. Spatial distribution of branching and spontaneous nucleation.

(A) Branching and spontaneous-nucleation layers, and the forbidden zone, defined as the space inside the membrane where the filaments cannot penetrate. (B) The Gaussian nucleation function, for k_nuc(r, y), Eq. (S33), defined in the black layer in (A). (C) The Gaussian branching function for k_br(r, y), Eq (3), defined in the red layer in (A).

Fig 5. Computational flow.

The four functions ρ, k_nuc, k_br and r_m have initial values at time t. Then ρ is updated according to Eq. S5. Next r_m is updated according Eq. S31, using force balance (Eq. S26). Finally k_nuc and k_br are updated according to Eqs. S33 and 3. The process is repeated at each time step.

Fig 6. Time evolution of the F-actin distribution and the membrane profile.

The F-actin density ρ(r, y) (red) at four time points during the time course of endocytosis in wild-type cells is shown by the heat map. The cell membrane is in green and the Las17 is in blue.

Fig 7. Time courses of F-actin (F) and Las17 (N) of wild-type and las17 pan1Δacidic (LPΔA) cells.

(A) Time courses of F and N obtained from the ME model, compared to experimental data [12] for wild-type cells. (B) Same comparison for LPΔA cells, in which ${\overline{k}}_{br}^{max}$ is reduced by 40%.

Fig 8. Time evolution of the pulling force per filament for different spontaneous nucleation rates, with representative ρ(r, y) plots.

(A) Time courses of the pulling force per filament for ${\overline{k}}_{nuc}^{max}=$ 33%, 66% and 100% of the default value (Table 1). (B)-(D) Representative ρ(r, y) for each case; (D) represents the default model. The representative time point is chosen when the pulling force per filament reaches its maximum value while the invagination is greater than zero. The actin network in the middle is significantly sparser in (B) than in (D), causing the pulling force per filament to exceed the rupture force measured in Ref. [42].

Fig 9. Responses of F_max and N_max to treatment with the drugs CK-666 and LatA, and the corresponding phase diagrams.

We assume that (A) CK-666 reduces ${\overline{k}}_{br}^{max}$, (B) LatA(1) reduces k_on G only, and (C) LatA(2) reduces both ${\overline{k}}_{br}^{max}$ and k_on G. The N_max−F_max plots in frames (D)-(F) can be directly compared to experiments. The wild-type data points are denoted by blue open circles, and decreasing ${\overline{k}}_{br}^{max}$ and/or k_on G corresponds to moving right in these frames.

Fig 10. Time evolution of the F-actin distribution and the membrane profile for low polymerization rate.

In this model, we have reduced ${\overline{l}}_{max}$ to 30, half its default value (see Table 1). This corresponds to a reduction of 50% in the polymerization rate. The actin network cannot effectively invaginate the membrane.