Model description.

<p>(A) Actin filaments are modeled as rigid rods made of subunits carrying an orientational vector <b>O</b>_<i>i</i> that represents the normal vector to the binding surface of the actin crosslinker. For visualization, each subunit is depicted as a sphere but is actually a disk-shape with a diameter of <i>b</i> = 6nm and a height of <i>δ</i> = 2.7nm. The orientation of a filament is described by the unit vector <b>N</b>, pointing from the pointed end (-) to the barbed end (+), and the first subunit’s orientational vector <b>M</b> = <b>O</b>₁. Two consecutive subunits have an angle of <i>π</i>14/13 in their orientations. (B) Crosslinker turnover is described by stochastic formation and breakage of bonds between two actin subunits in different filaments, with rate constants <i>k</i>_f and <i>k</i>_b, respectively. (C) Each crosslinker is modeled as a combination of three springs, one extensional spring with stiffness <i>κ</i>_ext, which characterizes the stretchiness <i>l</i>_<i>c</i> between the two actin binding domains, and two torsional springs with stiffness <i>κ</i>_tor, which characterizes the flexibility of the angles <i>θ</i>_<i>i</i> and <i>θ</i>_<i>j</i> between the axis of the crosslinker (which links the centers of each subunits it is attached to) and the vector normal to the binding surface of each actin subunit in each filament (<b>O</b>_<i>i</i> and <b>O</b>_<i>j</i>).</p

Influence of the crosslinker’s mechanical and kinetic properties on the organization of the actin network.

<p>(A-D) Organization of 135nm-long actin filaments at the end of the simulation (<i>t</i> = 50<i>s</i>) for different values of extensional stiffness <i>κ</i>_ext, torsional stiffness <i>κ</i>_tor, and breakage rate , as indicated above the figure. (E) Local nematic order parameter <i>S</i>_local as a function of the extensional stiffness <i>κ</i>_ext, with <i>κ</i>_tor = 10pN ⋅ nm ⋅ rad⁻¹ and . (F) Local nematic order parameter <i>S</i>_local as a function of the torsional stiffness <i>κ</i>_tor, with <i>κ</i>_ext = 0.1pN/nm and . (G) Local nematic order parameter <i>S</i>_local as a function of the strain-free breakage rate , with <i>κ</i>_tor = 10pN ⋅ nm and <i>κ</i>_ext = 1pN/nm. In (E-G), simulations were performed for filaments of various lengths: 81nm (blue), 135nm (red), 189nm (orange). For each simulation, the means of <i>S</i>_local were calculated from the data between <i>t</i> = 40<i>s</i> to 50<i>s</i> and the error bars indicate standard deviation over 10 simulations. <i>S</i>_local corresponding to the networks in panels (A-D) are identified by the red symbols with corresponding shapes.</p

Straining of crosslinkers and energy storage.

<p>(A-C) Probability density distribution (PDF) of the extensional strains (top) and torsional strains (bottom) of crosslinkers with various stiffness values as indicated above the figure, at the end of individual simulations (<i>t</i> = 50<i>s</i>). For comparison, red lines indicate the corresponding Boltzmann distributions of a free spring with the same stiffness, and , where <i>C</i>₁ and <i>C</i>₂ are the normalization constants for <i>ϵ</i> ∈ (−1, ∞) and <i>θ</i> ∈ (0, <i>π</i>) respectively. Note that for the extensional strain, when we calculate the Boltzmann distribution, the energy contribution from steric interactions, which lead to the empty region at highly negative strains in the histogram, is neglected. (D, E) Average absolute value of the extensional strain <i>ϵ</i> (D) and the corresponding extensional energy <i>E</i>_ext (E) as a function of the extensional stiffness <i>κ</i>_ext. (F, G) Average torsional strain <i>θ</i> (F) and the corresponding torsional energy <i>E</i>_tor (G) as a function of the torsional stiffness <i>κ</i>_tor. In (D-G), simulations are performed for filaments of various lengths: 81nm (blue), 135nm (red), and 189nm (orange). For each simulation, the means of the energy were calculated from the data between 40s to 50s and the error bars indicate standard deviation over 10 simulations. Energies corresponding to the networks in panels (A-C) are identified by the red symbols with corresponding shapes.</p

Structural organization and energy storage in crosslinked actin assemblies

<div><p>During clathrin-mediated endocytosis in yeast cells, short actin filaments (< 200nm) and crosslinking protein fimbrin assemble to drive the internalization of the plasma membrane. However, the organization of the actin meshwork during endocytosis remains largely unknown. In addition, only a small fraction of the force necessary to elongate and pinch off vesicles can be accounted for by actin polymerization alone. In this paper, we used mathematical modeling to study the self-organization of rigid actin filaments in the presence of elastic crosslinkers in conditions relevant to endocytosis. We found that actin filaments condense into either a disordered meshwork or an ordered bundle depending on filament length and the mechanical and kinetic properties of the crosslinkers. Our simulations also demonstrated that these nanometer-scale actin structures can store a large amount of elastic energy within the crosslinkers (up to 10<i>k</i>_B<i>T</i> per crosslinker). This conversion of binding energy into elastic energy is the consequence of geometric constraints created by the helical pitch of the actin filaments, which results in frustrated configurations of crosslinkers attached to filaments. We propose that this stored elastic energy can be used at a later time in the endocytic process. As a proof of principle, we presented a simple mechanism for sustained torque production by ordered detachment of crosslinkers from a pair of parallel filaments.</p></div

Phase diagram of actin network organization as a function of the crosslinking rate <i>k</i>_f and filament length <i>L</i>.

<p>(A, D) Local nematic order parameter <i>S</i>_local as a function of <i>k</i>_f and <i>L</i>. (B, E) Number of attached crosslinkers <i>N</i>_attach as a function of <i>k</i>_f and <i>L</i>. (C, F) Classification of actin network organizations as a function of <i>k</i>_f and <i>L</i>. The extensional stiffness <i>κ</i>_ext is 0.1pN/nm in panels (A-C), and 1pN/nm in panels (D-F). In panels (A, B, D, E), plots are constructed by interpolation of results obtained for increment Δ<i>L</i> = 27nm of <i>L</i> between 81nm and 218nm, and increment Δ<i>k</i>_f = 0.1<i>s</i>⁻¹ of <i>k</i>_f between 0.1<i>s</i>⁻¹ and 1<i>s</i>⁻¹. The value for each parameter set is an average over 10 simulations. In (C) and (F), the border separating bundle (light gray) from meshwork (gray) is defined by <i>S</i>_local = 0.75. The border separating meshwork (gray) from uncrosslinked (dark gray) is defined by <i>N</i>_attach = 300.</p

List of parameters.

<p>List of parameters.</p

Schematic illustration of a possible mechanism for torque generation by sequential detachment of crosslinkers.

<p>(A) Two filaments in parallel are crosslinked on every other subunit. From left to right, crosslinkers are detached from the pointed end (−) to the barbed end (+) sequentially. Upon detachment of a crosslinker (yellow symbols), both filaments rotate around their axes counterclockwise. (B) The angular displacement of the filament upon every detachment is <i>π</i>/13.</p

The structure of crosslinked actin networks depends on filament length.

<p>(A) Snapshots of an actin network formed by 81nm-long filaments in a 500nm-wide cubic box. Each filament is represented by a blue line and each crosslink by a red line. (B) Local (blue) and global (red) nematic order parameters of the actin network over the course of the simulation shown in (A). (C) Number of attached crosslinkers (magenta, left axis) and number of clusters (black, right axis) for the simulation in (A). (D-F) Similar figures as in (A-C) but for 216nm-long filaments. (G) Local (blue) and global (red) nematic order parameters as a function of filament length. (H) Number of attached crosslinkers (magenta, left axis) and number of clusters (black, right axis) as a function of filament length. In (G) and (H), for each simulation, the means of the metrics were calculated from the data between 40s to 50s and the error bars indicate standard deviation over 10 simulations.</p