Fig 1.
(A) Actin filaments are modeled as rigid rods made of subunits carrying an orientational vector Oi 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 b = 6nm and a height of δ = 2.7nm. The orientation of a filament is described by the unit vector N, pointing from the pointed end (-) to the barbed end (+), and the first subunit’s orientational vector M = O1. Two consecutive subunits have an angle of π14/13 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 kf and kb, respectively. (C) Each crosslinker is modeled as a combination of three springs, one extensional spring with stiffness κext, which characterizes the stretchiness lc between the two actin binding domains, and two torsional springs with stiffness κtor, which characterizes the flexibility of the angles θi and θj 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 (Oi and Oj).
Table 1.
List of parameters.
Fig 2.
The structure of crosslinked actin networks depends on filament length.
(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.
Fig 3.
Influence of the crosslinker’s mechanical and kinetic properties on the organization of the actin network.
(A-D) Organization of 135nm-long actin filaments at the end of the simulation (t = 50s) for different values of extensional stiffness κext, torsional stiffness κtor, and breakage rate , as indicated above the figure. (E) Local nematic order parameter Slocal as a function of the extensional stiffness κext, with κtor = 10pN ⋅ nm ⋅ rad−1 and
. (F) Local nematic order parameter Slocal as a function of the torsional stiffness κtor, with κext = 0.1pN/nm and
. (G) Local nematic order parameter Slocal as a function of the strain-free breakage rate
, with κtor = 10pN ⋅ nm and κext = 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 Slocal were calculated from the data between t = 40s to 50s and the error bars indicate standard deviation over 10 simulations. Slocal corresponding to the networks in panels (A-D) are identified by the red symbols with corresponding shapes.
Fig 4.
Phase diagram of actin network organization as a function of the crosslinking rate kf and filament length L.
(A, D) Local nematic order parameter Slocal as a function of kf and L. (B, E) Number of attached crosslinkers Nattach as a function of kf and L. (C, F) Classification of actin network organizations as a function of kf and L. The extensional stiffness κext 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 ΔL = 27nm of L between 81nm and 218nm, and increment Δkf = 0.1s−1 of kf between 0.1s−1 and 1s−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 Slocal = 0.75. The border separating meshwork (gray) from uncrosslinked (dark gray) is defined by Nattach = 300.
Fig 5.
Straining of crosslinkers and energy storage.
(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 (t = 50s). For comparison, red lines indicate the corresponding Boltzmann distributions of a free spring with the same stiffness, and
, where C1 and C2 are the normalization constants for ϵ ∈ (−1, ∞) and θ ∈ (0, π) 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 ϵ (D) and the corresponding extensional energy Eext (E) as a function of the extensional stiffness κext. (F, G) Average torsional strain θ (F) and the corresponding torsional energy Etor (G) as a function of the torsional stiffness κtor. 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.
Fig 6.
Schematic illustration of a possible mechanism for torque generation by sequential detachment of crosslinkers.
(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 π/13.