Force Production by a Bundle of Growing Actin Filaments Is Limited by Its Mechanical Properties

Abstract

Bundles of actin filaments are central to a large variety of cellular structures such as filopodia, stress fibers, cytokinetic rings, and focal adhesions. The mechanical properties of these bundles are critical for proper force transmission and force bearing. Previous mathematical modeling efforts have focused on bundles’ rigidity and shape. However, it remains unknown how bundle length and buckling are controlled by external physical factors. In this work, we present a biophysical model for dynamic bundles of actin filaments submitted to an external load. In combination with in vitro motility assays of beads coated with formins, our model allowed us to characterize conditions for bead movement and bundle buckling. From the deformation profiles, we determined key biophysical properties of tethered actin bundles such as their rigidity and filament density.

Significance

Groups of dynamic and parallel actin filaments, called bundles, can produce and sustain more force than isolated filaments. In this work, we perform experiments in which bundles of filaments grow from micrometric beads with their dynamic ends facing the load. We show that a straight bundle can steadily push the bead forward until the bundle becomes too long and bends, stopping the motion of the bead. We develop a mathematical model that recapitulates this behavior, explains the balance of the forces involved, and identifies the conditions under which movement is possible.

Introduction

Bundles of actin filaments are involved in a large variety of cellular structures such as filopodia (1), the cytokinetic ring (2,3), auditory hair cells (4), actin cables (5,6), stress fibers (7,8), focal adhesions (9), and adherens junctions (10) and are hijacked by some pathogens for their propulsion within and between host cells (11). Bundles also determine the shape of macroscopic structures like Drosophila bristles (12) and are the core of microvilli (13,14).

In motile cells, bundles of actin filaments can develop large amounts of force that deform the cell membrane at the leading edge (13,15,16), and are used to generate tension in stress fibers (17). Filopodial bundles are created by formin- and Ena/VASP-mediated assembly of parallel actin filaments. In the lamella, actin filaments from the lamellipodium are condensed into bundles under the action of the retrograde flow and motor proteins. Through these bundles, cells optimize and adapt their response to mechanical stress and disassemble the actin cytoskeleton at their trailing edge (18, 19, 20).

A limiting factor for force production is filament buckling (20, 21, 22). Buckling occurs when the force exerted between the ends of the filament reaches a critical value, which depends on the mechanical properties of the filament, its geometry, and the attachment conditions of its ends (23,24). When buckling occurs, the force produced by a filament (or a bundle) vanishes.

Theoretical and in vitro biophysical studies have determined the mechanical properties of bundles of actin filaments (15,25,26) and analyzed the effects of cross-linkers (27), motor proteins (21,22,28), active transport (29), or combinations thereof to mimic constricting cytokinetic rings (21,30, 31, 32, 33, 34). These studies essentially focused on the effect of bundle rigidity on its stability and shape and how filament dynamics and membrane tension control the bundle length. In particular, a careful analysis of the physics of protrusion determined the optimal number of filaments in a filopodium and its length (15). These quantitative predictions represent the first attempt to understand the dynamics of filopodia and classify their morphology (length, number of filaments in the bundle, and interfilopodium distance). However, despite these experimental and theoretical studies, it remains unknown how bundle length and buckling are controlled by external physical factors.

In this work, we present a biophysical model for bundles of dynamic actin filaments. The model is supported by experimental in vitro reconstitution of bead motility powered by bundles of actin filaments, which are nucleated on the bead surface by formins. Here, formins also play the role of tethers that maintain filament barbed ends on the bead. Using our experimental data, we quantified the bead movement and correlated it with the elongation and deformation of filament bundles. This allowed us to determine the bundles’ rigidity and average number of filaments in the bundles. Thus, using these two experimentally measured constraints, we propose an original, simple, and robust model for the movement and deformation of actin filaments that predicts how filament mechanics tunes the force that can be produced by dynamic actin-filament bundles.

Materials and Methods

Experimental data

Motility assay

Experiments performed in this study were carried out according to the procedure described in (35). In brief, beads grafted with the FH1-FH2 domains of mDia1 were mixed with a motility medium containing 8 μM F-actin, 4 μM profilin, and 10 μM human cofilin in X buffer (10 mM HEPES (pH 7.8), 0.1 M KCl, 1 mM MgCl2, 1 mM ATP, and 0.1 mM CaCl2), supplemented with 1% bovine serum albumin, 0.2% methylcellulose, 3 mM dithiothreitol, 1.8 mM ATP, and 0.1 mM DABCO (1,4-diazabicyclo[2.2.2]octane).

Microscopy

Motility assays were acquired with a Zeiss Axioplan microscope (Jena, Germany) equipped with a 63×/1.5 NA Plan-APOCHROMAT objective lens, a Hamamatsu ORCA CCD camera (Hamamatsu Photonics Deutschland, Herrsching, Germany) and Metavue version 6.2r6 (Universal Imaging, Bedford, NY).

Buckles of filaments were imaged using total internal reflection microscopy (TIRFM). Glass flow cells were cleaned and prepared according to (36). Rhodamine-actin and Alexa-532-labeled actin-filament polymerization was observed and acquired as specified in (37).

Theory and simulations

Mechanical equilibrium equations for an actin filament

We developed a model for the mechanical equilibrium of elastic filaments subjected to external forces and constraints based on a model we previously developed (38). In this section, we present the definitions of variables and the equations for force and moment balance. The orientation of the filament cross section at any point along the filament is given by a set of three unit vectors (d₁, d₂, d₃), which define the material frame associated with position r(s) (Fig. S1 A). The filament-bending or twisting strain κ, the density of force f, and moment m vectors are defined in the material frame as κ = κ₁d₁ + κ₂d₂ + κ₃d₃, f = f₁d₁ + f₂d₂ + f₃d₃, and m = m₁d₁ + m₂d₂ + md₃. The moment is proportional to the filament-bending strains m = Bκ where B is the bending rigidities diagonal matrix (C_B, C_B, C_T). The balance of force and moment reads

$$\begin{array}{c}\begin{array}{c}\frac{\partial \mathbf{f}}{\partial s}+\mathit{\kappa }\times \mathbf{f}={\mathbf{f}}_{ext}\\ \frac{\partial \mathbf{m}}{\partial s}+\mathit{\kappa }\times B\mathit{\kappa }+{\mathbf{d}}_3\times \mathbf{f}={\mathbf{m}}_{ext}\end{array}\end{array},$$ (1)

where f_ext and m_ext are the external force and moment densities applied to the filament at position s. All filament-filament or filament-medium interactions are modeled by adapting the expression for f_ext in Eq. 1. The unit tangent vector, denoted d₃, is given by

$$\frac{d\mathbf{r}}{ds}={\mathbf{d}}_3,$$ (2)

where r is a point along the filament (Fig. S1 A). Because filaments move at low Reynolds number (∼10⁻⁵), all force and moment terms proportional to linear or angular acceleration were eliminated from Eq. 1 (38). The components of the material frame (d₁, d₂, d₃) vectors are parametrized by Euler parameters (unit quaternions), which couple the change of the cross-sectional orientation to the force moment applied at s.

In Appendix A, we simplify this three-dimensional model to develop a general method for the analysis of filaments or bundles of filaments considered as two-dimensional (2D) elastic structures and the measurement of the polymerization and drag forces (Appendix A). In Appendix B, we adapt this model to the analysis of bead movement.

Results

Buckling of actin-filament bundles limits the motility of formin-grafted beads

When placed in a motility medium containing actin, profilin, and cofilin (35,39), beads coated with formin mDia1 nucleate actin filaments at their surface (Fig. 1 A). These elongating filaments rapidly form long bundles of actin filaments, which propel the bead at a constant velocity ∼0.25 μm ⋅ min⁻¹ for a duration that varied from a few minutes to over 1 h, when the experiment was stopped (Fig. 2). In most cases, beads eventually stalled and remained permanently stuck to the coverslip (Figs. 1 A and 2). During the motility phase, bead trajectories were rectilinear, their bundle remained straight (Fig. 1 A), and the bead velocity and the bundle elongation rate were identical (Fig. 2). After beads stalled, bundles continued growing at the same elongation rate while deforming to eventually form a large buckle (Fig. 3).

Figure 1.

Bead motility mediated by bundles of actin filaments nucleated by processive formins (mDia1). (A) Beads coated with formins mDia1 (20–160 formins/bead) are added to an F-actin motility medium (8 μM F-actin, 4 μM profilin, and 10 μM actin depolymerizing factor (ADF)/cofilin). Actin filaments are arrayed in parallel bundles with their barbed ends at the bead surface. In some cases, the force developed by actin polymerization propels the bead at constant velocity (∼0.25–0.27 μm ⋅ min⁻¹) for up to 1 h (bundle and bead at the top). In other cases, the driving force fails to move the bead after a certain time of steady displacement (e.g., 30 min for the bead at the bottom) while polymerization continues, as demonstrated by bundle deformation. Scale bar: 5 μm (B) A schematic of the mechanical model for bead motility and bundle deformations. We combine the mechanics of bundle with elongation at the barbed end. The polymerization rate is corrected by a term depending on ΔE, the mechanical work necessary to insert a monomer. To see this figure in color, go online.

Figure 2.

Bundle elongation and bead displacement kinetics. (A and B) The bundle length (blue dots) and the bead trajectory (red dots) for the two examples shown in Fig. 1A (A: bottom bead; B: top bead). The dashed curves represent linear interpolation of experimental points. (C and D) Simulation of the bead displacement and bundle elongation using Eqs. B.1, B.2, B.3, B.4, and B.5 for bundles with persistence lengths of 50 μm (C) and 150 μm (D). To see this figure in color, go online.

Figure 3.

Loops formed by individual filaments and bundles of filaments. (A and B) A folded filament forms a flat loop, which is stabilized by attractive forces along the stem (dashed green arrow, B) balancing the elastic forces along the loop (dashed blue arrow, B). (C) A geometric model of the filament configuration in (A). M is a point at distance s from the loop origin O; θ is the angle between the unit vector tangent to the filament at M (black arrow) and the horizontal axis. To simplify the model, we consider that the attraction force exerted on M is horizontal, along the line MM′. M′ is the point symmetric to M located at a distance L − s from O. Additionally, we assume that the attraction between M and M′ depends on |MM′| only. (D) A cartoon showing how attractive forces along the loop stem (blue arrows) balance the elastic forces caused by filament bending (green arrows). (E and F) Representative loops from a single filament (E) and a bundle (F) can be fitted with our 2D model for elastic filaments (superimposed red line). Scale bars, 5 μm. (G and H) The attractive force density can be determined by our model. (G) A schematic representation of the magnitude and sign of the attraction force. (H) Distribution of the repulsion/attraction horizontal force as a function of the arc-length position for the solution of the single filament shown in (E). To see this figure in color, go online.

What is the cause of bead movement and why do beads stall? Because filaments in bundles grow at the same rate when beads are moving and when beads are stalled, we deduced that actin monomer concentration and viscosity of the motility medium remained constant (Fig. 2). In consequence, these two factors cannot account for the changes of the bead velocity. We hypothesized that the variety of motility behavior observed experimentally is due to the variations in the mechanical properties of the bundle of actin filaments over time.

Determination of bundle rigidity and depletion forces from loop geometries

To test our hypothesis, we took advantage of loops formed by long filaments (Fig. 3, A, B, and E) and bundles of filaments (Fig. 3 F) at the end of our experiments. These loops are typical of semiflexible polymers in a solvent and appear because depletion forces tend to bundle both ends of the filaments (40). Because we do not have enough videos with the full time course of the dynamical formation of these loops, we analyzed pictures of already formed, stable, bundle loops. We assumed that these loops are elastic structures at equilibrium, at which the elastic bending force of the filament (or bundle) that tends to open the loop and the attractive depletion forces that keep the stem of the loop closed balanced each other (Eqs. A.1 and A.2; Fig. 3 D; Appendix A). To further simplify the analysis and simulations, we assume that the filaments in the bundle are intact (no internal gap) and run along the whole bundle length. In addition, in agreement with the experimental conditions, we assume that the filaments in the bundle remain independent and can slide without friction during the bundle deformation.

The persistence length of bundles can be directly deduced by fitting the shape of the loops to our model (Eqs. A.5 and A.6; Figs. 3, E and F and 4 A). As expected, the apparent persistence length of bundles increased with the number of formin molecules bound to the bead and plateaued at larger formin density, probably because friction between filaments becomes high but is not accounted for in our model (Fig. 4, B–D). However, our results show that the number of filaments is significantly smaller than the number of formins on the bead estimated experimentally and plateaus for beads with more than ∼100 formins. This discrepancy can be partly explained by the fact that not all the adsorbed formins are functional. This argument can also explain the large variability in the number of filaments nucleated by a bead (Fig. 4 D). In addition, as we show below, the number of filaments attached to the bead is limited by mechanical stress applied by elongating filaments on the formins. Overall, the variability in the number of filaments in bundles likely explains the diversity of bead trajectories or loops observed in the same experimental conditions (Figs. 1 A and 4 A).

Figure 4.

Estimation of bundles’ persistence length by the fitting of loops. (A) Representative bundle loops for different densities of formin. The blue curves are the best fit by Eqs. 7, 8, 9, and 10 (with β = 1) to the experimental loops, which are shown in red. Horizontal and vertical axis are in micrometers. (B and C) Persistence length of bundles determined using Eqs. 7, 8, 9, and 10, assuming the filaments are bound together (β = 2, B) or slide perfectly relatively to each other (β = 1, C). Each red dot represents the data for one loop, and the average and standard deviation are plotted in blue. (D) The number of actin filaments in the bundles estimated from the rigidity measurement assuming perfect sliding in the bundle (β = 1, blue dots) or cannot slide (β = 2, red dots). To see this figure in color, go online.

In addition to the persistence length, our model also allowed us to determine the magnitude of the depletion forces that keep the loop stems together. Indeed, our model predicts that the self-interacting force density k_S is attractive along the stem up to the junction between loop and stem sections (red arrows, Fig. 3, G and H). For single filaments, it peaks ∼100 pN ⋅ μm⁻¹ at the transition between the stem and the loop, at which elastic forces from the loop counterpoise attraction in the stem (Fig. 3, G and H). The forces we measure are larger than the forces that were estimated in previous experiments (40), but this difference could be explained by the nature and concentrations of the crowding agents used in each experiment being largely different (0.2% methylcellulose vs. ∼2% PEG (polyethylene glycol)). In addition, our model evaluates the force at each point along the filament, whereas the model in (40) only considered a point force where the loop closes.

Bead stalling as a transition between polymerization-dominated to elasticity-dominated regime

To test whether the difference in the mechanical properties of the bundle can explain our experimental results, we developed a model for bundle mechanics and bead movement (Fig. 1 B), in which the bundle is modeled as a single elastic rod with bending rigidity L_P and contour length L (Appendix A and B). The bundle elongates from its barbed end, which is attached to the bead via formins. We assume that the bead-filament attachment remains unchanged during the whole process. In particular, the formin-bead or formin-filament link does not break, whatever the magnitude of the force developed by the filament polymerization. The other end of the bundle is assumed to have a fixed position and orientation on the coverslip, as experimentally observed. The transition between bead movement and bundle deformation depends on the balance between two antagonistic forces, namely 1) the force produced by actin polymerization and 2) the viscous drag due to the viscosity of the medium on the bead (Fig. 5 A). The physical origin of these forces and their order of magnitude are given in Table 2. If the force required to deform a bundle is higher than the drag force, the bead is steadily pushed forward while the bundle remains straight (Fig. 5 A, left). Conversely, when the viscous drag is larger than the critical buckling force of the bundle, the bead stalls, and the force generated by the elongation deforms the bundle (Fig. 5 A, right). This critical buckling force scales as the reciprocal of the square bundle length (adapted from Eq. A.13)

$$F_{Buckling}=\gamma \frac{\left(kT\right){\pi }^2{L}_p{N}^{3/2}}{L^2},$$ (3)

where L_p is the persistence length of a single actin filament, N is the number of filaments in the bundle, L is the bundle length, and γ is a numerical factor that depends on the boundary conditions at the bundle ends; here, β = 1 (filaments free to slide in the bundle).

Figure 5.

Motility and bundle deformation. (A) The off-balance between polymerization and opposing forces (e.g., drag forces) on the bead controls the transition from bead movement to bundle deformation (left: polymerization overcomes the drag and other opposing forces; right: the drag and other forces stall the bead). (B) The bundle rigidity controls the transition between bead movement and bundle deformation. Shaded areas give conditions for which bundles generate enough force to propel the bead. (C and D) Phase diagrams for bead movement and bundle deformation. (C) The transition between bead motility and bundle deformation in the parameter space (bundle length versus bundle rigidity). The dashed and solid arrows illustrate the typical trajectory in the phase diagram space associated with, respectively, a stalling bead (Fig. 1A, bottom, Fig. 2, A and C) and a moving bead (Fig. 1A, top, Fig. 2, B and D). (D) Increasing the viscous drag force controls the position of the transition curve separating motility and bundle deformation. To see this figure in color, go online.

Therefore, in conditions of constant elongation, the bundle length will eventually meet the condition for buckling, and the exact time for this transition to occur depends on the bundle rigidity or, equivalently, the number of filaments in the bundle (Eq. A.15).

In the analysis presented so far, we simplified the transition phase, and we did not include the drag force applied to the bundle itself. The transition phase is complex because within a very short period of time, the bead velocity drops from a constant value to zero while the bundle starts deforming, and its transversal movement drags fluid around it. In principle, we cannot rule out postbuckling micromovement of the bead. However, our experimental data show that this period is extremely short (less than 1 s) and abrupt (Fig. 2). The bead stalls almost suddenly and, simultaneously, the bundle starts deforming upon actin polymerization.

Our mathematical model for bead movement (Eqs. B.1, B.2, B.3, B.4, and B.5) reproduces the transitions between both kinds of movement (Figs. 2, A and C and 5 B) and shows the dependence of this transition on bundle rigidity L_P. Bundles with relatively low persistence lengths (e.g., L_p = 50 μm) cannot propel the bead throughout the medium over a long period of time (Fig. 5 B). Once the critical buckling condition is reached, i.e., when the bundle reaches the critical length), $L_{Crit}=\sqrt{\left(\gamma {\pi }^2\left(kT\right)L_p/F_{Drag}\right)N^{3/2}}$ (adapted from Eq. A.15 with β = 1, as in the case of sliding filaments), the bead stops while the bundle continues its growth. This situation is persistent over time because the elastic force produced by the bent bundle diminishes with the bundle length as

$$F_{Buckling}=\frac{K}{L^2},$$ (4)

where K is a constant (Eq. A.13). In the case of high bundle rigidity (e.g., L_p = 150 μm), the bundle remains straight for about 1 h, during which the bead moves at a constant velocity without bending the bundle (Figs. 2 D and 5 B). The shape of the bundle when the bead is stalled (Fig. 5 B) is in very good agreement with experimental data (Fig. 1 A). Presently, in the absence of available experimental data, we were not able to test whether thicker bundles, made of several filaments, would buckle at longer length than bundles made of few (less than 5) filaments.

The expression for L_Crit also allowed us to draw a boundary between bead motility and bundle deformation (Fig. 5 C). During a typical experiment, the persistence length of the bundle for a given bead remains constant while its length increases. Therefore, it is represented by a trajectory along a vertical arrow in the phase diagram (Fig. 5 C). The bead starts with a short bundle (under the phase transition boundary), then elongates while pushing the bead forward until the bundle eventually reaches the critical length. Above this critical value, the bundle bends and the bead stalls (Figs. 1 A and 5 C, dashed arrow). Increasing viscosity of the medium increases the resistance to movement and shifts the phase transition boundary downwards in the phase space without altering its shape (Eq. A15; Fig. 5 D). If polymerization is stopped (by depleting the free actin monomers) and if friction of the bead and/or the bundle against the glass or the fluid is eliminated, the bundle should keep its straight configuration, whatever the bundle thickness or length. This prediction could be tested in future experiments.

Discussion

Buckling as a mechanism to control filopodia length

Our results suggest that formin-driven extension of bundles, similar to the bundles in filopodia, is mainly controlled by bundle rigidity (Fig. 5), which is proportional to the number of active barbed ends. For example, when a filopodia reaches a critical length, which is determined by its filament density and attachment to the plasma membrane via formins and/or Ena/VASP, it stops pushing the membrane and buckles (41,42). This transition between extension and buckling (Fig. 5, C and D) yields a simple and robust way to control the extension of filopodia in cells, as seen in vivo (13,43). This mechanism is also important for the creation and stability of cell-cell junctions in cell tissues, particularly in the formation of villi of same sizes that ensure tissue coherence (13,14).

Buckling is a way to release elastic energy in networks and accompany cytoskeleton deformation

Both in vitro experiments and models have shown the importance of buckling in the contraction of disordered stress fibers (21) and in the final dismantling of the network (20). Our study suggests this principle may be extended to the whole cell itself. When a bundle of filaments is subjected to mechanical forces—from other cells, obstacles, or external forces, for example—its buckling under a critical load may constitute an initial response that could trigger a more complex signaling pathway in the cell. Then, differences in the bundle composition (e.g., by modulating the nature or amount of cross-links, motor proteins, number of filaments), geometry (e.g., by modulating the bundle length or anchoring at the plasma membrane or on an organelle), and/or mechanics would modulate the cell response and yield different fates for the cell cytoskeleton. When bundles are stiff (e.g., if they are cross-linked or composed of a large number of filaments), the elastic restitution of the energy stored in bundle buckling could allow cells to resist external constraints. For soft bundles (long ones, without cross-linkers, or made of severed filaments), the stressed cytoskeleton would fall in rapidly. For example, it has been shown that spatial three-dimensional distribution of bundles and their interactions (either bundle-bundle or bundle-rest of the cytoskeleton junctions) is crucial for cells (44). Last, bundles are part of the fiber system allowing cells to communicate with other cells (e.g., bundles in villi), to sense the extracellular space (e.g., filopodia), or to couple to focal adhesions. Therefore, any biological condition that change either the geometry or the mechanics of bundles could exert control over cell dynamics.

Author Contributions

Designed the research: J.-L.M., L.B., and J.B.; performed the experiments: J.B., A.M., and R.B.-P.; developed the models: J.-L.M.; analyzed the data: J.-L.M. and J.B.; wrote the manuscript: J.-L.M. and J.B.

Editor: Michael Ostap.

Appendix A: Simplified Model for the Analysis of Experiments

Correct interpretation of experimental observations in Fig. 1 requires the determination of actin-filament bundle rigidity. We limit our model for bundle mechanics and deformation to 2D bending strain. This is dictated by the available experimental data (TIRFM images, Fig. 3), which gives access to 2D bundle deformations only. Therefore, we assume that bundle rigidity is given by a single parameter, the apparent rigidity modulus (or persistence length), which depends on two quantities: the bending rigidity of a single filament and the number of filaments in a bundle (15). We focus our analysis on bundles that self-interact to form a stem and a loop (Fig. 3). These configurations represent structures at mechanical equilibrium when elastic forces, which tend to straighten the bundle, are balanced by attractive (depletion) forces that bind distant sections of the bundle to form a stem (Fig. 3).

2D equations for mechanical equilibrium of filaments lying in a plane

To determine the rigidity of filament bundles (Fig. 1 A), we adapt Eqs. 1 and 2 to 2D actin-filament bundles as observed in TIRFM (Fig. S1 B). The material frame vectors (d₁, d₃) are in the plane (d₂ points out of the plane). The orientation of the material frame (d₁, d₃) requires a single parameter, θ, the angle between d₃ and the horizontal axis. The filament strain or curvature dθ/ds enters in the definition of bending strain vector as $\kappa =\left(d\theta /ds\right){\mathbf{d}}_2$. From Eq. 1, we derive the force and moment balance equation in its component-wise form

$$\begin{array}{c}\begin{array}{c}\frac{d{f}_1}{ds}+\frac{d\theta }{ds}{f}_3=f_{ext,1}\\ \frac{d{f}_3}{ds}-\frac{d\theta }{ds}{f}_1=f_{ext,3}\\ C_B\frac{d^2\theta }{d{s}^2}-f_1=0\end{array}\end{array},$$ (A.1)

where f_i is the component of the internal force along the director d_i, i = 1, 3, and C_B is the bending rigidity. Note that the bending rigidity parameter, C_B, and the persistence length, L_p, are related by C_B = kTL_p, where k is the Boltzmann constant and T the temperature in Kelvin. The twist strain, which is proportional to C_T in Eq. 1, is absent in Eq. A.1. This approximation is valid because twist energy is always lower than bending energy (38), and 2D filament mechanics is controlled by filament-bending curvature (24). The inextensibility condition (Eq. 2) gives two equations for the horizontal and vertical components of r(s):

$$\frac{dx}{ds}=\cos \theta ,\frac{dy}{ds}=\sin \theta .$$ (A.2)

For filaments or bundles forming a loop (Fig. 3), the right-hand side (rhs) of Eq. A.1 (first two lines) models the filament-filament interactions responsible for loop formation and its stabilization (Fig. 3, A and B). Because a loop is symmetric, we assume that the force between two points on the filament, M and M′, is directed along the line MM′ and proportional to |MM′| = |x| (Fig. 3 C). The force is attractive as long as the points are within a distance r₁ from each other and repulsive if the distance is smaller than r₀ (hard core repulsion). The force vanishes for an interfilament distance x larger than r₁. Using these assumptions, Eq. A.1 is changed into

$$\begin{array}{c}\begin{array}{c}\frac{d{f}_1}{ds}+\frac{d\theta }{ds}{f}_3=-2k_s\sin \theta \left|x\right|F\left(x\right)\\ \frac{d{f}_3}{ds}-\frac{d\theta }{ds}{f}_1=2k_s\cos \theta \left|x\right|F\left(x\right)\end{array}\end{array}$$ (A.3)

with

$$F\left(x\right)=\{\begin{array}{cc}-1& 0\le x\le r_0\\ 1& r_0\le x\le r_1\\ \exp \left(-\left|x-r_1\right|/r_2\right)& r_1\le x\end{array}.$$ (A.4)

After normalization of the position (x, y) and arc-length variable s by the filament length L, the final equations read

$$\begin{array}{c}\begin{array}{c}\frac{d\overline{x}}{d\overline{s}}=\cos \theta \\ \frac{d\overline{y}}{d\overline{s}}=\sin \theta \\ \frac{d^2\theta }{d{\overline{s}}^2}={\alpha }_1{n}_1\end{array}\end{array}$$ (A.5)

and

$$\begin{array}{c}\frac{d{n}_1}{d\overline{s}}+\frac{d\theta }{d\overline{s}}{n}_3=-{\alpha }_2\sin \theta \left|\overline{x}\right|F\left(\overline{x}\right)\\ \frac{d{n}_3}{d\overline{s}}-\frac{d\theta }{d\overline{s}}{n}_1=-{\alpha }_2\cos \theta \left|\overline{x}\right|F\left(\overline{x}\right)\end{array},$$

where $\left(\overline{x},\overline{y}\right)$ = (x/L, y/L), $\overline{s}$ = s/L, α₁ = (N₀L²/C_B), and α₂ = (2k_sL²/N₀); L is the total loop contour length; N₀ = (kT/L₀) is the natural force unit for the system (≈ 4.1 × 10⁻¹⁵ N) with L₀ = 1 μm; and n₁ and n₂ are the force components normalized by N₀. Note that α₁ and α₂ are dimensionless parameters. All the variables and parameters are summarized in Table 1. The vertical symmetry of the configuration (Fig. 3 C) gives the additional relations

$$\begin{array}{l}\begin{array}{cc}x\left(1-s\right)=-x\left(s\right),& y\left(1-s\right)=y\left(s\right),\end{array}\\ \theta \left(1-s\right)=-\theta \left(s\right),\\ \begin{array}{cc}{n}_1\left(1-s\right)=n_1\left(s\right),& n_3\left(1-s\right)=n_3\left(s\right).\end{array}\end{array}$$ (A.6)

Table 1.

Variables and Parameters Used in the Analysis of Bundle Rigidity

	Definition	Dimension	Typical Value
Variables
s	arc length	L	20–100 μm
(x, y)	point coordinates on the filament	L
θ	angle between the tangent and the horizontal axis
(f1,f3)	force along d1 and d3	M ⋅ L ⋅ T−2
Parameters
kT	thermal energy	M ⋅ L2 ⋅ T−2	4.1 × 10−21 J
kS	self-interaction force density	M ⋅ T−2	2.7 × 10−3 pN ⋅ μm−2
r1	cutoff distance	L	0.055 μm
r2	spatial decay for the self-interaction force	L	0.02 μm

All these conditions are a direct consequence of the loop symmetry with respect to the vertical y-axis.

Variables and parameters used in simulations of bundles

Boundary conditions

The solution of Eq. A.5 depends on the boundary conditions at the ends of the filament, i.e., at $\overline{s}$ = 0 or $\overline{s}$ = 1. Because the attraction force exerted along the stem balances the elastic force due to the loop rigidity (Fig. 3, A–C), the components of the internal force (n₁, n₂) should vanish at $\overline{s}$ = 0:

$$n_1\left(0\right)=0,n_2\left(0\right)=0.$$ (A.7)

In consequence, from Eq. A.6, the internal force should also vanish at $\overline{s}$ = 1. We complete the set of boundary conditions by specifying the position of the filament at s = 0 and s = 1:

$$\begin{array}{cc}\overline{x}\left(0\right)=0,& \overline{x}\left(1\right)=0,\\ \overline{y}\left(0\right)=0,& \overline{y}\left(1\right)=0.\end{array}$$ (A.8)

Determination of bundle bending rigidity

The shape of a loop is given by a solution of Eqs. A.5, A.6, A.7, and A.8, which are valid for a 2D elastic rod. However, because solutions of Eq. A.5 depend on the ratio (α₁/α₂) only, we cannot have access to bundle rigidity directly. Therefore, we first determined the force interaction in the case of a single filament for which the bending rigidity is known. Then, we used the same method to analyze the elasticity of loops formed by a bundle made of several filaments by scaling the parameters α₁ and α₂ appropriately.

Algorithm for the determination of the interaction parameter in the case of a single-filament loop

See Table 1.

• 1We extracted the configuration of the loop and the stem from the microscopy image (Fig. 3, A and B) and measured its contour length (L).
• 2We normalized the filament configuration and arc length with the filament contour length L.
• 3Using a persistence length of 10 μm for a single actin filament (or a bending modulus C_B = 4.1 × 10⁻²⁶ N ⋅ m) (45,46) and L, we estimated the parameter α₁ = (N₀L²/C_B) (with N₀ = 4.1 × 10⁻¹⁵ N).
• 4The best configuration fit (Fig. 3 E) was obtained by adjusting α₂, the unique free parameter remaining in Eq. A.5.
• 5The interactive depletion force k_S was then given by

$$k_s=\frac{1}{2}\frac{{\alpha }_2}{{\alpha }_1}\left(\frac{N_0^2}{C_B}\right).$$ (A.9)

Determination of bundle rigidity

We now focus on the determination of the rigidity of a bundle made of several filaments. Firstly, assuming a close packing of filaments, R, the radius of a bundle made of N filaments scales as $R/r=\sqrt{N}$, where r is the radius of a single filament (Fig. S2). Secondly, we assume that the interaction force is proportional to the perimeter of the bundle because we expect that only the filaments in the outer part of the bundle can exert force on filaments outside the bundle (Fig. S2). Therefore, the parameter k_S in Eq. 5 (or α₂ in Eq. A.5) scales as

$$\frac{k_{S,Bundle}}{k_{S,Filament}}=\sqrt{N}.$$ (A.10)

In addition, previous studies showed that simple geometrical arguments allow to approximate the apparent bundle rigidity as a power function of the number of filaments

$$\frac{C_{B,Bundle}}{C_{B,Filament}}=N^{\beta },$$ (A.11)

with 1 ≤ β ≤ 2 (15). The case β = 1 corresponds to filaments free to slide in the bundle, and β = 2 corresponds to totally cross-linked filaments (15). By combining Eqs. A.10 and A.11, we conclude that

$${\left(\frac{{\alpha }_1}{{\alpha }_2}\right)}_{Bundle}=\frac{1}{N^{1/2+\beta }}{\left(\frac{{\alpha }_1}{{\alpha }_2}\right)}_{Filament}.$$ (A.12)

Therefore, by comparing experimental images of bundles forming a loop obtained by TIRFM to a solution of Eqs. A.5, A.6, A.7, A.8, A.9, A.10, A.11, and A.12, we can deduce N, the number of filaments in a bundle, and, thanks to Eq. A.11, the bundle rigidity. Fig. 4 A shows typical observed closed loops (dotted lines) and the corresponding optimal solution to Eqs. A.3, A.4, A.5, and A.6 and the parameters listed in Table 1.

Polymerization kinetics and polymerization force

In the presence of profilin, the polymerization rate for actin filaments’ barbed ends capped by the processive formin mDia1 is 38 μM⁻¹ s⁻¹ (36). Assuming a depolymerization rate of 0.1 s⁻¹ (47), we predict the critical concentration for actin monomer to be [G]₀ = 0.037 μM (Table 2). The concentration of actin monomers, which yields a constant elongation of 0.2 μm ⋅ min⁻¹ for actin filaments (measured from Figs. 1 and 2), is [G] = 0.068 μM (Table 2). In consequence, the maximal force developed by actin polymerization, given by F_pol,max = (kT/δ)ln([G]/[G]₀), is 0.93 pN (defined in Table 2).

Table 2.

Kinetic Parameters, Actin Monomer Concentration, and Forces Used to Model Bead Displacement

Parameter	Definition	Value	Reference
r	bead radius	2 μm	N/A
CB0	single-actin-filament bending rigidity	4.1 × 10−26 N ⋅ m	N/A
Vmax	constant bead velocity	0.0033 μm ⋅ s−1	N/A
kon	polymerization rate (barbed end) in the presence of formin	38 μM−1 ⋅ s−1	(4)
koff	dissociation rate from the barbed end	1.4 s−1	(8)
C	drag coefficient, C = 6πηr	1.13 × 10−7 Pa ⋅ s ⋅ m	N/A
[G]0	critical actin monomer concentration (koff/kon)	0.037 μM	N/A
[G]	apparent monomer concentration [G] = [G]0 +$\left(V_{max}/k_{on}\delta \right)$	0.067 μM	N/A
FPol	polymerization force, Fpol,max = (kT/δ)ln([G]/[G]0)	0.93 pN	N/A
η	viscosity	0.08–0.26 Pa ⋅ s	N/A
Estimated drag force	Fdrag = (6πηr)Vmax	0.01–0.032 pN	N/A
Critical buckling forcea	$F_{Buckling}=\gamma \left(\left(kT\right){\pi }^2{L}_p{N}^{1/2+\beta }/L^2\right)$	unknown	N/A
Critical length at buckling (Eq. A. 15)	$\left(L_{Crit}^2/N^{1/2+\beta }\right)=\left(\gamma {\pi }^2\left(kT\right)L_p/F_{Drag}\right)$	40–12 μm2/filament1.5b	N/A

N/A, not applicable.

^aN is the number of actin filaments in the bundle; L_Crit is the bundle length at the onset of buckling; γ is a numerical factor.

^bUsing γ = 1 and β = 1.

Drag force

The presence of actin filaments in the bulk increases the apparent viscosity in the experiments close to values characteristic of that of a cell. Direct viscosity measurement in vivo reported values ranging from 0.08 to 0.26 Pa ⋅ s (48,49), which represents a 100-fold increase of viscosity compared to that of water (0.001 Pa ⋅ s). Given the polymerization force (0.93 pN) and drag force exerted on a sphere (Table 2), we predict that the drag force during bead movement is in the range 0.01–0.032 pN (Table 2).

Buckling force

The critical force required to buckle an elastic bundle is

$$F_{Buckling}=\gamma \frac{\left(kT\right){\pi }^2{L}_p{N}^{1/2+\beta }}{L^2},$$ (A.13)

where L_P is the persistence length of a single actin filament, N is the number of filaments in the bundle, L is the bundle length, and γ is a numerical factor that depends on the boundary conditions at the bundle ends (7). In consequence, when the drag force balances the critical force at buckling,

$$F_{Buckling}=F_{Drag},$$ (A.14)

and the bead is stalled while bundle elongation continues. This transition occurs at a critical bundle length given by solving Eq. A.14 with the help of Eq. A.13:

$$\frac{L_{Crit}^2}{N^{1/2+\beta }}=\frac{\gamma {\pi }^2\left(kT\right)L_p}{F_{Drag}}.$$ (A.15)

Note that the rhs of Eq. A.15 is a constant that does not depend on bundle characteristics. The estimated values (Table 2) yield critical lengths in the range of experimental observations. From Eq. A.15 and assuming constant elongation, one can predict that large bundles will sustain the bead propagation regime for longer time periods.

Appendix B: Model of Filament Bundles During Macroscopic Bead Movement

We adapt the set of Eqs. A.5, A.6, A.7, and A.8 to simulate the movement of beads driven by a growing elastic bundle, actin polymerization, and the friction that opposes the movement:

$$\begin{array}{c}\frac{d{f}_1}{ds}+\frac{d\theta }{ds}{f}_3=0\\ \frac{d{f}_3}{ds}-\frac{d\theta }{ds}{f}_1=0\\ C_B\frac{d^2\theta }{d{s}^2}-f_1=0\end{array}.$$ (B.1)

The arc-length variable, s, lies in the interval [0, L(t)], where L(t) is the time-dependent bundle length. Assuming that all filaments in the bundle experience the same stress at their barbed end, the global force-dependent elongation rate of the bundle is given by

$$\frac{dL\left(t\right)}{dt}=k_{on}\left(\left[G\right]-{\left[G\right]}_0\right)exp\left(-\frac{\Delta E}{kT}\right).$$ (B.2)

In the above equation, k_on is the polymerization rate for an actin filament capped by formins (36); [G] and [G]₀ are, respectively, the concentration of actin monomers and the critical concentration in presence of formin; δ is the radius of a monomer; and ΔE = δ|f₃| is the work against the elastic force required to insert one monomer between the bead and the barbed end. The final set of equation governs the bead position

$$\begin{array}{c}\frac{d{x}_B}{dt}=w\left(t\right)\text{cos}\theta ;\frac{d{y}_B}{dt}=w\left(t\right)\text{sin}\theta \\ \text{with}w\left(t\right)=max\left\{\begin{array}{cc}0,& \left(C\frac{dL\left(t\right)}{dt}-F\right)\end{array}\right\}\end{array},$$ (B.3)

where (x_B(t), y_B(t)) is the bead center, θ is the angle between the bundle and the horizontal axis at the junction between the bundle and the bead, F is the magnitude of the constant viscous drag or friction exerted on the bead, and C is the coefficient giving the drag exerted on the bead (Table 2).

Boundary conditions used to simulate bead movement

The pointed end (at s = 0) remains fixed in time with

$$\begin{array}{ccc}x\left(0\right)=0,& y\left(0\right)=0,& \theta \left(0\right)=0\end{array}.$$ (B.4)

The boundary at s = L(t) connects the bundle and the bead

$$\begin{array}{c}x\left(L\left(t\right)\right)=x_B\left(t\right)-r_B\left(t\right)\text{cos}\theta \left(L\left(t\right)\right)\\ y\left(L\left(t\right)\right)=y_B\left(t\right)-r_B\left(t\right)\text{sin}\theta \left(L\left(t\right)\right)\\ {\left(\frac{d\theta }{ds}\right)}_{L\left(t\right)}=0\end{array},$$ (B.5)

where r_B is the bead radius.