Twist response of actin filaments

Significance

How actin filaments respond to mechanical loads is central to understanding cellular force generation and mechanosensing. While there is consensus on the actin filament bending stiffness, reported values of the filament torsional stiffness vary by almost 2 orders of magnitude. We used magnetic tweezers and hydrodynamic flow to determine how filaments respond to applied twisting and pulling loads. Twisting causes filaments to adopt a supercoil conformation. Pulling forces inhibit supercoil formation and fragment filaments. These observations explain how contractile forces generated by myosin motors accelerate filament severing by cofilin regulatory proteins in cells.

Abstract

Actin cytoskeleton force generation, sensing, and adaptation are dictated by the bending and twisting mechanics of filaments. Here, we use magnetic tweezers and microfluidics to twist and pull individual actin filaments and evaluate their response to applied loads. Twisted filaments bend and dissipate torsional strain by adopting a supercoiled plectoneme. Pulling prevents plectoneme formation, which causes twisted filaments to sever. Analysis over a range of twisting and pulling forces and direct visualization of filament and single subunit twisting fluctuations yield an actin filament torsional persistence length of ~10 µm, similar to the bending persistence length. Filament severing by cofilin is driven by local twist strain at boundaries between bare and decorated segments and is accelerated by low pN pulling forces. This work explains how contractile forces generated by myosin motors accelerate filament severing by cofilin and establishes a role for filament twisting in the regulation of actin filament stability and assembly dynamics.

Cells sense, respond, and adapt to internal and external forces (1, 2). The actin cytoskeleton, a dynamic, branched, and cross-linked network of protein filaments (3) that behave as semiflexible polymers on cellular length scales (4–6), mediates many of these cellular responses. Cellular actin networks are pulled (7, 8), squeezed (9, 10), and twisted (11, 12) during growth and remodeling and through interactions with contractile and regulatory binding proteins (13). These physical forces can stall network growth (14), alter the filament structure (15, 16), modulate interactions among filaments (17) and with regulatory proteins (7, 8, 18, 19), and induce filament fragmentation (15, 19–23), all of which influence network remodeling and mediate cellular “mechanosensing” (24, 25).

The capacity for actin networks to respond to force is dictated by the mechanical properties of filaments. Relaxed (i.e., resting) filaments are straight but helical with an intrinsic twist (26). The forces required to twist and bend a filament scale with the filament mechanical properties (4, 27), specifically their bending and torsional stiffness, which are commonly represented in terms of bending and twisting persistence lengths (L_B and L_T; we note these are effective persistence lengths because filaments are not homogeneous, isotropic materials). Filaments with larger persistence lengths are stiffer and require more force to deform than those with shorter persistence lengths. Similarly, stiff filaments store more elastic strain energy for any given deformation than more compliant ones.

The elastic free energy (i.e., strain energy) stored in the filament shape (16) can generate force and work when relaxing to the resting configuration. It can also fragment filaments (15, 19–23, 28) and mediate interactions with binding partners (7, 8, 18, 19). Dissipation of elastic energy in bent filaments contributes to force generation at the leading edge of migrating cells (10, 29) and during essential cellular processes such as endocytosis (9). Twisted filaments are also strained. Such twisting has been implicated in symmetry breaking (30, 31), network chirality (11), and the buckling of actin networks in filopodia (12). Quantitative knowledge of filament bending and twisting mechanics is therefore critical to reliably account for and model complex cellular behaviors.

The bending mechanics of actin filaments have been extensively characterized. There is general agreement that filaments have a bending persistence length (L_B) of ~10 µm (32–34), which can be modulated by regulatory proteins (35, 36) and ligands (5, 33, 37, 38). A consensus on the filament twisting stiffness is lacking, with torsional persistence lengths reported from 0.5 to 20 µm (20, 39–44). In addition, it is not known how filament twisting and bending are coupled (16, 27) or how filaments respond to combinations of twisting, bending, and pulling forces, as experienced in cells.

Here, we use a magnetic tweezers apparatus coupled with microfluidics to evaluate how single actin filaments respond to applied twisting and pulling loads. Our results and analyses provide multiple, independent determinations of the filament bending and twisting stiffness, demonstrate how bending and twisting are coupled, and show how this coupling is affected by pulling and filament fragmentation. These findings have implications for actin cytoskeleton mechanosensing and network force generation and remodeling.

Results

Twisting and Pulling Actin Filaments.

We developed an assay to twist actin filaments about their long axis with magnetic tweezers while simultaneously visualizing them by a TIRF microscope (Fig. 1). Short, Alexa 488–labeled actin filament seeds were tethered to the surface of the microscope coverslip and elongated from the barbed end with purified, Alexa 647–labeled actin monomers. These filaments were further elongated from their barbed ends with digoxigenin-conjugated actin monomers to which we attached a paramagnetic bead. The barbed-end–conjugated paramagnetic bead was twisted at a constant rate (0.31 rot s⁻¹) with permanent magnets mounted on a stepper motor (Fig. 1B). Filament-attached beads and filaments rotated in phase with the permanent magnets (Fig. 1 B–D and Movies S1 and S2), indicating that no slipping occurs during manipulation. Filaments attached to two beads (Fig. 1B) were used only to determine whether rotations were in phase with the permanent magnet (Fig. 1C); filaments attached to a single bead were used in all subsequent experiments.

Fig. 1.

Twisting actin filaments with magnetic tweezers. (A) Cartoon schematic of the experimental setup. Alexa 488–labeled actin filament seeds (green) were attached to a Biotin-PEG-Silane surface through biotin (yellow circles) and neutravidin (black diamonds) interactions and elongated from the barbed ends with Alexa 647–labeled actin (red). Filaments were further elongated from their barbed ends with digoxigenin (DIG)-labeled actin (purple). Paramagnetic beads (2.8 μm in diameter; gray) coated with DIG antibodies (purple) were attached to filaments at or near their barbed ends. Filament-attached beads can be rotated by a permanent magnet (blue and red rectangles) and pulled by buffer flow (black arrow). Relative filament and bead sizes are not drawn to scale. Twisting clockwise or counterclockwise corresponds to under- or overtwisting, respectively. (B) Rotation of a phalloidin-decorated filament attached to two paramagnetic beads. (C) Cosine of the rotational angle of the second paramagnetic bead (black trace) and the magnet (red trace) indicates that the bead and magnet rotation are in phase. (D) Rotation of an Alexa 647–labeled actin filament with visible attachment to paramagnetic bead indicates that the bead and filament rotation are in phase.

Pulling forces exerted by the permanent magnet are negligible in our experimental setup (SI Appendix, Fig. S1), so they were applied with fluid flow using a microfluidic device. The force applied on the filament scales with the size of the bead (d = 2.8 µm) and the fluid velocity (SI Appendix). This force is constant throughout the filament (at a constant flow rate) and is independent of the filament length in contrast to the much smaller and negligible forces exerted by flow on tethered filaments without conjugated beads (17, 22, 45, 46).

Actin Filament Force–Extension Response.

To establish the mechanical properties of actin filaments can be reliably determined with our experimental conditions, we measured the bending persistence length (L_B) in the absence of twisting from the force–extension response of filaments undergoing thermally driven shape fluctuations (Fig. 2 and Movie S3). The pulling force required to straighten thermally bent filament scales with the bending stiffness (L_B). The filament end-to-end distance (R), defined as the linear distance from the bead and surface attachment points, depends on the long-axis pulling force (Fig. 2). In the absence of fluid flow (i.e., pulling force ~ 0) filaments undergo thermally driven bending, so R is shorter than the filament contour length (L), i.e., R/L < 1. At ~ 0.02 pN pulling force, the ratio of the filament end-to-end distance and the contour length (R/L) was ~0.92 (Fig. 2). The end-to-end distance approached the contour length R/L ~ 1 at >3 pN pulling force (Fig. 2). The best fit of the force dependence of the filament end-to-end distance (Fig. 2) to a worm-like chain model [Eq. 1; (47)] yields an actin filament bending persistence length (L_B) of 10.7 (±1.0) µm (Table 1), consistent with previous wet lab and computational model determinations of bare filaments [i.e., without phalloidin or other binding partners (19, 32–35, 42, 48–50)]. The forces in these experiments are calculated using a bead-center distance from the surface (d) equal to the bead radius (r; d = r) since the bead was at the surface of coverslip (SI Appendix).

Fig. 2.

Actin filament force–extension response. (A) Representative fluorescent images of Alexa 647–labeled actin filaments (magenta) conjugated to a paramagnetic bead (cyan) under fluid flow. No magnetic field is applied. (B) Force–extension curves for actin filaments of varying lengths (colored points) with the global best fit to Eq. 1 (colored lines) with pulling forces at d = r (Methods). (C) Average force–extension curves, normalized to filament lengths, for 7 actin filaments and corresponding theory with the global best fit persistence length of 10.7 (±1) µm (solid red line) (Eq. 1). Theoretical force–extension curves (Eq. 1) in descending order with L_B = 100, 5, 1, and 0.1 µm (dashed red lines). Uncertainty bars indicate standard error of the mean (SEM).

Table 1.

Actin filament bending and torsional persistence lengths

Persistence Length (µm)		Assay
Bending (LB)	10.7 (±1.0)	Force–extension, Fig. 2
	10.9 (±2.6)	Twist–extension, Fig. 3, f at d = r
Twisting (LT)	11.7 (±2.2)	Twist–extension, Fig. 3, f at d = r
	12.9 (±2.4)	Twist fluctuations, Fig. 4
	8.2 (±0.2)	Cryo-EM, refinement volume of 5 subunits, histogram fit, Fig. 5
	5.5 (±0.2)	Cryo-EM, refinement volume of 5 subunits, MLE*
	4.4 (±2.7)	Cryo-EM, refinement volume of 5 subunits, MCMC†
	5.8 (±0.3)	Cryo-EM, refinement volume of 1 subunit, histogram fit
	7.0 (±1.1)	Cryo-EM, refinement volume of 1 subunit, MLE*
	6.3 (±3.3)	Cryo-EM, refinement volume of 1 subunit, MCMC†

^*Maximum likelihood estimation (MLE, see Methods).

^†Markov chain Monte Carlo (MCMC, see Methods).

Twisted Filaments Bend and Supercoil.

Filaments undergo a series of shape transitions when continuously twisted with magnetic tweezers (Movie S4). In the absence of applied twist (and with or without pulling loads), filaments bend randomly due to thermally driven forces. Applied twisting (twist density <0.8 rot µm⁻¹) causes filaments to bend further in a nonrandom manner, provided long-axis pulling forces are weak (≤0.03 pN; Fig. 3A), and the value of R continues to shorten gradually with twisting until a critical twist density (σ_s) is reached, at which point filaments form a looped segment. Additional twisting causes the linear, nonlooped filament segments to wind up and twist around each other, yielding an interwound, actin filament supercoil (called a plectoneme) like those observed with twisted DNA (51, 52). This transition is detected as a dramatic and abrupt reduction in R (Fig. 3B). Only a single loop (i.e., one plectoneme) per filament was observed in our experiments. Plectoneme formation is reversible and relaxed (i.e., straight) filaments can be recovered with untwisting (Movie S5). We note that rhodamine phalloidin–decorated actin filaments supercoil when subjected to high-intensity laser light (53), presumably due to photo-induced torsional strain.

Fig. 3.

Actin filament twist–extension response and supercoiling. (A) Representative fluorescent images of actin filament twist–extension with 0.01 (Top), 0.03 (Middle), and 0.25 (Bottom) pN pulling force (SI Appendix, Eq. S1 with d = r). (Scale bar, 5 µm.) (B) Twist–extension curves for actin filaments under 0.01 (white circles), 0.03 (gray circles), and 0.25 (black circles) pN pulling forces with the global best fit to Eq. 3 (red lines). Solid and dashed red lines differentiate model before and after plectoneme formation, respectively. Rotations along the positive x-axis indicate filament overtwisting, and rotations along the negative x-axis indicate undertwisting. The complete dataset represents 50 filaments and n > 3 for each experimental condition. Uncertainty bars represent SEM. The asymmetric look of red dashed fitting lines for over- and undertwists under the same pulling force is due to the different fixed parameters $\overline{\left(\frac{1}{L}\right)}$ in the fitting (Methods) and not because of differences in response to applied over- and undertwist.

Actin filaments are helical and can be described as having an intrinsic right-handed twist. Therefore, counterclockwise rotations increase the intrinsic twist (“overtwisting”), whereas clockwise rotations lower the intrinsic twist (“undertwisting”). Binding of cofilin/ADF regulatory proteins, for example, also undertwists filaments (54, 55). Filament plectoneme formation depends on the pulling force but not on the twisting direction (i.e., over- versus undertwisting), as indicated by the symmetry of the twist–extension response curves (Fig. 3B). The lack of a detectable asymmetry with over- and undertwisting is surprising given the intrinsic filament twist. However, this observed behavior likely arises from the fact that the deformations associated with plectoneme formation are modest at the subunit level and not in the regime in which differences between the two directions could be detected (16). Filaments adopt a plectoneme configuration at low twist densities (~0.2 rot µm⁻¹) when the pulling force is low (<0.01 pN) but require more twist (~0.8 rot µm⁻¹) at higher pulling forces (0.03 pN; Fig. 3). Plectonemes did not form when the pulling force was 0.25 pN, even up to twist densities of ~1 rot µm⁻¹. The reversibility of plectoneme formation was also independent of the twisting direction.

The plectoneme loop size also depends on the pulling force (SI Appendix, Fig. S2). The average loop radius was ~400 to 500 nm under 0.01 pN, whereas it was ~200 nm under 0.03 pN. The local filament curvatures at these radii are small compared with those predicted to significantly accelerate filament fragmentation (15), consistent with these plectoneme loops being stable throughout the duration of our experiments.

Filament Twist–Extension Response.

The experimental data presented thus far demonstrate that twisted actin filaments bend and adopt supercoiled plectoneme structures when pulling forces are low (<0.25 pN). This response originates from the intrinsic filament bending and twisting mechanics and the coupling between these two deformations (16, 52, 56). Accordingly, the actin filament mechanical properties can be extracted from the data with appropriate theory, analysis, and modeling.

A two-state model used to describe plectoneme formation in DNA (52) (modified to include polymers with L~L_B, such as actin filaments; Methods; Eq. 3) accounts for the twist–extension response and plectoneme formation of actin filaments over the range of pulling forces evaluated here (Fig. 3B). This model considers actin filament segments as semiflexible rods in either “straight” or plectoneme states. Pulling forces favor the straight configuration. Twisting introduces strain energy, which is dissipated by bending and subsequent plectoneme formation. Prior to reaching a critical twist density (σ_s) for plectoneme formation, twisted filaments bend to dissipate torsional strain, shortening R. Once a plectoneme has formed, applied twisting strain is dissipated through conversion of strained, straight state segments to more relaxed plectoneme configurations (i.e., straight segments shorten, while plectoneme segments elongate linearly with applied twisting loads). Since only the linear, nonlooped filament segments contribute to the R value (plectoneme segments do not), the R value decreases linearly with the applied twist until R = 0, at which the entire filament is in a plectoneme configuration (middle line in Eq. 3; Fig. 3B). The applied twist at R = 0 is referred to as σ_p.

The best fit of the filament twist–extension data to this model (Eq. 3; Fig. 3B) yields actin filament persistence lengths for bending (L_B) and twisting (L_T) of 10.9 (±2.6) and 11.7 (±2.2) µm, respectively (Table 1). This value of the bending persistence length (L_B) is comparable with the value of 10.7 (±1.0) µm obtained from the force–extension response (Fig. 2). Filament torsional persistence lengths an order of magnitude longer or shorter do not account well for the observed experimental data (SI Appendix, Fig. S5).

We note that some twisted filaments “wobbled” slightly during twisting manipulations (Movie S2). To estimate the maximum possible error introduced by bead wobbling in these cases, we analyzed the data assuming a larger force as expected if the bead moved away from the surface during rotation. Deviations far greater than observed for wobbling (i.e., one full bead height deviation, such that d = 2r; SI Appendix) yield essentially identical L_T values (L_T = 11.7 ± 2.2 µm versus L_T = 11.4 ± 2.1 µm for beads wobbling an entire bead height) but ~twofold lower L_B values (10.9 ± 2.6 µm versus 6.5 ± 1.5 µm for beads wobbling an entire bead height).

Thermally Driven, Filament Twist Fluctuations.

We also determined the filament torsional stiffness by directly visualizing spontaneous, thermally driven, twist fluctuations (Fig. 4). A hollow, cylindrical magnet was positioned above the tethered filament to generate a magnetic field (perpendicular to the surface of the sample chamber) that held the filament orthogonally to the surface without constraining its rotation (57) (Fig. 4 A and B and Movie S6). The pulling force generated by the closely positioned hollow magnet maintains the filament relatively straight, thereby eliminating contributions from filament bending to the observed dynamics and allowing determination of the true filament torsional stiffness (SI Appendix, Fig. S3, (39, 56)).

Fig. 4.

Direct visualization of actin filament twisting fluctuations. (A) Cartoon schematic of the experimental setup. A cylindrical magnet with a center hole was positioned 5 mm above the coverslip surface–anchored actin filament (i.e., in the z direction perpendicular to the surface). A DIG-coated marker bead was added to the paramagnetic bead to track rotations. No flow was applied during experiments. Filament length and bending are not to scale. (B) (a) Assay is set up by identifying a filament attached to a paramagnetic bead (large dim bead) with identifiable marker beads (small bright bead) under fluid flow. (b) Fluid flow is turned off. (c) Cylindrical magnet is lowered into position. (d) Filament is pulled out of the focal plane. (e) Focal plane is adjusted to observe the rotational fluctuations of both beads. Images were taken every 5 s. (C) Example images of the angular fluctuations of the filament visualized by the absolute angle of a line connecting the two beads to the x direction. (D) The mean-subtracted absolute angle (SI Appendix, Eq. S25) of the marker beads over time. Black trace indicates the angular fluctuations from the filament tracked in (C). Gray traces represent four other sample traces from different experiments performed at different times. Histogram represents the distribution of absolute angles from the black trace. The actin filament torsional persistence length of 12.9 (±2.4) μm is an average (n = 5) of separate measurements, each was determined from the value of the variance at long times (see Panel E) and the filament length according to Eq. 4. (E) Time-dependent variance of the traces in D. It demonstrates that the measurement time of whole filament angular fluctuation has to be long enough for the variance to reach equilibrium such that experiencing all possibilities.

Filament twisting fluctuations were monitored by tracking a smaller fluorescent marker bead attached to the paramagnetic bead at the filament barbed end (Fig. 4 C and D and Movie S7). The angular fluctuations of the marker bead reflect the cumulative rotational fluctuation of all subunits between the surface and paramagnetic bead attachment points (L = 8 to 19 μm, ~365 subunits µm⁻¹; SI Appendix, Fig. S6).

The rotation angle of the marker beads around the filament center axis fluctuates randomly (Fig. 4 C and D). Time courses of the observed marker bead angle variance (Fig. 4E) plateau at times >1,000 s, indicating the entire accessible diffusive space of the filament-attached bead had been sampled (58). The twist persistence length for each set of data in Fig. 4D was determined from the angular variance calculated by directly averaging the set of data and the filament length according to Eq. 4 (Methods). The averaged (n = 5) value from separate experiments yields a filament torsional persistence length of 12.9 (±2.4) µm (Table 1) comparable with the value of 11.7 (±2.2) µm determined from the filament twist–extension response and plectoneme formation (Table 1) and with reported values of ~16 µm (20) and ~6 µm (39).

Single Subunit Twisting Fluctuations.

We measured the filament twisting persistence length a third way, from the variance of twisting angles between subunits, as visualized by electron cryomicroscopy (Fig. 5). Alignment parameters output from the 3D structure refinement yield estimates of filament subunit orientations and hence the twisting angle between them (54). Deviations of the observed intersubunit twist angle from the intrinsic (average) filament twist reflect thermally driven twist fluctuations. The distribution (i.e., width) of these deviations scales with the filament torsional stiffness, such that filaments displaying a narrow distribution are less compliant in twisting than those with a broader distribution.

Fig. 5.

Torsional persistence length of actin filaments determined by electron cryomicroscopy. (A) Cartoon schematic of measured filament subunit twisting fluctuations. The top cartoon depicts an actin filament (gray) with the average, intrinsic twist (Δφ_1intrinsic) between adjacent subunits i and i+1 illustrated as a red curved arrow. The observed twist deviates from the intrinsic twist, either over or under, because of thermal fluctuations. The middle and bottom cartoons illustrate an undertwisted filament (light gray) overlaying a canonical filament with an intrinsic twist (dark gray). Blue arrows illustrate the observed twist between subunits i and i+1 (Middle) or i and i+3 (Bottom), which differs from the intrinsic twist by Δφ’_n (illustrated by black arrows). (B) Histograms of actin filament subunit twist fluctuations (Δφ’_n, in degrees) estimated from cryo-EM alignment parameters (reference volume of 5 subunits) for n = 1, 10, and 50 subunits. Red lines represent fits to a normal distribution with mean zero and variance $({\sigma }_{obs,n+1}^2$). (C) n dependence of the twist variance (${\sigma }_{obs,n+1}^2$). The solid red lines represent the best fit to SI Appendix, Eq. S21.

A challenge with accurately calculating this angle distribution by cryo-EM is the low signal-to-noise ratio associated with the images. The noise introduces uncertainty in the angle measurements that can exceed the true intersubunit angle variance. Since the torsional stiffness and persistence length (L_T) are determined from the distribution variance (SI Appendix, Eq. S15), and large uncertainties in individual intersubunit angle measurements (SI Appendix, Eq. S17) yield a larger variance than the true variance, neglecting contributions from these uncertainties causes filaments to appear more compliant than they actually are.

We therefore developed an analysis method (SI Appendix, Eqs. S17–S21) that addresses the uncertainty in angle measurement to accurately measure L_T from cryo-EM micrographs. Because each filament subunit is independently subject to thermally driven torsional angle fluctuations in a scale dictated by the torsional stiffness, the width of the true angular distributions σ_n+1², measured across filament segments, increases linearly with the number of subunits according to ${\sigma }_{obs,n+1}^2 = n\frac{\mathrm{\Delta }s}{L_T}+{\sigma }_{\epsilon }^2$ (SI Appendix, Eq. S21). In contrast, the uncertainty in the estimated twist angle ${\sigma }_{\epsilon }^2$ remains constant.

Our cryo-EM images confirmed this behavior (Fig. 5C), yielding estimates of the true intersubunit torsional variance and persistence length (σ_n+1² and L_T, respectively; SI Appendix, Eq. S21) from the slope of the line relating the observed variance (${\sigma }_{\epsilon }^2$) to n (Fig. 5C). The intercepts of these lines reflect the contribution of noise to ${\sigma }_{\mathit{obs}}^2$ (SI Appendix, Eq. S21). The best linear fit of the n-dependent variance, obtained by Gaussian distribution fit to the angle Φ histogram measured from a refinement volume of 5 subunits, to SI Appendix, Eq. S21 yields an L_T of 8.2 (±0.2; ± indicates SDs of the fit) comparable with the L_T values of 11.7 (±2.2) and 12.9 (±2.4) µm determined by plectoneme formation (Fig. 3) and thermally driven filament torsional fluctuations (Fig. 4), respectively. We include two additional analysis methods, directly averaging (MLE) and MCMC (Methods), to independently determine n-dependent variance and thus L_T to see if one is more robust than the others (Table 1). The differences among the three different methods are not significant. We also repeated the analysis to the angles measured from a refinement volume of 1 subunit (Table 1). These values are not significantly different from those from a refinement volume of 5 subunits.

Twist-Induced Filament Fragmentation.

Filaments often fragmented during twist–extension manipulations (Fig. 3). At 0.25 pN pulling force, filaments fragmented before forming a plectoneme, indicating that intersubunit bonds rupture if strain from applied twisting is not dissipated (15, 19). Most filaments (83/96) fragmented at the bead or surface attachment sites (twist density = 1.2 (±0.6) rot µm⁻¹; discussed below), suggesting that the weakest mechanical elements are filament attachment points. The remaining events (13/96) could be reliably discerned as fragmentation within the filament. This occurred at a twist density of ~1.1 (±0.5) rot µm⁻¹ (Movie S8).

At low pulling forces (≤0.03 pN), filaments adopt a plectoneme configuration, which dissipates the torsional strain from applied twisting. Accordingly, no fragmentation was observed, and all filaments formed a plectoneme under 0.01 pN pulling force. At a pulling force of 0.03 pN, some fragmentation events were observed. These occurred at the onset or during plectoneme formation [twist density = 0.83 (±0.17); Fig. 6 and Movie S9], with 9/17 filaments fragmenting exclusively within the filament.

Fig. 6.

Twisted cofilactin filaments fragment more easily than twisted bare actin filaments. (A) Representative images of twist-induced fragmentation for undertwisted (UT) bare, overtwisted (OT) bare, undertwisted cofilin saturated, and overtwisted cofilin saturated filaments. (Scale bar, 4 µm.) (B) Survival analysis from the experiments in (A) at a pulling force of 0.03 pN. Log-rank test comparing UT bare to UT cofilin (P < 0.0001) and OT bare and OT cofilin (P = 0.0094). Both log-rank tests and Gehan–Breslow–Wilcoxon tests yielded similar P values, which conclude that the observed twisting response of bare and cofilin-decorated filaments is statistically different.

Cofilin Promotes Twist-Induced Filament Fragmentation.

Filaments saturated with the actin regulatory protein, cofilin, referred to as cofilactin [cofilin] = 2 μM, which is saturating for this yeast isoform under our conditions (35, 38), did not form a plectoneme, even at low (0.03 pN) pulling force, because they fragmented. Cofilactin filament fragmentation occurred at a lower twist density than fragmentation of bare filaments (Fig. 6 and Movies S10 and S11). The twist density dependence of the filament survival probability decay (Fig. 6) indicated a midpoint of ~ 1 rot µm⁻¹ for fragmentation of bare actin, while the midpoint for cofilactin filaments was significantly lower (0.64 (±0.28) and 0.43 (±0.12) rot µm⁻¹ for overtwisting and undertwisting, respectively; Fig. 6 and Movies S10 and S11). The observed fragmentation originates from twisting strain rather than flow-mediated forces as neither untwisted bare nor cofilactin filaments fragmented on the timescales of these experiments (SI Appendix, Fig. S4 and Movie S12).

Discussion

Actin Filament Torsional Persistence Length is ~10 μm.

Here, we have shown through twist–extension (Fig. 3), filament rotational fluctuations (Fig. 4), and single subunit fluctuations (Fig. 5) that actin filaments have a torsional persistence length of ~10 μm, comparable with their bending persistence lengths (Fig. 2) (19, 32–35, 38, 42, 48–50, 59). The reported torsional persistence length of actin filaments varies significantly from 0.5 to 20 µm (20, 22, 39–44). Our measured L_T is consistent with reported values determined in optical traps (20, 39), fluorescent polarization microscopy (22), and electron microscopy of filaments straightened with hydrodynamic flow (43) but differs from the shorter L_T values determined with fluorescent polarization microscopy (41), negative stain electron microscopy (44), phosphorescence anisotropy (40), and molecular dynamics simulations (42). Our actin persistence lengths are about two orders of magnitude more rigid than DNA, which has an L_T and L_B of 100 and 50 nm, respectively (52), although these values depend greatly on solution conditions (e.g., salt composition and concentration).

The large uncertainties in the measured rotation angles of individual actin subunits could contribute to the short L_T values determined by electron microscopy (44). Uncertainty in subunit rotation angles overestimates the angular fluctuations of adjacent filament subunits and yields an artificially short twisting persistence length. Using an analysis method as in this work (Methods) that accounts for these uncertainties in rotational angles yields a larger (i.e., stiffer) torsional persistence length (Fig. 5). The discrepancy in the filament twisting persistence length values determined by fluorescence polarization and phosphorescence anisotropy (40, 41) may be due to the independent movement of protein side chains or subunit domains to which the spectroscopic probe is conjugated.

Actin Filaments Fragment at a Twist Density of ~1 to 2 deg sub⁻¹.

In our twist–extension experiments with 0.25 pN pulling force, actin filaments fragmented at a twist density of 1 to 2 rot μm⁻¹ (Figs. 3B and 7A). A twist density of 1 rot μm⁻¹ is equivalent to ~1 deg rotation per actin subunit, which corresponds to a 1-deg change in relative twist between two laterally adjacent actin subunits and a 2-deg change in twist between two longitudinally adjacent actin subunits. This twist density introduces only 0.65 and 1.3 k_BT of strain energy (SI Appendix, Eq. S4) at the lateral and longitudinal contacts of actin subunits (Fig. 7B), respectively, which is considerably less than the estimated bond energies associated with lateral (4 to 8 k_BT) and longitudinal (12 to 20 k_BT) filament contacts (26).

Fig. 7.

Modeling twist-induced fragmentation of actin filaments. (A) Experimental actin filament survival curves for undertwisted (blue) and overtwisted (black) bare actin filaments at 0.25 pN pulling force (2 µL min⁻¹ flow rate) and a twisting rate of ω = 0.3 rot s⁻¹. Data include instances where fragmentation occurs close to the bead or surface interfaces. For comparison, the plot includes simulations of filament survival curves as a function of twist density (SI Appendix, Eq. S31) at the same twist rate of 0.3 rot s⁻¹ with a length of L = 15 µm (red trace), as that typical in our twist–extension experiments in this study, and a twist rate of ω = 2.2 rot s⁻¹ with a length L = 0.1 µm (gray trace). Inset image is an example of twist-induced fragmentation. Inset graph is the model-predicted filament torque (SI Appendix, Eq. S9). (B) Model-predicted twisting strain energy per subunit (left y-axis, SI Appendix, Eq. S4) and the relative increase in fragmentation rate constant of strained relative to relaxed, native filaments (right y-axis, SI Appendix, Eq. S28). Dashed lines indicate the model-predicted twisting strain energy for the twist density imposed at boundaries of human cofilin clusters (blue) and by singly isolated bound human cofilin (red).

Why then do actin filaments fragment at such low twist densities? For an actin filament to fragment, three intersubunit interfaces—two longitudinal and one lateral—must rupture simultaneously (15, 19). The combined twist strain energy in these three bonds of a filament twisted to a density of ~1 deg rotation per subunit is only ~3.2 k_BT, more than an order of magnitude lower than the 44 k_BT subunit⁻¹ activation energy for fragmentation (15). Although the imposed twist strain energy does not directly overcome the activation energy for filament fragmentation, it does accelerate the filament fragmentation rate constant ~twofold (calculated from exp(E_strain/k_BT); SI Appendix, Eq. S28) (15). While this effect may seem small, it accounts for the rapid fragmentation of twisted filaments observed in our experiments when other factors contributing to severing are considered.

Two additional factors that contribute to the rapid fragmentation of twisted filaments are the number of potential severing sites and the time duration of the applied twisting load. Filament severing occurs at subunit interfaces, so long filaments have more potential fragmentation sites than shorter ones. That is, the severing reaction is a microscopic process, but observed filament severing is a macroscopic process that scales with the filament length, a collective effect happening at individual subunits. A typical filament in our experiments is >10 µm in length (>3,700 subunits), which means the observed, macroscopic severing rate constant is the microscopic severing rate constant times 3,700. In terms of fragmentation probability, if P is the microscopic fragmentation probability expressed in units subunit⁻¹, the macroscopic probability for fragmentation of a filament that comprised n subunits is given by 1−(1−P)ⁿ ~ nP + 0(P²) (SI Appendix, Eq. S30).

The duration (Δt) of applied twist deformation is a second critical factor contributing to the observed fragmentation. Twist loads introduce strain energy and accelerate filament fragmentation according to the Arrhenius equation (SI Appendix, Eq. S28) (15) from which the twofold acceleration is calculated above. However, the filament fragmentation probability (P) scales with the fragmentation rate constant (k_frag) and the duration of the applied load (Δt) according to P = 1− exp(−k_fragΔt) (SI Appendix, Eq.  S29) (15). Therefore, fragmentation will not be significantly affected if the duration of the applied twist is short relative to the characteristic severing time (Δt  ≪ 1/k_frag).

We derived the filament survival probability as a function of time for a given rate of applied twisting strain (SI Appendix, Eq. S31). Survival curves, as a function of twist density, were simulated according to SI Appendix, Eq. S31 under two different conditions: 1) slow rotation (~0.3 rot s⁻¹ or 3.2 s for 1 rot) of a 15-μm filament, as carried out in our experiments, and 2) rapid rotation (2.2 rot s⁻¹) of a short filament (0.1 μm), as previously modeled (15) (Fig. 7A). Long filaments with slow rotation break at ~2 rot μm⁻¹ (duration 96 s), whereas the short filaments with fast rotation break at 5 rot μm⁻¹ (duration 0.22 s). This behavior explains why filaments do not break under thermal twist fluctuation of ~1 deg sub⁻¹ (Fig. 7A) and suggests that filaments can undergo rather large structural changes without fragmenting, provided the duration of these shape changes are short.

Local Twist Strain Drives Filament Severing by Cofilin.

The actin filament severing protein, cofilin, binds between two adjacent longitudinal actin subunits and undertwists the filament ~4.3 deg sub⁻¹ (54, 55). The filament twist changes abruptly, within ~1 to 2 subunits, at boundaries between bare and cofilin-decorated segments (54, 55, 60). Filaments preferentially sever at these boundaries (45, 61, 62) within the bare actin side of the boundary (60). If we assume that the ~4.3-deg twist strain spreads evenly (54) among the two subunits at the boundary such that each experiences a twist change of ~2.1 deg, a twist density of ~2.1 deg sub⁻¹ introduces strain energy of ~2.9 k_BT sub⁻¹ ($\frac{1}{2}{L}_{\mathit{Ts}}{\sigma }^2$, SI Appendix, Eq. S4, Fig. 7B). The strain energy of this magnitude is predicted to accelerate the bare actin intrinsic severing rate constant ~18-fold (exp(2.9) using SI Appendix, Eq. S28; Fig. 7B). This value agrees with estimates of a boundary severing rate constant that is ~10 to 25 times faster than that of bare actin (32, 35, 38).

It has been reported that boundary severing rate constants vary among cofilin isoforms [e.g., Saccharomyces cerevisiae severs more rapidly than human cofilin (32, 35, 38, 45, 61, 63, 64)]. Subtle differences in cofilin-induced twist could potentially account for large variability in severing, given the quadratic dependence of the severing rate constant on twist density (Fig. 7B). We note that the filament twist is constant and does not change within bare and cofilin-decorated segments and therefore does not introduce strain between subunits in those regions as it does at boundaries where twist discontinuities exist (62).

A singly bound [i.e., isolated (65)] cofilin also changes the filament twist, although less than at a boundary and only at the one subunit to which it directly binds (55). Accordingly, a singly bound cofilin severs filaments but does so more slowly than boundaries (32, 38). Assuming the twist induced by isolated bound cofilin is ~ 1 to 2 deg, the local twist strain should accelerate fragmentation ~2 to 18-fold (Fig. 7B), consistent with the reported ~fivefold acceleration (32).

Cofilin Renders Twisted Actin Filaments Brittle.

Cofilactin filaments (i.e., saturated with cofilin) break at lower twist densities than bare actin filaments (Fig. 6). Several factors can potentially contribute to this mechanical instability. A twisted cofilactin filament could store more elastic strain energy than a bare filament for a given twist load, thereby resulting in more rapid fragmentation. While conceivable, cofilactin filaments are thought to be more compliant in bending (16, 19, 34, 42) and twisting (40, 42) than bare actin filaments so they have less strain energy for an identical shape deformation.

It is also possible that cofilactin filaments are more fragile and fragment more easily than bare actin filaments. While this is also conceivable, cofilin bridging interactions stabilize cofilactin filaments and protect them from fragmentation (15, 60). These bridging interactions render fully decorated cofilactin filaments comparably stable to bare actin filaments (61, 62, 65–68).

A third more likely possibility is that cofilin dissociation from actin filaments transiently introduces a boundary, which fragments more easily under twisting (and bending) loads (15, 19). Spontaneous cofilin dissociation can introduce this effect but it would be more prominent if twisting loads weakened cofilin binding and/or accelerated cofilin dissociation, as predicted from modeling studies (19).

Pulling Accelerates Cofilactin Filament Fragmentation.

Surface tethering and cross-linking constrain filament bending and twisting, which prevents the dissipation of cofilin-induced torsional strain, thereby enhancing cofilin severing activity (15, 19, 22, 67). Long-axis pulling forces on actin filaments, of magnitude comparable with those exerted by myosin motors [3 to 5 pN per ATP hydrolyzed (69)], also dampen thermally driven (Fig. 2) and twist-induced filament bending (Fig. 3) and dramatically accelerate filament fragmentation.

This behavior supports models in which contractile forces generated by myosin motors rapidly sever twisted filaments such as those with bound cofilin (15, 19, 28). Recent studies show that contractile forces produced by myosin motors “catalyze” cofilin-mediated actin filament disassembly and turnover in Aplysia neurons (70) and also contribute to actin filament turnover during contractile ring constriction in S. pombe (71), demonstrating how combinations of pulling, bending, and twisting forces can dramatically accelerate actin filament network fragmentation and turnover in cells.

Twisting a Filament Bundle.

With the torsional persistence length measured in this study, it is possible to make predictions about the behavior of bundled actin filaments. We can model a perfectly cross-linked actin filament bundle as a filament with different dimensions such that the radius of the bundle cross-section (R) is determined by the radius of a single actin filament (r) and the number (n) of filaments comprising the bundle. The area of the cross-section of the bundle is the sum of the cross-sectional areas of each filament forming the bundle (i.e., πR² = nπr²). Therefore, R² = nr², and the bundle’s torsional and bending persistence lengths become L_T,bundle = G(πR⁴/2)/k_BT = n² G(πr⁴/2)/k_BT = n²L_T and L_B,bundle = E(πR⁴/4)/k_BT = n² E(πr⁴/4)/k_BT = n²L_B, where G is the shear modulus, and E is the Young’s modulus of an actin filament, respectively. Therefore, a filament bundle’s torsional and bending persistence lengths and corresponding strain energies scale with n². This suggests that twisted bundles are a result of very large applied torques as twisting a three-filament bundle requires nine times as much torque to twist compared with a single filament. The authors of this study (12) concluded that these torsional loads on actin bundles in filopodia are driven by myosin contractility, indicating the off-axis torques generated by myosin motors (72–74) are sufficiently strong to twist filament bundles.

Materials and Methods

A brief description of the experimental materials and methods used is provided here, and for more details, see SI Appendix.

Protein Purification.

Actin was purified from rabbit skeletal muscle and labeled on surface lysines with NHS ester derivatives of Alexa 488, Alexa 647, biotin, or digoxigenin (17). Alexa 488 phalloidin was purchased from Thermo Fisher (catalog# A12379). Ca²⁺-actin monomers (5 µM) were converted to Mg²⁺-actin by addition of 50 µM MgCl₂ and 0.2 mM EGTA and equilibrated for 5 min on ice immediately before use (75). Saccharomyces cerevisiae cofilin with a surface-engineered cysteine was purified and labeled with Alexa 488 (61).

Microscope Sample Preparation.

Surface functionalization and passivation of microscope coverslips with 2 to 5% Biotin-labeled PEG-Silane slides were adapted from elsewhere (76). Microfluidic chambers were assembled as described (17, 77).

Superparamagnetic Dynabeads™ M-270 Epoxy (2.8 µm in diameter, Thermo Fisher catalog #14301) were conjugated to antidigoxigenin antibody following the company-provided protocol.

Samples were prepared and experiments carried out in KMI buffer (10 mM fluorescence-grade imidazole pH 7.0, 50 mM KCl, 2 mM MgCl₂, 0.2 mM ATP, and 2 mM DTT) supplemented with 15 mM glucose, 0.02 mg mL⁻¹ catalase, and 0.1 mg mL⁻¹ glucose oxidase. Actin polymerization was done as previously described (65). Filament–bead conjugation in sample chambers is described in detail in SI Appendix.

Microscopy.

Imaging was conducted on a Till iMic digital total internal reflection fluorescence (TIRF) microscope equipped with a 100× objective (Olympus) and an Andor iXon897 electron-multiplying charge-coupled device (EMCCD) camera (17, 33, 38).

Filament Force–Extension.

In the force–extension experiments without twist, filament end‐to‐end length (R) was measured as a function of the tensile force (f) applied with buffer flow. Images were recorded at each unique buffer flow (tensile force) with an acquisition rate of 1 frame s ⁻¹. The positions of the paramagnetic beads were tracked with the TrackMate plugin in ImageJ (NIH, USA) (78). The value of R was determined as the direct straight-line distance between the two attachment points at the bead and surface. A filament contour length (L) was measured as R at a high flow rate to make the filament straight.

The measured force dependence of R (i.e., force–extension curve) was fitted to the following equation describing the force–extension behavior of semiflexible polymers (6, 47):

$$\begin{array}{c}R=L-\text{Δ}L=L\left(1-\frac{1}{2}\sqrt{\frac{k_{B}T}{L_{B}f}}\left(\coth \left(L\sqrt{\frac{f}{L_B{k}_{B}T}}\right)-\frac{1}{L}\sqrt{\frac{L_B{k}_{B}T}{f}}\right)\right)\hfill \\ =\left\{\begin{array}{c}\stackrel{L\to 0}{\to }L\\ \stackrel{L\to \infty }{\to }L\left(1-\frac{1}{2}\sqrt{\frac{k_{B}T}{L_{B}f}}\right)\end{array}\right.,\end{array}$$ [1]

where L_B is the filament bending persistence length, k_B is Boltzmann constant, and T is the room temperature (296 K). ΔL is the deviation of R from L of a semiflexible polymer due to thermally driven random bending, and it does not scale with L linearly. For the polymers with very short L, ΔL ~ 0, the deviation per L is negligible and R ~ L, whereas with very long L, the deviation of R from L reaches the maximum per unit length $\frac{1}{2}\sqrt{\frac{k_{B}T}{L_{B}f}}$. This nonlinear L dependence of ΔL is consistent with the conclusion by another study (79), which claims a popular force–extension equation (80) with $\frac{\mathrm{\Delta }L}{L}=const$ is overly simplified.

Using Origin software (Originlab, Northampton, MA, USA), experimental replicates were globally fitted with L_B as a global fitting parameter and L unique to each individual filament dataset (Fig. 2B).

In addition to globally fitting individual filament force–extension curves, we compiled and fitted data for filaments of different lengths by averaging the normalized filament end‐to-end length (R/L), which is given as follows:

$$\begin{array}{c}\overline{\left(\frac{R}{L}\right)} = \frac{1}{n}{\displaystyle \sum }_{i=1}^n\frac{R_i}{L_i}= 1-\frac{1}{2}\sqrt{\frac{k_{B}T}{L_{B}f}}\left(1+\frac{2}{n}{\displaystyle \sum }_{i=1}^n\frac{e^{-2L_i\sqrt{\frac{f}{L_B{k}_{B}T}}}}{1-e^{-2L_i\sqrt{\frac{f}{L_B{k}_{B}T}}}}\right.\hfill \\ \left.-\sqrt{\frac{L_B{k}_{B}T}{f}}\frac{1}{n}{\displaystyle \sum }_{i=1}^n\frac{1}{L_i}\right)\end{array},$$ [2]

$\sim 1-\frac{1}{2}\sqrt{\frac{k_{B}T}{L_{B}f}}+\frac{k_{B}T}{2f}\overline{\left(\frac{1}{L}\right),}$ because the term ${\displaystyle \raisebox{1ex}{${2e}^{-2L_i\sqrt{\frac{f}{L_B{k}_{B}T}}}$}\!\left/ \!\raisebox{-1ex}{$1-e^{-2L_i\sqrt{\frac{f}{L_B{k}_{B}T}}}$}\right.}$ is <0.08 and can be ignored since it is ≪1 and more than 1 order of magnitude smaller than $\sqrt{\frac{L_B{k}_{B}T}{f}}\frac{1}{L_i}$ in our experimental force and filament length ranges.

Filament Twist–Extension.

Filament images in the twist–extension experiments were recorded with an acquisition rate of 1 frame s ⁻¹ while twisting filament at a constant rate of 0.31 rot s⁻¹ and applying a given tensile force (f) with hydrodynamic flow. The position of the paramagnetic beads was tracked with the same procedure, and filament end‐to‐end distance (R) and contour length (L) were determined in the same manner as those in the preceding Force–Extension section. R/L is a function of twist density, σ (unit: rot μm⁻¹), and L. Filament contour lengths in our experiments range from 7 to 20 μm. Normalized R/L data were averaged by binning according to the filament twist density (rot μm ⁻¹) with a bin size of 0.025 rot µm ⁻¹.

To describe the twist–extension behavior of actin filaments, an analytical two-state model developed for DNA (52) was modified for actin filaments. In the model, it is assumed that a filament subunit exists in one of two states when a filament is twisted: plectonemic or nonplectonemic (linear). A filament with a plectoneme consists of a mixture of plectonemic and nonplectonemic regions. The fraction of plectonemic regions increases linearly with twist density and does not contribute to R/L. Eq. 3 (SI Appendix, Eq. S10) gives the force dependence of R/L contributed from only the fraction in the nonplectonemic state (52) (SI Appendix) as follows:

$$\begin{array}{c}\frac{R}{L}=\left\{\begin{array}{c}1-\frac{1}{2}\sqrt{\frac{k_{B}T}{L_{B}f}}+\frac{k_{B}T}{2f}\overline{\left(\frac{1}{L}\right)}\hfill \\ -\frac{{\omega }_0{}^2{\sigma }^2}{{\left(\frac{4L_B}{L_T}\sqrt{\frac{f{L}_B}{k_{B}T}}+1\right)}^2}\sqrt{\frac{k_{B}T{L}_B{}^3}{f}}, \left|\sigma \right|<\left|{\sigma }_{\text{s}}\right|\\ \frac{\left|{\sigma }_p\right|-\left|\sigma \right|}{\left|{\sigma }_p\right|-\left|{\sigma }_s\right|}\left(1-\frac{1}{2}\sqrt{\frac{k_{B}T}{L_{B}f}}+\frac{k_{B}T}{2f}\overline{\left(\frac{1}{L}\right)}\right.\\ \left.-\frac{{\omega }_0{}^2{\sigma }_s{}^2}{{\left(\frac{4L_B}{L_T}\sqrt{\frac{f{L}_B}{k_{B}T}}+1\right)}^2}\sqrt{\frac{k_{B}T{L}_B{}^3}{f}}\right), \left|{\sigma }_s\right|\le \left|\sigma \right|<\left|{\sigma }_P\right|\\ 0, \left|{\sigma }_{\text{p}}\right|\le \left|\sigma \right|\end{array}\right..\hfill \end{array}$$ [3]

In the equation, σ_s is a special applied twist density at which filaments form plectonemes, and σ_p is applied twist density when the entire filament is plectonemic. The fraction of subunits in the nonplectonemic conformation is given by ${\displaystyle \raisebox{1ex}{$\left|{\sigma }_p\right|-\left|\sigma \right|$}\!\left/ \!\raisebox{-1ex}{$\left|{\sigma }_p\right|-\left|{\sigma }_s\right|$}\right.}$.

A custom routine was written with Origin software to globally fit the experimental data of R/L as a function of applied twist at different pulling forces to Eq. 3 (Fig. 3) with L_B and L_T as shared unconstrained parameters. σ_s for pulling force of 0.01 and 0.03 pN were fixed parameters from averaging experimentally observed values, but the other σ_s for pulling force of 0.25 pN and value of σ_p for all forces were also unconstrained during the fitting procedure but were not shared because they depend on and vary with the pulling force. The value of $\overline{\left(\frac{1}{L}\right)}$ was constrained during fitting to the values calculated from the actual filament contour lengths measured under a high pulling force (to straighten).

Design of Freely Orbiting Magnetic Tweezers.

The rotational fluctuations of actin filaments were measured with freely orbiting magnetic tweezers (57). Five cylindrical magnets (R422-N52, K&J Magnetics) arranged in series were mounted on a linear XY micrometer stage (XR25, Thorlabs) and centered directly above the microscope objective while visualizing with wide-field microscopy. The procedure for conjugating antidigoxigenin paramagnetic beads to digoxigenin marker beads for these experiments is given in detail in SI Appendix.

Analysis of Filament Rotational Fluctuations.

In whole filament twisting fluctuation experiments (Fig. 4A), we monitor how the angle θ(t) between the two filament attachment points (at the surface and on the bead, length L) changes over time, and the variance (σ_θ²) of the angle fluctuation is given by Eq. 4 (SI Appendix, Eq. S22):

$$\begin{array}{c}{\sigma }_{\theta }^2 = {<{\left(\theta \left(t\right)-<\theta \left(t\right)>\right)}^2>}_{t\to \infty } =\frac{L}{L_T}+{\sigma }_{\epsilon }^2\sim \frac{L}{L_T}.\hfill \end{array}$$ [4]

Here, σ_ε² is the variance of the uniform randomly distributed noise in a relative twist angle between two adjacent filament subunits, and we neglect it because it is much smaller than L/L_T (SI Appendix). Eq. 4 indicates that at times sufficiently long to sample the equilibrium distribution, indicated by plateau of the variance (Fig. 4E), the variance of the angle fluctuations at equilibrium is inversely proportional to L_T, and scales with L linearly (SI Appendix, Fig. S6), which has been demonstrated for actin filaments by experimental observation (20, 39).

Analysis of Filament Subunit Angular Fluctuations from cryo-EM Data.

From the electron micrographs of actin filaments, which were classified as described previously (54), we defined Δs as the length of an actin subunit and Φ as the rotational Euler angle around the filament centerline. Any rotation angle difference (twist) between two subunits spaced n subunits apart (SI Appendix, Eq. S20) was calculated by the sum of all the observed relative twists between 2 adjacent subunits and it displays a Gaussian distribution with a variance that scales linearly with n (SI Appendix, Eq. S21). Fitting SI Appendix, Eq. S21 to the n dependence of the observed variance (${\sigma }_{obs,n+1}^2$) yields Δs/L_T from the slope of the best linear fit of the data and the noise variance (${\sigma }_{\epsilon }^2$) from the y intercept.

The variance of the angle distribution was extracted from the data using three different analysis procedures. In one approach, the histogram of ΔΦ_I,obs,n+1 for every value of n was independently fitted to a normal distribution, yielding σ²_i,obs,n+1. Then, σ²_i,obs,n+1as a function of n was fitted to SI Appendix, Eq. S21. In addition, in maximum likelihood estimation (MLE), each variance with spacing of n was directly calculated from the square of the SD as follows:

$$\begin{array}{c}{\sigma }_{obs,n+1}^2=\langle {\left(\mathrm{\Delta }{\mathrm{\varnothing }}_{i,obs,n+1}-\langle {\mathrm{\Delta \theta }}_{obs,n+1}\rangle \right)}^2\rangle .\hfill \end{array}$$ [5]

The n dependence of σ²_i,obs,n+1 was then fitted to SI Appendix, Eq. S21. In the third method, we applied Bayesian inference using Markov chain Monte Carlo (MCMC; Metropolis–Hastings algorithm) to sample the posterior probability distribution of the true variance (81). The method was coded in language R (www.r-project.org), and the most likely $L_T$ and ${\sigma }_{\epsilon }^2$ values were determined from the peaks of their probability distributions. These three methods of analyses were repeated for the rotation Euler angle Φ determined from a refinement volume of 1 subunit instead of 5 with comparable results (Table 1) (54, 55).

Supplementary Material

Appendix 01 (PDF)

Movie S1.

Phalloidin decorated actin filament attached to two paramagnetic beads is rotated (left). Cosine of the rotation angle of bead (black dots) and the magnet (red line) are in phase (right). Note that all movies are not played in real time.

Movie S2.

Alexa-647 actin filament (magenta) conjugated to a paramagnetic bead (cyan) rotates around the paramagnetic bead as the permanent magnet is rotated.

Movie S3.

Alexa-647 actin filament (magenta) conjugated to a paramagnetic bead (cyan) is subjected to a range of flow induced pulling forces, 0.007pN to 2.9pN.

Movie S4.

Alexa-647 actin filament (magenta) conjugated to a paramagnetic bead (cyan) is pulled with 0.03 pN of tension and twisted (left). Filament end-to-end distance is determined by tracking of the paramagnetic bead (right).

Movie S5.

Alexa-647 actin filament (magenta) conjugated to a paramagnetic bead (cyan) is pulled with 0.03 pN of tension and over-twisted until it supercoils. After supercoiling filament is under twisted until the filament twist is neutral.

Movie S6.

Alexa-647 actin filament (magenta) conjugated to a paramagnetic bead (cyan) is pulled with 0.03 pN of tension and does not fragment.

Movie S7.

Alexa-647 actin filament (magenta) conjugated to a paramagnetic bead (dim magenta) and a nonmagnetic marker bead (bright magenta). Flow is turned off, a cylindrical magnet is lowered into place, and the focal plane is adjusted to visualize the paramagnetic and marker beads.

Movie S8.

Alexa-647 actin filament (magenta) conjugated to a paramagnetic bead (cyan) is pulled with 0.25 pN of tension and over-twisted until it fragments.

Movie S9.

Alexa-647 actin filament (magenta) conjugated to a paramagnetic bead (cyan) is pulled with 0.03 pN of tension and over-twisted until it fragments.

Movie S10.

Alexa-647 actin filament (magenta) saturated with cofilin (cyan) is conjugated to a paramagnetic bead (cyan) and over-twisted until the filament fragments.

Movie S11.

Alexa-647 actin filament (magenta) saturated with cofilin (cyan) is conjugated to a paramagnetic bead (cyan) and under-twisted until the filament fragments.

Movie S12.

Rotational fluctuations of a surface tethered Alexa-647 actin filament conjugated to a paramagnetic bead (dim magenta) and a nonmagnetic marker bead (bright magenta). Filament is pulled straight perpendicular to the coverslip.

Data, Materials, and Software Availability

All study data are included in the article and/or SI Appendix.