Direct Measurement of the Torsional Rigidity of Single Actin Filaments

TLDR: This paper shows that the torsional rigidity can be measured directly by visualizing the tORSional Brownian motion of a single actin filament with a novel methodology based on an optical trapping technique that is one to two orders of magnitude greater than previous experimental estimates.

About: This article is published in Journal of Molecular Biology.The article was published on 1996-10-25 and is currently open access. It has received 128 citations till now. The article focuses on the topics: Protein filament & Flexural rigidity.

Figures

Figure 2. (a) and (b), Fluctuation of the rotational angle, u, of a bead duplex sampled at intervals of 33 ms (video rate). In (a) the duplex was bound to a Ca2+-actin filament at L1 = 3.5 mm and L2 = 2.8 mm (see Figure 1(a)), and in (b) to a Mg2+-actin filament at L1 = 2.9 mm and L2 = 2.4 mm. (c) Distribution of u for the F-Ca2+-actin in (a) (hatched bars) and for F-Mg2+-actin in (b) (filled bars). Continuous lines indicate a Gaussian fit to each distribution. Variance of the angular distribution, (u − u )2 , was 142 deg2 for the F-Ca2+-actin and 470 deg2 for the F-Mg2-actin. To estimate the reliability of these values, the data in (a) and (b) were divided into halves; the first and second halves respectively gave 139 and 127 deg2 for the F-Ca2+-actin, and 496 and 427 deg2 for the F-Mg2+-actin. (d) Autocorrelation function of the data in (a) and (b). Continuous lines show a theoretical fit with exp(−t/t), where t = 0.42 s for the F-Ca2+-actin (open circles) and t = 1.0 s for the F-Mg2+-actin (filled circles). Division of (a) and (b) into halves yielded t = 0.50 and 0.33 s for the F-Ca2+-actin and t = 0.97 and 0.91 s for the F-Mg2actin.

Figure 4. CCF for the bending Brownian motion of F-Ca2+ (w q) and F-Mg2+-actin (W Q). To estimate reliability of the measurement, filaments 9 to 10 mm (w W) and 11 to 19 mm (q Q) long were analyzed separately. Each symbol shows the average over 112 to 128 images (7 or 8 filaments), the bar being standard error (n = 7 or 8). CCF for each symbol was fitted with the function exp(−LkBT/2kF ), giving the flexural rigidity, kF , of (5.77 (w) and 5.84 (q)) × 10−26 N m2 for F-Ca2+-actin and (6.10 (W) and 6.35 (Q)) × 10−26 N m2 for F-Mg2+-actin. From the two values of kF in the same type of actin, the accuracy was estimated as 00.1 × 10−26 N m2. Lines indicate the averages of the two exponential functions, continuous line for F-Ca2+-actin and broken line for F-Mg2+-actin.

Figure 1. Direct observation of the torsional Brownian motion of an actin filament. (a) Schematic illustration. The striped beads are fixed on a surface (the surface is not shown). x and y axes are respectively perpendicular and parallel to the filament axis in the horizontal (image) plane, and z is a vertical axis. xr and zr are the coordinates of the upper bead relative to the lower bead. u is the tilt angle of the bead duplex in the x–z plane. (b) Fluorescence image (the bead duplex at the center is out of focus). (c) Bright-field image. The microscope focus was adjusted so that the upper bead appeared brighter. Bar, 5 mm. (d) Snapshots of the fluctuating bead duplex at intervals of 0.13 s. The tilt angle, u, in these images was determined as 37°, 36°, 27°, 47°, 50°, and 53°. (e) Motions of the upper (thin line) and lower (thick line) beads in the duplex projected onto the x–y plane. y-axis is parallel to the filament axis. Bead coordinates were read by fitting a circle to the bead image by eye. (f) Motion of the upper bead relative to the lower bead, projected onto the x–y plane. The relative coordinates (xr, yr) were calculated as (xu − xl , yu − yl ) where (xu , yu ) and (xl , yl ) are, respectively, the coordinates of the upper and lower beads in (e).

Figure 3. Reciprocal of the spring constant, 1/k, versus the effective length, Le, of F-Ca2+ (w) and F-Mg2+-actin (W). Each point represents one filament, from which 64 values of u were measured at intervals of 0.53 seconds (total 34 seconds). Points with a white arrowhead show filaments under a tension of at least 1.5 to 1.8 pN, and points with a black arrowhead under a tension of at most 0.2 to 0.3 pN (see Materials and Methods). Continuous lines show linear regression lines. Samples in which u exceeded (280°) or u showed a slow drift over several seconds (presumably due to medium flow) were discarded. In the calculation of Le , the lengths L1 and L2 were taken as in Figure 1(a). These lengths are overestimates if the filament was wound around the lower bead and are underestimates if the attachment to an end bead was at a further point. The errors are, at most, 0.44 mm (the radius of the beads), and thus the error in Le is 00.2 mm.

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Direct measurement of the torsional rigidity of single actin filaments" ?

This paper shows that the torsional rigidity can be measured directly by visualizing the torsional Brownian motion of a single actin filament with a novel methodology based on an optical trapping technique. 

Q2. What was the light source for the opticaltweezers?

An infrared microlaser (1053 nm, 1 W, OEM 1053-1000 p, Amoco Laser Co., Naperville, IL, USA) was the light source for the opticaltweezers. 

Q3. What was the effect of the optical trap on the filament?

After binding, the optical trap was turned off, and the fluctuation of the bead duplex due to the bending and torsional motion of the filament started. 

Q4. What is the effect of the twist on the actin filament?

A twist of the myosin head around an actin filament, counteracted by the rigid torsional spring of the actin filament, would help strain the slack joints. 

Q5. Why was the u in the x–z plane overestimated?

Because of the limited time resolution in the video analysis, the variance was underestimated, hence k was overestimated, by 1 to 3% (see equation (6) in Materials and Methods). 

Q6. What is the theory for a homogeneous isotropic rod?

In addition their torsional and flexural rigidity values, whether of F-Ca2+-actin or of F-Mg2+-actin, do not rigorously conform to the theory for a homogeneous isotropic rod (Landau & Lifshitz, 1970) which predicts that the ratio of the flexural rigidity to the torsional rigidity should be between 1 and 1.5. 

Q7. What was the effect of the bending Brownian motion between the fixed ends?

The actin filaments in their system were slightly under tension, as a result of the bending Brownian motion between the fixed ends. 

Q8. What is the ratio of bending in the y–z plane?

Bending in the y–z plane also contributes to the y-spread, but the vertical bending was negligible in this sample as was also the case for horizontal bending (as indicated by the approximately 3:1 ratio in the x-excursions of the upper and lower beads in Figure 1(e)). 

Q9. What is the strategy for detecting anisotropic rigidity?

A system inwhich only one end of a filament is fixed (e.g. Suzuki et al., 1996; Tsuda et al., 1996) would be slightly better for the detection of anisotropic rigidity, but the best strategy will be to apply a much larger torque with, e.g. optical tweezers. 

Q10. What is the uncertainty in the bending amplitude of the filament?

The bending amplitude in each filament was thus estimated as d = 4L2x (K2 − U2x /L2x )/(K2 − 1)51/2 with the uncertainty in K of 3 < K < 5.