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Actin Filament Guidance on a Chip: Toward High-Throughput Assays and Lab-on-a-Chip Applications Mark Sundberg,† Richard Bunk,‡ Nuria Albet-Torres,† Anders Kvennefors,‡ Fredrik Persson,‡ Lars Montelius,‡ Ian A. Nicholls,† Sara Ghatnekar-Nilsson,‡ Pa¨r Omling,‡ Sven Tågerud,† and Alf Månsson*,† Department of Chemistry and Biomedical Sciences, UniVersity of Kalmar, SE-391 82 Kalmar, Sweden, and DiVision of Solid State Physics and The Nanometer Consortium, UniVersity of Lund, Box 118, SE-221 00 Lund, Sweden ReceiVed March 30, 2006. In Final Form: June 7, 2006 Biological molecular motors that are constrained so that function is effectively limited to predefined nanosized tracks may be used as molecular shuttles in nanotechnological applications. For these applications and in highthroughput functional assays (e.g., drug screening), it is important that the motors propel their cytoskeletal filaments unidirectionally along the tracks with a minimal number of escape events. We here analyze the requirements for achieving this for actin filaments that are propelled by myosin II motor fragments (heavy meromyosin; HMM). First, we tested the guidance of HMM-propelled actin filaments along chemically defined borders. Here, trimethylchlorosilane (TMCS)-derivatized areas with high-quality HMM function were surrounded by SiO2 domains where HMM did not bind actin. Guidance along the TMCS-SiO2 border was almost 100% for filament approach angles between 0 and 20° but only about 10% at approach angles near 90°. A model (Clemmens, J.; Hess, H.; Lipscomb, R.; Hanein, Y.; Bohringer, K. F.; Matzke, C. M.; Bachand, G. D.; Bunker, B. C.; Vogel, V. Langmuir 2003, 19, 10967-10974) accounted for essential aspects of the data and also correctly predicted a more efficient guidance of actin filaments than previously shown for kinesin-propelled microtubules. Despite the efficient guidance at low approach angles, nanosized (65 µm. However, eventually all filaments detached. As expected from the dependence of guiding probability on approach angle (Figure 3), the filaments also detached when they reached the curved parts of the tracks (Figure 4). Guidance of HMM-Induced Actin Filament Sliding by Topographically Defined Tracks. There is evidence from previous work (see Introduction) that topographical barriers in the form of walls are not sufficient to fully guide the sliding of myosin-propelled actin filaments along tracks in the absence of chemically based selectivity. In most previous work with actin filaments, the topographical barriers consisted of walls without an overhang as used here (Figure 2C). It was therefore of interest to clarify if the inclusion of the overhang would allow guidance in the absence of chemically based selectivity. Tests of this hypothesis were performed by applying conditions without complete selectivity in function between the topographically defined tracks and their surroundings. Such conditions were achieved if there was no TMCS functionalization of the channel floors and if methylcellulose was included in the assay solution. Under these conditions, several actin filaments were found sliding both on the SiO2 floors of the channels and on the surrounding PMMA. Several filaments were also found to escape from the channels by climbing the channel walls and continuing their sliding on PMMA (cf. Figure 5). Such climbing of channel walls was never observed for the channels with TMCSderivatized floors when there was no methylcellulose in the assay solution. Guidance and Velocity of HMM-Induced Actin Filament Sliding on Tracks That Are Both Topographically and Chemically Defined. In a40 assay solution, the nanostructured surfaces with TMCS tracks surrounded by PMMA always exhibited motility only on TMCS areas with no motility on PMMA (six different surfaces studied; see also ref 15). None of the filaments sliding on the TMCS-derivatized loading
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Figure 2. Design of nanostructured surfaces used in this study. (A) Schematic illustration of the nanostructure used in a majority of the experiments. Two actin filament loading zones (rectangles; solid lines) are connected by ten 1.2-mm-long tracks (only half of the tracks are shown) with widths in the range of 50-600 nm (before etching of the LOR layer) and curvatures of different radii (see text). The area within dashed square refers to Figure 2B. Drawing not to scale. (B) AFM image of the TMCS-derivatized loading zone, the surrounding PMMA layer, and the opening of 700 nm track/channel (area within the dashed rectangle in Figure 2A). Tip convolution artifacts distort the track cross section and loading zone walls, but note that the channel is continuous with the TMCS loading zone. Imaged using Topometrix Explorer AFM (Thermo-Microscopes, CA) with a high aspect ratio super tip (10:1, ThermoMicroscopes) in contact mode. (C) Scanning electron microscopy15 image illustrating the cross section of a channel with a top channel width of ∼400 nm. SiO2 in the channel floors was derivatized with TMCS. Walls are made of polymer resist LOR (bottom layer) and PMMA (top layer). Nanofabrication by electron beam lithography and specific etching of the LOR polymer resist layer. (D) EBL resist mask illustrating closed-loop bilayer resist channels where the individual channels are similar to those illustrated in Figure 2B and C but with a bottom channel width (track width) of 200-250 nm. Top: part of the gold pattern that aids in the localization of the nanopatterns in the fluorescence microscope. Bottom: three pear-shaped loading zones to the right, and three tracks for closed-loop transport to the left. Dashed square illustrated in greater detail in E. (E) Scanning electron micrograph (acceleration voltage 15 kV; cf. ref 15) showing details of the junction between two nanosized tracks (dashed square in D). A 10-nm-thick film of gold was sputtered onto the sample before obtaining the image.
zone (>100 filaments on three different surfaces) escaped upon hitting the PMMA/LOR walls, irrespective of approach angle. This suggests that this dually (chemically and topographically) defined border exhibits close to 100% guidance irrespective of approach angle. Close to total guidance was also achieved along nanosized tracks as exemplified in Figure 6. Here, one filament is observed to enter the nanosized channel from a loading zone (Figure 6A), and two other filaments are observed to be guided along curved parts of a channel (Figure 6B). Guidance along curved domains was observed for a radius down to 1 µm. A quantitative study was performed of the degree of guidance along
different tracks on three different surfaces with curvatures of radii in the range of 1 to 5 µm. Results from track widths from 150 to 700 nm were pooled. These results showed that only 5 filaments escaped for 100 min of filament sliding (i.e., for 100 filaments‚min; 3 surfaces). Further evidence for efficient guiding without escape events is the long-term trapping of actin filaments on closed-loop tracks dually defined by topographical barriers and chemical selectivity (see below; Figure 8). A comparison between sliding velocity on the loading zones and on tracks of different widths was performed in one experiment. The velocity of smoothly sliding actin filaments on the narrowest
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Figure 3. Guidance probability versus approach angle: comparison of experimental data and theoretical predictions obtained by the approximate method of Clemmens et al.17 Mean values (filled circles) ( SE of experimental data from three micropatterned surfaces. The value of SE in this case is obtained as the standard deviation/x3 (for the three replicate experiments) whereas SE for each data point in the individual experiments (Supporting Information) was obtained as described in Experimental Methods. Data from different HMM densities and sliding velocities are pooled. Squares connected by dashed lines represent theoretical predictions for the two most different experimental conditions encountered: open squares (Vf ) 10 µm s-1, average filament length ) 1.7 µm) and filled squares (Vf ) 5 µm s-1, average filament length ) 0.6 µm). The full curved line represents the fitting of the third-degree polynomial to the data. Triangles and full lines are the data of Clemmens et al.17 replotted after approximate measurements from their Figure 6. Triangles and the dashed line indicate theoretical predictions for microtubule guidance by Clemmens et al.17. The line that was extended to the abscissa for clarity does not indicate the guidance probability between 0 and 8° but is included to simplify visualization.
Figure 4. Sequence of background-subtracted images of an actin filament (encircled in sequence 1) sliding on a curved TMCS track (indicated by the dashed line in sequence 1). The track was surrounded by bare SiO2. The positions of actin filaments were recorded at 1.2 s intervals (exposure 0.3 s) to illustrate the filament path. The direction of sliding is illustrated by arrow. Note the loss of focus in sequences 8 and 9 due to the escape of the filament into solution. Assay solution a40. Scale bar 10 µm. Temperature 24 °C. Images slightly retouched for clarity.
tracks (150-200 nm) was 4.6 ( 0.1 µm s-1 (n ) 19 filaments) compared to 4.7 ( 0.2 µm s-1 (n ) 9) for 300-nm-wide tracks and 4.4 ( 0.2 µm/s (n ) 9) in the loading zone (24 °C). Conditions for Unidirectional Sliding of HMM-Propelled Filaments along Nanosized Tracks. Because tracks that were dually defined by chemistry and topography allowed almost complete guidance, such tracks could be used to analyze the conditions that were required to achieve unidirectional sliding of actin filaments. On 300-700-nm-wide tracks, filaments were able to make
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Figure 5. Escape of HMM-propelled actin filament from topographically defined nanosized track in aMC130 assay solution. The track was similar to that in Figure 2A-C but without TMCS derivatization of the channel floor. Sixty superimposed images of an actin filament (at 0.2 s interval) sliding from left to right on the second track from below (sliding direction indicated by arrow) are shown. Note that the filament first slides a substantial distance on the track before climbing the track wall/roof and thereafter continues sliding on PMMA. Scale bar 10 µm. Temperature 24 °C.
Figure 6. Actin filament sliding on a nanostructured surface with TMCS-derivatized areas surrounded by PMMA barriers (cf. Figure 2). (A) Actin filament (arrows) entering a channel (cf. Figure 2B and C) from the loading zone (dark region on top; cf. Figure 2B). The loading zone and channel floor were derivatized with TMCS, and the channel was surrounded by PMMA (grey). (B) Two actin filaments guided by the curved region of a 150-nm-wide track whose presence is indicated by weak fluorescence. The time interval is 1 s between individual images in the image sequences in both A and B. The filaments are moving downward in each image. Temperature 24 °C.
U-turns on three tested surfaces. On tracks of width 26 mm). On one nanostructured surface, motility was observed for 2 h after the first infusion of assay solution (i.e., close to 2.5 h after
Figure 8. HMM-induced sliding of actin filaments on a closedloop track similar to that described in Figure 2D and E (width 20 filament paths unless otherwise indicated within parentheses. The fraction of motile filaments was >0.9. Note that the same three closed loops were observed during the first 45 min. Thereafter, because of image deterioration, the observed area had to be changed to other loops Filled symbols: HMM incubation concentration 200 µg mL-1 for 5 min. Fluorescent actin filaments (2 nM) were added. Open symbols: HMM incubation concentration 50 µg mL-1 for 5 min. Fluorescent actin filaments (25 nM) were added. The temperature was 20-22 °C with an estimated 2 °C increase during the first 10 min explaining the small initial increase in velocity. (C) For comparison, the sliding velocity (circles) and total number of filaments (squares) and number of motile filaments (triangles) are plotted against time after the addition of assay solution to a nonpatterned TMCS surface. This surface was preincubated with HMM at a concentration of 100 µg mL-1 for a time period (2 min) that has been shown to produce saturation of the surface with HMM. The temperature was 20.4-24.6 °C with an early rapid increase followed by a decrease from 24.6 to 21.8 °C in the first hour. Variability in the number of filaments over time was attributed to a combination of several factors (e.g., fragmentation first leading to an increase in the number of filaments but later to a decrease because filaments that are too small are not counted by the automatic analysis program). In all experiments, a40 assay solution was used without the addition of creatine phosphate and creatine kinase.
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incubation with HMM). It is possible that motility was maintained for an even longer time, but this was not verified. In this case, new actin filaments and new assay solution had been added. Under these conditions, there was no decrease in the sliding velocity of smoothly sliding filaments compared to the situation immediately after the first infusion of assay solution, but the fraction of stationary filaments increased. Using the same HMM preparation and assay solution as in the observation of closed-loop transport in Figure 8A and B, a flow cell with a nonpatterned TMCS surface was also closed to prevent evaporation in order to allow long-term observation. On this TMCS surface, the sliding velocity, the number of filaments, and the fraction of motile filaments changed with time after a40 infusion as illustrated in Figure 8C. The sliding of actin filaments on the closed loop was observed only intermittently, and the escape of filaments from the tracks could therefore not be directly verified. However, we have indirect evidence that the escape is related to a lowered density over time of HMM with actin binding capability (possibly due to HMM desorption). This is in accordance with the detachment of filaments rather early after the infusion of the assay solution if the HMM incubation concentration was lowered to 50 µg mL-1 (Figure 8B). That the loss of actin filaments from the loops was due to lowered HMM density on the tracks is also consistent with observations in the experiment, mentioned above, when motility was restored by the addition of new actin filaments almost 2 h after the first addition of assay solution. In this case, a considerably larger fraction of observed filaments (seven filaments in 6 filaments‚min) detached compared to the situation normally seen early after the first addition of assay solution (see above). For the case where filaments were not guided, it is important to note that they detached from the track and diffused into the solution rather than climbing up the walls.
Discussion Mechanism of Actin Filament Guidance at Chemically Defined Borders. When HMM-propelled actin filaments on TMCS approached a TMCS/SiO2 border under conditions with no motility on SiO2, they either detached from the surface or were guided by the border, thus remaining on the TMCS area. Such guidance occurred frequently even if there was no topographical barrier at the TMCS/SiO2 border. This is similar to the guidance of kinesin-propelled microtubules17 at borders between regions with and without kinesin function. In agreement with the present work, Clemmens et al.17 found a guidance probability that varied with the angle of approach to the border. As for the myosin-propelled actin filaments, there was almost 100% guidance for the kinesin-propelled microtubules at approach angles near 0° and a guidance probability close to zero at an approach angles of 90°. As shown in Supporting Information, the probability of guidance for myosin-propelled actin filaments as a function of approach angle can be partially predicted by an analysis similar to that of Clemmens et al. for microtubule guidance.17 In this analysis, it is assumed that the actomyosin binding and motility differs in an all-or-nothing manner (in a40 solution) between the TMCS and the SiO2 areas. Whereas we assumed that there was no actin binding on SiO2 in the absence of methylcellulose in the assay solution, we assumed a 100% probability of binding if an actin filament touched the HMMcoated surface of TMCS. The validity of the latter assumption was tested by studying the guidance at two different HMM incubation concentrations. These studies supported the hypothesis of 100% actin binding because the effectiveness of guidance of the TMCS/SiO2 border was not reduced if the HMM surface
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density was reduced by about 50%. In the theoretical analysis in the Supporting Information (cf. ref 17), it is furthermore assumed that guidance is a consequence of thermal fluctuations of the part of an actin filament that overhangs the SiO2 area. If the thermal fluctuations are large enough to bring the overhanging part back to the TMCS area, then the filament will bind to myosin heads and continue to slide, thus being guided. The theoretical treatment (Figure 3) predicts guidance for myosin-propelled actin filaments for considerably higher approach angles than for kinesinpropelled microtubules, and this is also what we have found. The better guidance is, according to theory, explained by the considerably greater flexibility of the actin filaments allowing larger amplitudes of the thermal fluctuations. As for the kinesinmicrotubule guidance, the guidance of actin filaments was considerably more effective at intermediate approach angles (between 20 and 80°) than predicted by the theory. This can be accounted for by simplifying assumptions in the theoretical treatment (see Supporting Information) and in the experimental analysis. One important simplification in the theoretical treatment is the assumption that the overhanging part of the actin filament is cantilevered to HMM molecules on the TMCS area with a given fixed angle with respect to the TMCS/SiO2 border (see Supporting Information). That this assumption is correct could not be verified with certainty. Thus, variations in angle that were reasonably localized (e.g., short trailing part of a filament) may not have been resolved (in time and space) by the microscopeCCD system. In accordance with the good correspondence between theory and experimental data for low approach angles, the problem with changes in angle with time is likely to be much less serious under these conditions. A possible reason is that the filaments under these conditions would be guided before any appreciable changes in angle between the HMM-tethered part and the border have had time to develop. A factor that may have caused higher guidance than predicted by the theory is guidance by alternative mechanisms. Two possibilities are (1) swiveling of an actin filament around rigor heads at the TMCS/SiO2 border and (2) actual detachment events followed by reattachment by part of the actin filament to the TMCS surface. Whereas the possibility of such events has been indicated in our recordings, the limited time and space resolution of the CCD-microscopy system could not, beyond doubt, verify their existence. For comparison, we used the same approximate approach as Clemmens et al.17 to obtain predictions for the guidance probability at different approach angles. Thus, an average length was assumed for the actin filaments. Furthermore, guidance for the average filament was assumed to occur if the time spent by the filament overhanging the SiO2 area exceeded the first passage time for thermal fluctuation to bring the leading edge of the filament back to TMCS. One could argue that this approximate approach might introduce significant errors accounting for parts of the discrepancy between model predictions and experiments. However, considerably more detailed calculations for one experiment (Supporting Information) gave quite similar results to the simplified approach of Clemmens et al.17 In these more detailed calculations, the distribution of actin filament lengths was taken into account, and the probability of guidance was integrated from the time the filament crossed the border until the time that it had been sliding its entire length to detach from the surface. Definition of Nanosized Tracks by Chemical Differences or Topographical Barriers Alone Are Not Sufficient to Realize Efficient Guidance of Actin Filaments. Highly efficient guidance for low approach angles toward a TMCS/SiO2 border seems to predict efficient guidance of actin filament sliding along
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nanometer-sized TMCS tracks that are surrounded by SiO2. Although reasonable guidance along such tracks was observed, the guidance was not effective for keeping actin filaments on the tracks for long distances. This is most likely accounted for by two factors. First, the narrow tracks and the short persistence length of the actin filaments (or of the actin filament paths) lead to frequent border collisions25 for each event with a finite probability of escape. Second, and possibly even more important, even at track widths down to 200 nm the approach angle of the leading end of actin filaments may be close to 90° because of the short persistence length of actin and the bending action of the myosin heads (see below and Supporting Information). Thus, even though the high flexibility of actin filaments is a prerequisite for efficient guidance, at large approach angles it may actually contribute to the eventual detachment of filaments from chemically defined nanosized TMCS tracks. It has previously been shown that tracks surrounded by topographical barriers in the form of walls that are a few hundred nanometers high do not provide full guidance for actin filaments in the absence of complete chemical selectivity.4 To improve the guidance, we have recently15 designed topographical barriers that were created in the form of a bilayer resist system with the top layer (PMMA) overhanging the tracks, about 50 nm on either side (Figure 2C). However, even if the overhang improves the guidance, the present results show that the barrier is not sufficient to achieve complete guidance in the absence of chemical selectivity between the tracks and surroundings. Thus, when sliding was also possible on the surrounding PMMA by the inclusion of methylcellulose in the assay solution, filaments could escape from the tracks by sliding up the PMMA walls. This lack of guidance was unlike the situation for microtubules sliding in microsized channels with an overhang but without chemical selectivity.19 The lack of guidance in our experiments is most likely attributable to the lower flexural rigidity of the actin filaments. Guidance and Velocity of HMM-Induced Actin Filament Sliding on Dually Defined Nanosized Tracks. We have, in three previous studies,4,8,16 tested a combination of chemically and topographically defined nanosized tracks to achieve guidance. In this earlier work, we used a bilayer resist system composed of MRL-6000 (for protein tracks) and PMMA for the suppression of myosin function outside the tracks. Whereas this system was relatively successful in providing guidance of actin filaments, complete guidance was not achieved, presumably because some motility was supported on the PMMA surface with the assay solution that was used. Unlike the situation for TMCS surfaces, the sliding speed was lower on MRL-6000 than on nitrocellulose. In our more recent work,15 we have therefore replaced MRL6000 with vapor-deposited TMCS for better actomyosin function whereas the areas surrounding the HMM-binding tracks were covered with oxygen-plasma-treated (hydrophilic) PMMA to suppress motility.24 Furthermore, we used an assay solution without methylcellulose to enhance the selectivity between TMCS and PMMA.24 As demonstrated here, this approach greatly improves the guiding of filaments (to near 100%). This can be easily understood because in order for a filament to escape from tracks such as those in Figure 2C it has to bend around three approximately 90° corners. On a track of 100-300 nm width, this would require the build up of bending energy in the actin filaments that is not readily achievable without the action of active motors (cf. ref 17). This is most likely not available in the absence of methylcellulose in the assay solution because there (25) Clemmens, J.; Hess, H.; Howard, J.; Vogel, V. Langmuir 2003, 19, 17381744.
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was no actin propelling function of the HMM molecules bound to PMMA (the channel roof) under these conditions. As the track width approaches the size of individual HMM molecules (up to ∼90 nm26), one may envisage that the probability of binding in certain orientations will be affected and as a result the velocity on the tracks might differ from that on larger areas where there is no preferential orientation for HMM binding.27-29 However, in conflict with this idea there was no such difference in sliding velocity between different track widths and between tracks and the TMCS loading zones. Furthermore, there was no difference in sliding velocity for different directions along a given track (Figure 7). Conditions for Unidirectional Motor Protein-Induced Filament Sliding on Nanosized Tracks and Breaking of Filaments during U-Turns. The path for a myosin-propelled actin filament that is most likely to lead to a U-turn on a track of width w would be one where the filament first moves along one wall in the channel (cf. Figure 7). The limiting conditions for a U-turn would then be that the leading edge of the filament traces out a path corresponding to a quarter of a circle (with radius w), thereby approaching the other wall at an infinitesmally greater angle than 90°. The smallest value of w (wmin) that allows this to happen can approximately be derived by equating the elastic energy in the actin filament when tracing out the circular path with the binding energy due to the myosin heads under these conditions. Such a treatment (Supporting Information) suggests the following expression for wmin:
wmin ≈
{
kTLp 2dFf GAM
}
1/2
(1)
Here, kT is the Boltzmann factor, and Lp represents the persistence length of the actin filament (15-19 µm; refs 30 and 31). The quantity d is the width of a band surrounding the filament where interactions with adsorbed motor proteins is possible (26-70 nm; refs 32 and 33), F is the motor protein surface density (400013 000 myosin heads corresponding to 2000-6500 HMM molecules µm-2; ref 24), and f is the duty ratio (0.02-0.06; refs 32 and 33). The quantity GAM (10kT-20kT; ref 34) represents the free energy of binding between one myosin head and actin. The highest value used in the calculations for the HMM density was approximately equal to that obtained for the HMM incubation concentration of 200 µg mL-1 on nonpatterned surfaces.24 The difficulty that the HMM molecules encountered when diffusing into the bilayer resist channels could mean that the HMM density was lower in these channels despite a high incubation concentration and long incubation times. Another possibility for a lower HMM density would also be the detachment of HMM from the surfaces during prolonged observation times (see discussion below). However, the HMM density was unlikely to be lower than about 2000 HMM molecules µm-2. Thus, in experiments using nonpatterned TMCS surfaces and a similar assay solution, we found that the actin filaments detached rapidly from the surface (26) Cardinaud, R.; Bernengo, J. C. Biophys. J. 1985, 48, 751-763. (27) Yamada, A.; Yoshio, M.; Nakayama, H. FEBS Lett. 1997, 409, 380-384. (28) Harada, Y.; Noguchi, A.; Kishino, A.; Yanagida, T. Nature 1987, 326, 805-808. (29) Scholz, T.; Brenner, B. J. Muscle Res. Cell Motil. 2003, 24, 77-86. (30) Gittes, F.; Mickey, B.; Nettleton, J.; Howard, J. J. Cell Biol. 1993, 120, 923-934. (31) Yanagida, T.; Nakase, M.; Nishiyama, K.; Oosawa, F. Nature 1984, 307, 58-60. (32) Uyeda, T. Q.; Kron, S. J.; Spudich, J. A. J. Mol. Biol. 1990, 214, 699710. (33) Harris, D. E.; Warshaw, D. M. J. Biol. Chem. 1993, 268, 14764-14768. (34) Karatzaferi, C.; Chinn, M. K.; Cooke, R. Biophys. J. 2004, 87, 25322544.
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at these HMM densities (HMM incubation concentrations e30 µg mL-1, ref 24). Such rapid detachment was not observed from the nanosized tracks/channels (incubated with HMM at 200 µg mL-1) that were used for studies of unidirectional actin filament sliding. The value of wmin may be regarded as the lowest channel width where there is any possibility at all that a filament may follow a path that allows U-turns to occur. For even a small reduction in width below wmin, there is a very rapid increase in the difference between the elastic energy in the actin filament and the available energy provided by binding to myosin heads (see Supporting Information). For this reason, the value of wmin is sharply defined by eq 1. Inserting the extreme estimates for the parameter values given above suggest that wmin should be between 80 and 960 nm. Experimentally, we observed U-turns for track widths of 300-700 nm with a tendency toward an increased probability with increased widths. No U-turns were observed for track widths e300 nm. This is in good agreement with previous results showing full rectification of filament sliding on electron-beam-lithography-produced tracks of 150 nm width.4 Collectively, the above data suggest that wmin is less than 300 nm. Interestingly, eq 1 yields wmin ≈ 184 nm if the numerical values of all parameters in eq 1 are chosen to be in the middle of the ranges given above (Lp ) 17 µm, d ) 48 nm, F ) 8 500 µm-2, f ) 0.04, and GAM ) 15kT). This result is consistent with the view presented in eq 1 that the numerical value of wmin reports specific functional information about actomyosin (see ref 35 for a related situation). The results in this section also suggest that a track width less than ∼200 nm ensures unidirectional sliding of actin filaments. However, for kinetic reasons (see Supporting Information) it is very unlikely, in reality, to observe U-turns at track widths lower than 300 nm. For track widths close to the limit for observed U-turns (300-400 nm; radius of curvature of the filament during a U-turn is 150-200 nm), and occasionally for higher track widths (Figure 7), there was often breakage of the filaments if a U-turn actually occurred. Actin filaments have been shown to break when knots of radius
