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Nonlinear Elasticity and an 8-nm Working Stroke of Single Myosin Molecules in Myofilaments

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Science  06 Aug 2010:
Vol. 329, Issue 5992, pp. 686-689
DOI: 10.1126/science.1191484

Abstract

Using optical trapping and fluorescence imaging techniques, we measured the step size and stiffness of single skeletal myosins interacting with actin filaments and arranged on myosin-rod cofilaments that approximate myosin mechanics during muscle contraction. Stiffness is dramatically lower for negatively compared to positively strained myosins, consistent with buckling of myosin’s subfragment 2 rod domain. Low stiffness minimizes drag of negatively strained myosins during contraction at loaded conditions. Myosin's elastic portion is stretched during active force generation, reducing apparent step size with increasing load, even though the working stroke is approximately constant at about 8 nanometers. Taking account of the nonlinear nature of myosin elasticity is essential to relate myosin’s internal structural changes to physiological force generation and filament sliding.

Molecular motors, such as muscle myosins, axonemal dyneins, and mitotic spindle kinesins, function as assemblies of motors (1). To achieve their motile activities efficiently, it is crucial for these motors to generate force collectively and to minimize the interference between motors (2, 3). In Huxley’s model of muscle contraction (4), it is postulated that an attached myosin generates the sliding movement of an actin filament and is subsequently pulled into a drag region, resulting in drag (negative) forces. Because the contribution of individual myosin giving negative force cannot be separated from the integrated forces of all attached myosins in muscle fiber measurements (46), the molecular mechanism of how dragged myosins affect the dynamics of muscle contraction remains unclear. To elucidate this mechanism, we focus on the changes in elasticity of single myosins for both the positive and negative strain directions, because it directly affects both the active and drag force generation when myosins are stretched or shortened (2, 4).

Streptavidin-coated quantum dots (QDs) were attached to biotinylated actin filaments to minimize the uncertainty in the bead-actin linkage stiffness (7, 8). The stiffness of actin filaments between myosin binding and QDs positions was ~20 pN/nm [supporting online material (SOM) text 1] (9), which was much greater than observed for single myosins. An actin filament bound to QDs was suspended between two streptavidin-coated beads held in two optical traps (Fig. 1A). For no nucleotide or 1 mM adenosine diphosphate (ADP) conditions, single myosins embedded in myosin-rod cofilaments were tightly bound to the actin filaments. The positions of the two trapped beads were then triangularly moved along the longitudinal axis of the cofilament (Fig. 1B). The total force exerted by a single myosin was then calculated as the sum of the forces exerted on the two trapped beads (fig. S5). By taking the ensemble average on 50 tracking traces, the precision was greatly improved, up to 0.3 nm for QDs bound to actin filaments at 2-ms exposure during the stiffness measurements (fig. S6).

Fig. 1

Measurement of single-myosin stiffness. (A) Schematic diagram of the optical trap system for a single-myosin stiffness measurement. (B) Time course of displacement of beads and a quantum dot on an actin filament bound to a single myosin during oscillation of the optical trap. (C) Force-displacement curve of single myosins in no nucleotide (black) and the presence of 1 mM ADP (red). Data points (black and red circles, mean ± SE) were combined from the force-displacement curves of individual myosins (n = 8 for no nucleotide, and n = 7 for ADP) based on the ensemble-averaged data (fig. S5). Green diamonds show the linear force-displacement relationship for myosin-rod cofilaments (n = 10) (fig. S7). The dotted lines are the fitting curves on the original data for displacements above –10 nm, and the solid lines are the fitting curves adjusted by accounting for the compliance of myosin-rod cofilaments and actin filaments (SOM text 1). The gray dotted and solid lines for no nucleotide are the corresponding fitting curves for displacements below –10 nm. As shown by the black arrows in the illustration of myosins on the top, plots with displacements either positive or below –80 nm characterize the stiffness of myosin S1, whereas those with displacements between 0 and –80 nm characterize the stiffness of myosin S2. The right panel shows the force-displacement curve in a zoomed range of strain.

The force-displacement curve for a single myosin showed the nonlinear elasticity for both no nucleotide and 1 mM ADP conditions (Fig. 1C). Myosin-rod cofilaments attached to the glass surface were slightly displaced relative to the surface during the stretching and shortening of the single myosins (fig. S7). The stiffness of the myosin-rod cofilaments was 9.2 pN/nm (Fig. 1C). The displacements of the myosins were obtained by subtracting the displacements of the myosin-rod cofilaments and the estimated elongation of the actin filament from those of the QDs (SOM text 1). The stiffness of a single myosin was greater (>2.5 pN/nm) when stretched by a force >~2 pN compared with (0.02 to 0.5 pN/nm) when it is compressed (fig. S1). The biphasic force response between the positive and negative strain regions was consistent for both no nucleotide and 1 mM ADP conditions (Fig. 1C). In contrast to the linear elasticity, the nonlinear elasticity allows for large active force when myosins are actively stretched but reduces negative force when myosins are compressed (SOM text 2).

What part of the myosin head is responsible for the biphasic elastic response? The low stiffness region was observed for displacements from 0 to –80 nm. Eighty nanometers is approximately double the length of a myosin subfragment 2 (S2) (10), implying that the low stiffness characterizes the elastic response of the myosin S2. The stiffness value of 0.02 pN/nm (fig. S1) is consistent with the theoretically calculated bending stiffness of myosin S2 (~0.01 pN/nm) (11). The large stiffness observed in the positive displacement region and for displacements of lower than –80 nm characterizes the elasticity of myosin S1 when the myosin S2 portion is fully stretched in either direction. The maximum stiffness of 2.6 (ADP) to 2.9 pN/nm (fig. S1)(no nucleotide) is consistent with both the stretching and bending stiffness of myosin S1 (2 pN/nm) (8, 12) but not with the bending (~0.01 pN/nm) or stretching (60 to 80 pN/nm) stiffness of myosin S2 (11).

The stiffness of 2.6 to 2.9 pN/nm obtained here is higher than most previously reported values (table S1). We may have obtained higher stiffness values in this study for three reasons: First, the uncertainties of the bead-linkage stiffness and myofilament compliance were reduced by the direct measurement of the actin and the myosin-rod displacements; second, the stiffness of two-headed myosins here might be higher than that of single-headed myosins; third, the cooperative behavior of the two heads may have led to an optimal orientation of one head for stiffness production (13, 14) (SOM text 3).

Next, we investigated how the observed elasticity of single myosins affects the displacement of single myosins during actomyosin interaction. The values of unitary displacement of single myosins have varied from 3 to 17 nm (table S1), probably because of the nonprocessivity of the myosin II motor, different myosin preparations, and the random orientations of myosin heads relative to an actin filament. Here, displacements and forces of single skeletal myosins during processive movement were generated by a few myosin molecules synthesized into myosin-rod cofilaments (fig. S8) so that myosins could interact processively with actin filaments in a more natural environment. Myosin molecules were allowed to interact with biotinylated actin filaments attached to single streptavidin beads (Fig. 2A). The displacements and forces generated by myosins were measured by an optical trap (15) (Fig. 2A). Similar to the stiffness measurements, we assumed cooperativity between the two heads (SOM text 3).

Fig. 2

Processive movements generated by synthesized myosin-rod cofilaments. (A) A schematic diagram of the optical trap system with dark-field illumination. (B) Time course of bead displacements during the unbinding of the acto-myosin interactions in the absence of ATP. Arrowheads show individual unbinding points associated with sharp deviations of bead displacement. (C) Estimated numbers of interacting myosin molecules as a function of the half-length of myosin-rod cofilaments. The lengths were estimated from fluorescent images (fig. S8). The linear regression indicates ~5 interacting molecules per half-length of 450 nm (r2 = 0.63), as illustrated by the myosin heads in red on the top. Myosins in pink illustrate existing myosins that cannot interact with an actin. (D and E) Time courses of bead displacements generated by short (415 nm, gray) and long (900 nm, blue) myosin-rod cofilaments, respectively, in an ATP concentration of 20 μM.

To estimate the maximum number of myosin molecules interacting with an actin filament, the number of interacting molecules as a function of the half-length of the myosin-rod cofilament was obtained from the measurement of unbinding events of rigor bonds (16) (Fig. 2B). Five myosins per half cofilament length of 450 nm, on average, were able to interact with an actin (Fig. 2C), which corresponds to approximately 1.5 interacting molecules for each 43-nm pitch in a native thick filament, consistent with the structural configuration of native thick and thin filaments (SOM text 4).

The time course of bead displacements for short and long cofilaments showed stepwise movements of actin filaments over 100 to 300 nm, corresponding to 3 to 11 pN of loads in 20 μM ATP (Fig. 2, D and E, and Fig. 3, A to C). A histogram of the step sizes, detected by a step-finding algorithm (17, 18), was well approximated by a Gaussian curve, with a peak at 4 to 7 nm (Fig. 3, D to I), depending on the applied load. For three cofilament lengths, the elementary step size consistently decreased from 7 to 4 nm with increasing load (Fig. 4A). The similarity of the step sizes at low and high loads was confirmed using a pair-wise distance analysis procedure (fig. S9). The forward steps are presumably generated by the attachment of active myosins, because the detachment of drag motors associated with low stiffness should not be evident in the forward-step generation, while the backward steps (e.g., Fig. 3, D to I) are the consequence of the detachment of active motors.

Fig. 3

Stepwise displacements generated by myosin-rod cofilaments. (A to C) Raw bead displacement data (gray line) generated by short, middle, and long myosin-rod cofilaments at low [(A) and (B)] and high (C) loads. The individual steps (black line) were detected by the step-finding algorithm. (D to I) Histograms of step sizes for long (880 ± 18 nm, n = 3), middle (660 nm, n = 1), and short (410 ± 7 nm, n = 2) myosin-rod cofilaments at different loads. The mean step sizes were determined by the values at the central positions of single Gaussian curves.

Fig. 4

(A) Mean step sizes, working stroke sizes, and the amounts of stretch in the elastic portion of single myosins as functions of the loads. The amount of stretch for no nucleotide condition (triangles) is estimated from the nonlinear fitting curve with forces on the force-displacement curve depicted in Fig. 1C. The working stroke sizes (diamonds) were given as sums of the observed step sizes (circles, mean±SE) and the corresponding amounts of stretch in the elastic portion. The horizontal dashed line indicates a mean working stroke size of 7.6 nm, independent of loads. The black linear regression line on the step size represents the trend of the load-dependent step size. (B) Dwell times between steps at no load as a function of the number of interacting heads. The number of interacting myosin heads (N) is estimated from Fig. 2C by assuming the cooperativity between the two heads (SOM text 3). Dwell times for the n = 2.5, 4, and 5 myosins at no load were estimated from the dwell time-force relationships (fig. S10). The dwell time for a single myosin head (n = 1) corresponds to the turnover time in 20 μM ATP, which is calculated as the sum of the measured attachment lifetime (τon = 33 ms) and the estimated detachment lifetime (τoff = 25 ms) (SOM text 5). Dwell times (blue circles) fit well with an inverse function of N (blue line). The rate constants (gray circles), given as inverse dwell times, also fit well with a linear function of N (black line) (SOM text 6).

How many myosins contribute to step generation? Dwell times for adjacent steps and no load for three cofilament lengths were estimated by extrapolating curves fitted to the relationship between dwell times and loads (fig. S10). The turnover time of a single myosin in 20 μM ATP was calculated as the sum of the measured lifetime of attachment (τon = 33 ms) and the estimated lifetime of detachment (τoff = 25 ms) (SOM text 5). These estimated values were combined and plotted against the number of interacting heads (N) (Fig. 4B). The relationships fit well to an inverse function of N, suggesting that individual steps were generated by the turnover of one myosin head, which randomly bound with an actin within the turnover time of single myosin heads (SOM text 6 and fig. S3). Thus, the observed step sizes characterize the mechanical properties of single myosin heads, suggesting that the step sizes of single myosin heads vary from 7 to 4 nm in a load-dependent manner (Fig. 4A). Load-dependent step size has also been shown in experiments using stepwise length releases in single muscle fibers (1922). Thus, the load-dependent changes in the step size are an essential property of skeletal myosin.

The load-dependent step size can be interpreted in the following manner (fig. S11). First, the detached myosin reattaches to an actin filament and then performs the working stroke (dw), which is generated by the conformational changes of the myosin head (23). When no load is applied, the size of the working stroke is directly translated into the sliding movement of the actin filament, that is, the observed step size (ds). However, when a load is applied, the working stroke is limited by the stretch of the elastic portion of myosin head (de). Consequently, the observed step size (ds = dwde) decreases. Thus, the working stroke size, calculated as the sum of the observed step sizes and the corresponding stretch of the elastic portion of the myosin head estimated from the force-displacement curve (Fig. 1C), appears to be load-independent and is approximately 8 nm (Fig. 4A). A working stroke of 8 nm is slightly larger than the typical size of 5 nm for S1 (24, 25) and smaller than the values of 10 to 12 nm estimated in structural studies on single myosins (23), but is more consistent with the 7 to 10 nm observed in studies on two-headed myosins (14) and muscle fibers (21). Thus, the discrepancy of working stroke size between the present study and the S1 measurement might be attributed to the difference in the number of myosin heads interacting with actin (or in structure) (14).

Similar step sizes were observed for myosin-rod cofilaments of three different lengths, which consist of different numbers of interacting molecules (Fig. 4A). This result can be explained by the nonlinear elasticity of single myosins. Because the duty ratio [τon / (τon + τoff)] is 0.57 for no load or higher for loaded conditions (fig. S10), more than half of the interacting myosin molecules are in the strongly bound state. If the elasticity of myosins were linear even in the negative strain region (2, 4), the drag forces would become substantially higher with an increasing population of the strongly bound myosins for longer cofilament (SOM text 2 and fig. S2), resulting in smaller step sizes. In contrast, the nonlinear elasticity obtained here ensures small drag forces even for long cofilaments with more drag motors. Thus, the similar step sizes observed for different cofilament lengths are well explained by the concept of nonlinear elasticity (SOM text 2 and fig. S2).

By measuring the step sizes of single myosins detected from the processive movements of the synthetic myosin-rod cofilaments, we found that step size is load-dependent and working size is load-independent on the single-molecule level. We also found a nonlinear elasticity of skeletal myosins by measuring the elastic property of “full-length” myosins for a wide range of positive and negative strains. The nonlinear elasticity implies that the attachment and working stroke of active myosins with high stiffness is primarily responsible for the forward step generation, whereas the detachment of drag myosins with low force due to low stiffness does not primarily contribute to the forward steps. Based on these findings, we propose a model for the molecular mechanism of muscle contraction (fig. S2 and fig. S11). During force generation, individual myosins repeatedly perform the working distance of 8 nm. After completion of force generation, myosins become much softer as they shorten and possibly buckle, resulting in a reduction of the drag force against the subsequent force generation performed by other neighboring myosins. Such molecular properties may be inherent in the assembly of molecular motors and may reduce molecular interference, leading to the high mechanical efficiency of muscle contraction (26).

Supporting Online Material

www.sciencemag.org/cgi/content/full/329/5992/686/DC1

Materials and Methods

SOM Text

Figs. S1 to S11

Table S1

References

References and Notes

  1. We are grateful to J. Kerssemakers and M. Dogterom for generously providing their step-finding algorithm, W. Herzog and Y. Goldman for valuable discussion and critical reading of this manuscript, and T. Kambara for comments. This work has been supported by Grants-in-Aid for Scientific Research in Priority Areas from the Japan Ministry of Education, Culture, Sports, Science, and Technology (H.H.), Core Research for Evolutional Science and Technology of the Japan Science and Technology Agency (H.H.), and Young Scientists from the Japan Society for the Promotion of Science (M.K.).
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