Impacts of structural properties of myosin II filaments on force generation

Figure 1.. Modeling setups. (A) In the model, F-actin (blue), actin cross-linking protein (ACP, purple), and motor (red) are simplified by cylindrical segments. F-actin has polarity defined by barbed and pointed ends. Motors consist of a backbone with motor arms that can bind to and walk along F-actin. ACPs comprise two segments connected at the center point. κs and κb represent extensional and bending stiffnesses, respectively. (B) Two-filament system consisting of two F-actins whose barbed ends are clamped to rigid boundaries (gray). (C) The disorganized bundle system with 2NF2 F-actins randomly located and oriented in the presence of the periodic boundary condition (PBC) in the z direction, where NF is a parameter defining bundle thickness. (D) The two-dimensional network system consisting of F-actins with random positions and orientations with the PBC in x and y directions. (E) A variation in the motor structure in three different ways: (i) increasing the number of motor arms (Na↑), (ii) increasing the bare zone length (Lbz↑), and (iii) increasing a spacing between motor arms (Lsp↑).

Figure 1—figure supplement 1.. Measurement of the bundle force. For each cross-section located every 200 nm across the domain, segments crossing the cross-section are identified. In this example, four actin segments, two ACP segments, and one motor backbone segment cross the cross-section. Then, the z component of spring forces acting on those identified segments is summed as Fi. The average of Fi over all 100 cross-sections at a steady state is considered the total force acting on the bundle, Ftot.

Figure 2.. Interactions between motors and actin cross-linking proteins (ACPs) regulate force generation. (A) Time evolution of the force generated by two motors in the two-filament system. The upper and lower dashed lines indicate ideal upper and lower limits of a force that two motors can generate, respectively. Different colors represent distinct cases repeated 20 times. (B) Without any ACP between two motors, they can generate a force close to the upper limit which is twofold larger than a force that one motor can generate (=FMmax)\begin{document}$(=F_{\text{M}}^{\,max})$\end{document}. (C) With ACP(s) between two motors, ACPs counterbalance a force generated by one of the motors. Thus, they can generate a force close to the lower limit which is equal to the force that one motor can generate (=FMmax/2)\begin{document}$(=F_{\text{M}}^{\,max}/2)$\end{document}. Part of F-actins is buckled due to two forces with opposite directions. (D) If two motors are close to each other, ACP can counterbalance a fraction of the force generated by one motor. Then, two motors can generate a force between the upper and lower limits. (E, F) Initial and final configurations (E) without or (F) with ACPs…

Figure 3.. With fixed motor density, thicker bundles generate larger force in a less efficient manner. (A) An example of disorganized bundles with NF = 7 visualized at the beginning of the simulation, where NF is a parameter defining bundle thickness. F-actins are visualized as transparent elements to show the positions of motors. (B) Bundle-level force and (C) the efficiency of force generation measured at a steady state with different NF. In thicker bundles (NF > 2), larger forces were generated, but the efficiency was lower than that in the thinnest bundle (NF = 2). Data represent the mean ± standard deviation (SD) calculated from n = 5 independent simulation runs. Statistical significance was assessed using paired t-test. Significance levels are denoted as follows: *** p ≤ 0.001, and n.s., not significant (p>0.05).

Figure 4.. An increase in the number of motors (NM) in disorganized bundles results in larger forces but smaller force generation efficiency. The thickest bundle (NF = 7) was used for all cases. (A) Bundle-level force (blue circles) and the efficiency of force generation (red triangles) with a wide range of NM between 1 and 1045. With more motors, a larger force (Ftot) was generated, but the efficiency (η) was lower. (B) Configuration of motors with different NM. Ξ represents the estimated number of motors in the strongest contractile unit. With small NM, Ftot and Ξ are unlikely to increase significantly until an entire bundle is occupied by motors, so η is roughly 1 /NM. By contrast, with high NM, an increase NM directly enhances Ftot and Ξ, and η almost remains constant. (C) Prediction of a force using the positions of motors. To find the estimated force (Fest), Ξ is calculated first, and Equation 7 is used. Data represent the mean ± standard deviation (SD) calculated from n = 5 independent simulation runs.

Figure 4—figure supplement 1.. Examples of calculating the maximal number of cooperatively overlapping motors. ξ indicates the extent of an overlap between two motors, and it is defined on the left (L) and right (R) sides of each motor. If two motors have a fully cooperative overlap, ξ is 1 on both sides of the two motors. If there is no overlap at all, ξ is 0 on both sides of the two motors. If two motors have a partially cooperative overlap, ξ is greater than 0 only on one side of the two motors, Lov / Lc, where Lov is an overlap distance between two motors, and Lc is the critical length required for the fully cooperative overlap. Ξ is the maximal number of cooperatively overlapping motors obtained by comparing all the combinations of motors in the system. Four examples are provided here for better understanding of this analysis method.

Figure 5.. Motor distribution affects the force generation in disorganized bundles. We varied the relative size of a region where motors were initially located, f. f=1 means that motors can be located at any part of the bundle. (A) Bundle-level force (Ftot) and efficiency (η) depending on f. As motors were localized more closely to the center (i.e., smaller f), the force and the efficiency were higher. (B) Configuration of motors with different f. Ξ indicates the estimated number of motors in the strongest contractile unit. Given the number of motors, smaller f results in more cooperative overlaps (i.e., higher Ξ) and thus leads to higher Ftot and η. (C) Prediction of the bundle-level force with different f. Data represent the mean ± standard deviation (SD) calulated from n = 5 independent simulation runs.

Figure 6.. The architecture of motors impacts the force generation process in disorganized bundles. (A) Bundle-level force (Ftot) depending on LM. The motor length (LM) is varied between 42 nm and 966 nm by changing either of the number of motor arms (Na, red circles) between 4 and 48, the bare zone length (Lbz, blue triangles) between 42 nm and 714 nm, or the spacing of motor arms (Lsp, green squares) between 42 nm and 138 nm. (B) Prediction of the bundle-level force using the positions of motors. (C) As each motor has more arms (Na↑), Ftot and the efficiency of force generation (η) become higher because forces generated by motor arms are counterbalanced to a lesser extent. (D) Possible overlaps between two motors with different structures. A dark gray color indicates a fully cooperative overlap, and light gray indicates a partially cooperative overlap. To have the fully cooperative overlap, motors with many arms need to be located very closely, whereas motors with the long bare zone can overlap in the fully cooperative manner with a relatively long distance between them. Data represent the mean ± standard deviation (SD) calculated from n = 5 independent simulation runs.

Figure 6—figure supplement 1.. Further analyses for cases with different motor length (LM) varied by three methods. (A) The efficiency of force generation. (B) The maximum number of cooperatively overlapping motors. Data represent the mean ± standard deviation (SD) calculated from n = 5 independent simulation runs.

Figure 7.. Force generation in two-dimensional actomyosin networks is governed by similar mechanisms. (A) Examples of networks at the initial state under the reference condition, with a smaller or larger number of motors, and with longer motors. These snapshots show only a quarter of networks to better visualize individual motors. (B) Network-level tension (Ftot) and the efficiency of force generation (η) with a different number of motors between 10 and 2006. (C) Ftot depending on motor length varied by changing either of the number of motor arms (Na, red circles) between 4 and 48, the bare zone length (Lbz, blue triangles) between 42 nm and 714 nm, or the spacing of motor arms (Lsp, green squares) between 42 nm and 138 nm. Data represent the mean ± standard deviation (SD) calculated from n = 5 independent simulation runs.

Figure 7—figure supplement 1.. The efficiency of force generation in two-dimensional actomyosin networks with different motor length varied by three methods. Data represent the mean ± standard deviation (SD) calculated from n = 5 independent simulation runs.

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