Figure 1.
Cross-bridge types and kinetic scheme.
(A)–(C) The three cross-bridge models, plotted against a myosin crystal structure for comparison (structure image generated from Gourinath et al. (2003) [40] with PyMol [41]). The energy landscape of each cross-bridge and the free energy at rest lattice spacing are shown adjacent to the cross-bridge schematic. (A) The 1sXB introduced in Huxley (1957) [11]. (B) The 2sXB which uses a torsional/angular spring () and an extensional spring (
). (C) The 4sXB with two torsional and two extensional springs. Of the 4sXB's springs,
corresponds to the point at which the S2 region rejoins the thick filament backbone,
to the S2 region itself,
to the area linking the S2 and the light chain domains, and
to the light chain domain itself.
replicates the change in angle accompanying the power stroke by applying torque to the freely moving joint representing the converter domain. (D) The three state kinetic system. The three states represent (1) an unbound state, (2) a pre-power stroke state, and (3) a post-power stroke state. The rate of transition between states
and
is represented as
. The forward and reverse transition rate constants are functions of energy stored in the cross-bridge.
Figure 2.
Forces, energy, and kinetics of the 1sXB, 2sXB, and 4sXB models at resting lattice spacing.
(A)–(F) show the energy, transition rate constants, and forces of the 1sXB model (black), 2sXB model (green), and 4sXB model (red) at resting lattice spacing. The 1sXB model values shown for comparison are derived from those of Daniel et al. (1998) and Tanner et al. (2007), [12], [14], shifted axially so the resting location of the cross-bridge head in each case is aligned with the resting locations of the 2sXB model and 4sXB model allowing easier comparison. The free energy of the cross-bridges in state two is shown in (A), where the multi-spring cross-bridges' shifts from a purely parabolic trajectory is visible. The explicit two-dimensional thermal forcing of the multi-spring cross-bridge heads in (B) results in binding probabilities that are more distributed than those of the single spring cross-bridge. The rate of power strokes (C) remains least changed between the single and the multi-spring cross-bridge models. The energy-based kinetics of the multi-spring cross-bridges are unable to fully replicate the biased detachment rate of the 1sXB model in (D). (E) and (F) show the 1sXB's sharp discontinuities in axial force and lack of any radial force.
Figure 3.
Energy and kinetics of the multi-spring cross-bridge models change with axial offset and lattice spacing.
Axial offset is the distance between the current axial location of the cross-bridge tip and the location where the cross-bridge attaches to the thick filament. Lattice spacing () is defined as in Millman (1998) [3], with an offset to account for filament thicknesses so the cross-bridge spans the filaments at a rest lattice spacing of 34 nm. (A)–(H) The properties of the 4sXB model (A, C, E, and G) and the 2sXB model (B, D, F, and H) as they change with binding site offset and lattice spacing. (A) depicts the free energy of the 4sXB model at various lattice spacings, with the head stretched to an axial offset from the thick filament attachment point. The free energy of the 2sXB model is shown in (B). (C) and (D) show
, the probability that the 4sXB and 2sXB models will transition from an unbound state to a bound state. (E) and (F) show
, the probability of transition from a pre-power stroke state to a post-power stroke state, for the same cross-bridges, axes, and scales as (C) and (D) show
. (G) and (H) show
, the probability of unbinding from a post-power stroke state. The reverse rate constants,
,
, and
are back-calculated from the forward rate constants.
Figure 4.
Overview and detail of the forces exerted by the 2sXB and 4sXB models in the post-power stroke state.
(A)–(D) show the post-power stroke forces exerted by the 4sXB and the 2sXB models as vector fields of reaction forces. The reaction force is that necessary to retain the cross-bridge head in a given location, thus the vectors for a compressed cross-bridge orient upwards and those for an extended cross-bridge orient downwards. Positions in which the cross-bridge is unlikely to generate force are omitted; these unlikely locations are determined by the sum of and the inverse of
. (A) and (B) show overviews of the forces exerted, respectively, by the 4sXB model and the 2sXB model over lattice spacings and axial offsets that vary as in Figure 2. The forces exerted by the two cross-bridges have radial components which frequently equal or exceed their axial components. A more detailed view of the region surrounding the rest position of the cross-bridges is shown in (C) and (D), where the large radial components of the cross-bridge forces, particularly for the 2sXB model, is especially evident.
Figure 5.
Axial and radial post-power stroke forces as separate components.
(A)–(D) show, separated, the axial and radial components of the forces produced by the 4sXB and the 2sXB models in the post-power stroke state.
Figure 6.
Changes in step size with lattice spacing.
Step size varies as lattice spacing diverges from its rest value. Step size is defined as the change in the rest axial offset between the pre- and post-power stroke states. The step size of the 4sXB model and 2sXB model produce different absolute step sizes as lattice spacing change. However, both models exhibit a local maximum step size at a specific lattice spacing with a decreasing step size as lattice spacing diverges from that point.
Table 1.
Model parameters and their sources.
Figure 7.
Changes in cross-bridge resting geometry with the power stroke.
(A)–(C) show, schematically, the change in the rest lengths and angles of the single and multi-spring cross-bridges. The rest length and angle of the 2sXB's extensional and torsional springs are set, in both the pre- and post-power stroke states so as to match the tip position of the 2sXB in each condition to that of the 4sXB (in Table 1). The change in the unstressed radial distance from the thick filament to the tip of the multi-spring cross-bridges that occurs with the power stroke is particularly visible in (B) and (C) when compared to the single spring cross-bridge (A). The effects of the universal joint attaching the springs of the 4sXB and the 2sXB to the globular domain, and the globular domain's own fixed angle to the thin filament, are shown by the continued radial orientation of the globular domain after the power stroke occurs.