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The correlation of biochemistry and mechanical data on the cross bridge cycle.

Our approach provides structurally-based, rather than empirical, theoretical constraints for strain-dependent actomyosin interaction kinetics. In striated muscle, including cardiac muscle, myosin and actin filaments are packed into a regular three-dimensional lattice. Each myosin filament is decorated with crowns of myosin dimers, spaced by ~14.3 nm along the filament. Crown orientations are repeated every 42.9 nm, i.e. every third crown has the same orientation as the initial. The actin filaments form a double-helix, associated with its regulatory proteins tropomyosin and troponin with binding sites by ~5.5 nm on each strand, with a half-period of ~35.75 nm in the relaxed state. The difference in periodicities between actin binding sites (35.75 nm) and myosin crowns (42.9 nm) creates a vernier of longitudinal head-site spacings which control strain dependent binding of myosin to actin filament. The 3D geometry of myosin head binding domains and actin sites in a sarcomere requires both longitudinal position matching and angular matching in the azimuthal plane. A myosin head and closest actin site form the most probable pair of these molecules which can create a crossbridge interconnecting actin and myosin filaments. Thus, structural changes along the reaction pathway must be responsible for the spatial strain dependence of the force-generating and product-release steps in actomyosin in the 3D sarcomere lattice. Our model therefore, incorporates of azimuthal weight factors and precise (nonlinear) elasticity of crossbridge in order to construct realistic energy landscapes.

The three-dimensional sarcomere lattice is composed of thick and thin filaments arranged in an overlapping hexagonal lattice with a geometry that is consistent with the average spacing measured in vertebrate striated muscle. (A) Myosin filaments extend from the central M-band towards Z-lines and actin filaments extending from Z-lines forming a hexagonal arrangement in cross-section; (B) The actin monomers are helically arranged in a double strand helical structure and orientationally favorable myosin binding sites on actin filament or target zones associated with myosin heads are show in red. Each myosin molecule is attached to the trunk of myosin filament via the S2 rod and has two heads (S1 fragments) at the free end, but only one head per dimer is shown. The pairs of myosin heads form a triple helix along myosin filament. The myosin heads are arranged in layers and at each layer forms a “crown” with three pairs of heads. The crowns       =1, 2, 3 are axially separated by 14.3 nm and rotated by 40° forming different angular arrangements with actin filaments but only those which might interact with the actin filament are shown. In the axial direction, each pair of heads and multiple binding sites (target zones) on surrounding actin filaments form a large number of arrangements defined by the relative axial distances,     , between the unstrained position of the myosin head or crossbridge and the nearest actin binding site, and azimuthal angles       and       as defined in Fig. 4. (C) The hexagonal sarcomere lattice with 2:1 actin to myosin filament ratio shows in the azimuthal plane that up to three myosins can attach to each actin filament. The spatial arrangement of crowns       = 1 interacting with six surrounding actin filaments is shown. (D) The heads in crowns         = 2 and 3 have different azimuthal spatial arrangement relative to binding sites on the actin filaments displayed by azimuthal angles         and        .

Azimuthal binding weight factors and probabilities. Our model defines myosin-actin in the context of target zones, which are prescribed by the 3D spatial arrangement of actin filaments in the sarcomeric hexagonal lattice. The density of myosin head binding to actin is determined by this distributed affinity through the simultaneous interactions involving myosin and actin in the 3D sarcomere. We have developed two methods of defining probabilities; (i) using a mapping method (described in Smith et al., 2008) which is used in a semi-probabilistic approach; and (ii) exact 3D lattice where we have derived two weighting factors taking into account the azimuthal departure of actin binding sites from perfect azimuthal alignment with myosin head positions. These factors are derived from the distributed affinity of myosin for actin as a function of axial position along the thin filament (Fig. 4). By analogy with these findings, we adapted the same distribution function in order to define Cb as a function of axial departure from perfect matching, and the corresponding angle of active actin site relative to alignment plane between myosin and actin filament. In similar fashion we have developed a metric, Ca, which takes into account azimuthal misalignment between myosin crowns and actin filaments. The maximum value of these weight functions is set by a normalization factor so the overall probability of binding remains equivalent to in the one dimensional approach. We have incorporated these functions into a model of stochastic myosin binding with extensible filaments taking in account the instantaneous geometry of a 3D sarcomere (MUSICO).

Azimuthal weight factors       and       myosin binding in 3D sarcomere lattice. When myosin heads in Crown I are directly aligned with three actin filaments than       = 1 and         weights azimuthal departure of an actin site from the plane passing through myosin and actin longitudinal axes. The angle       is function of axial departure from perfect matching,                     , as shown in A, resembles preference for myosin heads to bind to “target zones” of  other favorably oriented sites on the helices of the actin filament similar to measured in single molecule studies (B).  When myosin heads are not directly aligned to the surrounding actin filaments, such as in Crowns 2 and 3, the weight factor       takes in account departure for angle       as shown in C.

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