Axial and Radial Forces of Cross-Bridges Depend on Lattice Spacing

Citation: Williams CD, Regnier M, Daniel TL ( Axial and Radial Forces of Cross-Bridges Depend on Lattice Spacing C. David Williams 0 Michael Regnier 0 Thomas L. Daniel 0 Andrew D. McCulloch, University of California San Diego, United States of America 0 1 Department of Physiology and Biophysics, University of Washington, Seattle, Washington, United States of America, 2 Department of Bioengineering, University of Washington, Seattle, Washington, United States of America, 3 Department of Biology, University of Washington , Seattle, Washington , United States of America Nearly all mechanochemical models of the cross-bridge treat myosin as a simple linear spring arranged parallel to the contractile filaments. These single-spring models cannot account for the radial force that muscle generates (orthogonal to the long axis of the myofilaments) or the effects of changes in filament lattice spacing. We describe a more complex myosin cross-bridge model that uses multiple springs to replicate myosin's force-generating power stroke and account for the effects of lattice spacing and radial force. The four springs which comprise this model (the 4sXB) correspond to the mechanically relevant portions of myosin's structure. As occurs in vivo, the 4sXB's state-transition kinetics and forceproduction dynamics vary with lattice spacing. Additionally, we describe a simpler two-spring cross-bridge (2sXB) model which produces results similar to those of the 4sXB model. Unlike the 4sXB model, the 2sXB model requires no iterative techniques, making it more computationally efficient. The rate at which both multi-spring cross-bridges bind and generate force decreases as lattice spacing grows. The axial force generated by each cross-bridge as it undergoes a power stroke increases as lattice spacing grows. The radial force that a cross-bridge produces as it undergoes a power stroke varies from expansive to compressive as lattice spacing increases. Importantly, these results mirror those for intact, contracting muscle force production. - Funding: This work was supported by funds from the Joan and Richard Komen Endowed Chair to Thomas Daniel, a Heart Lung and Blood Institute (NHLBI) Project Grant (R01 HL65497-05-09) to Michael Regnier, and an NIH pre-doctoral training grant (T32 EB001650) to David Williams. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist. Radial forces are the same order of magnitude as axial forces in contracting muscles [13]. These forces, along with axial force acting in the direction of muscle contraction, depend on myofilament lattice spacing [4,5]. At the same time, structural information about myosin cross-bridges suggests that they generate force by applying torque to a lever arm [68]. This lever arm generates the strain accompanying the power stroke via a change in the rest angle at which the lever is attached to S1 region [8,9]. This change in angle occurs at the converter region, a flexible area in myosin S1 which acts as a torsional spring. These phenomena may be related: the radial forces a cross-bridge creates are results of the lever arm geometry (as suggested by Schoenberg [10]). Existing theoretical and computational models of cross-bridge force generation at the level of the half-sarcomere assume that force is generated by a simple extensional linear spring oriented parallel to the long axis of the myofilaments (Figure 1A). This assumption has persisted from the earliest fundamental models of muscle contraction to more elaborate and spatially explicit models [1115]. These single-spring models yielded insight into the processes that regulate production of force in the direction of contraction, parallel to the long axis of the myofilaments. However, these prior models of muscle contraction have paid less attention to radial forces and the effects of changes in filament lattice spacing. As a result, geometries of the single spring crossbridge models have changed little while kinetic schemes governing transitions between conformational states have increased in complexity [11,12,16,17]. To analyze the radial forces that occur during muscle contraction, a different cross-bridge geometry is needed: a geometry that produces both forces aligned with and forces orthogonal to the long axis of the myofilaments. A lever arm of several springs can: (1) simulate the deformations a cross-bridge undergoes as it generates force through the power stroke, (2) provide a geometry which is practical for use in cross-bridge models, and (3) account for both axial and radial forces [9]. Here we detail two models of cross-bridges that use multiple springs to replicate the lever arm mechanism and capture its biologically relevant effects (Figure 1BC). Both models are affected by changes in lattice spacing as well as axial offset from binding sites along the thin filament, and both account for the radial component of force produced during the power stroke. The first model (referred to as the 4sXB model) simulates the crossbridge as a system of four linearly elastic springs arranged in a geometry based upon the structure of the S1 and S2 regions of myosin II (Figure 1C). Our second model (referred to as the 2sXB model) consists of two linearly elastic springs and provides greater computational efficiency than the 4sXB model while replicating many of the more complex models behaviors (Figure 1B). A prior two spring cross-bridge model was proposed by Schoenberg (1980), with the S2 arm represented as an extensional spring and the S2-S1 junction as a torsional spring [10,18]. Both the 4sXB The molecular motor myosin drives the contraction of muscle, but doesnt just produce force in the axis of shortening. Models of muscle contraction have primarily treated myosin as a simple spring oriented parallel to its direction of movement. This assumption does not allow prediction of the relationship between the forces produced and the spacing between contractile filaments or of radial forces, perpendicular to the axis of shortening, all of which are observed during muscle contraction. We develop an alternative model, still computationally efficient enough to be used in simulations of the sarcomere, that incorporates both extensional and torsional (angle dependent, like those found in a watch) springs. Our model captures much of the spacing-dependent kinetics and forces that are missing from single-spring models of the cross-bridge. model and the 2sXB model use a three-state model of cross-bridge cycling kinetics, consisting of an unbound state, a low-force prepower stroke state, and a force-producing post-power stroke state. The kinetics of transition from one state to another in our models are similar to those used previously but are generalized for use in two dimensions; our kinetics calculate transition probabilities using the free ener (...truncated)