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 force-production 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.
Radial forces are the same order of magnitude as axial forces in contracting muscles [1]–[3]. 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 [6]–[8]. 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 [11]–[15]. 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 cross-bridge 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 1B–C). 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 cross-bridge 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 model' s 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 model and the 2sXB model use a three-state model of cross-bridge cycling kinetics, consisting of an unbound state, a low-force pre-power 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 energy landscape of the cross-bridges instead of the offset of the cross-bridge head (Figure 1D and Figure S1) [12], [14], [16], [19]. We compare the 4sXB and 2sXB models to a single spring model of the cross-bridge (referred to as the 1sXB model), similar to those used previously. We quantify both the axial and the radial forces of our two cross-bridge models. Additionally, we show how changes in lattice spacing and axial offset affect kinetics and forces in our multiple-spring models.
At rest lattice spacing, the free energies and kinetics of the of the single- and multi-spring cross-bridge models are largely similar, as seen in Figure 2 (where the 1sXB values used are calculated as in Figure 10 of Tanner et al. (2007) [14]). These properties share a common base that is intentionally conserved, where possible, between the multiple-spring and single-spring cross-bridges [16]. The free energies of the multi-spring cross-bridges are a result of both extensional springs that are at an angle to the thick filament and torsional springs sensitive to the angle they make with the thick filament. As the multi-spring cross-bridges move in the axial direction, their angles to the thick filament backbone change. This angle dependence skews the free energies of the multi-spring cross-bridges from the symmetric hyperbola of the 1sXB (Figure 2A). The two-dimensional diffusion-based binding probability function that governs the multi-spring cross-bridges (as described in the binding rate calculation section) causes the likely binding areas to occupy a greater range of axial positions than those of the single-spring cross-bridge (Figure 2B) [20], [21]. Multi-spring cross-bridges are thus less likely than the 1sXB model to bind near their rest position, but are more likely to bind than the 1sXB at greater offsets from their rest position. This flattening and spreading of the binding probability function is a result of the extra degrees of freedom of motion in the two-dimensional models. The power stroke rate constants of the multi-spring cross-bridges are the same as those of the single-spring cross-bridge, with energy-dependent terms using the sum of the free energy of every spring comprising a cross-bridge (Figure 2C). The detachment rate constant of the 1sXB explicitly relies on cross-bridge head position as well as energy. This position dependence was removed in adapting the 1sXB model' s detachment rate constant for the multi-spring cross-bridges. The detachment rate constant thus loses the intentional asymmetry that the position term provided and retains only the asymmetry created by the spring geometries of the 2sXB and 4sXB models (Figure 2D). The rate of detachment and the other cross-bridge kinetic rate constants remain close to those of the 1sXB, even though the kinetics of the multi-spring cross-bridges are based not on axial position but on the free energy of the cross-bridge in multiple dimensions. The axial offset of a cross-bridge property is the axial distance from the point where the cross-bridge attaches to the thick filament to the point where the cross-bridge property reaches an extreme value or inflection point. These axial offsets are depicted in Figure 3 and Figure S2 where, for example, the axial offset of the 2sXB attachment rate constant at 34 nm is approximately 12 nm. As lattice spacing increases, the axial offsets of most multi-spring cross-bridge kinetic rates and free energies grows smaller. This relationship is shown in Figures 3A and B and Figure S2 A and B, where the axial offset of the 4sXB or 2sXB model' s lowest energy point is more than 3 nm greater at a lattice spacing of 32 nm than at a lattice spacing of 38 nm. The positions where cross-bridges are most likely to bind shift to smaller axial offsets at larger lattice spacings, decreasing how extended a cross-bridge is likely to be upon binding (Figures 3C–D and Figure S2C–D). Similarly, as lattice spacing increases, decreases in the axial offset of the power stroke rate constant inflection point cause the size of the power stroke to change with lattice spacing (Figures 3E–F and Figure S2E–F). The 4sXB model' s rate of detachment is the only cross-bridge property whose axial offset is predominately invariant with changes in lattice spacing (Figure 3G and Figure S2G). This exception is explained by the largely radially aligned post-power stroke orientation of, the 4sXB model' s final spring. Combined, these effects reduce the axial force a cross-bridge generates at larger lattice spacings with implications for the sarcomere length dependence of force production and relaxation. These multi-spring cross-bridge models are the first to be capable of reproducing these lattice spacing dependent effects on force production and kinetics. The number of cross-bridges in a force generating state depends on lattice spacing. At any axial location, as lattice spacing diverges from its 34 nm rest value, the rate of attachment decreases while the rate of detachment increases (Figure 3C–D and 3G–H). These kinetic rate constants change with lattice spacing because they depend on the difference in free energy between the unbound state and the pre- or post-power stroke state, a difference which increases with lattice spacing. This increase in energy makes a cross-bridge increasingly likely to transition to the unbound state and remain there (Figure 3C–D and 3G–H). An example of the decrease in the likelihood of a cross-bridge remaining bound can be seen in the 4sXB model, where the slowest rate of detachment is 20/sec at a lattice spacing of 34 nm but rises to 260/sec at 38 nm (Figure 3G). As a result of these changes, individual cross-bridges spend less time in a bound state and are less likely to generate force as lattice spacing diverges from its rest value. The axial and radial forces at a given axial offset correlate with lattice spacing (Figures 4 and 5). When lattice spacing is compressed, more expansive radial forces and smaller axial forces are produced. When lattice spacing is expanded, more compressive radial forces and larger axial forces are produced. An example of increased forces with increased lattice spacing is seen in the 4sXB model which, at a 10 nm axial offset, produces half the radial and half the axial force at 35 nm as it does at 38 nm (Figure 5A–B). Similarly with the 2sXB model at a 12 nm axial offset, a lattice spacing of 35 nm produces two thirds of the axial and radial forces as does a lattice spacing of 38 nm (Figure 5C–D). At large lattice spacings, this greater force per cross-bridge competes with the decreased probability a cross-bridge will bind and generate force, an interaction that requires a model of the half-sarcomere using multi-spring cross-bridges to fully evaluate [22]. The force landscapes of Figure 5 also show that no lattice spacing is free of radial force at all axial offsets. The radial force produced by a cross-bridge, even at rest lattice spacing, increases in magnitude as the cross-bridge tip moves away from its unstrained axial offset. The step size of both multi-spring models varies with lattice spacing (Figure 6). We define step size at a given lattice spacing as the axial distance between the pre- and post-power stroke positions of the myosin head. Put another way, step size at one lattice spacing is the distance from the axial offset with the lowest free energy in the pre-power stroke state, to the axial offset with the least amount of energy in the post-power stroke state. Both models have a peak step size at a relatively uncompressed lattice spacing, with decreasing step size as lattice spacing diverges from that value. The 4sXB model has a maximum step size of 5. 0nm near 34nm lattice spacing and the 2sXB model has a maximum step size of 6. 1nm near 36nm lattice spacing. The radial and axial components of force, produced by a 4sXB model or 2sXB model moved from its rest position to an axial offset, are of the same order of magnitude (Figures 2E–F and 4A–D). The values of the axial and radial forces produced by the multiple-spring cross-bridge models at rest lattice spacing are compared to those produced by the single-spring cross-bridge model in Figure 2E–F. The relative values of the radial and axial forces are visualized as the angles of the force vectors in Figure 4A–D. Axial locations and lattice spacings with balanced axial and radial forces produce force vectors which are neither vertical nor horizontal, but in some intermediate orientation. Most axial and radial offsets are populated by such vectors, particularly regions a cross-bridge would be most likely to occupy (unlikely regions are not shown in the vector plots). The few regions dominated by one force, notably some small offset positions in the 2sXB model (Figure 4D), are dominated by radial forces. This presence of large radial forces suggests that, in all but the least strained locations at the smallest axial offsets, radial forces will be present in magnitudes comparable to those of axial forces.
The lattice spacing of the filaments around an attached multi-spring cross-bridge determine the energy landscape of the cross-bridge and thus the force it can generate. The forces and strains a cross-bridge produces at most axial offsets grow more positive as lattice spacing increases (Figure 4E–H). While this increased cross-bridge strain translates into greater axial and radial force per post-power stroke cross-bridge, the probability that these cross-bridges will bind decreases as lattice spacing increases (Figure 3C–D). The decrease in attachment rate constants at extreme lattice spacings, while power stroke rate constants remain unchanged (Figure 3E–F), suggests lattice spacing influences muscle fiber force generation by altering the rate of cross-bridge attachment rather than the power stroke rate [22]. Spatially explicit effects in the compliant sarcomere, such as cross-bridge induced realignment of binding sites, may act to balance the decreased binding and increased detachment at larger lattice spacings. The energies, kinetics, and forces generated by the 2sXB model are subject to the same governing trends as those of the 4sXB model, and can be made similar by deliberate parameter choice (Table 1 and Figures 2,3, 4, and 5). That the 2sXB model can replicate the results of the 4sXB model indicates two things: first, the 2sXB can be used in place of the 4sXB in larger simulations, enabling work that would otherwise require prohibitive resources, and second, a feature shared between our two models is responsible for the interesting properties of our simulations, the use of a lever arm which undergoes an angle change to generate force. While the energies, binding rate constants, and power stroke rate constants of the multi-spring cross-bridges are almost identical, there are some smaller differences between the two models. The rate constant of detachment is rotated by approximately 20 between the two systems due to differences in the way the post-power stroke position is achieved (Figure 3). The 4sXB model and the 2sXB model generate somewhat different forces; the axial force produced by each model increases with lattice spacing, but that produced by the 4sXB does so more steeply (Figure 5A, C). In a reversal of this pattern, the 2sXB model' s radial force is more dependent on lattice spacing (Figure 5B, D). In each of these cases, the forces generated by both multi-spring cross-bridges are subject to the same trend. The close agreement between the forces and other properties of the two cross-bridge representations supports the position that the key feature of our multi-spring models is the use of a lever arm to generate force, rather than a factor unique to the 4sXB model, such as the simulation of interaction between the lever arm and the S2 domain. Substituting the 2sXB model for the 4sXB model reduces the runtime of a simulation by two orders of magnitude and puts multi-spring cross-bridge simulations of the half-sarcomere within reach. The geometries of the multi-spring models require a change in step size accompany a change in lattice spacing. This is because, while the length of the lever arm changes as lattice spacing varies, the pre- and post-power stroke angles do not. Step size varies more in the 4sXB model as the 4sXB model' s spring configuration causes the pre- and post-power stroke free energies to differ more than in the 2sXB model. As the detachment rate constant is a product of the post-power stroke free energy, the greater rotation in the 4sXB' s post-power stroke free energy, relative to that of the 2sXB model, can be seen in Figure 3 G–H. Experimental measurements of step size vary, and it has been postulated that this is due to more than experimental error, but to our knowledge these results are the first prediction of a step size that varies with lattice spacing [24]. Experimental confirmation of these predictions is not possible with current literature: existing in vivo measurements of step size are from isolated myosin preparations which are unable to simulate a change in muscle lattice spacing [25], [26]. While our single cross-bridge models lack the predictive power of a multi-filament model, the dependence of step size on lattice spacing offers insight into unloaded shortening velocity. Maximum unloaded shortening velocity is commonly interpreted as a function of both myosin' s step size and drag from attached post-power stroke cross-bridges [27]. A decrease in unloaded shortening velocity is observed when lattice spacing is compressed via dextran [28], [29]. This slower unloaded shortening is supported by the multi-spring models: their step size exhibits a similar decrease as lattice spacing shrinks (Figure 6). However, a moderate increase in the rate of detachment at highly compressed lattice spacings, seen in Figure 3 G–H, may balance smaller steps sizes. This increased detachment rate is due to the greater post-power stroke strain that is present with greater radial displacement of the cross-bridge. Changes in modeled detachment rates and step size are both likely to be needed, along with changes in filament overlap, to explain the complicated dependence of unloaded shortening velocity on sarcomere length [30]. The 4sXB and the 2sXB produce radial forces of the same order of magnitude as the axial forces generated by a cross-bridge. These forces range between 10% and 50% of the axial force at the least strained axial and radial offsets where a cross-bridge is most likely to enter the post-power stroke state (Figure 4). Muscle fibers display these radial forces by resisting width changes as osmotic pressure is applied [1]. Direct measurement of lattice spacing by X-ray diffraction has confirmed fiber width estimates of radial force [31]. Checchi et al. (1990) [2] observed large radial forces by examining lattice spacing during redevelopment of tension following length changes. A spatially explicit model, even one using multiple thick and thin filaments arranged in a lattice, is insensitive to lattice spacing if it uses a version of the 1sXB model. Embedding multi-spring cross-bridges in a multi-filament model allows the simulation of radial force regulation in a lattice of thick and thin filaments. The inclusion of radial forces in a multi-filament model permits examination of previously unavailable kinds of cooperativity, ones where radial force can be transmitted through the backbone lattice to affect the kinetics of other cross-bridges. Radial force is a potential regulator of lattice spacing and of sensitivity as lattice spacing and sarcomere length vary [3]. A multi-filament model using the 4sXB or 2sXB can simulate the interaction of radial force generated by a cross-bridge with radial forces provided by other mechanisms, e. g. titin or electrostatic repulsion [3], [22], [32]. Thus multi-spring cross-bridges make it possible to evaluate the influence of these radial forces, posited to be regulators of lattice spacing, and processes which may depend on lattice spacing or myosin head to thin filament distance, such as the Frank-Starling mechanism; something not possible with a 1sXB model [33]. In future studies, these models will permit the investigation of radial forces and lattice spacing in multi-filament models, and will allow us to examine disease states that alter myosin compliance. The inclusion of radial forces and lattice spacing in half-sarcomere models will illuminate regulatory mechanisms of shortening velocity and length-dependent axial force generation. Other efforts may use existing studies of how disease-related mutations alter myosin compliance to produce disease state mimicking cross-bridge models [34]. Multi-filament simulations using these altered cross-bridge models have the potential to explain how symptoms of disease states such as hypertrophic cardiomyopathy arise from myosin-level changes.
To describe the kinetics we use a simplified three-state model of the cross-bridge cycle originally described by Pate and Cooke (1989) [16] and modified by Tanner et al. (2007) [14]. This relatively simple scheme directly links the cross-bridge' s kinetics and mechanics; the three kinetic states are directly comparable to the myosin configurations described in Houdusse (2000) [36]. The kinetic rates are independent of the number of springs used in a model cross-bridge, allowing the 4sXB and the 2sXB models to use the same system. The three states represented in the kinetic scheme are (1) an unbound state: Myosin- (2) a loosely-bound state: Actin-Myosin- and (3) a force-generating post-power stroke state: Actin-Myosin-ADP (Figure 1D). These kinetics replicate those of a generic cross-bridge, and are aimed at reproducing properties shared between cardiac, skeletal, and insect myosin types. The kinetics of both the 4sXB and the 2sXB models are strain dependent and are essentially transforms of the free energy landscapes experienced by the cross-bridges in their different states. These free energies are a function of the distortion necessary to move the point representing the simulated myosin head' s tip to the proposed binding site. Examples of these free energy landscapes are shown in Figure 3A and B, with cuts through them at the rest lattice spacing visible in Figure 2A. As the free energies of the cross-bridges are functions of their spring rest values and stiffnesses, changing the geometry and stiffness of the springs used by the model also changes the kinetics of the model. The binding probabilities of both the 4sXB and the 2sXB models are determined by Monte-Carlo simulations of their diffusion as a result of being perturbed by Boltzmann-derived energy distributions [21]. After a new head location is found, a binding probability is calculated which decreases exponentially with distance from the potential binding site. This probability is tested against a random number from a uniform distribution to determine if binding occurs in our chosen time step of 1 ms.
The molecular motor myosin drives the contraction of muscle, but doesn' t 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.
biophysics/theory and simulation
computational biology
Axial and Radial Forces of Cross-Bridges Depend on Lattice Spacing