Hill’s equation of muscle performance and its hidden insight on molecular mechanisms

Abstract

Muscles shorten faster against light loads than they do against heavy loads. The hyperbolic equation first used by A.V. Hill over seven decades ago to illustrate the relationship between shortening velocity and load is still the predominant method used to characterize muscle performance, even though it has been regarded as purely empirical and lacking precision in predicting velocities at high and low loads. Popularity of the Hill equation has been sustained perhaps because of historical reasons, but its simplicity is certainly attractive. The descriptive nature of the equation does not diminish its role as a useful tool in our quest to understand animal locomotion and optimal design of muscle-powered devices like bicycles. In this Review, an analysis is presented to illustrate the connection between the historic Hill equation and the kinetics of myosin cross-bridge cycle based on the latest findings on myosin motor interaction with actin filaments within the structural confines of a sarcomere. In light of the new data and perspective, some previous studies of force–velocity relations of muscle are revisited to further our understanding of muscle mechanics and the underlying biochemical events, specifically how extracellular and intracellular environment, protein isoform expression, and posttranslational modification of contractile and regulatory proteins change the interaction between myosin and actin that in turn alter muscle force, shortening velocity, and the relationship between them.

Smooth muscle function and myosin polymerization

“…3) during length adaptation. The fact that the force-velocity data obtained at short and long (relative to L ref ) adapted lengths could be fitted well by scaling the velocity values (without a change in the curvature of the force-velocity curve) is also predicted by the proposed model, which assumes no change in the intrinsic properties of individual contractile units [for further information, see a review by Seow (2013) on the relationship between force-velocity curvature and the intrinsic cross-bridge kinetics]. Activation-induced and basal MLC phosphorylation at different adapted muscle lengths As discussed above, results from force-velocity measurements in the present study (Figs 2, 3 and Fig.…”

“…The method of fitting by scaling the velocity values is valid only if the curvature of the force-velocity curve does not change at different adapted lengths (Pratusevich et al, 1995). The good fits obtained at 0.75 and 1.5L ref indicate no change in the curvature of the force-velocity relationship at the three lengths, suggesting that there was no change in the kinetics of actomyosin interaction at the level of cross-bridge cycle (Seow, 2013). This information from curve fitting was used later in the interpretation of the force-velocity data.…”

Mechanics of Vascular Smooth Muscle

“…23D). However, the force-velocity relationship is not precisely continuous [not shown here, but see, e.g., (71,411)]. Notably, studies in which velocity is measured at force values near f 0 reveal that a break point, and thus, a deviation from the Hill hyperbola, occurs in the region near ∼0.8-fold f 0 (85,116,244).…”

“…Notably, studies in which velocity is measured at force values near f 0 reveal that a break point, and thus, a deviation from the Hill hyperbola, occurs in the region near ∼0.8-fold f 0 (85,116,244). Like the active stress-strain relationship in skeletal muscle that was originally empirically derived and later found to have a firm basis in the molecular, subcellular, structure of CC proteins, there is now evidence that the force-velocity relationship is a reflection of molecular-level cross-bridge dynamics (71,354,411). Muscles that maintain isometric force with high energy economy, such as tonic VSM, will have less force deviation from the Hill hyperbola in the low velocity, high force range because such muscles fit more closely than others the assumption that, when the muscle is isometric, the cross-bridge detachment rate is ∼zero (411).…”

“…The force-velocity curve identifies V 0 , the velocity extrapolated to zero load, and f 0 , force at the velocity = zero intercept. Reduced force at increased shortening velocities as revealed by the Hill hyperbola is due, in part, to a decrease in the number of attached cross-bridges (354,411). The degree of curvature of the rectangular hyperbola is specified by the a/f 0 quotient; an increase in this quotient reflects a decrease in the curvature of the Hill hyperbola.…”