In isolated muscle preparations, force–velocity data are usually obtained using a method called isotonic quick release. Typically, during an isometric contraction, a step (or isotonic) release is applied so that the muscle shortens under a constant load, as illustrated in Figure 1A. It is known that the length change in response to the force step consists of four phases (Civan and Podolsky, 1966) and that each can be traced to certain cellular and subcellular origins (Ford et al., 1977). Phase 1 is primarily an elastic recoil immediately following the step release of tension. In striated muscle preparations with little stray compliance, phase 1 response stems mainly from the crossbridge and actin filament elasticity. Muscle stiffness assessed from the phase 1 response can therefore be used as an approximate index of the number of attached crossbridges. The phase 2 response originates mainly from the synchronous power strokes of some crossbridges after the step release (Dobbie et al., 1998; Suzuki et al., 1998). Phase 3 occurs when some of the crossbridges reach the end of their power stroke, and others start to detach from actin filaments. Phases 1 to 3 are transient responses, unlike phase 4, which occurs when the crossbridges enter steady-state asynchronous cycles. In force–velocity measurements, the initial slope of the phase 4 response is taken as the steady-state shortening velocity of the muscle under the externally applied isotonic load. The force–velocity properties of the muscle are therefore steady-state properties.

Many muscle preparations used in experiments contain a substantial amount of stray compliance, which usually stems from the crushed ends of the muscle preparation. The stray compliance is viscoelastic in nature (Seow and Ford, 1992), and its recoil during the step release can obscure the transient response of the muscle. Figure 1B shows a length response from a skinned (membrane-permeabilized) rabbit psoas fiber (cell), and a sarcomere length response from the same fiber. The muscle length response contains stray compliance which obscures the transients, whereas the sarcomere length response is free of the influence of the stray compliance and the transient phases are evident. It is clear from the sarcomere length record where to measure the steady-state shortening velocity (phase 4). It is crucial that this velocity is measured at the right time after the quick release; measurements made too early will overestimate the velocity, while measurements made too late will underestimate the velocity.

In fitting force–velocity data with the hyperbolic Hill equation, equation 1 is usually transformed to an explicit function, that is, shortening velocity as an explicit function of muscle load, as follows:

Alternatively, the Hill equation can be expressed as an explicit function of force:

A non-linear fit of force–velocity data is performed to obtain Hill’s constants a, b, and c. An example is shown in Figure 2. Curve fitting of force–velocity data by the Hill equation usually produces an excellent agreement between the data and the curve, especially in the force range of 5–80% of maximal isometric force (F_max). Significant deviations of data from the curve can occur in the force ranges of 0–5% F_max and 80–100% F_max. Reasons for the deviation have been discussed in detail by Seow (2013).

Because power equals force times velocity, the muscle’s force–power relationship can be derived from the force–velocity relationship, as shown in Figure 2 (dotted curve). Maximal power typically occurs in the force range of 10–50% F_max. In this force range, curve fitting of the data is by interpolation. The maximal power output (P_max) obtained from curve fitting is usually more reliable than the extrapolated values of maximal shortening velocity (V_max) and F_max.

The basic contractile unit in striated muscles is the half-sarcomere. In a muscle cell, all sarcomeres have the same structure and function. During contraction, each half-sarcomere contributes equally to the generation of force, shortening, and power by the muscle. Measured mechanical output of a muscle cell can vary depending on the cell size and how the sarcomeres are arranged within the cell. A muscle cell will generate more force if it has more sarcomeres packed in parallel. Because the cell’s cross-sectional area is proportional to the number of sarcomeres in parallel, muscle force is normalized by the cross-sectional area of the muscle preparation. Normalized force is called stress, which has the same unit as pressure, for example, kilopascal (kPa) which is also the same as mN/mm².

The number of sarcomeres in series is proportional to the muscle length, therefore muscle length or shortening velocity is normalized by the resting length of the muscle preparation, if the sarcomere length is not known. The unit for the normalized shortening velocity is typically muscle length/s. If the sarcomere length is known, then the number of sarcomeres in series can be obtained by dividing the muscle length by the sarcomere length. Of course, the number of half-sarcomeres is twice that of sarcomeres. To normalize shortening velocity, the measured velocity is divided by the number of half-sarcomeres. The unit for normalized shortening velocity is therefore typically μm s⁻¹ half-sarcomere⁻¹.

The power output of a muscle is proportional to the total number of sarcomeres in the muscle, regardless of their arrangement (i.e., in series or in parallel). The total number of sarcomeres in a muscle is proportional to the volume of the muscle, that is, cross-sectional area of the muscle multiplied by the muscle length. Because muscle volume is more difficult to measure than muscle weight, the latter is typically used to normalize muscle power, since muscle weight is linearly proportional to muscle volume, with the muscle density (~1.06 g/cm³) being the constant of proportionality. The unit for normalized power is typically milliwatts per gram of muscle tissue, or mW/g.

The following section further illustrates the rationale for normalization of force–velocity properties.

We have found that in airway smooth muscle, partial activation (defined as a reduction in F_max, P_max, and the degree of phosphorylation of the regulatory myosin light chain, but without a change in V_max) does not result in a change in the curvature of the force–velocity curve (Luo et al., 2019). Mathematically, the curvature of Hill’s hyperbola is determined by the ratio of maximal isometric force over the Hill constant a, or the ratio of maximal velocity over the Hill constant b, that is:

Figure 3 illustrates how curvature of a force–velocity curve could be altered by changes in the ratio of F_max/a or V_max/b.

For interventions that do not alter the kinetics of actomyosin interaction, the force–velocity curvature will not be affected by such interventions. For example, in fully and partially activated muscles where the change in activation is strictly limited to a change in the number of activated crossbridges, with no change in the kinetics of actomyosin interaction, the curvature is not expected to change. The Hill equation for the fully and partially activated muscle can be written as follows:

In equations 5 and 6, V_f and V_p denote shortening velocity of fully and partially activated muscles, respectively, b_p and a_p are Hill’s constants for the partially activated muscle, and n is an index of activation with n = 1 for full activation and n < 1 for partial activation. We have shown that if partial activation only results in a reduction in the number of attached crossbridges, the Hill constant b remains unchanged (Luo et al., 2019). That is, b_p = b. Because V_max is the same in either full or partial activation, and V_max occurs when F = 0, from equations 5 and 6:

From equation 7 one can derive: a_p = na. The curvature of the force–velocity curve for the partially activated muscle is nF_max/a_p or nF_max/na, or simply F_max/a. The curvatures as defined by equation 4 (F_max/a) for both fully and partially activated muscles are therefore the same.

Hypertrophy or atrophy of a striated muscle cell does not alter the structure or function of individual sarcomeres, but changes their number. Figure 4A shows a simplified illustration of a sarcomere, and Figure 4B shows three different arrangements of four such sarcomeres. Figure 4C shows force–velocity curves for each of the arrangements: a, bi, bii, and biii. Note that, graphically, curve bi is obtained by scaling the velocity values of curve a by a factor of 4; curve bii is obtained by scaling both force and velocity values of curve a by a factor of 2; and curve biii is obtained by scaling the force value of curve a by a factor of 4. Because the crossbridge kinetics is not altered by different geometric arrangements, the shape of the force–velocity curves after normalization by the number of sarcomeres in parallel and in series, bi, bii, and biii will all superimpose exactly on curve a. In other words, scaling of force and velocity in each of the cases, bi, bii, and biii, does not change the curvature of the force–velocity curves. The conservation of curvature can be illustrated mathematically. From equation 7 we learn that scaling the force by a factor n will lead to a change in the Hill constant a by the same factor. For example, in the case of partial activation, a_p = na. It can be shown that n can be any positive number and it does not have to be less than 1 as in the case of partial activation. If we use a_n to denote the Hill constant a in a force–velocity relationship where force can be scaled up or down by a factor of n, then a_n = na. In a similar manner, it can be shown that b_m = mb, where m is the scaling factor for velocity. An interesting property of the Hill equation is that the constants a and b are mutually exclusive in that changing the force scale only affects constant a, and changing the velocity scale only affects constant b. Therefore, for case bi (Figure 4C), b_m = 4b. The curvature of curve bi, as defined by equation 4 is 4V_max/b_m or 4V_max/4b or simply V_max/b, is the same curvature as the unscaled curve a. For the case bii in Figure 4C, a_n = 2a and b_m = 2b. The curvature does not change in this case either, because nF_max/a_n = 2F_max/2a = F_max/a and mV_max/b_m = 2V_max/2b = V_max/b. Finally, for case biii in Figure 4C, nF_max/a_n = 4F_max/4a = F_max/a. Again, scaling force does not lead to a change in the curvature of the force–velocity curve. It should be pointed out that, in any case, the scaling factor has to be a constant, and not a function.

For the examples shown in Figure 4C, shortening velocity is proportional to the number of sarcomeres in series and force is proportional to the number of sarcomeres in parallel. From Figure 4D, it is clear that power output of a muscle is proportional to the number of sarcomeres, regardless of their arrangements. For curves bi, bii, and biii, the maximal power is the same, and it is 4 times the maximal power for curve a because they have 4 times more sarcomeres. The examples shown in Figure 4 support the rationale for how force, velocity, and power are normalized, as described earlier in the section Normalization of force, velocity, and power above.

Force in a force–velocity relationship usually refers to externally applied force or load against which the muscle shortens. The viscous and compressive loads within the muscle tissue are not part of the external load. In the absence of an internal load, the externally applied force is the total force “seen” by the muscle. In most studies, internal loads are neglected because they are relatively small compared to the maximal load (F_max). However, even with an internal load less than 5% of F_max, V_max, and a/F_max can be significantly altered. Internal loads are ubiquitous in muscle preparations. They stem from any loci within the muscle that offer impediments to shortening. They are present in smooth (Wang et al., 2002; Luo et al., 2019) and cardiac (Chiu et al., 1982; Walley et al., 1991) muscles. The internal load in skeletal muscle is relatively small and may be justifiably neglected, but under some experimental conditions, such as acidosis (Seow and Ford, 1993), the internal load can be significant.

Figure 5 graphically illustrates what an internal load is. Hill’s constants a and b are the asymptotes of the Hill hyperbola. In a typical force–velocity curve only the portion between V_max and F_max of the curve is plotted (Figure 5, solid curve). If an internal load is present, the force–velocity curve is truncated and only the portion of the curve between V’_max and F’_max is plotted. In order for a muscle to shorten, the contractile force developed by the muscle has to overcome both the internal and external loads. An internal load therefore functions to reduce the contractile force of the muscle, similar to partial activation. As in partial activation, an internal load affects the force–velocity relationship by changing the Hill constant a but not b (Seow and Stephens, 1986; Seow and Ford, 1993; Luo et al., 2019). The curvature of the force–velocity curve within the force range of F’_max can be derived from:

where a’ is the new asymptote for the curve with an internal load (F_i), and F’_max is the maximal isometric force with an internal load. Because F_i is always positive, it follows that a’/F’_max will always be greater than a/F_max, and the greater the internal load, or ratio of F_i/F_max, the greater the value of a’/F’_max will be. Therefore, even though an internal load does not actually alter the shape of the force–velocity curve (Figure 5), the apparent curvature indicated by F’_max/a’ is decreased.

The inset in Figure 5 illustrates a method with which internal loads can be determined. Without an internal load, V_max will not be altered by partial activation of the muscle where only the number of activated crossbridges is reduced without changing the crossbridge kinetics (Luo et al., 2019). Therefore, by extending the force–velocity curve pass the vertical dot-dash line (Figure 5) to where the true zero load is (zero origin of the force axis), all curves with different amounts of internal load will converge to one point on the graph. That point is where both the internal and external loads are zero and where the true maximal shortening velocity (V_max) occurs. The inset of Figure 5 shows two curves, one for full activation and the other for half-activation. Without taking internal loads into consideration, one would conclude that the maximal velocity obtained under full activation (V’_max) is greater than that under half-activation (V″_max). However, if the internal load is considered, we would conclude that the true maximal velocity (V_max) is not changed, and the apparent difference among V_max, V’_max, and V″_max is just an artifice created by the internal load and partial activation.

After a force–velocity curve has been properly normalized and internal loads taken into account, if its curvature is still different before and after an intervention, one can conclude that the crossbridge kinetics have been altered by the intervention. The intervention can come from Nature during evolution of species where the biochemical properties of isoforms of myosin and actin have been changed. The intervention can also come from changes in the chemical milieu in which actomyosin interaction occurs. The changes in chemical milieu in turn can be initiated by environmental factors, such as temperature and oxygen level, or pathological conditions, such as acidosis and alkalosis. A broader discussion on how interventions modify the force–velocity properties of muscle can be found in a review by Seow (2013). The following discussion focuses on interventions that specifically alter the force–velocity curvature.

It has been shown in earlier discussions that changes in the arrangement of sarcomeres, partial activation, and internal loads do not fundamentally alter the force–velocity relationship, because after normalization of force and velocity, the force–velocity curvature remains the same. Fast and slow muscle fibers contain different myosin isoforms. Observations from many studies comparing the force–velocity relationships in fast and slow muscles, revealed that fast muscles, besides having higher V_max, also have less curvature (lower values for F_max/a; Katz, 1939; Close, 1964; Woledge, 1968; Cecchi et al., 1978; Lännergren, 1978; Luff, 1981; Lännergren et al., 1982; Ranatunga, 1982; Brooks and Faulkner, 1988; Stienen et al., 1988; Barclay et al., 1993; Wahr and Metzger, 1998; Gilliver et al., 2009). This means that fast muscles possess a greater power output than slow muscles do, not only by having a faster shortening velocity, but also by having less curvature in their force–velocity curves. Cardiac myocytes containing the faster isoform α-MHC have a higher shortening velocity and power output than those containing the slower isoform β-MHC. The faster isoform α-MHC is also associated with a reduction in the curvature of the force–velocity curve, F_max/a (Herron et al., 2001).

Because shortening velocity is proportional to the apparent rate of crossbridge detachment (g_APP), and from equation 13, K = f_APP/k, we can deduce that the molecular mechanisms conferring the phenotype of fast myosin isoforms are (1) a faster rate of crossbridge detachment (g_APP), likely due to an increased rate of dissociation of ADP from its myosin binding site (Siemankowski et al., 1985; Rayment et al., 1993; Elangovan et al., 2012), (2) an increase in the rate of crossbridge attachment (f_APP) and/or (3) a decrease in the velocity–dependent rate of detachment (k). The first mechanism gives rise to higher shortening velocities and the second and third mechanisms, individually or together, can explain the increase in K value observed in muscles with fast myosin.

It has been observed that the slow tortoise muscle is more energetically efficient than the faster frog muscle and that the former also possesses a greater curvature in its force–velocity curve than the latter (Woledge, 1968). It is possible that a greater value for k is associated with the slow tortoise muscle. This makes the dissociation rate of crossbridges from their actin binding sites faster as the shortening velocity increases, reaching its maximal value at V_max. This could lead to less bridges being dragged into the negative-force region (Huxley, 1957). The improvement in energetic efficiency may come as a result of fewer negatively strained bridges.

It has been found that V_max and a/F_max both increase with temperature, with a strong correlation between V_max and a/F_max over a temperature range of 10°–35° (Ranatunga, 1984). Furthermore, V_max and a/F_max in fast muscles, compared with those in slow muscles, have higher temperature sensitivity. F_max has also been found to be temperature-dependent, and the augmentation of F_max by temperature appears to be due to enhancement of a step transition of crossbridges from the detached state to a force-generating state before the release of inorganic phosphate from the myosin head (Ranatunga, 2010). The temperature-dependent increase in V_max, F_max, and a/F_max will tend to increase the power output of the muscle. Interestingly, analysis by Ranatunga (1998) showed that among the force–velocity parameters, P_max has the highest temperature sensitivity. This perhaps can be understood as a consequence of a synergistic effect from the contribution of V_max, F_max, and a/F_max to P_max. For example, if velocity and force both increase by 10% due to temperature increase, this alone would lead to an increase in power output by 21% because power equals velocity times force, assuming no change in a/F_max. An increase in a/F_max with temperature will further contribute to the increase in P_max.

From equation 13 we know that a/F_max (or K) is determined by f_APP and k. A study on the effects of temperature on the crossbridge cycle has revealed that the fraction of detached bridges decreases as temperature increases (Zhao and Kawai, 1994), indicating that f_APP is enhanced and/or k is diminished by increasing temperature. In homeothermic species, the temperature sensitivity of a/F_max is perhaps physiologically less relevant. For ectotherms, such as fish living in water with large, depth-dependent temperature gradient, a large change in a/F_max (hence P_max) could significantly impact their function. This is perhaps why, in some cold-water fish, a/F_max is temperature insensitive (Johnston and Salamonski, 1984; Johnston and Sidell, 1984). In carp red muscle, the temperature-dependent change in a/F_max is opposite to that in mammalian skeletal muscle (Rome and Sosnicki, 1990). This is perhaps a compensatory mechanism for ectotherms to reduce power loss in muscle at low temperature by increasing a/F_max (decreasing the curvature).

In muscle contraction, the immediate energy source is ATP hydrolysis. When an ATP molecule is hydrolyzed, P_i (inorganic phosphate), ADP (adenosine diphosphate), and H⁺ (hydrogen ion) are produced. The hydrolysis products are known to alter the kinetics of the crossbridge cycle and hence the force–velocity relationship.

In experiments using membrane-permeabilized muscle fibers, it has been shown that when ADP release is partially blocked by high concentrations of ADP in the intracellular milieu, F_max is increased and V_max is decreased (Cooke and Pate, 1985; Seow and Ford, 1997). Furthermore, the curvature of the force–velocity curve is reduced by high concentrations of ADP (Seow and Ford, 1997). The increase in a/F_max (reduced curvature) is likely due to a decrease in the sensitivity of g_APP on shortening velocity, that is, a reduction in the value of k in equation 13. The effect of slowing ADP release in the crossbridge cycle can also be demonstrated in experiments using a bipyridine compound (amrinone), which is known to increase ADP affinity to myosin. The presence of amrinone enhances F_max, diminishes V_max, and augments a/F_max (Albet-Torres et al., 2009). Low [ATP] has a very similar effect as high [ADP] on the crossbridge cycle and hence the force–velocity relationship (Ferenczi et al., 1984; Cooke and Pate, 1985; Stienen et al., 1988; Seow and Ford, 1997). Furthermore, low [ATP] has been found to cause an increase in a/F_max (Ferenczi et al., 1984; Stienen et al., 1988; Seow and Ford, 1997; Wakayama and Yamada, 2000; Cheng et al., 2020).

The effect of high [H⁺] on force–velocity curvature is more complicated, because it affects multiple points in the crossbridge cycle. Detention of crossbridges in the low-force states by high (H⁺; Seow and Ford, 1993) tends to decrease f_APP, which would increase the curvature (Overgaard et al., 2010). On the other hand, slowing of ADP release (Debold et al., 2008) would tend to decrease the value of k and hence curvature (Albet-Torres et al., 2009). This may explain the mixed reports on the effects of high [H⁺] on the force–velocity curvature. In addition, the high [H⁺] effect on the force–velocity curvature is dependent on temperature and muscle type. For example, at low temperature (15°C), Knuth et al. (2006) found that in type I muscle, a/F_max increased at low pH and decreased at high pH, but at a higher temperature (30°C), a/F_max decreased at low pH and increased at high pH. In type II muscle at 15°C, a/F_max increased at low pH and decreased at high pH, just like that in type I muscle, but at 30°C, pH had no significant effect on a/F_max in type II muscle. In rabbit psoas (fast) muscle at 10°C, Cooke et al. (1988) found no significant change in a/F_max due to pH change. For the same muscle at 1.5°C, Seow and Ford (1993) found a significant increase in a/F_max at low pH. From the scattered results one can conclude that to understand the effect of low pH on physiological function of muscle, the force–velocity parameters should be measured at or near body temperature.

At the muscle cell level, fatigue is often a consequence of energy demand exceeding supply. The immediate energy supply for muscle contraction is derived from the intracellular pool of ATP, normally at a concentration of about 5 mM. Metabolism of ATP produces metabolites that include ADP, P_i, and H⁺. The intracellular concentration of ATP is usually maintained at a constant level by buffering systems that involve phosphocreatine, glycolysis, and oxidative phosphorylation. When oxygen supply lags demand, lactic acid accumulates, leading to acidosis. To make things worse, both P_i and H⁺ reduce myofibrillar Ca²⁺ sensitivity (Debold et al., 2006; Nelson and Fitts, 2014). Effects of muscle fatigue on the force–velocity relationship are therefore multifaceted due to the many contributing factors.

The consequence of muscle fatigue involves a reduction in the ability of the muscle to generate force and reduced shortening velocity. However, there are mixed reports regarding changes in the curvature of the force–velocity curve (Barclay, 1996; Fitts, 2008; Jones, 2010; Devrome and MacIntosh, 2018; Kristensen et al., 2019, 2020). The ratio of ADP/ATP is usually well maintained and is thought to play little role in causing fatigue (Cooke, 2007). Reduction in myofibrillar Ca²⁺ sensitivity leads to a reduced number of activated crossbridges, similar to the situation in partial activation. If lower myofibrillar Ca²⁺ sensitivity is equivalent to partial activation (discussed above), it would decrease F_max but have no effect on a/F_max, as reported by Langeron et al. (1999). However, other studies showed that lowering intracellular [Ca²⁺] led to a slight increase in a/F_max, and augmenting intracellular [Ca²⁺] led to a decrease in a/F_max (Kristensen et al., 2018). The effect of intracellular [Ca²⁺] on the force–velocity curvature is therefore inconclusive. Note that the presence of an internal load tends to increase the ratio a/F_max, as discussed above. Taking internal loads into account in the determination of force–velocity properties may help to resolve some of the discrepancies.

In a muscle cell, ADP produced from ATP hydrolysis is used in the phosphocreatine buffering system to replenish ATP, but P_i produced by ATP hydrolysis is not used in this buffering system and therefore can accumulate inside the muscle cell. As discussed above, high [P_i] depresses F_max but has little effect on V_max. In mild fatigue, a small decrease in F_max occurs without a significant decrease in V_max (Edman and Mattiazzi, 1981; Jones et al., 2006). It is likely that P_i accumulation is responsible for the initial phase of muscle fatigue (Wilson et al., 1988). In later phases of fatigue, accumulation of H⁺ and lactic acid, combined with high [P_i] likely lead to a large depression of F_max and V_max due to the direct effects of low pH and high [P_i], and indirect effect of low myofibrillar Ca²⁺ sensitivity. Low pH at near body temperature has been shown to increase the curvature (decrease in a/F_max; Knuth et al., 2006). Low pH is therefore one of the contributors to the increase in force–velocity curvature observed in muscle fatigue. As discussed above, high [H⁺] detains crossbridges in the low-force states and reduces the apparent rate of attachment (f_APP). A similar conclusion is reached by Jones (2010) in that there must be a substantial decrease in the rate constant for attachment in the Huxley (1957) model to account for the observed decrease in power and increase in force–velocity curvature in fatigued muscle.