# The Force Velocity Relationship The Hill Model

One of the earliest models for a muscle is due to A.V. Hill (1938) and was constructed before the details of the sarcomere anatomy were known. Hill observed that when a muscle contracts against a constant load (an isotonic contraction), the relationship

Figure 18.6 A: Isometric tension as a function of the length of the sarcomere. B: schematic diagrams of the arrangement of the thick and thin filaments for the six different places indicated in panel A. (Gordon et al., 1966, reproduced in White and Thorson, 1975, Fig. 14.)

between the constant rate of shortening v and the load p is well described by the forcevelocity equation

Figure 18.7 The relationship between the load on a muscle and the velocity of contraction (Hill, 1938; Fig. 12). The symbols are the data points, while the smooth curve is calculated from (18.1) using the parameter values a = 357 grams (of weight) per square centimeter of muscle fiber (g-wt/cm2), a/p0 = 0.22, b = 0.27 muscle lengths per second.

Figure 18.7 The relationship between the load on a muscle and the velocity of contraction (Hill, 1938; Fig. 12). The symbols are the data points, while the smooth curve is calculated from (18.1) using the parameter values a = 357 grams (of weight) per square centimeter of muscle fiber (g-wt/cm2), a/p0 = 0.22, b = 0.27 muscle lengths per second.

where a and b are constants that are determined by fitting to experimental data in a way that we discuss presently. A typical force-velocity curve is plotted in Fig. 18.7. When v = 0, then p = p0, and thus p0 represents the force generated by the muscle when the length is held fixed; i.e., p0 is the isometric force. As we discussed above, the tension generated by a skeletal muscle in isometric tetanus is approximately independent of length, and thus p0 is approximately independent of length also. Whenp = 0, v = bp0/a, which is the maximum speed at which a muscle is able to shorten.

elastic

contractile 1

Figure 18.8 Schematic diagram of Hill's two-element model for skeletal muscle. The muscle is assumed to consist of an elastic element in series with a contractile element with a given force-velocity relationship.

In an attempt to explain these observations, we model a muscle fiber as a contractile element with the given force-velocity relationship, in series with an elastic element (Fig. 18.8). In some versions of the model a parallel elastic element is included (see Exercise 1), but as it plays no essential role in the following discussion, it is omitted here. As shown in Fig. 18.8, we let l denote the length of the contractile element, we let x denote the length of the elastic element, and we let L = l+x denote the total length of the fiber. Then, letting v denote the velocity of contraction of the contractile element, we have

where, by assumption, v is related to the load on the muscle by the force-velocity equation (18.1). To derive a differential equation for the time dependence of p, we note that because the elastic element is in series with the contractile element, the two experience the same force. We assume that the load on the elastic element is a function of its length p = P(x) and then use the chain rule and the force-velocity equation to obtain dp dt dPdx dx dt dP

dx dP

dx dP

dx dL dl dt dt dL

It remains to determine dP/dx.

Hill made the simplest possible assumption, that the elastic element is linear, and thus

where x0 is its resting length. Thus, dP/dx = a, and the differential equation for p is dp dt dL b(po — p) ' dt p + a