THE MOLECULAR BASIS OF MUSCLE CONTRACTION 



279 



Now if speed, v, of shortening is always proportional to restoring force 

 (this is equivalent to assuming the spring is embedded in a plastic or highly 

 viscous mass, and that the spring is critically damped) then: 



Integration gives 



k(s - s f ) 



s, + (s () - s f )e- 



•7 1 v J 7. 



where s is the initial, or starting, length. From this the shortening speed 

 can be expressed as a function of time by finding the derivative. It is 



v = k(s — s,)e~ kt 



The fraction shortened,/, defined as (s Q — ^)/(.r — Sj), at any time reduces 

 to 



/= 1 - 



-*/ 



and k becomes known as the shortening constant. This expression is illustrated 

 in Figure 10-9, in which the fraction shortened during shortening is plotted 

 for both the case discussed and for muscle. Elasticity in the muscle, which 

 lowers the initial rate of shortening (df/dt), and recovery following full con- 

 traction are the chief differences. Note that the ^-shaped curve in the case of 

 muscle can appear to be linear, especially if sensitivity of measurement is not 

 high enough; and hence the shortening rate ( — ds/dt) is often considered to 

 be constant. 



T ime 



0,5 sec 



Figure 10-9. Fraction Shortened as Function of Time During Shortening. 



The larger the load, m, the smaller is the shortening constant, k. This is 

 to say that the muscle can contract quickly if the load is light, and only 

 slowly if the load is heavy. It is found that k varies with m in such a way 



