Hill's force-velocity relation for a fully activated muscle (or for a single…

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Problem

Hill's force-velocity relation for a fully activated muscle (or for a single actomyosin cross-bridge ensemble) can be written

v(F)=\frac{b\,(F_0-F)}{F+a},

where

F_0

is the isometric (zero-velocity) force,

F

is the load, and

a

(a force) and

b

(a velocity) are positive constants. This relation is an empirical fit to measurements, quoted here. ((a)) The mechanical power delivered against the load is

P(F)=Fv(F)

. Find the load

F^*

at which this power is largest, in terms of

a

and

F_0

. ((b)) Find how

F^*/F_0

behaves as

a/F_0\to0

and as

a/F_0\to\infty

. In which limit does Hill's relation become a straight line from its top speed at zero load to zero speed at

F_0

, and why does the power then peak at exactly half the isometric force? ((c)) A different muscle has an isometric force

F_0=150\,\mathrm{N}

and a Hill constant

a=60\,\mathrm{N}

(so

a/F_0=0.40

). Find, in newtons and as a fraction of

F_0

, the load at which this muscle delivers its maximum mechanical power.

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