The equation of Hill. Power of single reduction.

Dependence of speed of shortening on loading Р is the major at studying work of a muscle as allows to reveal laws of muscular reduction and its power. Hill studied it in detail at different modes of reductions and presented on fig. 8.

He has offered the equation describing this curve

This expression refers to as the equation of Hill and is the basic equation of mechanics of muscular reduction.

Р₀ - the maximal isometric pressure developed by a muscle, or the maximal cargo which is held with a muscle without lengthening; b - a constant which has dimension of speed, and - a constant, dimensional forces.

F rom equation we get, that speed becomes maximal at Р = 0,

At Р = Р₀ it is received V = 0, that is shortening does not occur. Work A, made by a muscle at single shortening on size is equal to А= Р .

Dependence of capacity on developed force Р

Capacity is equal to zero at Р =Р₀ and Р = 0 and reaches the maximal value at optimum size of loading

The overall performance of a muscle at reduction can be certain as the attitude of the perfect work to the spent energy : .

Development of the greatest capacity and efficiency of reduction is reached at efforts

0,3 - 0,4 from maximal isometric loadings Р₀, for the given muscle. It is used, for example, by sportsmen-bicycle racers: at transition from plain on a mountain site loading on muscles increases also the sportsman switches speed to the lowest transfer, reducing Р, approaching it to Р_max. Practical efficiency can reach values of 40-60 % for different types of muscles. The highest efficiency is observed at muscles of the turtle, reaching 73 - 80 %.

Modelling of muscular reduction

The equation of Hill has been received at generalization of a plenty of data. The kind of this equation specifies existence in a muscle internal viscous (depending from speed) the friction interfering its shortening. Force of reduction of a fibre is equal to the sum of the forces generated by bridges in a layer, equal to half sarcomere since sarcomere on thickness of a fibre are included in parallel. Speed of change of length of fibre Vв: Vв = 2NV, where N - number of sarcomeres in a fibre, V - relative speed of sliding of strings. At sliding strings the bridge can be in one of three possible conditions: opened, closed pulling when the head generates the force directed to the center of sarcomere, and closed braking when actin string has passed coordinate of the center of an attachment of a head and the attached bridge creates negative force in direction F, after that it is disconnected. Transitions from one condition in another, are defined by corresponding constants of speeds. The full cycle of the bridge is accompanied by disintegration of molecules АТP.

For the general number pulling (x) and braking (z) bridges developed by sarcomere force

The first composed-it the force generated closed, and second - braking bridges. Then the system of the kinetic differential equations for conditions of bridges can be written down in the form of:

The first equation the left part - speed of change of quantity of pulling bridges. In the right part the first composed - the general number of bridges, a minus quantity pulling х and braking z bridges, i.e. in square brackets - quantity of the remained opened bridges. Multiplying this quantity on constant , we receive speed of increase in quantity of pulling bridges; the second composed - speed of reduction of quantity of pulling bridges due to their transition in a brake condition. The difference between growth rates and reduction of quantity of pulling bridges gives required speed of change of their quantity.

Then from system of the equations and conditions, that x, z and U do not vary in time is received:

Thus: