Muscle contraction is most easily studied in skeletal muscle because of its regular structure, i.e., (a) whole muscles comprise more of less parallel fibres, and (b) each fibre is striated with regular bands of contractile elements.
In the following discussion we’ll consider a single muscle fibre or a muscle with all its fibres acting in concert. In other words we'll ignore the mechanisms of recruitment and variable activation (variable neuronal firing rate).
Passive muscle fibres behave like natural elastic fibres. This means that as you stretch a fibre, it resists stretching and this resisting force increases with increasing stretch length. The relationship between the fibre length and the passive resisting force is not linear (in contrast to a Hookean spring), but instead is approximately exponential as in the well known passive length-tension curve. Therefore, the natural tendency of passive muscle fibres is to adopt a short resting length.
When activated muscle fibres produce a greater resisting force to stretch. The additional resisting force due to activation is the familiar “active length-tension curve”. This curve is hill shaped reaching a maximum at some optimum muscle length and tapering off at both shorter and longer lengths. The activated muscle’s total resisting force to stretch is, of course, the sum of the “passive” and “active” resisting forces. This is the total or observed length-tension curve of active muscle.
The “force-length” or “length-tension” curves are obtained by isometric measurements when the muscle length is kept fixed during the measurement of the resisting force. These isometric length dependent properties are readily observed in daily life – when lifting a load with the biceps one can notice that the greatest strength is available when the elbow is approximately at right angles. This is when the biceps length is “optimum” and corresponds to the peak in the active muscle’s length-tension curve.
In addition to these isometric, length-tension properties muscles also exhibit distinct dynamic properties during movement. For example, during many jumping and carrying activities we notice that muscles act like shock absorbers. Shock absorbers present a resisting force proportional to the speed at which they are moved. This means that a muscle will change length more easily if it is moved slowly than if it is moved quickly. In other words, a rapidly moving muscle has less force available to move an external load. To describe the behaviour of moving muscle, Hill postulated the following mechanical equivalent. If two independent idealised mechanical components, namely, an ideal force generator and a “dashpot” or “shock-absorber” when combined can mimic the behaviour of moving muscle (at least under the limited condition of shortening muscle). This can be represented by the familiar Hill model for isotonic muscle contraction.
Hill devised experiments to determine the actual velocity dependent property of skeletal muscle and found that it behaved like a non-linear dashpot with profile as shown by the dotted curve in the above figure. Combined with his experiments on muscle energetics and heat production during shortening movements, Hill used the following hyperbolic expression to describe the force-velocity behaviour of skeletal muscle (V=velocity of muscle shortening, F=muscle force and a, b, Fmax are constants:
These natural observations – the elastic resistance to stretch and velocity dependent resistance to movement, can be combined as follows for a more complete rheological model of skeletal muscle.
The change in active force with muscle length corresponds to the change in the overlap of the bands of Actin and Myosin molecules in striated muscle. Therefore, it appears that the force production mechanism in muscle depends on the interaction between the two molecules.
It can be further observed that in the presence of Calcium the molecules Actin and Myosin attach to each other (actomyosin). This attached state can be reversed using energy from ATP. This can be summarised as:
The attachment of Myosin to Actin is termed as the formation of cross-bridges. Using the data from these muscle properties, Huxley proposed a biophysical model of muscle contraction based on the idea:
The stiffness of an active muscle is proportional to the number of Myosin molecules in the thick filament attached to Actin molecules in the thin filament, i.e., the number of attached cross-bridges
Huxley’s idea of muscle contraction based on the attachment of “cross-bridges” can be understood using the analogy of a millipede walking on a stick. The strength with which the millipede can advance on the stick will be proportional to the number of legs that grip the stick.
Figure 1: Millipede on a stick :– Millipede=Myosin and Stick=Actin
The shortening of a sarcomere can now be pictured as two millipedes tied together facing away and walking on two different sticks as shown in the next figure.
Figure 2: Two opposing millipedes – pulling on the sticks as they try to crawl over them
Looking at this figure it is possible to imagine how the sticks will be drawn closer as the legs of the millipedes pull on them. This can be extended to the case of many sarcomeres in series as shown below:
Figure 3: Sarcomeres in series: millipedes crawling on sticks with millipedes pulling sticks with millipedes crawling … and so on.
A.F.Huxley assembled the idea of Myosin interacting with Actin into a formal biophysical model as follows:
Let f be the rate constant of the reaction combining Myosin with Actin and let g be the rate constant of the reverse reaction detaching Myosin from Actin
At any point in time, of all the possible Myosin-Actin pairs, if n is the fraction that is actually attached, then (1-n) is the fraction that is unattached
New pairs are formed from the (1-n) unattached Myosin-Actin pairs at the rate f and from the n attached cross-bridges some detach at the rate g
This can be written as a differential equation:
In order to simplify the above differential equation to a solvable form, Huxley only considered the case where the muscle is shortening at a constanct velocity – the case of the force-velocity experiments. In our analogy this is when all the millipedes are moving at a fixed speed. If all the millipedes are walking rapidly, then the speed at which the legs extend and flex will determine the speed at which the millipedes pull themselves and the sticks together – i.e., the speed of overall shortening. From the millipede analogy, it is also easy to picture that when extended the legs go to hold the stick and when flexed the legs detach in order to extend and reattach. If the amount of flexion-extension is indicate by the displacement x of each leg, then, the rate at which the cross-bridges form, or the rate at which the millipedes’ legs form new holds on the sticks will depend on the distribution of flexion-extensionsas well as the rate at which the sticks+millipedes are pulled together (the speed of shortening, -v). That is,
The differential equation describing the formation of cross-bridges can be rewritten as:
The only independent variable is x which is the displacement of “individual myosin heads” which is analogous to the amount of “flexion-extension” on each millipede leg. It is assumed (and supported by experiment for isotonic or isovelocity conditions) that during steady movement no variation with time occurs and therefore, time is not a variable.
If the values of f(x) and g(x) are known (or assumed as Huxley did), then the above differential equation can be solved for any speed of movement (velocity of shortening v).
The solution of this equation will give the distribution of n(x), i.e., the distribution of the state of flexion-extension of all the millipede legs holding the stick. As mentioned at the beginning of this discussion, knowing the amount of flexion-extension of all the attached legs, Huxley could calculate the force produced by the muscle. The following figure shows the distribution of n(x) as a function of displacement (flexion-extension) for 4 different speeds of movement.
Figure 4: Distribution of cross-bridge distensions for different shortening velocities
If the pulling strength of each leg is proportional to the amount of extension (with proportionality constant, k) then the force can be calculated as:
where k is the proportionality constant for an indivual leg/cross-bridge.
The force produced by a skeletal muscle to move an external load depends on: (a) the level of activation (firing rate and recruitment), (b) muscle length and (c) speed of movement.
Huxley’s cross-bridge model of contraction provides a mathematical formulation to calculate force production at constant velocity muscle shortening if the rate constants of actin-myosin interaction are known or assumed.
However it may be worth noting that the properties of length dependence and velocity dependence can be explained by other theories as well. Indeed there are several competing models of muscle contraction exist using alternative ideas like electrostatic repulsion between molecules, etc.
The use of simulations was introduced in the last lecture. A mathematical model is a set of mathematical equations whose behaviour is in some sense analogous to the behaviour of a system of interest. Such models can be divided into two broad classes: (a) black-box models and (b) biophysical models.
Perhaps the best known biophysical model is the Hodgkin-Huxley model of the nerve action potential. In that model, the opening and closing of the sodium and potassium channels is modelled in terms of chemical kinetics of sub-channels. This leads to an ordinary differential equation for each of the sub-channels. Using experimental data, the coefficients of the reaction rates were estimated. The differential equations can then be solved numerically and the instantaneous sodium and potassium conductances and currents can be calculated. From this the membrane potential can be calculated.
The basic computation in the Hodgkin-Huxley model is the numerical solution of a set of ordinary differential equations.
The original computation of the Hodgkin-Huxley model was done manually. The widespread availability of digital computers with graphical display, of course, makes the HH simulation more appealing to students. Moreover, the HH model can be extended to include spatial propagation of the AP.
The length-tension curve is often idealised as a set of straight lines corresponding to changes in the overlap of the thick and thin filaments. The experimental data of the length-tension behaviour can be described by the following empirical functions.
An exponential function for passive muscle force:
And a simple parabolic function for active force:
where the constants a, b, and c are determined from experimental data as:
,
,
where Fmax is the maximum active force, Lfmax is the length at Fmax and Lw is the change in length over which the active force is non-zero. The maximum active force, Fmax, depends on the level of muscle activation which in turn depends on two factors, (a) the number of active fibres (recruitment), and (b) the action potential rate (the force-frequency relationship) of the active fibres.
Use the following simulation to perform computer experiments on length-tension measurements: Length-tension experiment
Time varying simulations
Isotonic Experiment Simulation
Recording force – transduction and amplification.
Recording length – a simple length transducer
Stimulation of electrical tissue – pulse characteristics. Synchronisation for evoked responses.
Acquisition of data into the computer – A/D conversion
© Suresh Devasahayam, Department of Bioengineering, Christian Medical College, Vellore
