Actin-myosin crossbridge cycling in skeletal muscle
(and Hill's equation, and two demos...)

Terminology: By "muscle" we mean whole muscle, such as the gastrocnemius or biceps or deltoid muscle.

Whole muscle is made of many muscle fibers in parallel. One fiber may be as long as the whole muscle, but only about 10 microns in diameter. We will discuss the orientation of muscle fibers in whole muscle-pennate muscle has diagonally oriented fibers to increase strength at the expense of maximum length change. A muscle fiber is a muscle cell. During development many myoblasts coalesce into one fiber, resulting in one cell with many nuclei.

T. A. McMahon, Muscles Reflexes and Locomotion, Princeton Univ. Press (1984). see Chpt 1.
NOTE. Tom McMahon, a famous biomechanics professor at Harvard, inventor of the tuned track, died in early 1999, at age of 55.

Mechanical use of muscle
Passive muscle has the length-tension properties of a nonlinear spring. Active muscle contracts, or creates isometric tension. When it contracts (shortens) it may reduce the angle of a joint, or force blood out of the heart, or rotate the eyes, or generate speech. In the skeletal system muscle is fundamentally a torque or moment generator.

If the load a muscle pulls against is at an equilibrium torque value T0, muscle (+tendon) won't contract, but will develop tension (isometric). How long active muscle can maintain tension depends on how well supplied with energy it is; muscle which develops tension most quickly is the type of muscle most likely to fatigue during sustained activity.

Muscle joins to bone by the series connection of tendons. Tendons are passive collagenous tissues which cannot contract. It will turn out that various experiments which demonstrating muscle properties utilize muscle disconnected from any bones and held by tendon, connected to a force gauge.

History. I imagine cavemen scrutinized muscle when they cooked it over fire. Heat-treatment of muscle in rigor mortis further stiffens the tissue by cross-linking proteins. Scientific study of muscle probably began with the Greeks, and Aristotle. See Martha Nussbaum's translation of Aristotle's De Moto Animalium [On The Motion of Animals]. Aristotle made the analogy that animals moving were like puppets, controlled by "strings" from what centuries later would be known as nerve fibers of the central nervous system.

Passive muscle
Passive muscle has a nonlinear stress-strain curve. The first equation in TAM, on page 8, says that the rate of stress change as a function of strain change in unstimulated muscle is proportional to stress.

more detail:

taking care to look at the limits of integration of the natural logarithm, derived from

McMahon makes an analogy between stress-strain in passive muscle and stress-strain in yarn. Pull slowly on loose yarn. As more strands of yarn are brought into tension, the yarn resists increased pull in the nonlinear way suggested by the equation above. In the reading of TAM chpt 1, see Fig. 1.6. The bottom curve is for passive muscle. Below rest length L0 muscle offers no resistance to being stretched. Loose yarn doesn't resist until at least a few strands are pulled to their full length. Passive stress continues out to 1.5 L0, before nearing the limit of elastic deformation.

The upper curve in TAM Fig. 1.6 shows stress-strain for active muscle. Notice stress now starts at 0.5 L0. The curve is a combination of active and passive effects. If the passive curve of Fig 1.6 is subtracted from the stimulated curve, then a bell-shaped curve in Fig. 1.7, labeled"Developed" by TAM, results. Note that the maximum tension of active muscle is not at the greatest length. TAM explains the bell shape of the active curve later, in his chapter 3. See Fig. 3.7. At the sarcomere length where all myosin fibers can make contact with actin, muscle exerts maximum tension. Otherwise muscle is on one side or the other of the bell curve which represents a lack of actin-myosin crossbridge coupling. We discuss below the crossbridge cycling mechanism which accounts for the increase in stress of active muscle.

Actin and myosin crossbridge cycling:
Actin and myosin are filamentous proteins which interlock and overlap in a way to produce length change and tension development in skeletal muscle. First we discuss the arrangement of myosin and actin in muscle sarcomeres. In the schematic below approximate relationship between thick myosin and thin actin fibers is illustrated. Not shown is the inhibitory tropomyosin molecule. Histological labels are shown in capital letters. The A label marks the myosin position, not the actin! The M line marks the midpoint of the myosin fibers, the Z line marks the midpoint of the actin fibers. H is the distance between the ends of actin filaments.

End-on, myosin and actin form overlapping hex and triangular arrays, as shown below.
The ratio of thin to thick filaments is 2:1
Myosin filaments are about 1.66 microns in length; actin filaments are about 1.0 microns. Bundles of myosin and actin group together in myofibrils, and the fibrils group into muscle fibers. Remember: A fiber is one (multi-nucleated) muscle cell.

The structure of myosin is now known in great detail, amino acid by amino acid. In 1993 Science (24th December issue) selected myosin as 1st runner-up for Molecule of the Year. The complete 3-D crystal structure of myosin, to a resolution of 2.8 , was published, and specific locations where the myosin head could flex and where ATP could bind were suggested. A similar structure for actin was published in 1992.

TAM chapters 3-5 go into detail about the mechanism of crossbridge attachment and force development in muscle fibers. Here we outline what is known.

Explaining the "inverted U" shape of muscle Length-Tension curve. See Fig 3.7, p. 67, of McMahon. 1μ actin fiber strand on either side of 1.67μ myosin strand; eventually the actin fibers overlap and at the depth of contraction distort myosin and reduce tension.

Gordon, Huxley & Julian (1966) Thick and thin filaments overlap. Least tension for 1 and 6. Most tension 2 and 3.

Before actin-myosin crossbridge cycling can occur tropomyosin must be moved out of the way. Troponin is a component of tropomyosin; it blocks myosin from its binding site. Tropomyosin is inhibited (moved) by calcium, which is released from sacroplasmic reticulum in muscle. Sarcoplasmic reticulum is induced to release calcium ions when a voltage reversal occurs at the muscle cell membrane. Neurotransmitter (acetylcholine, ACh) from a motoneuron axon interacts with a receptor at the neuromuscular synapse to open sodium channels which lead to the transient membrane voltage reversal. And all that happens BEFORE crossbridges can cycle! Here is a narrative of the events:
1. Pulse travels down motoneuron axon to neuromuscular synapse.
2. Depolarization of pulse at presynaptic terminal causes packets of ACh to be released.
3. ACh diffuses across 10 angstrom synaptic gap to cause conformation change in postsynaptic receptor protein.
4. Channels in receptors open and allow Na+ ions to flow in the muscle cell.
5. Inward sodium current depolarizes muscle cell from negative potential.
6. Depolarized state causes sacroplasmic reticulum to release calcium in muscle cell.
7. Elevated Ca++ level in muscle cell forces tropomyosin to change conformation and pull away from actin and myosin.

As long as tropomyosin is out of the way actin and myosin can interact.
1. The head of a myosin molecule binds to actin. There is ADP still on the myosin from a previous cycle, and this ADP is now released, resulting in...
2. Myosin heads flex in tandem, pulling the actin strand 50-100 angstroms into the myosin channel and contracting the muscle. If the load on the muscle is too great, no movement occurs, but tension develops. The myosin head flex is an example of protein conformation change. The energy required to flex the myosin head has been previously stored in the myosin molecule and is now expended.
3. The conformation change in myosin exposes a site where ATP can bind. If ATP is present it binds on the flexed myosin. Binding causes release of myosin from actin.
4. The myosin head releases from the actin; Hydrolysis of ATP to ADP causes the myosin to straighten out. The split of ATP (into ADP + phosphate) stores energy in the myosin head and releases some heat. Phosphate leaves the reaction site.
5. The cycle starts again if the myosin head finds a new actin binding site.

If ATP is not present and tropomyosin is inhibited, the myosin stays bound to actin. If ATP never returns, then a state of rigor mortis ensues. If the load is such that actin doesn't move with respect to myosin, the myosin stays linked to the actin for a longer time. The ATP doesn't have a chance to latch onto the myosin and release its energy and heat. As a consequence, isometric muscle releases less heat than contracting muscle. If the load is so great that muscle is stretched during crossbridge cycling, then crossbridges are prematurely torn apart and micro-injury occurs in the muscle. The micro-injury is a stimulus for muscle to "repair" itself into a strengthened state. (Thus muscle-building during weight-lifting occurs only during the "negatives." A negative occurs when the weight is slowly lowered by the lifter. As you lower yourself during a push-up, you do a negative. So too with a chin-up.)

Nature 352 (25 July, 1991): pages 284 & 352, "Biomechanics goes quantum," The power stroke is 10 nm; the myosin head is about 20 nm long.

The more overlap of myosin and actin during crossbridge cycling, the greater the tension in the muscle. Why? more myosin heads can make contact with actin! At rest length L0 actin and myosin overlap to a maximum degree, thus the peak in the active length-tension curve seen in TAM Fig. 1.7.

Overall muscle strength depends on length of the muscle. Try sensing strength of grip with various wrist angles. At what wrist angle is grip strongest? Why?

Twitch and tetanus of muscle
Hill's equation
Years before the protein constituents of muscle were known, experiments were done on the mechanical properties of isolated whole muscle. On page 10 of TAM, dynamic ("quick-release") experiments on isolated muscle described. A muscle is stretched to a constant length while stimulated to tetanus, then it is released against a variable load. Below are shown a series of quick release length changes as suggested by the single graph of TAM's fig. 1.8(b). The length Δx1 decreases more in the top graph because the muscle has the opportunity to contract against the light load. As TAM says, pay particular attention to the (negative) slope of the length change at time of release; that's the velocity of shortening, v.

A series of quick release experiments will show an inverse relationship between load and velocity of shortening. A. V. Hill fitted the data empirically to the equation

where v is shortening velocity, T is the variable load and T0 is the particular load at which the the muscle doesn't change length, and a and b are empirical constants. A curve of v vs T has a hyperbolic shape. (What are the asymptotes of the hyperbola?) It turns out that a wide variety of muscles tested by quick release result in about the same values of a and b, and that is the importance of Hill's equation.

When velocity v is zero then T = T0.
Maximum velocity of shortening occurs when T=0.

TAM says on p. 24, "For purposes of comparing different muscles, Hill's equation can be written in a normalized form,"

Muscle power
Power is energy/sec. Energy is force X distance.
In Hill's equation the term Tv has units of power.
Here's one way to isolate the power Tv in terms of v:
Divide both sides of Hill's equation by

which is equation 1.5 in TAM with all the steps shown.
See that the graph of Power vs Velocity has a maximum (fig in TAM). Given typical values of a and b, maximum power is delivered when velocity of shortening is about half of

Look at the plot of T vs v of Hill's equation if v becomes negative. The force continues to increase. What does negative velocity mean? What is a "negative" in muscle building? It means the muscle is being stretched during stimulation, instead of being able to contract.

Muscle mechanics around joints
Why is muscle so long? Why isn't muscle short and stubby, near the joint (b)? On the other hand, why is muscle so short? Why not be longer, and stretch from one end to the other? (c) What mechanical advantage (or disadvantage) occurs for the way muscle acts to rotate bones at joints? Keep in mind that muscle has a limited range of contraction, compared to the large amount of movement required at the ends of the limbs.

The situation in (a) means that a long muscle has a chance for a significant shortening ll, whereas (b) is too short to create much rotation. The great leverage of (c) is impossible unless a pulley re-directs force, or the muscle extends across webbing.

Types of striated muscle
So far we have been assuming that striated muscle is of one kind. Yes, all striated muscles use the same actin-myosin ratcheting mechanism for contraction and development of tension, but in fact there are two major types of striated muscle fiber. The distinction is greatest in non-mammalian vertebrates, where the fibers are called FAST and SLOW. On a fiber-by-fiber basis FAST and SLOW compare in the following ways:

large maximum tension small maximum tension
short delay from stimulus to tension longer delay
easily fatigued greater endurance
white appearance the "dark meat"
anaerobic metabolism aerobic

Swordfish fillet demo
In mammals the distinction is between fast and slow twitch muscle, which are more properly called FG, for fast-glycolytic and SO for slow-oxidative. Muscle fibers remain segregated in regard to which motoneurons innervate them. By virtue of their performance as strong, easily fatigued fibers the FG fibers are used for jumping, kicking and other movements which are intense and brief (predator, like swordfish, going after prey...). On the other hand SO fibers may be involved in posture and sustained activity such migratory bird flight, where endurance is at a premium.

Muscle fatigue, strengthening and injury
Muscle changes. Over the short term it contracts and warms up. If exercised too much, it can fatigue. Certainly white fast-twitch muscle can fatigue biochemically after a dozen twitches because ATP is depleted. Another kind of "fatigue" results from the micro-injury of active muscle that is lengthened; this kind of fatigue results, over a day or so, in strengthening. Just what is the biochemical stimulus for strengthening due to the micro-injury of "negatives"? It's not clear to researchers. See Chpt 2 of TAM for some thoughts. It is known that increase in muscle bulk comes from a thickening of existing fibers, instead of growth of new fibers.

If muscle is not used, it atrophies. If muscle is in weightlessness, it atrophies a lot! Unused muscle around bone which has healed from a fracture is smaller than the muscle around active bone. Soviet cosmonauts learned the hard way about the atrophy of disuse; they were unable to walk away from their space vehicle when it returned to earth after months of weightless orbit! American astronauts exercise on treadmills in the Space Shuttle while it is in orbit, to minimize the atrophy of muscle not required to resist gravity.

As far as I know, tendon and ligament cannot be strengthened by use or weakened by disuse. Bone? I think studies have shown that limb bone unused for long periods of time loses some of it capacity to "remodel" after stress.

A muscle is thickest in its middle, thinning as it approaches the tendons. Since, for a muscle in static equilibrium, the tension is uniform along its length, the thin sections of muscle bear the greatest stress. The points of contact with tendon are therefore the most likely places for injury. You'd think Nature would have made the injury threshold of muscle greater than it is possible for an animal to exert, so self-inflicted injury could not occur. However, it is sometimes possible to tear or "pull" muscle by exerting too much if the muscle has not been "warmed up." Some muscles, such as the hamstring, are especially susceptible. If I dive cold into the pool and start a strenuous breaststroke frog kick immediately, I could pull a hamstring.

Muscle injuries most often occur when an external force [often contact with the ground] causes an unnatural pull. Muscles, however, are not the weakest links in the musculo-skeletal system. As sports medicine specialists will testify, ligaments tear before muscles pull. Thin tendons are generally more susceptible than muscle itself, too.

Muscle can have problems other than fatigue. Cramps occur when an errant hormonal or neural signal causes all the fibers in a muscle to contract at once. An epileptic seizure starting in the cortex can send synchronous excitation down through the motor system, contracting muscles simultaneously.

%---------------archive Notes-----------------------