The human machine: (thermo)dynamics of muscles

The previous post in this series can be found here.

The following is a guest post from Björn Malte Schäfer
(blog: cosmology question of the week, personal webpage)

How do muscles work?

Physics students learn the definition of work and mechanical energy in the first course on classical mechanics. Mechanical work is performed when a test body is moved against a force, and the work performed is equal to the (vectorial) product of the distance covered times the force, if it stays constant, otherwise you would have to evaluate an integral. If the test particle is stationary, no work is performed. But what what happens when you hold a heavy object with your arm? Even if you don't move the arm and don't perform any work from a physical point of view, it proves exhausting and after a while the arm starts aching. You at least have the feeling of having performed work, although this contradicts the physical definition of work.

Mechanical vs. molecular engines

What's wrong here? Are muscles different compared to engines? Clearly, there's a contradiction. It turns out that this example can be explained via the mechanism of molecular engines, which work very differently compared to the mechanical engines we're familiar with. And one needs to understand a bit of non-equilibrium thermodynamics!

Actin and myosin proteines

Muscles consist of two proteins called actin and myosin. Actin is in fact a very old "invention" of Nature, it is almost identical in yeast and in humans, and serves the purpose of cytokinesis, i.e. the separation of cells as well as locomotion. It consists of amino-acids and has a helical shape. Myosin is a protein that is able to change its shape under the influence of adenosin-triphosphate (ATP). It resembles a q-tip with a head that can carry out a nodding movement. The energy for changes to myosin's shape is provided by consuming an ATP molecule and dissociating it into adenosin-diphosphate (ADP). Fresh ATP is generated in mitochondria, which are small cellular organelles, by oxidation of sugars.

illustration of the actin-myosin protein assembly inside muscle cells

The actin-myosin engine

The actin-myosin engine proceeds by 5 steps:

1. actin has a "docking spot" for myosin, which gets activated by calcium ions. These ions are released by the nerve cells into the muscle as a signal for contraction.
2. myosin and ATP connect and myosin assumes a stretched, "cocked" configuration
3. myosin latches onto the docking spot on the actin molecule by molecular adhesion
4. in the combined system (consisting of myosin, actin and calcium) ATP is split into ADP and a phosphate ion, while myosin carries out a nodding movement jumping back to its non-activated configuration, and moves by about 7 nanometers.
5. the phosphate ion and ADP are released and actin moves by another nanometer.

sequence of processes involved in the motion of the actin-myosin system. credit: National University of Singapore

This cycles continues as long as the nerve cells pour enough calcium ions into the muscles and as long as there is a fresh supply of ATP. The fact that ATP is used to separate actin and myosin can be seen in rigor mortis, the stiffness of dead bodies. If no new ATP is provided by the metabolism, the molecules stay locked and the muscles don't move.

animation showing the motion of of myosin along an actin filament powered by consuming ATP. source: Vale Labs

What's wrong about this picture?

Although the motion of actin and myosin powered by the consumption by ATP looks quite convincing, there is in fact a serious objection to this picture. Clearly, the proteins are subjected to thermal motion: at temperatures slightly larger than room temperature, the molecules exchange about 1e-21 Joule of thermal energy with the surrounding. The time scale for this process is set by thermal relaxation and amounts to about 1e-13 seconds, meaning that the molecules are moved randomly by 1e-8 Watts of thermal power, much more than the 1e-16 Watts of power provided by the dissociation of 100 to 1000 ATP molecules per second at each myosin molecule. This seems to suggest that the motion of actin and myosin should be completely random, and the molecules should in fact be carrying out random walks along each other, because the power provided by ATP is very small compared to the thermal motion. Imagine to walk along a street while every second a 10-ton jet fighter bumps into you at three times the speed of sound (which is about 3.000 km/h, just for visualising the factor of 1e8 between the energies)!

Directing diffusion

The working principle of the actin-myosin engine is such that, in fact, the above argumentation does not apply! What happens is a very ingenious mechanism: the origin of motion is separated from the origin of mechanical work, as opposed to what happens in artificial engines. The actin-myosin system is a way to convert thermal, unordered motion into directed motion, and the power provided by ATP is only used for controlling the thermal random walk of the myosin relative to the actin molecules. The way Nature has achieved that is by effectively implementing a molecular ratchet, which allows thermal motion (or diffusion) in one direction, and locks the system if thermal motion wants to move the molecules into the opposite direction.

Violation of the second law of thermodynamics?

But wait - how is that be possible? Isn't that effectively a violation of the second law of thermodynamics? It would imply that there exists a system which performs work by cooling down, and physics students know this cannot be the case. Has Nature found a way to do that? Of course not, as Eddington stated: "...but if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation" (source: wikipedia).

An unusual Carnot-engine

In his famous series of lectures, Feynman has discussed a very unusual design of a Carnot-engine: a thermodynamic engine which is able to generate mechanical work from the temperature difference of two reservoirs. Let's follow his argumentation and let's then see how this idea applies to the actin-myosin system.

Imagine a propeller on an axle with ratchet mechanism which allows the propeller to rotate in one direction but locks if the propeller is turned into the opposite direction. To be exact, let's assume that one needs energy (E) to disengage the ratchet mechanism. If air molecules impact on the propeller, their kinetic energy can be used to disengage the ratchet, turn the propeller and for performing work (W).

If the propeller and the ratchet are at identical temperatures, nothing happens. But if they're not at equal temperatures, the machine is a normal thermodynamical engine, which is able to generate mechanical work from the contact with heat reservoirs at different temperatures. To move a step forward, you need an amount of energy equivalent to E+W, taken from the propeller at temperature T1, which is then split up to perform the mechanical work W (for instance, lifting a weight) and E for unlocking the ratchet, where E is dissipated. The rate at which this happens is proportional to the Boltzmann-factor exp(-(E+W)/(kT1)). To move a step backward, you need the an amount E of energy to disengage the ratchet, which is at temperature T2, and this releases an amount W of mechanical work, providing an amount of energy equivalent to E+W to the propeller and ultimately heating up the air. This happens at the rate exp(-E/(kT2)).

If the machine is reversible, the two rates are equal and one immediately obtains the efficiency of a Carnot engine from the Boltzmann factors: (E+W)/T1 = E/T2. In summary, there are two important ingredients: the asymmetric teeth of the ratchet and a non-equilibrium between the ratchet mechanism and the wheel.

Feynman's ratchet engine: gas molecules impacting on the propeller try to turn it, and the motion is directed by the ratchet mechanism. source: wikipedia

Analogy with the actin-myosin-system

The actin-myosin system is in fact the implementation of a ratchet on the molecular scale. If you could move the myosin molecule along the actin molecule, you would notice a periodic but asymmetric potential, very much like the teeth of a ratchet. ATP provides energy for lifting the ratchet and the thermal motion moves the molecules once they're unlocked. We've understood how such an unusual engine can perform work if the ratchet mechanism is not in equilibrium with the part performing work - in muscles it is not the thermal non-equilibrium but chemical non-equilibrium (by the constant supply of new ATP), but otherwise the analogy is perfect.

Conclusion

In summary, muscles work by directed diffusion. This has the consequence that even if the muscles don't lift anything, they still consume energy because the chemical non-equilibrium needs to be maintained and the mitochondria need to supply ATP so that the actin-myosin system can control the thermal motion. In this context, biological engines differ a lot from macroscopic mechanical machines. But isn't it amazing how Usain Bolt is able to cover the 100m in 9.58 seconds by thermal diffusion?

Usain Bolt dashes the 100m in less than 10 seconds by using directed diffusion in his muscles! source: The Sun

(acknowledgements: James Felce and Edna Mode helped clarifying the article and spotted many typos.)

The next post in this series can be found here.