26 December 2013

The efficiency of molecular motors

We know since the work of Sadi Carnot that the efficiency \(\eta\) (the fraction of heat converted into work) of a thermal engine cannot exceed a surprisingly simple maximum value, \(\eta_{\text{max}} = 1 - T_c/T_h\), defined in terms of the absolute temperatures of the cold and hot heat sources, \(T_c\) and \(T_h\). This limitation applies to a wide variety of devices, from combustion engines to solar cells: in the latter case, \(T_h \simeq 5800 \, \text{K}\) is that of the Sun and \(T_c\) is the ambient temperature, yielding \(\eta_{\text{max}} = 93\%\) [1].
We also know that living organisms can operate at (or even below) the temperature of their environment. In these conditions, Carnot's formula would yield zero efficiency, and thus no work production. And yet, our muscles can reach an efficiency of above 50% [2], higher than that of our cars! How can we solve this paradox?
Obviously, muscles are not heat engines, although both use chemical energy. The question is then how to formalize that distinction and show that Carnot's formula applies in one case, but not the other.

A (characteristically bold) answer was put forward by E. T. Jaynes [3]:  in muscles, the energy is converted from chemical to mechanical form without first being "degraded into heat". The system is not in equilibrium, the energy \(E\) being concentrated in only a few degrees of freedom \(N\). Jaynes argues that \(E/N\) is of the order of a few thousand Kelvin and replaces the temperature by this ratio  in the efficiency formula. His conclusion is that:
 [A] muscle is able to work efficiently not because it violates any laws of thermodynamics, but because it is powered by tiny "hot spots" of molecular size, as hot as the sun [3b].
However, Jaynes' description lacks an essential ingredient: any such energetic degrees of freedom are hot spots of nanometer size that would equilibrate over nanoseconds, much faster than the typical time scales of molecular motors. In practice, they must be prevented from leaking their energy into the environment until the opportune time by something like a ratchet, but a ratchet at equilibrium cannot beat the Carnot limit, as shown by Feynman [4]. Nonequilibrium fluctuations must be present in the system.

to be continued...


[1] C. H. Henry, Limiting efficiencies of ideal single and multiple energy gap terrestrial solar cells, J. Appl. Phys. 51, 4494 (1980). In practice, the black body spectrum must be corrected for atmospherical absorption.
[2] Frank E. Nelson et al., High efficiency in human muscle: an anomaly and an opportunity?, J. Exper. Biol. 214, 2649 (2011).
[3] E. T. Jaynes,(a)The muscle as an engine (1984?); (b)Clearing up mysteries - the original goal (1989).
[4] R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics (Chap. 46), Addison Wesley (1963).

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