R.S.MacKay*, Mathematics Institute, University of Warwick, Coventry CV4 7AL, U.K.
D.J.C.MacKay, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, U.K.
R.S.MacKay@warwick.ac.uk, FAX: +44: 24 765 24182
Much is known about the structure and operation of myosin. The basic cycle is believed to be
Extended M.D.P -> A.M.D.P
^ +2 | -12
| -2 A.M.D
-8 -2 | -3
Hooked M.T <- A.M.T <- A.M
where M=myosin, A=actin, T=ATP, D=ADP, P=phosphate ion, and the numbers indicate approximate free energy changes in units of kT per molecule (for rabbit skeletal muscle under no tension, from Howard). The 25kT free energy drop per cycle is enough to explain the work done by muscle, estimated at a maximum of 15kT per cycle. This can be presumed to come mainly from the “power stroke”, the right hand part of the cycle where the myosin is attached to the actin and changes conformation from extended to hooked (though the 10kT decrease during ATP binding and actin detachment also looks too good to waste and merits consideration). But what is the mechanism for the conversion of free energy to work?
A large contribution to the 15kT free energy drop in the power stroke comes from the increase in accessible volume for the P and ADP from trapped in the binding pocket to free in the cellular fluid. We propose that this entropy increase is converted to work by “ergodic pumping” of the conformation change by the P and ADP. Ergodic pumping is the time-averaged force exerted on some slow degrees of freedom by some degrees of freedom of intermediate timescale, maintained close to thermal equilibrium by many faster ones.
Firstly, based on Howard’s figures of [P]=2mM, [ADP]= 0.02mM, the volume per P in the cellular fluid is about 800 nm3 and per ADP is 80000 nm3. Supposing they start in the binding pocket with effective accessible volumes of say 1 and 10 A3, their liberation gives an entropy increase of about 27k. Part may be used to unbind the P and ADP (though compensated by strengthening of actin binding), and part is dissipated, but ample remains to do 15kT of work.
Secondly, a particle in a trap exerts a pressure on the sides of the trap, which does work by expanding the volume of the trap. The translational energy of a particle in a trap is of order only 3kT, but maintaining it at constant temperature T allows it to do work equal to T times the logarithm of the volume increase factor if the potential against which it works is well matched. Dynamical equations for the conformation change can be derived in toy models. To use the full entropy increase, ergodic pumping would need to continue until the effective volumes of the traps for P and ADP approach those per molecule in solution. This might be achieved by using electrostatic attraction to extend the traps along appropriate parts of the myosin surface.
Ergodic pumping is likely to be a significant part of the mechanism of other biomotors and could be a good design principle for force generation in nanobiotechnology.