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Abstract by Robert S. MacKay

Ergodic pumping: a proposed mechanism for the power stroke of myosin

R.S.MacKay, Mathematics Institute, University of Warwick, Coventry CV4 7AL, U.K.

mackay@maths.warwick.ac.uk, FAX: +44: 24 765 24182

Thanks to many ingenious experiments, much is known about the structure and operation of myosin. In particular the basic cycle is believed to be

            Detached               Attached

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 for each step in units of kT per molecule (for rabbit skeletal muscle under no tension, adapted 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 (called UP and DOWN by Geeves) (though the 8kT liberated on detachment also looks too good to waste and merits consideration). But what is the mechanism for the conversion of free energy to work?

I propose that most of the 15kT free energy drop in the power stroke corresponds to increase in accessible volume for the P and ADP from trapped in the binding pocket to free in the cellular fluid, and that this is converted to work by "ergodic pumping" of the conformation change by the P and the ADP. Ergodic pumping is the time-averaged force (in this case pressure) exerted by degrees of freedom of intermediate timescale between slow ones on which they act and fast ones maintaining them close to thermal equilibrium. This proposal is justified as follows.

Firstly, based on Howard's figures of [P]=2mM, [ADP]= 0.02mM, the volume per P in the cellular fluid is about 800,000 A3 and per ADP is 80,000,000 A3. So an entropy increase of 15k (i.e. volume factor e15 = 6x106) would be consistent with their starting in the binding pocket with effective accessible volumes of say 1 and 10 A3 (perhaps they start with less accessible volume but pay a corresponding energy to unbind).

[If instead we take [ADP] = 1 mM and [Pi] = 8 mM (from another source), the volume per P is 200,000 A3 and per ADP is 1.6x10^6 A3.]

Secondly, a particle in a trap begins to exert a pressure when the trap diameter exceeds the particle's de Broglie wavelength h/sqrt{mkT} = 0.3 A for P at 300K. The pressure of a particle at 300K in a trap of size 1 A3 for example is of the order of 5x109 Pa, though this is partly balanced by pressures from outside. The translational energy of a particle in a trap is of order only (3/2)kT, but maintaining it at constant temperature allows it to do work equal to T times the logarithm of the volume increase factor: the particle sucks heat out of the surroundings. Effective equations of motion for the conformation change can be derived in toy models.

Ergodic pumping is likely also to be the mechanism of motion of kinesin and could also be a good design principle for force generation in bionanotechnology.

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