J. theor. Biol. (1984) 110, 131-133

LETTER TO THE EDITOR

A Comment on "Some Aspects of the Role of Quantum Mechanics in the Theory of Muscle Contraction"

The letter by Brugman et al. (1984) on our early work on muscle contraction (Gray & Gonda, 1977a, b) contains so many errors that a reply is called for, particularly as the original theory has been expanded in later work on muscle (Cheung & Gray, 1981a, b, 1983), mitochondrial coupling (Blumenfeld, 1981), enzyme catalysis (Volkenstein, 1981), photosynthesis (Rich & Hill, 1982), and general biophysics (Gonda & Gray, 1980).

Firstly, with respect to the "particle in a box" model of a molecular machine, which was included for purely pedagogical reasons, one cannot calculate the force it can exert in two ways as stated by Brugman et al. (1984). The statistical method is not available because we are not dealing with large numbers of systems, or a pressure P arising from collisions of many particles with the wall.

One has to use the first or purely quantum mechanical method, as we did, as that is the only one available for a single quantal system.

We did not do the calculation erroneously as they state, in fact on p. 170 (Gray & Gonda, 1977a, p. 170) we state: "defining force as we would classically i.e.

Ov

Ox

t h e n . . . " In their Appendix, Method 1, Brugman et al. have pointlessly repeated

this calculation and got the same result that we did i.e.

2E Fx = ( f ) =~o.

The second "problem" they discuss (Gray & Oonda, 1977a, p. 172) is entirely of their own making arising from their fundamental error in defining work as S P d V where P is the internal pressure (force) exerted by the particle on the wall. In an irreversible change when the external force F~xt is not identically equal to the internal force (or pressure) per unit area P, the work done by the system should be calculated correctly as (Gray &

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132 B . F . G R A Y A N D I. G O N D A

Gonda, 1977a, 1982)

f Fex t d V = F e x t A V

for constant external load. J P d V is simply the change in internal energy of the system, and has nothing whatever to do with work done by the system except in a reversible change, which cannot be achieved in a molecular machine (or at least it would not be efficient; Gray, 1975).

There was no "energy discrepancy" in our work, we simply showed that in the necessarily irreversible operation of a molecular machine, some energy is dissipated as heat. This heat is in fact due to the inevitable mismatch between the external and internal forces as explained above.

There is no need to identify the force acting on the wall of the box with gravity as Brugman et al. (1984) did. A better analogy is to regard the position of the "wall" as a nuclear coordinate and the "particle" to be an electronic cloud. The position of the nuclei is then governed by the internal "pressure" of the particle ("electrons") and the external force which, in the simplest case, is constant. Under these circumstances, the net force on the barrier may be approximated by a force proportional to the displacement from the equilibrium position, and the energy levels of the barrier are therefore approximated by those of a harmonic oscillator. We have shown previously (Gray & Gonda, 1977a; Gonda & Gray, 1980) that the motion of the barrier can be adiabatic in a realistic molecular machine, and therefore the spontaneous loss of energy as envisaged by Brugman et aL would not OCCUr.

We have been careful to explain (Gray & Gonda, 1977b) that there are many misconceptions about the nature of different vibronic states. Thus, two conformational isomers of a protein are different vibronic states of the same molecule. Therefore, the criticism of Brugman et al. (1984) regarding the lifetime of photochemically excited states does not apply to our general theory. Moreover, the probability of a vibronic transition is dependent upon the overlap of nuclear wave functions. In a molecular machine, a high probability of transition would exist only at the beginning and the end of a working stroke (Gray & Gonda, 1977b).

As for the final paragraph in Brugman et al. (1984) calling for a quantum theoretical approach but in the direction of absolute reaction rate theory, we would simply point out that: absolute reaction rate theory is classical, not quantum mechanical (Wigner, 1937, 1938); and that absolute reaction rate theory has not yet given an adequate, semi-quantitative description of even a simple solution reaction such as the boat-chair isomerization of cyciohexane.

LETTER TO THE EDITOR 133

A different app roach is called for, perhaps , a long the l ines previous ly suggested by us ( G o n d a & Gray, 1980).

School of Chemistry, Macquarie University and School of Pharmacy, University of Sydney, Sydney, N.S. W. Australia

(Received 14 February 1984)

B. F. GRAY

AND

I. GONDA

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vol. 2, p. 609. (Srinivasan R. ed.) GRAy, B. F. & GONDA, I. (1977a). J. theor. Biol. 69, 167. GRAY, B. F. & GONDA, I. (1977b). J. theor. Biol. 69, 187. HILL, R. & RICH, P. R. (1982) t Proc. natn. Acad. Sci. U.S.A. 80, 978. VOLKENSTEIN, M. (1981). J. theor. Biol. 89, 45. WIGNER, E. P. (1937). Z chem. Phys. 5, 720. WIGNER, E. P. (1938). Trans. Faraday Soc. 34, 29.