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The Energetics and Distribution of Molecular Machines in Y Space

Our next task is to determine the distribution of all possible machine configurations at a given ambient temperature. From here on we will no longer be discussing just one molecular velocity configuration, but rather the entire set of configurations that satisfy the distribution given by (20). This collection is called a molecular machine ``ensemble'' or a ``state'' of the machine. The probability density in Y space is the product ( $$\prod$$) of the individual independent probabilities:

$$f(y_{1} , \ldots, y_j, \ldots, y_{D}) = \prod_{j=1}^{D} f(y_j) .$$ (22)

Using equation (20) this becomes

$$f(y_{1} , \ldots, y_j, \ldots, y_{D}) = {\exp(- \beta \sum_{j=1}^{D} {E}_{j} )} / z^{D} = {\exp(- \beta N_y )} / z^{D} ,$$ (23)

where N_y is the total thermal noise energy in the ``pins'' of the molecular machine. If instead of using (20), we combine (21) with (22) we obtain

$$f(y_{1} , \ldots, y_j, \ldots, y_{D}) = {\exp(- \beta \sum_{... ... {y_j}^{2} )} / z^{D} = {\exp(- \beta {r_y}^{2} )} / z^{D} ,$$ (24)

where we have used the Pythagorean theorem to collapse the D orthogonal y_j² values into a single variable, r_y, which is the radial distance from the origin to one of the possible points describing the motions of the machine. Comparing equation (23) to (24) shows that:

$${r_y} = \sqrt{ N_y } .$$ (25)

The Boltzmann distribution we have assumed for the ``pins'' implies that the parts of the machine are at equilibrium with each other. At equilibrium the machine is not dissipating energy and the thermal noise N_y is roughly the same for every possible configuration. So r_y is also roughly constant (Appendix 22). Since a constant distance does not imply any particular direction in space, the set of possible motions of the machine form a sphere in Y space.

Shannon called such spheres ``sharply defined billiard balls'' [Shannon, 1949], but perhaps the ping-pong ball is a more apt analogy because at high dimensions most of the density of a sphere is close to the surface. This is demonstrated in Fig. 4, where one can see that at higher dimensions the sphere density becomes tightly focused. The derivation of the distributions for the higher dimensions is given in Appendix 22.

Brillouin [Brillouin, 1962,Callen, 1985] gave the following simple proof of this curious property. Just as the area of a circle is proportional to the radius squared, and the volume of a sphere is proportional to the radius cubed, the volume of a D-dimensional sphere of radius r is proportional to the radius raised to the dimension D:

$$V = \frac {\pi^{\frac {D} {2}}} {\Gamma \left( \frac{D}{2} + 1 \right) } r^{D} ,$$ (26)

where $\Gamma$ is the gamma function [Sommerville, 1929,Shannon, 1949,Kendall, 1961]. Taking the derivative gives us

$$dV = \frac {\pi^{\frac {D} {2}}} {\Gamma \left( \frac{D}{2} + 1 \right) } D r^{D-1}dr$$ (27)

and dividing (27) by (26) gives

$$\frac{dV}{V} = D \frac{dr}{r} .$$ (28)

This equation means that a fractional change in the radius (dr/r) is magnified by the dimension (D) to get the fractional change in the volume (dV/V).

Even for a small molecule, D can be enormous. For example, Warshel [Warshel, 1976] modeled the light-activated ``switch'' in rhodopsin,11-cis retinal, with 200 vibrational modes. To emphasize the potential for high dimensions, we will find a minimum for the number of dimensions needed to describe the motion of this vitamin A derivative. A minimum number of atoms to model would be the 20-carbon backbone of retinal, so n = 20, d_space= 54 and D = 108.

Is this enough to create a sharply defined sphere? Suppose the radius of the sphere describing retinal is 11 in Y space units. (It doesn't matter what these units are, since they cancel in Equation (28).) Then as the radius increases from 10 to 11 units, the volume increases by $\frac{dV}{V} = 108 \cdot\frac{1}{10} \approx$ 11 fold. More than $\frac{dV}{V + dV} = 90$% of the volume is already concentrated in the outer 10% of the sphere.

Our estimate for n is conservative because we did not include the 28 hydrogen atoms in retinal nor any part of the 39,048 Dalton opsin protein to which retinal is attached [Ovchinnikov, 1982,Nathans & Hogness, 1983], nor the surrounding water and membrane (which undoubtedly have important effects on molecular motions [Levitt & Sharon, 1988,Brooks III & Karplus, 1989]). There are 5511 atoms in rhodopsin, so for rhodopsin alone D could be as high as 33,054. Not all of the atoms can be directly involved in the mechanism (Assumption 1), but it is clearly possible for the number of dimensions used by the machine to be large.

Exact calculation of the sphere density as a function of radius (Appendix 22) shows that the sphere surface becomes sharply defined at higher dimensions (Fig. 4), so the entire set of possible motions of even a small molecular machine are well depicted by a ping-pong ball.