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Machine Operations in Y Space

Once the machine dissipates energy, the vector $\vec{P_y}$ becomes a specific direction relating the before and after states. Referring to Fig. 8, we see that the before sphere has its center at point O, while the after sphere has its center at point B. In this two-dimensional diagram, the after sphere is represented by the line segment that extends from C to A. (The after state is still spherical, but the two-dimensional diagram cannot show it. In three dimensions, the after sphere would be represented by a circle at a particular latitude on a globe.) Because of the high dimensionality most of the after sphere is ``flattened'' at $90^\circ$with respect to the specific direction of $\vec{P_y}$, which is shown as $\vec{OB}$ in the figure.

Therefore, the machine operation corresponds to the motion of the sphere center from O to B with a concomitant collapse of the radius, and loss of energy P_y to the surrounding medium.

Since $\vec{s}$ represents the average configuration of the machine, a change in the sphere center, $\Delta \vec{s} = \vec{P_y}$, corresponds to a change in the average physical configuration of the molecular machine, and different directions and magnitudes of $\vec{P_y}$ in Yspace correspond to different state changes. Furthermore, the location of a small after noise sphere within the larger before sphere represents only one of several possible states of the machine since there can be several non-intersecting after spheres [Perutz, 1970,Chothia & Lesk, 1985,Porter et al., 1983]. Placement of the spheres according to equation (33) is called the molecular machine's coding scheme because the packing of spheres in space corresponds to the arrangement of code words in a communications system [Sloane, 1984]. The total dimensionality, D, determines how sharp-edged the spheres are, and so this controls the intensity of the threshold effects if two spheres overlap [Shannon, 1949]. Thus, the precision of a molecular machine depends on its size. If the machine is big enough ($n \gg 1$), then the noise is predictable because D may become so large that spheres are sharp-edged. By evolving to be big, even single molecules can have macroscopic stability. If the machine contains enough independent components, then the spheres may also be placed accurately in the space of Ddimensions so that they just barely miss contacting each other. Thus, the machine can have distinct after states. However, since the spheres are defined by a smooth analytic function (f_D(r), equation (48)), they always overlap and there is always a small probability that a machine in one after state can jump into another after state. The rate of such transitions (or incorrect transitions from before to after) is the error rate.

Of course, simply increasing the number of atoms in order to raise the dimensionality does not guarantee accurate placement of the spheres. However, the number of ``pins'' can be increased during evolution of the machine, so the placement could be refined. This suggests, for example, that many of the amino acids in a large protein could have subtle effects on the sphere placement and coding [Needels et al., 1989]. These effects could be missed by conventional genetic approaches that are based on the premise of finding ``the'' major recognition factor. For example, recent X-ray crystal structure determination of a tRNA synthetase bound to its cognate tRNA [Rould et al., 1989b,Rould et al., 1989a,Perona et al., 1988] suggests that the complete set of tRNA recognition factors is spread over a large surface of both molecules [Hou et al., 1989] (as one would expect from this theory) rather than concentrated in the anti-codon or other small regions.

We should emphasize that the configurations (points in Y space) that we have been considering are in either the before or the after states. We have not looked at configurations during the operation. Since the energy changes during an operation, a set of such configurations must connect the before to the after spheres. As we will see in the next section, it is to our advantage to focus only on the simple spherical before and after states, for together these characterize what the machine is able to do.