>

Location of Spheres in Y Space

The square of a distance in Y space is equal to the energy required to traverse that distance. Suppose that there are two after states of the machine. (That is, two gumball spheres.) If the distance from the first state to the second state is big enough, then the velocity configuration of the machine--which is represented by a point--would almost never be able to jump from one sphere to the other and the two spheres would be separated by an energy ``barrier''. On the other hand, if the distance between the sphere centers were small, the two spheres would be connected and the machine would often have enough thermal energy to make the transition to the other state.

An analogy is useful to see how the states could become connected. Suppose that we have a coin lying in a tub. There are two states, heads up and tails up. A certain fixed minimum amount of energy is required to lift the edge of the coin in order to flip it over. If we start to vibrate and shake the tub, then the probability that the coin will switch to the other side increases. If we successively replace the coin with each of the 5 regular Platonic solids --tetrahedron (4 sides), cube (6 sides), octahedron (8 sides), dodecahedron (12 sides), and icosahedron (20 sides)-- while keeping the mass the same, then switching between sides (states) becomes increasingly easy. With more intense shaking, the states also become less and less distinct.

The tub vibrations correspond to the temperature, which determines the radius of the molecular machine's spheres according to equation (32). Thus, at higher temperatures the sharply defined spheres overlap and the states are no longer distinct. A molecular example is the heat denaturation of double stranded DNA.

Specifying the location of the center of a sphere in Y space specifies the average configuration of the molecule relative to other possible configurations. To be able to discuss several spheres at once, we can represent the shape of the Y space ensemble with a vector notation:

$$\vec{y} = \vec{s} + \vec{N_y} .$$ (33)

The center of the sphere is defined by a vector, $\vec{s} = (s_1, \ldots, s_j, \ldots, s_{D})$, while the instantaneous radius of the current point on the sphere is defined by the vector $\vec{N_y} = (y_1, \ldots, y_j, \ldots, y_{D})$. The magnitude of $\vec{N_y}$ is given by any of the relations (25), (29), (32) or

$$\vert \vec{N_y} \vert = \sqrt{N_y}.$$ (34)

The s_j and y_j variables play important roles in this paper, since they correspond to the signal samples Shannon used in his theory. The set of variables that define the center of each sphere, s_j, plays the part of DC voltages, while the y_j correspond to AC voltages due to thermal noise (Appendix 21).