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The before and after states in Y Space

Let us now consider the energetics of the two states of EcoRI. When the molecule is bound to its sites in the after state, its ``pins'' have an energy determined by the thermal noise. Each possible configuration of the machine is represented by a point in Y space, and the set of all such points forms a sphere of radius

$$r_{after} = \sqrt{N_y}$$ (35)

according to equation (25).

In the before state, EcoRI must have an internal energy higher than it does in the after state, or it could not stick to the binding site in the after state. We will call the extra energy P_y, so that the total energy before is P_y + N_y. P_y is the energy difference between the states. We will assume that the energy P_y + N_yis equally partitioned between all the degrees of freedom open to the molecule (Assumption 4). This is reasonable for EcoRI since in the before state EcoRI wanders by Brownian motion along the DNA. Only when EcoRI encounters the sequence GAATTC can the energy P_y be dissipated (Assumption 5).

In Y space, the configuration of the machine is represented as a noisy vector displacement from the sphere center (equation (33)). If we add an energy P_y to the machine, the effect in Y space is to add a vector $\vec{P_y}$of magnitude $\sqrt{P_y}$ to the noise vector $\vec{N_y}$. But in the high dimensional Y space, most of this additional noisy energy will be directed at $90^\circ$ to the original noise energy because there are so many possible directions in the space. For example, if one were in the center of a three dimensional globe looking north, 2/3 of the noise would be at $90^\circ$ to the direction of sight. Likewise, if D = 100, then 99% of the noise would be at right angles to any given direction.

Therefore, as shown in Fig. 8, the two vectors $\vec{P_y}$ and $\vec{N_y}$ form a right triangle, whose hypotenuse is $\sqrt{P_y + N_y}$ according to the Pythagorean theorem. Since both $\vec{P_y}$ and $\vec{N_y}$ may point in any direction, the before state is represented by a sphere of radius

$$r_{before} = \sqrt{P_y + N_y}$$ (36)

with an energy r_before² = P_y + N_y, which is the total energy that we defined initially. It is difficult to see this geometry in three dimensions.

  

Figure 8: Geometry of Thermal Noise Spheres in High Dimensional Space.

The before sphere is represented by the outer circle, while the after sphere is represented by the line segment $\overline{CA}$. $\vec{N_y}$ is $\vec{BA}$ or $\vec{BC}$, with $\vert\vec{N_y}\vert = \sqrt{N_y}$. $\vec{P_y}$ is $\vec{OB}$ with $\vert\vec{P_y}\vert = \sqrt{P_y}$. See the main text for further description. The figure was derived from Shannon [Shannon, 1949].