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Y Space: A High Dimensional Model of Molecular Machines

By placing the magnitudes of the independent y_j numbers at right angles to one another, we form the coordinates of a single point in a space of D orthogonal dimensions. To paraphrase Shannon: ``Essentially we have replaced a complex entity (the velocity configuration of a macromolecule) in a simple environment (three dimensional space) by a simple entity (a point) in a complex environment (D dimensional space)'' [Shannon, 1949,Callen, 1985]. The space defined by all possible values of y_jis called Y space.

The reader may feel that such a high dimensional space is difficult to think about. Fortunately, it is always possible to visualize the two or three dimensional cases. We have already done this for the 2D oscillator. It is also worth keeping in mind that a point in a D-dimensional space is defined by nothing more than a list of D numbers [Conway & Sloane, 1988]. For example, a lock with 10 pins is a 20 dimensional machine because 20 numbers are needed to define the positions and velocities of the pins. Using a high dimensional space enormously simplifies the problem of understanding molecular machines because in such a space both the before and after states of the machine are represented by hollow spheres.