>

Lock-and-Key Model of a Molecular Machine

The state of a molecule is defined by the positions and motions of its atoms. To determine the locations of the n atoms in a molecular machine, we first define a coordinate system. Three spatial coordinates are needed to locate each atom, so we need 3n numbers. In many cases we won't care if the molecule is tumbling or moving through space, so we can affix the coordinate system to the molecule's center of mass and ignore the six numbers that describe the coordinate system's orientation and position in space. So for the positions we need no more than:

 

d_space= 3n - 6 (1)

coordinate numbers (Assumption 1). ³ These coordinates are called ``degrees of freedom''. We also need d_space numbers to describe the velocities.

A molecular machine can only use a few of these degrees of freedom because many of the atoms are required as structural components. In this context it is useful to extend the lock-and-key analogy of biological interactions [Rastetter, 1983,Gilbert & Greenberg, 1984]. A key opens a pin-tumbler lock by moving a set of two-part pins to positions which allow the two parts to separate when the key is turned [Roper, 1976,Macaulay, 1988]. The wrong key will leave one or more pins in a position that blocks the turning, and this will prevent the bolt from being released. Assumption 1 is that we only need to account for the motions of clusters of atoms--the molecular machine's ``pins''--in order to describe its operation. Likewise, it is not necessary to keep track of the individual atoms in a lock in order to understand how it works.

A second, closely related assumption is that the parts of a molecular machine move independently (Assumption 2). Likewise the pins in a lock move independently. Yet because of the design of a lock, the bolt can only move if the pins are all aligned correctly by the key. Thus, although the individual pins are independent, they must ``cooperate'' for the lock to open. If two pins were not independent, then it would be easier to pick the lock, and it would not carry as much ``protective'' information because one pin could be set and the position of the other would be determined. For example, two pins fused together would act as one pin. Thus, in this analogy, d_space refers to the number of ``pins'' used by the molecular machine, which is quite likely to be much smaller than the degrees of freedom:

$$d_{space}\ll 3n - 6 .$$ (2)

That is, the important degrees of freedom are not all of the degrees of freedom of the molecule, but only those directly involved in the machine operation. We only need to account for these to describe the machine's operation. Estimates of n and d_space for rhodopsin will be discussed later.