10.3. Molecular implementation of mechanical principles[[Student version, January 17, 2003]] 371
a A B
DC
b A B
D
C
Figure 10.13:(Diagram.) (a)Afully connected reaction diagram. (b)Asparsely connected reaction diagram.the helix–coil transition (Section 9.5.1), or the binding of oxygen by hemoglobin (Section 9.6.1)
surprisingly sharp. Similarly, a macromolecule can be quite specific about what small molecules
it binds, rejecting imposters by the cooperative effects of many charged or H-bonding groups in a
precise geometrical arrangement (see Idea 7.17 on page 231).
10.3.2 The reaction coordinate gives a useful reduced description of a chemical event
chemical event
The idea of multistability (the third point in Section 10.3.1), sometimes justifies us in writing
extremely simple kinetic diagrams (or “reaction graphs”) for the reactions of huge, complex macro-
molecules, as if they were simple molecules jumping between just a few well-defined configurations.
The reaction graphs we write will consist of discrete symbols (ornodes)joined by arrows, just like
many we have already written in Chapter 8, for example the isomerization reactionABstudied
in Section 8.2.1. A crucial point is that such reaction graphs are in generalsparsely connected.
That is, many of the arrows one could imagine drawing between nodes will in fact be missing,
reflecting the fact that the corresponding rates are negligibly small (Figure 10.13). Thus in many
cases reactions can proceed only in sequential steps, rarely if ever taking shortcuts on the reaction
graph. Usually we can’t work out the details of the reaction graph from explicit calculations of
molecular dynamics, but sometimes it’s enough to frame guesses about a system from experience
with similar systems, then look for quantitative predictions to test the guesses.
What exactly happens along those arrows in a reaction graph? To get from one configuration
to the next, the atoms composing the molecule must rearrange their relative positions and angles.
Wecould imagine listing the coordinates of every atom; then the starting and ending configurations
are points on the many-dimensional space of these coordinates. In fact they are special points, for
which the free energy is much lower than elsewhere. This property gives those points a special,
nearly stable status, entitling them to be singled out as nodes on the reaction graph. If we could
reach in and push individual atoms around, we’d have to do work on the molecule to move it away
from either of these points. But we can instead wait for thermal motion to do the pushing for us:
Chemical reactions reflect random walks on an energy landscape in the space
of molecular configurations.(10.10)
Unfortunately the size of the molecular configuration space is daunting, even for small molecules.
Toget a tractable example, consider an ultra-simple reaction: A hydrogen atom, called Ha,collides
with a hydrogen molecule, picking up one of the molecule’s two atoms, Hb.Todescribe the spatial
relation of the three H atoms, we can specify the three distances between pairs of atoms. Consider
for example the configurations in which all three atoms lie on a single line, so that the two distances
dabanddbcfully specify the geometry. Then Figure 10.14 shows schematically the energy surface for