372 Chapter 10. Enzymes and molecular machines[[Student version, January 17, 2003]]
dbc
dabdbc
dabreaction
coordinateoutgoingab cabcincomingenergyenergyHa - Hb DistanceHa - Hb
distancea b
Figure 10.14:(Schematic; sketch graph.) (a)Asimple chemical reaction: a hydrogen molecule transfers one of its
atoms to a lone H atom, H + H 2 →H 2 +H. (b)Imagined energy landscape for this reaction, assuming that the atoms
travel on one straight line. The dashed line is the lowest path joining the starting and ending configurations shown
in (a); it’s like a path through a mountain pass. The reaction coordinate can be thought of as distance along this
path. The highest point on this path is called the transition state.
the reaction. The energy is minimum at each end of the dashed line, where two H atoms are at the
usual bond distance and the third is far away. The dashed line represents a path in configuration
space that joins these two minima while climbing the energy landscape as little as possible. The
barrier that must be surmounted in such a walk corresponds to the bump in the middle of the
dashed line, representing an intermediate configuration withdab=dbc.
When an energy landscape has a well-defined mountain pass, as in Figure 10.14, it makes sense
to think of our problem approximately as just a one-dimensional walkalong this curve,and to
think in terms of the one-dimensional energy landscape seen along this walk. Chemists refer to
the distance along the path as thereaction coordinate,and to the highest point along it as the
transition state.We’ll denote the height of this point on the graph by the symbolG‡.
Remarkably, the utility of the reaction coordinate idea has proven not to be limited to small,
simple molecules. Even macromolecules described by thousands of atomic coordinates often admit
auseful reduced description with just one or two reaction coordinates. Section 10.2.3 showed how
the rate of barrier passage for a random walk on a one-dimensional potential is controlled by an
Arrhenius exponential factor, involving the activation barrier; in our present notation this factor
takes the form e−G‡/kBT.Totest the idea that a given reaction is effectively a random walk on
aone-dimensional free energy landscape, we write^7 G‡/kBT =(E‡/kBT)−(S‡/kB)Then our
prediction is that the reaction rate should depend on temperature as
rate∝e−E
‡/kBT. (10.11)
Indeed, many reactions among macromolecules obey such relations (see Figure 10.15). Section 10.3.3
will show how these ideas can help explain the enormous catalytic power of enzymes.
T 2 Section 10.3.2′on page 399 gives some more comments about the energy landscape concept.
(^7) T 2 More precisely, we should use the enthalpy in place ofE‡.