334 Free energy calculations
two dihedral angles contains four minima, then in a protein of modestsize containing
50 such pairs, the number of possible conformations would be 4^50 = 2^100. From this
simple exercise, we have a crude measure of the roughness of an energy landscape.
The large number of free energy minima exhibited by our modest protein is far
more than can be sampled in a typical computer simulation, and in fact, many of
these minima tend to be high enough in energy as to contribute little toan ensemble
average. How, then, can we limit sampling to the most important regions of an energy
landscape? There is no definitive answer at present, and the question remains an active
area of research. In many cases, however, it is possible to identifya subset of generalized
coordinates that is particularly well suited for characterizing a given process. In the
next few sections, we will describe how to make use of such variablesin enhanced
sampling schemes for mapping out their corresponding free energysurfaces.
8.6 Reaction coordinates
It is frequently the case that the progress of some chemical, mechanical, or thermody-
namics process can be characterized using a small set of generalized coordinates in a
system. When generalized coordinates are used in this manner, they are typically re-
ferred to asreaction coordinates,collective variables, ororder parameters, depending
on the context and type of system. Whenever referring to thesecoordinates, we will
refer to them asreaction coordinates, although the reader should be aware that the
other two designations are also used in the literature.
As an example of a reaction coordinate, consider a simple gas-phasediatomic dis-
sociation process AB−→A+B. IfrAandrBdenote the Cartesian coordinates of
atoms A and B, respectively, then a useful generalized coordinatefor describing a dis-
sociation reaction is simply the distancer=|rB−rA|. As we saw in Section 1.4.2, a
set of generalized coordinates that containsras one of the coordinates is the center-
of-massR= (mArA+mBrB)/(mA+mB), the magnitude of the relative coordinate
r=|rB−rA|, and the two anglesφ= tan−^1 (y/x) andθ= tan−^1 (
√
x^2 +y^2 /z), where
x,y, andzare the components ofr=rB−rA. Of course, in the gas phase,ris the most
relevant coordinate when the potential depends only onr. If the dissociation reaction
takes place in solution, then some thought is needed as to whether asimple coordinate
likeris sufficient to describe the reaction. If the solvent drives or hinders the reaction
by some mechanism, then a more complex coordinate that involves solvent degrees of
freedom is likely needed. If the role of the solvent is a more “passive”one, then the
free energyA(r), obtained by “integrating out” all other degrees of freedom, willyield
important information about the thermodynamics of the solution-phase reaction.
Another example is the gas-phase proton transfer reaction A–H···B−→A···H–B.
Here, although the two distances|rH−rA|and|rH−rB|can be used to monitor
the progress of the proton away from A and toward B, respectively, neither distance
alone is sufficient for following the progress of the reaction. However, the differenceδ=
|rH−rB|−|rH−rA|, which is positive (negative) when the proton is near A (B) and zero
when the proton is equidistant between A and B, is a reasonable choice for describing
the process. A complete set of generalized coordinates involvingδcan be constructed
as follows. IfrA,rBandrHdenote the Cartesian coordinates of the three atoms, then
first introduce the center-of-massR= (mArA+mBrB+mHrH)/(mA+mB+mH),