Reaction coordinates 335the relative coordinate between A and B,r=rB−rA, and a third relative coordinate
sbetween H and the center-of-mass of A and B,s=r−(mArA+mBrB)/(mA+mB).
Finally,ris transformed into spherical polar coordinates (r,θ,φ), and fromrands,
three more coordinates are formed as
σ=|s+mB
mA+mBr|+|s−mA
mA+mBr|,δ=|s+mB
mA+mBr|−|s−mA
mA+mBr|, (8.6.1)and the angleα, which measures the “tilt” of the plane containing the three atoms
from the vertical. These coordinates could also be useful if the reaction takes place in
solution, but as noted above, some careful thought about relevant solvent coordinates
is needed.
As a third example, conformational changes in small peptides, suchas di- and
tripeptides can be described in terms of the Ramachandran backbone dihedral angles
φandψ(see Fig. 3.8, for example). For longer oligopeptides that can fold into protein
secondary structure elements such as helices andβ-sheets, other coordinates such as
the radius of gyration ofNbheavy backbone atoms, given by
RG=
√
√
√
√
√
1
Nb∑Nbi=1
ri−^1
Nb∑Nbj=1rj
2
, (8.6.2)or the number of hydrogen bonds of length approximatelyd 0 betweennOoxygens and
nHhydrogens, which can be expressed as
NH=
∑nOi=1∑nHj=11 −[(ri−rj)/d 0 ]^6
1 −[(ri−rj)/d 0 ]^12, (8.6.3)
could be used in addition to the Ramachandran angles. These coordinates have been
shown to be useful in characterizing both the folded and unfolded states of polypep-
tides (Bussiet al., 2006). In all of these examples, the reaction coordinates are functions
of the primitive Cartesian coordinates of some or all of the atoms in the system.
While reaction coordinates or collective variables are potentially veryuseful con-
structs and intuitively appealing, they must be used with care. Enhanced sampling
approaches applied to poorly chosen reaction coordinates can biasthe system mislead-
ing ways and generate erroneous predictions of free energy barriers, transition states,
and mechanisms. A dramatic illustration of this can be seen with the autodissociation
of liquid water according to the classic reaction 2H 2 O(l)−→H 3 O+(aq) + OH−(aq),
as discussed in Section 7.7. The reaction ostensibly only requires transferring a proton
from one water molecule to another. If we pursue this simple picture, a seemingly
sensible reaction coordinate might simply be the distance between the oxygen and
the transferring proton or the number of hydrogens covalently bonded to one of the
oxygens. As it turns out, these coordinates are inadequate to describe the true nature
of the reaction and consequently fail to yield an accurate free energy or autoionization