Essentials of Computational Chemistry

12.2 COMPUTING FREE-ENERGY DIFFERENCES 439
E (s) + S (s)
E (s) + S′ (s)
∆G° aq (1)
E•S (s)
0 ∆∆G° mut(S) ∆∆G° mut(E•S)
∆G° aq (2)
E•S′ (s)
Figure 12.4 Differential binding free-energy cycle. The difference in binding free energies for two
different substrates, S and S ′ , is equal to the difference in mutation free energies for changing S into
S ′ in solution, and EžS intoEžS ′ in solution. The leftmost vertical free-energy change is zero, since
the free enzyme is a constant independent of substrate
12.2.5 Potentials of Mean Force
When free energy is expressed as a function of coordinate, it is referred to as a potential of
mean force (PMF). The PMF W can be determined as
W(q) =−kBT ln π(q) (12.22)
where q is the coordinate, and π is the probability of the coordinate taking on a particular
value, i.e.,
δ[q ′ (q) − q]e −E(q,p)/kBT
dqdp (12.23)
π(q) = Q −1
where Q is the (normalizing) full partition function, δ is the Dirac delta function, and q ′
is the value of the PMF coordinate for any arbitrary point in phase space having positional
coordinates q.
In practice, one may evaluate these probabilities following a histogram approach like
those outlined in Chapter 3. Over the course of a MC or MD simulation, the value of q ′
is collected and binned, and the probability of different ranges of values can be determined
upon completion of the simulation based on the number of points in a bin compared to
the total number of points. For example, we might be interested in the PMF for rotation
about the C−O bond in fluoromethanol (see Figure 2.3). Over the course of a simulation,
the torsional angle would be saved at every step, and with good sampling a probability
histogram would permit conversion to a PMF accurately reflecting the true potential. In the
case of fluoromethanol, the difference in energy between the lowest and highest points on
the potential energy curve is about 3 kcal mol −1 . At 298 K, we would thus expect to sample
points in the highest energy region about 100 times less frequently than points in the lowest
energy region. Of course, if the width of a bin is, say, one degree, there are many other
possibilities for bins to fill, and ultimately roughly one point in every 10 000 or so would
be statistically expected to fall into the highest energy bin. To obtain reliable statistics, we