Computational Drug Discovery and Design

QNPT¼

1

N!h^3 N

∫∫∫e⽊HðÞþx;p pVdVdxdp ð 3 Þ

withNbeing the number of particles,β¼1/(kBT),xandpthe

coordinates and momenta,Hthe Hamiltonian,pthe pressure,

Vthe volume, andhthe Planck’s constant. The Hamiltonian of a

system consists of a potential energy termU(x) that depends on the

particles’ coordinates, and a kinetic termK(p) that depends on their

momenta. However, since we are considering the free energy dif-

ference of a process in which neither the temperature or particles’

masses change, the kinetic contribution to the Hamiltonian is

constant. Therefore, only the configurational part of the partition

function needs to be considered in this case, and the Gibbs binding

free energy can be expressed as follows.

ΔGb¼kBTln

Ð

V 1

Ð

Γ 1 e

⽊U 1 ðÞþx pV (^1) dVdx
Ð
V 0
Ð
Γ 0 e
⽊U 0 ðÞþx pV (^0) dVdx ð^4 Þ
whereV 1 andV 0 are the container volumes, whileΓ 1 andΓ 2 are the
phase space volumesof the bound and unbound states, respectively.
However, due to the limited compressibility of water, at 1 atm the
effect of changes in average volume on the binding free energy is
negligible [11, 14]. This means that thepVcomponent of the free
energy can be ignored without major effects on the results, and that
the Helmholtz free energy closely approximates the Gibbs free
energy.
ΔGbffiΔAb¼kBTln
Ð
Γ 1 e
βUðÞxdx
Ð
Γ 0 e
βUðÞxdx ð^5 Þ
This definition assumes the existence of separate and well-
defined “bound” and “unbound” states, which is valid for tight
and specific binders, but might not be justified in the case of very
weak and nonspecific binders [11]. The partition function for a
complex system has no analytical solution and thus simulations
need to be used to sample the accessible phase space. The whole
phase space is computationally difficult to sample but, when calcu-
lating free energydifferences, inaccessible high-energy regions will
quite often not be sampled for either state of interest, resulting in a
cancellation of errors that allowΔGto be calculated.
When comparing binding free energies, it is important to keep
in mind that, as mentioned previously, the binding constant
depends upon a reference concentration. This dependence is due
to the connection between available volume and entropy [15]. It is
therefore necessary to refer to the same standard state when
comparing binding free energies. A standard concentration
c¼1 mol/L corresponds to a standard volumeV¼1660 A ̊^3
(the volume occupied by one molecule at the concentration of
202 Matteo Aldeghi et al.