Computational Drug Discovery and Design

2 Theory

Here we review some theoretical and methodological concepts that

underlie the use of computer simulations for the calculation of

binding free energies. We provide only a brief overview of some

key concepts and, due to the practical nature of the chapter, this

section is not meant to be exhaustive. The reader can find a more

extensive appraisal of theoretical aspects in one of the many excel-

lent reviews and textbooks written on the subject and here refer-

enced [9–13]. The book by Chipot and Pohorille [9] is particularly

comprehensive.

2.1 Definition of
Binding Free Energy

The reversible binding of a ligand (L) to a protein (P) to form a

complex (PL) can be described by the following chemical reaction:

PþL⇋PL

At equilibrium, the binding constantKbdefines the ratio of

the product and reactants concentrations in solution:

Kb¼c

½ŠPL

½ŠL½ŠP

ð 1 Þ

where square brackets indicate a concentration, andcis the stan-

dard state concentration, which is typically defined as 1 mol/L.

Since this definition of standard state does not change the numeri-

cal value ofKb(as it is multiplied by one) it is customary to omitc

when discussing equilibrium constants. However, it is important to

remember that the binding constant is a dimensionless quantity

(withoutcit would have units of inverse concentration), and it is

dependent on the chosen standard state.For a single ligand binding

event,Kbis associated to the binding free energy by the following

well-known equation.

ΔGb¼kBTlnKb ð 2 Þ

wherekBis the Boltzmann constant, andTis the temperature. For a

mole of ligand, the gas constant (R¼NAkB, whereNA is the

Avogadro constant) is used instead ofkB. From the above discussion

of the standard state, it follows that the binding free energy is also

thus defined with respect to the chosen standard state.

The equilibrium constant can also be expressed as the ratio of

probabilities ðÞP for the system being in either the bound or

unbound state, so that Kb¼P 1 =P 0 , where “1” denotes the

bound (PL) state, and “0” the unbound (P+L) state. The

probability of finding the system in the bound versus unbound

state is determined by the ratio of their partition functionsQ 1 and

Q 0 , where the partition function of a system in the isothermal-

isobaric (NPT) ensemble is defined as follows.

Absolute Alchemical Free Energy 201