Chemistry Reference
In-Depth Information
Recently a dynamic λ parameter was incorporated into the adiabatic free energy dynamics (AFED) method for
generating free energy profiles along reaction paths by choosing switching functions that generate a high barrier
between the endpoints and by carefully assigning the temperature and mass of λ [74]. Another approach use λ as a
self-regulating sampling variable to efficiently traverse high-energy barriers and to thoroughly explore low-energy
biasing. By using multiple copies of a subset of the system and intermittently varying their associated λ values a
copy in a high-energy region will adopt a small λ value and will sample more broadly until it finds a low-energy
region [75]. However, λ-dynamics have focused on simulating differences at single sites in a given system including
explicitly represented ligands to amino acid side chains within peptides as well as chemical moieties attached to a
specific substituent site on a common core compound [28,62,64,76].
The theoretical foundation of binding free energy predictions based on atomistic representations of the protein-
ligand complex is based on statistical thermodynamics whereas at equilibrium we can write the biomolecular
reaction for the standard free energy of association as -RTln(1/K
i
). Classical statistical thermodynamics relationships
connect the macroscopic thermodynamic properties to microscopic properties via molecular, canonical ensemble
partition functions of the complex, the individual species and the solvent. Although the partition functions, in
principle, enumerate all the possible microscopic states of the molecules, the direct calculation of the partition
function, in practice, for complex systems such as solvated proteins, is unfeasible due to the configurational integral.
Although there have been ingenious efforts to decompose the free energy into numerous components in order to
approximate the calculations, the difficulties persist mainly due to the error in the calculations of the free energy
components which is larger than the actual absolute value of the binding strength [28-55].
It is necessary to sometimes inject empirically derived corrections or simplifying assumptions. In the path-integral
method for predicting relative binding affinities of protein-ligand complexes a different approach is presented based
on a stochastic kinetic formalism. Inspired by Feynman's path integral formulation there was an extension of the
theory to classical interacting systems whereas the ligand is modeled as a Brownian particle subjected to the
effective nonbonding interaction potential of the receptor which allows the calculation of the relative binding
affinities of interacting biomolecules in water to be computed as a function of the ligand's diffusivity and the
curvature of the potential surface in the vicinity of the binding minimum. These calculations are reportedly
exceedingly rapid and in test cases, the correlation coefficient between actual and computed free energies > 0.93 for
accurate data sets [30].
In rational drug design the lack of sufficient computational resources remains one of the factors preventing the
general application of free energy methods. In recent years many nonequilibrium (NE) free energy methods have
been made parallel and more accessible regarding computational resources [33, 36, 45, 71]. The nonequilibrium
methods do not use the potential energy differences used in FEP calculations, instead, λ is incremented n times
(where n is a constant for the calculation ) from 0 to 1, with simulation sampling allowed between increments and
the work performed, as a consequence of each λ increment, is summed. Estimating the free energy difference
through the work, determined from the potential energy differences, invariably produces a systematic error due to
the nonequilibrium nature of the perturbation process. The Hamiltonian lag, i.e the simulation lags behind the
changing potential, contributes positively yielding the dissipated work, whose average is positive and is associated
with the increase of entropy during an irreversible process [33,72].
Various NE free energy methods use a basic rule of thermodynamics which states that over the course of an
isothermal, reversible process linking two equilibrium states, the work performed on the system is equal to the free
energy difference between the two states. However, for a process linking two states (switch) to be truly reversible it
must in principle be infinitely long, thus switches cannot be truly reversible and the NE free energy calculations
depend on the closeness between switch and reversible limit. There are a number of switching methodologies. In
practice, the number of switches needed to produce an accurate estimate of the free energy difference depends on the
nature of the distribution of work values produced by the nonequilibrium calculation [33, 49, 72].
The Bennett's acceptance ratio (BAR) is an interesting development of the switching methodology, with switches in
both forward and backward directions used to calculate the free energy differences. This is estimated from the lower
bound found through the average work and slowly increased iteratively until the difference is satisfied. The free
