λ-dynamics is a new method developed on the basis of the widely used free energy perturbation
method (FEP) and the umbrella sampling method. It differs from the FEP method by introducing
multiple parameters λi, instead of only one λ, which are assigned to different types of energy
components. In order to surmount the possible energy barriers and thus get more samples,
special biasing potentials are used based on the idea of umbrella sampling. The forms of these
biasing potentials are quite innovative, which look like kinetic energy and potential energy for
factious particles where λi, instead of r, serve as the coordinates and a set of fake masses of
particles are introduced. The benefit from introducing more λ is that more reaction pathways are
accessible during the simulation. On the other hand, the umbrella potential term is useful in
flattening the energy curve of the reaction pathway, preventing the boundary crossing of λi (e.g.
λi < 0 or λi > 1) and limiting the range of {λi}. Additionally, the fake masses make the control of
λi more flexible. In a word, these new variables and terms make the simulation more adjustable
and easier to control for different purposes. The results of this new method are at least as
accurate as those by the single λ FEP method, which are verified by simulations of relative free
energies of hydration and relative binding affinity simulations.

        Constant pH molecular dynamics (CPHMD) is one of the applications of the λ-dynamics
approach. In CPHMD method, n λi are assigned to n titrating residues, which varies from 0 to 1
when the titrating residue change from protonated form to deprotonated form and it takes the
mathematical form as an sine function. A two-dimensional λ-dynamics method is used here for
the purpose of take the proton tautomerism effect into consideration. The two dimensions are
due to the two reaction coordinates: one is for the deprotonation process (λ); the other is for the
tautomeric effect (x). These two coordinates are used throughout the potential energy
calculation via λ-dynamics method. When calculating van der Waals energy, pairwise
interactions are divided into two classes: interactions between titrating residues and other
resides and those between two titrating residues. For coulomb and GB electrostatic energies,
atomic partial charges on titrating residues are changed along the reaction coordinates. Biasing
potentials are also used to facilitate the simulation. One is like harmonic umbrella potential and
the other one originates from the calculation of free energy based on experimental pKa value.
In CPHMD, the unprotonated form has λ > 0.9, while that of the protonated form is smaller than
0.1. In both cases, the x value is either larger than 0 or smaller than 1 by a maximum magnitude
of 0.1, in other words, the tautomer state must be sufficiently pure. The enhance the sampling
near the two ends of the reaction coordinates, a barrier potential is introduced. This potential
function is symmetric considering the middle point of the reaction coordinate. The result is that
the energy is lower when the titrating residues are in pure states than that of the titrating
residues in mixed states and therefore more pure sate residues are sampled.

         In my opinion, the CPHMD method makes a best use of the advantage of the λ-dynamics
method, making the titration simulation more flexible and closer to the real experiments. Thus , I
expect more repeatable and accurate data will come from this method. But there are still some
minor problems. The authors assume the van der Waals energy changes linearly or
quadratically during the change of reaction coordinates, which may not be true compared with
that in the real world. Besides, the rate of the changes of van der Waals energy is coherent with
that of electrostatic energy in this paper. It is possible that assigning different reaction
coordinates to them may produce better results, just like what is done in the first paper.

References:
Kong and Brooks III.1996. λ−dynamics: A new approach to free energy
calculations. J.Chem. Phys. 105:2414-2423

Khandogin and Brooks III. 2005. Constant pH Molecular Dynamics with Proton
Tautermerism. Biophysical Journal. 89:141-157