1549380323-Statistical Mechanics Theory and Molecular Simulation

Metadynamics 355

atoms in an arbitrary dihedral angle be denotedrk+1,...,rk+4. The transformation
can be carried out in the following simple steps:

1. Transformrk+4into a coordinate system whose origin is located atrk+3:

r′k+1=rk+1, r′k+2=rk+2, r′k+3=rk+3, r′k+4=rk+4−rk+3. (8.10.23)

2. Rotater′k+4such that the vectorr′k+2−r′k+1lies along thex-axis:

r′′k+1=r′k+1, r′′k+2=r′k+2, r′′k+3=r′k+3

r′′k+4=R(r′k+1,r′k+2,r′k+3)r′k+4, (8.10.24)

whereR(r′k+1,r′k+2,r′k+3) is a rotation matrix given by

R(r′k+1,r′k+2,r′k+3) =

       

(r′k+3−r′k+2)×(r′k+1−r′k+2)

|(r′k+3−r′k+2)×(r′k+1−r′k+2)|×

r′k+3−r′k+2

|r′k+3−r′k+2|

(r′k+3−r′k+2)×(r′k+1−r′k+2)

|(r′k+3−r′k+2)×(r′k+1−r′k+2)|

r′k+3−r′k+2

|r′k+3−r′k+2|

       

(8.10.25)

The rows of this matrix are thex,y, andzcomponents of the three vectors shown.

3. The vectorsr′′k+4 is resolved into spherical polar coordinatesr′′k+4,θ′′k+4,φ′′k+4.
   When this is done, the angleφ′′k+4is the dihedral angle.

The free energy surface in Fig. 8.5 is expressed in terms of the Ramachandran dihedral
anglesφandψ, which characterize rotations about the bonds between the alpha-
carbon and the amide nitrogen and the alpha- and carbonyl carbons, respectively.
The surface was generated in an adiabatic dynamics calculation (Rossoet al., 2005)
usingm(φ,ψ)= 50mC,T(φ,ψ)= 1500 K, in a periodic box of length 25.64 ̊A, which
contains one alanine-dipeptide and 558 water molecules. The simulation was performed
using the CHARMM22 force field (MacKerellet al., 1998). Data were collected over
4.7 ns. Note that in order to obtain the same level of convergence with two-dimensional
umbrella sampling, a total of 35 ns would be needed. Fig. 8.5 shows four local minima,
corresponding to the most favored conformations, which are known asαRat (φ,ψ) =
(− 81 ,−63), C7eq(alsoβor C5) at (φ,ψ) = (− 90 ,170), C7axat (φ,ψ) = (60,−115),
andαLat (φ,ψ) = (50,63). These minima are ordered energetically such that ifαRis
at zero free energy, then C7eqis 0.2 kcal/mol above it, followed by C7axat 4.6 kcal/mol,
andαLat 8.2 kcal/mol. These minima are extended and helical motifs characteristic
of those found in protein folds.

8.11 Metadynamics

The last method we will describe for computing a free energy hypersurface is akin to a
dynamical version of the Wang–Landau approach from Section 7.6.Themetadynamics
method (Laio and Parrinello, 2002) is a dynamical scheme in which energy basins are
“filled in” using a time-dependent potential that depends on the history of the system’s
trajectory. Once a basin is filled in, the system is driven into the nextbasin, which