Normal mode partitioning of langevin dynamics for biomolecules
supplementary information
Additional evidence related to NML
Comparison of NMI discretization and NML
Comparison of speedup for different models
Simulation and implementation details
Directions to download executable
Implementation of NML in ProtoMol
List of configuration file options
There are three main aims of this supplementary information. First, we present additional support for some claims related to NML, the normal mode partitioning of Langevin dynamics integrator. Second, we present a comparison of NML and the normal mode impulse NMI method of Eqn. (14) in the paper. The main conclusions, reported in the paper, are that NMI is able to sample correctly, but is unable to reproduce kinetics. This is evidence that the minimization of the energy due to the fast frequency modes effectively eliminates coupling between spaces, and thus points out the advantage of the approach taken by NML as advocated in the paper. Additionally, NMI is less efficient than NML. Third, we produce details needed for anyone to reproduce our results. In this document we explain the testing methodology, a set of input files needed for one of the simulations, and the analysis methodology. Then we explain the structure of the software implementation of these methods. A table of URLs to full sets of simulation input and outputs as downloadable tar files are provided. Further information on the method is available at http://www.normalmodes.info .
Comparison of Ramachandran plots for alanine dipeptide propagated with the Langevin Impulse (Figure S1) and 'Subspace Molecular Dynamics' (SMD) method (Figure S2). The dynamics of the SMD method is localized by the subspace coupling, even after a 100ns trajectory.
Figure S1. LI propagator. 
Figure S2. SMD propagator. 
Figure S3 Number of remaining negative eigenvalues with number of steps the Hessian is averaged over.
Probability plots for alanine dipeptide, 100ns trajectory and BPTI, 10ns trajectory, for different numbers of propagated modes. The BPTI Ramachandran plot is constructed using a subset of the 57 backbone residue dihedrals which excludes the residue set {1 4 5 6 12 28 29 36 37 47 49 50 56 57}.


Figure S4 (a) 
Figure S4 (b) 


Figure S5 (a) 
Figure S5 (b) 


Figure S5 (c) 
Figure S5 (d) 
The NMI discretization is able to sample correctly, similarly to NML, as expected.
Figure S6(a). LI sampling. 
Figure S6(b). NMI sampling large γ. 


Figure S6(c). NMI
sampling, optimum γ.
The NMI discretization is unable to recover kinetics, unlike NML.
For Normal Mode Impulse (NMI) we see in Figure S7 that the optimal damping coefficient is close to the reciprocal of the largest eigenvalue λ_{3N}≈950, which would be the linesearch solution for a minimizer for the discreet quadratic approximation. The method was unstable for Δτ/γ of greater than 2.16x10^{3}, indicating that the propagator for high frequency degrees of freedom must be in the minimization region for stability. For comparison the Langevin Impulse rate (assuming a 0.83 population) was 2.75ns^{1}. The results for NML are presented in Figure S8, showing excellent rates when compared to LI.


The implicit solvent model used for the alanine dipeptide tests is a sigmoidal distancedependent dielectric to account for screening of electrostatic interactions due to solvent. For BPTI the distance dependent dielectric gave unsatisfactory results: for instance, the Ramachadran plots were severely distorted compared to explicit solvent, the molecule kept expanding in volume, and these results were sensitive to the value of S. Thus, we used a more accurate implicit solvent model, the screened Coulomb Potential implicit solvent model (SCPISM). SCPISM uses the relation between the physically measurable dielectric function ε(r) and the screening function D(r).
The derivation of the SCPISM Hessian, required for correct diagonalization, can be found here.
The following figure and table compares the speedup that can be achieved for different protein models.



Figure S9. 
Table S1 
Alanine
dipeptide input files for SAMPLING, RATES, EFFICIENCY and KINETIC ENERGY.
alan.nm.conf
NML config
file for sampling.
alan.nm2.conf
NML config file for rates.
alan.nm3.conf
NML config file for efficiency.
alan.nm3.conf
NML config file for kinetic energy.
alan.lang.conf
Langevin Impulse config file.
par_all27_prot_lipid.inp
par input file.
alan_mineq.psf
psf input file.
minC7eq.pdb
pdb input file.
eigVmC7eq
eigenvector file.
BPTI
input files for SAMPLING and EFFICIENCY.
bpti.nm.conf
NML config file for sampling.
bpti.nm1.conf
NML config file for efficiency.
bpti.lang.conf
Langevin Impulse config file.
par_all27_prot_lipid.inp
par input file.
cbpti.psf
psf input file.
cbpti.min.pdb pdb
input file.
eigVmBPTISCP eigenvector
file.
VMD
and Matlab analysis scripts.
proc_dihedrals_alan.sh
Script for extracting dihedral data for alanine dipeptide.
dihedrals_alan.tcl
Dihedral definition file for proc_dihedrals_alan.sh.
proc_dihedrals_bpti.sh
Script for extracting dihedral data for BPTI.
dihedrals_BPTI.tcl
Dihedral definition file for proc_dihedrals_bpti.sh.
RamachandranFe.m
Ramachandran Matlab script for alanine dipeptide.
RamachandranAll.m
Ramachandran Matlab script for BPTI.
RamachandranAll43.m Ramachandran
Matlab script for BPTI, 43 of 57 residues considered.
MyColormaps.mat
Colormap for Ramachandran Matlab scripts.
GetTPRates.m
Rate calculation for alanine dipeptide.
TransitionPaths.m Matlab script required by GetTPRates.m, finds
all transition paths.
TheState.m
Matlab script required by GetTPRates.m, finds the current state of the
molecule.
Directions to download executable
Implementation of NML in ProtoMol
NML is implemented in the open source molecular dynamics application Protomol, which can be found at:
http://protomol.sourceforge.net/
NML
propagator: framework/integrators/NormModeInt.cpp/h
NML minimizer: framework/integrators/NormModeMin.cpp/h
NML simple minimizer: framework/integrators/NormModeSmplMin.cpp/h
List of configuration file options
NML
propagator:
cyclelength
1 # Legacy MTS parameter,
always 1
fixmodes
44 # Number of high frequency modes
constrained
gamma
91 # Langevin Gamma
seed
1234 # Langevin random seed
temperature 300
# Langevin temperature
nve
0 # NVE simulation if not 0
Berendsen
0 # Berendsen tau in fs
fdof
0 # Fixed degrees of freedom
NML
minimizer
timestep
16 # timestep for propagates
modes (legacy MTS position)
minimlim
0.1 # Minimizer target PE difference kcal
mole^{1}
forcePEcheck true
# Force PE/calcForces check at end of loop, always set
true
massweight true # mass
weighted minimization, always se true.
randforce
true
# Add random force, always set true
NML
simple minimizer
As NML minimizer options.
The ‘C2’ switch implemented in Protomol is a C^{1 }switch defined as
where is the switchon value and is the cutoff value.