Supplemental Material
Recovering position-dependent diffusion from biased molecular dynamics simulations
Ajasja Ljubetič, Iztok Urbančič, Janez Štrancar.
FIG. S1
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Supporting information: Recovering position-dependent diffusion from biased molecular dynamics simulations
FIG. S1: Different energy surfaces and binning methods compared.
(a) Contour plots for different energy surfaces tested with the synthetic diffusive model system (Eqn. 17). The equations for the energy
surfaces are given above each plot, where Lx and Ly are the widths of the unit cell. The color of the frame corresponds to the color of the
curves in panels b and c.
(b) Root mean square deviation (RMSD) of difference between fitted and model diffusion surfaces as defined by Eqn. (19) versus sampling
timescale. The energy surfaces are “No Energy” (blue), “Wave Energy” (orange), “Energy Max” (green) and “Energy min” (red). The binning
methods top to bottom are “w/o padding”, “padding” and “midpoint”. Since the RMSD curves overlap, it is safe to conclude that energy
surfaces with barriers smaller than 3kbT do not significantly affect the analysis methods.
(c) Average p-values versus sampling timescale. The p-values are the result of the Anderson-Darling test applied to the distribution of steps
in each bin and averaged over bins. All distributions are normal according to the test.
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Supporting information: Recovering position-dependent diffusion from biased molecular dynamics simulations
FIG. S2
timescale = 1 fs
(a)
timescale = 100 fs
(b)
FIG. S2: Smooth and ballistic regime at different sampling timescales. (a) If the colvars are sampled each fs (sampling timescale 1 fs), the
motion of the colvars is ballistic, i.e., it follows a smooth trajectory. (b) At longer sampling timescales (here 100fs) the motion becomes
diffusive. Note that the panels are plotted at different scales.
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Supporting information: Recovering position-dependent diffusion from biased molecular dynamics simulations
FIG. S3
FIG. S3: Projection of the two dimensional energy F(, ) and diffusion D(, ) surface onto  (blue lines) in comparison with data from
Hummer1 with or without smoothing (green dashed and yellow lines, respectively). The data from Hummer was read from FIG. 3 in the
cited paper.
The one dimensional energy profile F() was calculated using

F ( ) 
180
180
F ( , )e  F ( , ) d

180
180
F ( , )d
.
(1)
The diffusion tensor was first projected onto  by
D ( , )  D( , )  j
(2)
and projected analogously to the energy surface:

D ( ) 

180
180
D ( , )e F ( , ) d

180
180
F ( , )d
.
(3)
The values of E ( , ) and D( , ) were interpolated using spline interpolation before doing numerical integration.
Hummer determined much higher energy barriers. This might be due to insufficient sampling in the  dimension.
ABF MD ensures good sampling in both dimensions, so the 1D projection is probably a better estimate. The
diffusions obtained are comparable. From the unsmoothed diffusion points it is obvious that diffusion in regions with
high is very uncertain due to poor sampling. ABF MD avoids this problem.
1
Hummer, G. Position-dependent diffusion coefficients and free energies from Bayesian analysis of equilibrium and replica
molecular dynamics simulations. New J. Phys. 7, 34–34 (2005).
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