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Chemical Reactions and Equilibria

(Note: These tutorials are meant to provide illustrative examples of how to use the AMBER software suite to carry out simulations that can be run on a simple workstation in a reasonable period of time. They do not necessarily provide the optimal choice of parameters or methods for the particular application area.)
Copyright Jason Swails and T. Dwight McGee, 2013

Analyzing the Results:

This section describes how to analyze the results obtained from the CpHMD simulations.

Using cphstats

The various protonation states that are sampled throughout the course of the CpHMD simulations are written to a cpout file. The program cphstats can be used to parse the cpout file and for calculating the predicted pKa values. Use the -h flag to get a full listing of the available command-line flags for cphstats.

cphstats -i cpin [cpout1 cpout2 cpout3 cpoutN] [-o output] [--population population_ouput]

For example, the command

cphstats -i 4LYT.equil.cpin 0/4LYT.md1.cpout -o pH0_calcpka.dat --population pH0_populations.dat

will create the following two files for the simulations at pH 0: pH0_calcpka.dat and pH0_populations.dat

You can use a small shell command to analyze all results at once. For instance:

for ph in 0 1 2 3 4 5 6 7; do
   cphstats -i 4LYT.equil.cpin ${ph}/4LYT.md1.cpout -o pH${ph}_calcpka.dat --population pH${ph}_populations.dat

will create files with the titration statistics for all simulations. We recommend you generate these files yourself, but a tarball containing these files are available here: pH_statistics.tar.bz2

Solvent pH is    0.000
GL4 7   : Offset  3.387  Pred  3.387  Frac Prot 1.000  Transitions        11
HIP 15  : Offset    Inf  Pred    Inf  Frac Prot 1.000  Transitions         0
AS4 18  : Offset  1.992  Pred  1.992  Frac Prot 0.990  Transitions       179
GL4 35  : Offset    Inf  Pred    Inf  Frac Prot 1.000  Transitions         0
AS4 48  : Offset  1.635  Pred  1.635  Frac Prot 0.977  Transitions        37
AS4 52  : Offset  1.395  Pred  1.395  Frac Prot 0.961  Transitions         1
AS4 66  : Offset   -Inf  Pred   -Inf  Frac Prot 0.000  Transitions         0
AS4 87  : Offset  2.365  Pred  2.365  Frac Prot 0.996  Transitions        13
AS4 101 : Offset  3.979  Pred  3.979  Frac Prot 1.000  Transitions         5
AS4 119 : Offset  1.840  Pred  1.840  Frac Prot 0.986  Transitions       191

Average total molecular protonation:   9.909


Offset: is the difference between the predicted pKa and the system pH
Pred: is the predicted pKa
Frac Prot: is the fraction of time the residue spends protonated
Transitions: gives the number of accpeted protonations state transitions
Average total molecular protonation: is the sum of the fractional protonations

cphstats also printed another file with the populations of every state for every titratable residue, pH0_populations.dat. This file prints the fraction of snapshots that the system spent in each state for each residue. For example, His 15 was protonated 100% of the time, whereas Asp 119 was deprotonated 1.42% of the time, syn- protonated ~48% of the time on each oxygen (this is good since they are rotationally degenerate!), and anti-protonated ~0.7% of the time.

   Residue Number     State  0     State  1     State  2     State  3     State  4
Residue: GL4 7    0.000410 (0) 0.453512 (1) 0.004275 (1) 0.535908 (1) 0.005895 (1)
Residue: HIP 15   1.000000 (2) 0.000000 (1) 0.000000 (1)
Residue: AS4 18   0.010085 (0) 0.486042 (1) 0.004335 (1) 0.495922 (1) 0.003615 (1)
Residue: GL4 35   0.000000 (0) 0.581343 (1) 0.000360 (1) 0.368002 (1) 0.050295 (1)
Residue: AS4 48   0.022640 (0) 0.520073 (1) 0.018440 (1) 0.425152 (1) 0.013695 (1)
Residue: AS4 52   0.038725 (0) 0.517533 (1) 0.113676 (1) 0.280601 (1) 0.049465 (1)
Residue: AS4 66   1.000000 (0) 0.000000 (1) 0.000000 (1) 0.000000 (1) 0.000000 (1)
Residue: AS4 87   0.004295 (0) 0.439747 (1) 0.010535 (1) 0.524678 (1) 0.020745 (1)
Residue: AS4 101  0.000105 (0) 0.504673 (1) 0.013110 (1) 0.478517 (1) 0.003595 (1)
Residue: AS4 119  0.014240 (0) 0.488767 (1) 0.007100 (1) 0.483272 (1) 0.006620 (1)

NOTE: The parentheses (*), indicates the number protons present in that state.

The cpout file can be divided into chunks of time using calcpka. This feature is enabled when a dump_interval, is specified (using the -t flag on the command-line). The information is printed to the file pKa_evolution.dat if the -ao flag is not given.

Calculating the pKas

Here we will look at a couple ways of computing pKa values using the output from calcpka. The predicted pKas printed in the calcpka output assumes ideal, Hendersen-Hasselbalch titration behavior obeying the equation:

pK_a=pH-\log\frac{[A^-]}{[HA]}"       [1]

In proteins, titration behavior for each titratable residue is not always this straight-forward, and two or more residues may impact each others' titration curves. Therefore, the Hill equation, shown below, is typically used to fit titration curves and calculate pKas.

pK_a = pH - n \log \frac {[A^-]} {[HA]}       [2]

The trick now becomes to plot our titration curve—typically plotted as the deprotonated fraction (which is directly related to [A-]) versus pH. Knowing that f_d = \frac{[A^-]} {[HA] + [A^-]}, where fd is the fraction deprotonated, we can rearrange Equation 2 to obtain:

f_d = \frac 1 {1 + 10 ^ {n (pK_a - pH)}}       [3]

Now the trick is to extract the deprotonated fraction from the calcpka output shown above (which is just 1 − fraction protonated) for each residue as a function of the pH. The data is shown below, where the first column is the pH and the subsequent columns are the deprotonated fraction for each residue.

#     pH     GLU7    HIS15    ASP18    GLU35    ASP48    ASP52    ASP66    ASP87   ASP101   ASP119
   0.000    0.000    0.000    0.010    0.000    0.023    0.039    1.000    0.004    0.000    0.014
   1.000    0.006    0.000    0.098    0.000    0.276    0.312    1.000    0.015    0.002    0.069
   2.000    0.021    0.000    0.342    0.000    0.977    0.387    0.999    0.009    0.014    0.546
   3.000    0.186    0.001    0.789    0.000    0.985    0.918    1.000    0.116    0.089    0.598
   4.000    0.774    0.004    0.981    0.002    0.998    0.999    1.000    0.633    0.516    0.994
   5.000    0.973    0.055    0.999    0.002    1.000    1.000    1.000    0.954    0.933    0.999
   6.000    0.999    0.405    0.999    0.037    1.000    1.000    1.000    1.000    0.989    1.000
   7.000    0.999    0.826    1.000    0.330    1.000    1.000    1.000    0.999    0.999    1.000

In this case, I use gnuplot to fit all of this data to Equation 3 using gnuplot's built-in fitting functionality. This function needs to be fit using a non-linear, least-squares fitting algorithm (e.g., the Marquardt-Levenberg algorithm). As a result, you should choose a reasonable starting guess for the fit (the predicted pKa printed out by calcpka for the pH with the lowest offset value is a good starting point). The gnuplot script shown below will fit all of the titration curves, plot them, and print the calculated pKa and Hill coefficients. I wrote a gnuplot tutorial (available here) if you wish to better understand what each line in the script is doing.

set terminal png font "Helvetica,16" enhanced size 1680,1200
set output "titration_curves.png"

# Define the titration curve Hill equation
f(x) = 1 / (1 + 10**(n*(pka - x)))

# Set up the x- and y-ranges
set xrange[0:7]
set yrange[0:1]

# Turn on grid
set grid linewidth 0

# Turn off the key, we will use labels instead
unset key

# Set up a multiplot
set multiplot layout 3,4

# Fit the first residue
set title 'Glu 7'
pka = 3.5; n = 1.0 # initial guesses
fit f(x) 'titration_curves.dat' using 1:2 via n,pka
set label 1 sprintf("pK_a = %.2f\nn = %.2f", pka, n) at graph 0.05,0.3
plot 'titration_curves.dat' using 1:2 with points pointsize 2, \
     f(x) with lines linewidth 2

# Fit the second residue
set title 'His 15'
pka = 6.5; n = 1.0 # initial guesses
fit f(x) 'titration_curves.dat' using 1:3 via n,pka
unset label 1
set label 2 sprintf("pK_a = %.2f\nn = %.2f", pka, n) at graph 0.05,0.3
plot 'titration_curves.dat' using 1:3 with points pointsize 2, \
     f(x) with lines linewidth 2

# Fit the third residue
set title 'Asp 18'
pka = 2.5; n = 1.0 # initial guesses
fit f(x) 'titration_curves.dat' using 1:4 via n,pka
unset label 2
set label 3 sprintf("pK_a = %.2f\nn = %.2f", pka, n) at graph 0.5,0.3
plot 'titration_curves.dat' using 1:4 with points pointsize 2, \
     f(x) with lines linewidth 2

# Fit the fourth residue
set title 'Glu 35'
pka = 7.5; n = 1.0 # initial guesses
fit f(x) 'titration_curves.dat' using 1:5 via n,pka
unset label 3
set label 4 sprintf("pK_a = %.2f\nn = %.2f", pka, n) at graph 0.05,0.3
plot 'titration_curves.dat' using 1:5 with points pointsize 2, \
     f(x) with lines linewidth 2

# Fit the fifth residue
set title 'Asp 48'
pka = 1.5; n = 1.0 # initial guesses
fit f(x) 'titration_curves.dat' using 1:6 via n,pka
unset label 4
set label 5 sprintf("pK_a = %.2f\nn = %.2f", pka, n) at graph 0.5,0.3
plot 'titration_curves.dat' using 1:6 with points pointsize 2, \
     f(x) with lines linewidth 2

# Fit the sixth residue
set title 'Asp 52'
pka = 2.5; n = 1.0 # initial guesses
fit f(x) 'titration_curves.dat' using 1:7 via n,pka
unset label 5
set label 6 sprintf("pK_a = %.2f\nn = %.2f", pka, n) at graph 0.5,0.3
plot 'titration_curves.dat' using 1:7 with points pointsize 2, \
     f(x) with lines linewidth 2

# Plot the seventh residue (do not fit, since it does not titrate)
set title 'Asp 66'
unset label 6
plot 'titration_curves.dat' using 1:8 with points pointsize 2

# Fit the eight residue
set title 'Asp 87'
pka = 3.5; n = 1.0 # initial guesses
fit f(x) 'titration_curves.dat' using 1:9 via n,pka
set label 8 sprintf("pK_a = %.2f\nn = %.2f", pka, n) at graph 0.05,0.3
plot 'titration_curves.dat' using 1:9 with points pointsize 2, \
     f(x) with lines linewidth 2

# Fit the ninth residue
set title 'Asp 101'
pka = 4.0; n = 1.0 # initial guesses
fit f(x) 'titration_curves.dat' using 1:10 via n,pka
unset label 8
set label 9 sprintf("pK_a = %.2f\nn = %.2f", pka, n) at graph 0.05,0.3
plot 'titration_curves.dat' using 1:10 with points pointsize 2, \
     f(x) with lines linewidth 2

# Fit the tenth residue
set title 'Asp 119'
pka = 2.5; n = 1.0 # initial guesses
fit f(x) 'titration_curves.dat' using 1:11 via n,pka
unset label 9
set label 10 sprintf("pK_a = %.2f\nn = %.2f", pka, n) at graph 0.5,0.3
plot 'titration_curves.dat' using 1:11 with points pointsize 2, \
     f(x) with lines linewidth 2

unset multiplot

The calculated pKas here show very good qualitative agreement with experimental measurements for HEWL pKas. Glu 35, whose pKa has been measured near 6.1, is correctly predicted to have a large shift away from the model compound pKa of 4.4 for an isolated Glutamate. In each case, this method correctly predicts the direction of the pKa shift from the model compound, and even gives quantitatively reasonable results for most residues as well. Keep in mind, though, that we ran only 1 ns simulations at each pH, which is most likely insufficient for extensive conformational sampling. It remains, however, illustrative in demonstrating how to carry out CpHMD simulations and calculate resulting pKas.

Comparing the Root Mean Squared Deviations of the CpHMD Simulations

Next, we calculate the rmsd of each the CpHMD simulations and histogram the data using cpptraj.
#calculate the rmsd and histogram the data
trajin 0/
reference 4LYT.equil.rst7
rmsd rmsph mass reference @CA,C,N,O out pH0.4LYT.rmsd.dat
hist rmsph,*,*,.1,* norm out pH0.4LYT.rmsd.histogram.dat

The input shown above is using the trajectory for pH 0 and can be modified to work with any of the other pH values. The coordinates from 4LYT.equil.rst7 were used as the reference structure and the rmsd of only the backbone atoms were considered. The hist function in cpptraj was used to histogram the data. The bin width is set to .1, and the wildcards (*) tell cpptraj to use the smallest and largest values of the data set to set the smallest and highest bin, respectively. The last wild-card tells cpptraj to use the number of bins necessary to go from the data set minimum to the data set maximum using a step size of 0.1 Angstroms. See the AmberTools manual for more detailed instructions on using cpptraj.

The data can be generated by executing the following command:

cpptraj 4LYT.parm7

A tarball containing all of the rmsd and histogram files can be downloaded here: rmsd_data.tar.bz2

Because our simulations were relatively short and we only have 1000 data points, the histograms appear pretty noisy. However, we can qualitatively say that the high pH and low pH simulations were more flexible and explored conformations farther from the crystal structure. This makes sense, since HEWL is most active near pH 4, and 4LYT was solved at pH 4.8. [2]

Visualizing the trajectories in VMD

If you do not already know how to use VMD with Amber files, please see Basic Tutorial 2 for instructions

The trajectories of the CpHMD simulations can be viewed using VMD. The program (along with get_creates a tcl script to load into VMD with your topology and trajectory file. This will script will allow you to see the different protonation states sampled by the protein during the CpHMD simulations. is inefficient for large trajectories, and was written quickly with limited knowledge of VMD's scripting environment. should be used as: cpin cpout1 cpout2 cpout3 cpoutN...

where the cpin file is just parsed for which residues are titrating (so it should typically be the cpin file written by The cpout files must be listed in the same order as the trajectories are loaded into VMD (the full records in the cpout file are matched to each frame of the trajectory). For large numbers of snapshots, this script becomes very inefficient.

Perfoming the command below will create a file named pHVMDscript.tcl that can be loaded into VMD. 4LYT.equil.cpin 0/4LYT.md1.cpout

This particular script identifies the titratable protons that are "gone" for each residue in each frame, then moves those hydrogen atoms 99 Angstroms away in each direction. Therefore, we need to use the Dynamic Bonds representation in VMD to avoid drawing all of the bonds to the H-atoms we have moved away from the protein.

Loading the files in VMD

  1. Open VMD
  2. In the console titled VMD Main, click File, followed by New Molecule
  3. Load the topology file, 4LYT.parm7, (AMBER 7 Parm) format, followed by the trajectory file, 0/, (NetCDF AMBER,MMTK) format
  4. In the vmd terminal type source pHVMDscript.tcl
  5. In the console titled VMD Main click Graphics, followed by Representations
  6. Change the drawing method in Graphical Representations console to New Cartoon
  7. Click the tab Create Rep and under Selected Atoms type resid 7 15 18 35 48 52 66 87 101 119
  8. Change the drawing method to Dynamic Bonds
  9. Click the tab Create Rep and change the drawing method to VDW and reduce the Sphere Scale to 0.4

Click here to go back to the Introduction

Click here to go back to Section 3


[2] A. C. M. Young, J. C. Dewan, C. Nave, and R. F. Tilton, "Comparison of radiation-induced decay and structure refinement from X-ray data collected from lysozyme crystals at low and ambient temperatures", J. Appl. Cryst., 1993, 26, pp. 309-319

(Note: These tutorials are meant to provide illustrative examples of how to use the AMBER software suite to carry out simulations that can be run on a simple workstation in a reasonable period of time. They do not necessarily provide the optimal choice of parameters or methods for the particular application area.)
Copyright Jason Swails and T. Dwight McGee, 2013