Analysis of Helical Axis

• This tutorial discuss the analyses that can be performed using the dnaMD Python module included in the do_x3dna package. The tutorial is prepared using Jupyter Notebook and this notebook tutorial file could be downloaded from this link.
• Download the input files that are used in the tutorial from this link.
• HelAxis_odna.dat input file is required in this tutorial. This file is generated from do_x3dna using the trajectory, which contains the free DNA. The input file should be present in the current/present working directory.
• The Python APIs should be only used when do_x3dna is executed with -ref option.
• Additional Requirement: SciPy package
• Detailed documentation is provided here.

Importing Python Modules

• numpy: Required for the calculations involving large arrays
• matplotlib: Required to plot the results
• dnaMD: Python module to analyze DNA/RNA structures from the do_x3dna output files.

import numpy as np
import matplotlib.pyplot as plt
import dnaMD
%matplotlib inline

Initializing DNA object and storing data to it

• DNA object is initialized by using the total number of base-pairs
• Local helical axis is formed by two adjacent base-pairs. Therefore, total number of Helical-axis postion is less than one of total number of base-pairs.
• X, Y and Z positions (Helical X-axis, Helical Y-axis and Helical Z-axis) of the Helical axis can be read and stored in DNA object from the input file using function set_helical_axis(…).

## Initialization
fdna = dnaMD.DNA(60)     #Initialization for 60 base-pairs free DNA

## Loading data from input files in respective DNA object
# Number of local helical axis positions = Number of base-pairs - one
# Number of helcial steps in a 60 base-pairs DNA = 59
fdna.set_helical_axis('tutorial_data/HelAxis_odna.dat')

Reading file : tutorial_data/HelAxis_odna.dat
Reading frame 1000
Finishid reading.... Total number of frame read =  1001

Smoothening of Helical Axis

• The helical axis postions calculated from do_x3dna are localized and do not form a smooth global axis upon joining these positions.
• The helical axis could be smoothed by spline interpolation using method generate_smooth_axis(…).
• A smooth three dimensional curve is fitted along the positions of local helical axis by spline interpolation method implemented in SciPy package.
• This procedure might take long time on large trajectory due to the fitting procedure.

Warning: Lower value of smooth may lead to an artifact of local sharp kink in the smoothed axis. Higher value of smooth may lead to the calculation of wrong helical axis.

---

### Smoothening of helical axis
fdna.generate_smooth_axis(smooth=500, spline=3, fill_point=6)

Fitting spline curve on helcial axis of frame 1000 out of 1001 frames
Finished spline curve fitting...

Extraction of original and smoothed helical axis postions as a function of time (manually)

• Psotions of original and smoothed helical axis for the given base-steps range can be extracted from the DNA obejct using function dnaMD.DNA.get_parameters(…).
• Following example shows, how to extract the data.

### Extraction of original helical axis for 5-55 base-steps segment

# X-axis original
RawX, bp_idx = fdna.get_parameters('helical x-axis', [5, 55], bp_range=True)
# Y-axis original
RawY, bp_idx = fdna.get_parameters('helical y-axis', [5, 55], bp_range=True)
# Z-axis original
RawZ, bp_idx = fdna.get_parameters('helical z-axis', [5, 55], bp_range=True)

# X-axis smoothed
SmoothX, bp_idx = fdna.get_parameters('helical x-axis smooth', [5, 55], bp_range=True)
# Y-axis smoothed
SmoothY, bp_idx = fdna.get_parameters('helical y-axis smooth', [5, 55], bp_range=True)
# Z-axis smoothed
SmoothZ, bp_idx = fdna.get_parameters('helical z-axis smooth', [5, 55], bp_range=True)


# Here RawX is a 2D array of shape (base-step, nframes)

# Some examples
## x, y, z coordinates of nth base-step in mth frame: base-step index = (n - 5), frame index = (m - 1)
print ( "\n====== Some Examples ======" )
print ("Original coordinates of 8th base-step in 15th frame  : [ %8.3f %8.3f %8.3f ]" %
       (RawX[3][14], RawY[3][14], RawZ[3][14]))
print ("Smoothened coordinates of 8th base-step in 15th frame: [ %8.3f %8.3f %8.3f ]" %
       (SmoothX[3][14], SmoothY[3][14], SmoothZ[3][14]))

print ("\nOriginal coordinates of 40th base-step in 900th frame  : [ %8.3f %8.3f %8.3f ]" %
       (RawX[37][899], RawY[37][899], RawZ[37][899]))
print ("Smoothened coordinates of 40th base-step in 900th frame: [ %8.3f %8.3f %8.3f ]\n" %
       (SmoothX[37][899], SmoothY[37][899], SmoothZ[37][899]))

====== Some Examples ======
Original coordinates of 8th base-step in 15th frame  : [  101.280  173.630   84.670 ]
Smoothened coordinates of 8th base-step in 15th frame: [  106.443  172.187   85.432 ]

Original coordinates of 40th base-step in 900th frame  : [  208.140  171.130   79.730 ]
Smoothened coordinates of 40th base-step in 900th frame: [  205.639  172.066   80.286 ]

To calculate curvature and tangent vectors along helical axis

• The bending at a specifc position on the helical axis can be quantifed by its curvature.
• Approximate bending of a DNA segment could be also quantified by the angle between the tangent vectors of the segment’s end points
• Both curvature and tangent vectors could be calculated using dnaMD.DNA.calculate_curvature_tangent(…) function.
• Curvature and tangent vectors should be calculated after generating the smooth helical axis.

### Calculating curvature and tangent vectors
# If store_tangent=True; then tangent vectors will be stored for later use, otherwise it will be discarded
fdna.calculate_curvature_tangent(store_tangent=True)
fdna.calculate_angle_bw_tangents([5,50])


# Curvature vs Time for 22nd bp
plt.title('Curvature for 22nd bp')
time, value = fdna.time_vs_parameter('helical axis curvature', [22])
plt.plot(time, value)
plt.xlabel('Time (ps)')
plt.ylabel('Curvature ($\AA^{-1}$)')
plt.show()

# Total Curvature vs Time for 10-50 bp segment
plt.title('Total Curvature for 10-50 bp segment')
# Bound DNA
# Here, Total Curvature is considered as the sum over the local curvatures of the base-steps
time, value = fdna.time_vs_parameter('helical axis curvature', [10, 50], merge=True, merge_method='sum')
plt.plot(time, value)

plt.xlabel('Time (ps)')
plt.ylabel('Curvature ($\AA^{-1}$)')
plt.show()

Writing trajectory of Helical Axis

• Calculated helical axis from do_x3dna and smoothed helical axis could be written as trajectory in PDB format file using dnaMD.DNA.write_haxis_pdb(…)
• If write_smooth_axis=True: Coordinates of smoothed helical axis will be written
• If write_orig_axis=True: Coordinates of original helical axis will be written
• If write_curv=True: The calculated curvature of smooth helical axis will be written in B-factor field of PDB file. This might be helpful to visualize the helical axis according to its bending in color scale.
• The values of curvature might be very small ( < 0.01 ). Therefore, scaling up of curvature by 100 to 1000 times may help in better visualization on color scale.

# Only smoothed helical axis
fdna.write_haxis_pdb(filename='only_smoothed_axis.pdb', write_smooth_axis=True)

# Only original helical axis
fdna.write_haxis_pdb(filename='only_original_axis.pdb', write_smooth_axis=False, write_orig_axis=True)

# Both original and smoothed axis
fdna.write_haxis_pdb(filename='original_smoothed_axis.pdb', write_smooth_axis=True, write_orig_axis=True)

# Both original and smoothed axis with curvature scaled-up by 1000 times
fdna.write_haxis_pdb(filename='original_smoothed_axis_curvature_.pdb', write_smooth_axis=True, write_orig_axis=True,
                     write_curv=True, scale_curv=1000)

Angle between tangent vectors

• Approximate bending of a DNA segment could be also quantified by the angle between the tangent vectors of the segment’s end points
• Angle between the tangent vectors can be calculated using dnaMD.DNA.calculate_angle_bw_tangents(…)

# Angle vs Time for 28-32 bp
plt.title('Bending Angle for 28-32 bp')

# Calculating angle between the tangent vectors of 38th and 32nd base-steps
angle = fdna.calculate_angle_bw_tangents([28,32])
# Change to Degree
angle = np.degrees(angle)
# Plotting
plt.plot(fdna.time, angle)
plt.xlabel('Time (ps)')
plt.ylabel('Angle ( $^o$)')
plt.show()


# Angle vs Time for 25-35 bp
plt.title('Bending Angle for 25-35 bp')

# Calculating angle between the tangent vectors of 25th and 35th base-steps
angle = fdna.calculate_angle_bw_tangents([25,35])
# Change to Degree
angle = np.degrees(angle)
# Plotting
plt.plot(fdna.time, angle)
plt.xlabel('Time (ps)')
plt.ylabel('Angle ( $^o$)')
plt.show()

# Angle vs Time for 20-40 bp
plt.title('Bending Angle for 20-40 bp')

# Calculating angle between the tangent vectors of 20th and 40th base-steps
angle = fdna.calculate_angle_bw_tangents([20,40])
# Change to Degree
angle = np.degrees(angle)
# Plotting
plt.plot(fdna.time, angle)
plt.xlabel('Time (ps)')
plt.ylabel('Angle ( $^o$)')
plt.show()

# Angle vs Time for 15-45 bp
plt.title('Bending Angle for 15-45 bp')

# Calculating angle between the tangent vectors of 15th and 45th base-steps
angle = fdna.calculate_angle_bw_tangents([15,45])
# Change to Degree
angle = np.degrees(angle)
# Plotting
plt.plot(fdna.time, angle)
plt.xlabel('Time (ps)')
plt.ylabel('Angle ( $^o$)')
plt.show()

# Angle vs Time for 5-55 bp
plt.title('Bending Angle for 5-55 bp')

# Calculating angle between the tangent vectors of 5th and 55th base-steps
angle = fdna.calculate_angle_bw_tangents([5,55])
# Change to Degree
angle = np.degrees(angle)
# Plotting
plt.plot(fdna.time, angle)
plt.xlabel('Time (ps)')
plt.ylabel('Angle ( $^o$)')
plt.show()