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Dendrogram of a substitution matrix#
In this example a dendrogram is created, that displays the similarity of amino acids in the BLOSUM62 substitution matrix. The amino acids are clustered with the UPGMA method.
# Code source: Patrick Kunzmann
# License: BSD 3 clause
import matplotlib.pyplot as plt
import numpy as np
import biotite.sequence as seq
import biotite.sequence.align as align
import biotite.sequence.graphics as graphics
import biotite.sequence.phylo as phylo
# Obtain BLOSUM62
matrix = align.SubstitutionMatrix.std_protein_matrix()
print(matrix)
A C D E F G H I K L M N P Q R S T V W Y B Z X *
A 4 0 -2 -1 -2 0 -2 -1 -1 -1 -1 -2 -1 -1 -1 1 0 0 -3 -2 -2 -1 0 -4
C 0 9 -3 -4 -2 -3 -3 -1 -3 -1 -1 -3 -3 -3 -3 -1 -1 -1 -2 -2 -3 -3 -2 -4
D -2 -3 6 2 -3 -1 -1 -3 -1 -4 -3 1 -1 0 -2 0 -1 -3 -4 -3 4 1 -1 -4
E -1 -4 2 5 -3 -2 0 -3 1 -3 -2 0 -1 2 0 0 -1 -2 -3 -2 1 4 -1 -4
F -2 -2 -3 -3 6 -3 -1 0 -3 0 0 -3 -4 -3 -3 -2 -2 -1 1 3 -3 -3 -1 -4
G 0 -3 -1 -2 -3 6 -2 -4 -2 -4 -3 0 -2 -2 -2 0 -2 -3 -2 -3 -1 -2 -1 -4
H -2 -3 -1 0 -1 -2 8 -3 -1 -3 -2 1 -2 0 0 -1 -2 -3 -2 2 0 0 -1 -4
I -1 -1 -3 -3 0 -4 -3 4 -3 2 1 -3 -3 -3 -3 -2 -1 3 -3 -1 -3 -3 -1 -4
K -1 -3 -1 1 -3 -2 -1 -3 5 -2 -1 0 -1 1 2 0 -1 -2 -3 -2 0 1 -1 -4
L -1 -1 -4 -3 0 -4 -3 2 -2 4 2 -3 -3 -2 -2 -2 -1 1 -2 -1 -4 -3 -1 -4
M -1 -1 -3 -2 0 -3 -2 1 -1 2 5 -2 -2 0 -1 -1 -1 1 -1 -1 -3 -1 -1 -4
N -2 -3 1 0 -3 0 1 -3 0 -3 -2 6 -2 0 0 1 0 -3 -4 -2 3 0 -1 -4
P -1 -3 -1 -1 -4 -2 -2 -3 -1 -3 -2 -2 7 -1 -2 -1 -1 -2 -4 -3 -2 -1 -2 -4
Q -1 -3 0 2 -3 -2 0 -3 1 -2 0 0 -1 5 1 0 -1 -2 -2 -1 0 3 -1 -4
R -1 -3 -2 0 -3 -2 0 -3 2 -2 -1 0 -2 1 5 -1 -1 -3 -3 -2 -1 0 -1 -4
S 1 -1 0 0 -2 0 -1 -2 0 -2 -1 1 -1 0 -1 4 1 -2 -3 -2 0 0 0 -4
T 0 -1 -1 -1 -2 -2 -2 -1 -1 -1 -1 0 -1 -1 -1 1 5 0 -2 -2 -1 -1 0 -4
V 0 -1 -3 -2 -1 -3 -3 3 -2 1 1 -3 -2 -2 -3 -2 0 4 -3 -1 -3 -2 -1 -4
W -3 -2 -4 -3 1 -2 -2 -3 -3 -2 -1 -4 -4 -2 -3 -3 -2 -3 11 2 -4 -3 -2 -4
Y -2 -2 -3 -2 3 -3 2 -1 -2 -1 -1 -2 -3 -1 -2 -2 -2 -1 2 7 -3 -2 -1 -4
B -2 -3 4 1 -3 -1 0 -3 0 -4 -3 3 -2 0 -1 0 -1 -3 -4 -3 4 1 -1 -4
Z -1 -3 1 4 -3 -2 0 -3 1 -3 -1 0 -1 3 0 0 -1 -2 -3 -2 1 4 -1 -4
X 0 -2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -2 -1 -1 0 0 -1 -2 -1 -1 -1 -1 -4
* -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 1
The original BLOSUM62 contains symbols for ambiguous amino acids and the stop signal. As these are not actual amino acids, a new substitution matrix is created, where these symbols are are removed.
# Matrix should not contain ambiguous symbols or stop signal
matrix = align.SubstitutionMatrix(
seq.Alphabet(matrix.get_alphabet1().get_symbols()[:-4]),
seq.Alphabet(matrix.get_alphabet2().get_symbols()[:-4]),
matrix.score_matrix()[:-4, :-4],
)
similarities = matrix.score_matrix()
print(matrix)
A C D E F G H I K L M N P Q R S T V W Y
A 4 0 -2 -1 -2 0 -2 -1 -1 -1 -1 -2 -1 -1 -1 1 0 0 -3 -2
C 0 9 -3 -4 -2 -3 -3 -1 -3 -1 -1 -3 -3 -3 -3 -1 -1 -1 -2 -2
D -2 -3 6 2 -3 -1 -1 -3 -1 -4 -3 1 -1 0 -2 0 -1 -3 -4 -3
E -1 -4 2 5 -3 -2 0 -3 1 -3 -2 0 -1 2 0 0 -1 -2 -3 -2
F -2 -2 -3 -3 6 -3 -1 0 -3 0 0 -3 -4 -3 -3 -2 -2 -1 1 3
G 0 -3 -1 -2 -3 6 -2 -4 -2 -4 -3 0 -2 -2 -2 0 -2 -3 -2 -3
H -2 -3 -1 0 -1 -2 8 -3 -1 -3 -2 1 -2 0 0 -1 -2 -3 -2 2
I -1 -1 -3 -3 0 -4 -3 4 -3 2 1 -3 -3 -3 -3 -2 -1 3 -3 -1
K -1 -3 -1 1 -3 -2 -1 -3 5 -2 -1 0 -1 1 2 0 -1 -2 -3 -2
L -1 -1 -4 -3 0 -4 -3 2 -2 4 2 -3 -3 -2 -2 -2 -1 1 -2 -1
M -1 -1 -3 -2 0 -3 -2 1 -1 2 5 -2 -2 0 -1 -1 -1 1 -1 -1
N -2 -3 1 0 -3 0 1 -3 0 -3 -2 6 -2 0 0 1 0 -3 -4 -2
P -1 -3 -1 -1 -4 -2 -2 -3 -1 -3 -2 -2 7 -1 -2 -1 -1 -2 -4 -3
Q -1 -3 0 2 -3 -2 0 -3 1 -2 0 0 -1 5 1 0 -1 -2 -2 -1
R -1 -3 -2 0 -3 -2 0 -3 2 -2 -1 0 -2 1 5 -1 -1 -3 -3 -2
S 1 -1 0 0 -2 0 -1 -2 0 -2 -1 1 -1 0 -1 4 1 -2 -3 -2
T 0 -1 -1 -1 -2 -2 -2 -1 -1 -1 -1 0 -1 -1 -1 1 5 0 -2 -2
V 0 -1 -3 -2 -1 -3 -3 3 -2 1 1 -3 -2 -2 -3 -2 0 4 -3 -1
W -3 -2 -4 -3 1 -2 -2 -3 -3 -2 -1 -4 -4 -2 -3 -3 -2 -3 11 2
Y -2 -2 -3 -2 3 -3 2 -1 -2 -1 -1 -2 -3 -1 -2 -2 -2 -1 2 7
Now a function must be defined, that converts the similarity depicted by a substitution matrix into a distance required by the UPGMA method. In this case, the distance is defined as the difference between the similarity of the two symbols and the average maximum similarity of the symbols to themselves.
Finally the obtained (phylogenetic) tree is plotted as dendrogram.
def get_distance(similarities, i, j):
s_max = (similarities[i, i] + similarities[j, j]) / 2
return s_max - similarities[i, j]
distances = np.zeros(similarities.shape)
for i in range(distances.shape[0]):
for j in range(distances.shape[1]):
distances[i, j] = get_distance(similarities, i, j)
tree = phylo.upgma(distances)
fig = plt.figure(figsize=(8.0, 5.0))
ax = fig.add_subplot(111)
# Use the 3-letter amino acid code aa label
labels = [
seq.ProteinSequence.convert_letter_1to3(letter).capitalize()
for letter in matrix.get_alphabet1()
]
graphics.plot_dendrogram(ax, tree, orientation="top", labels=labels)
ax.set_ylabel("Distance")
# Add grid for clearer distance perception
ax.yaxis.grid(color="lightgray")
plt.show()
