Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2024 / Paper 2 / C8 / c / Solution 2026-10-20
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For each alignment column, ignore gaps, form the empirical distribution of its amino acids, and calculate its Shannon entropy in bits. The requested alignment column information content is
(1)
Thus it is the relative entropy from the column distribution to the uniform distribution on twenty amino acids, not the ordinary Shannon entropy itself. A column containing one residue type has the maximum value bits.
information = []
for j in range(A.shape[1]):
    counts, n = emission_counts(A[:, j])
    p = np.array(list(counts.values()), dtype=float) / n
    H = -np.sum(p * np.log2(p))
    information.append(float(np.log2(20) - H))
print(np.round(information, 4))
The values for all 79 alignment columns are as follows, in bits. Each line lists its one-based column interval followed by the corresponding values in order:
01–10: 1.9025, 1.5964, 1.9025, 3.8187, 1.9025, 1.5964, 1.9025, 3.5577, 1.9025, 1.9025
11–20: 2.4420, 3.3355, 1.5964, 4.3219, 3.8187, 1.6803, 3.0975, 3.3355, 3.0975, 2.4420
21–30: 2.5692, 2.8219, 3.5106, 3.3219, 4.3219, 4.3219, 2.8000, 2.0703, 2.6574, 1.8003
31–40: 2.1938, 1.9163, 3.7784, 3.2606, 1.6803, 1.6803, 2.6642, 2.5692, 2.8753, 2.2631
41–50: 2.8753, 3.3355, 1.6803, 2.9703, 3.3355, 2.7914, 2.2086, 2.5692, 4.3219, 1.3742
51–60: 2.2631, 4.3219, 4.3219, 3.8187, 4.3219, 3.5577, 2.8753, 1.9025, 2.7914, 2.0409
61–70: 3.3355, 2.7370, 2.4309, 2.4309, 2.5692, 2.0409, 2.5692, 3.3355, 2.9298, 3.5577
71–79: 1.9025, 1.8187, 4.3219, 2.2631, 3.8187, 2.4309, 1.9025, 1.9025, 1.9025
A high score indicates conservation among the observed nongap letters, but does not itself imply low gap fraction. For example, an insertion column represented by only one observed nongap residue has maximal information by this calculation; its tiny sample size still makes that estimate uncertain.
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