Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\).
Let \(p_w(n)\) be the cardinality of \(P_w(n)\).
Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).
\(|P_{f(1YIH_1)}(2) \setminus P_{f(3WYB_1)}(2)|=28\),
\(|P_{f(3WYB_1)}(2) \setminus P_{f(1YIH_1)}(2)|=128\).
Let
\(
Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)|
\)
be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:110110000101111011101100110110011101100000110101001010101010011011001110100110110110010100101011010110001110111011101011101010011101001100000
Pair
\(Z_2\)
Length of longest common subsequence
1YIH_1,3WYB_1
156
4
1YIH_1,7VZS_1
184
3
3WYB_1,7VZS_1
138
4
Newick tree
[
1YIH_1:90.06,
[
3WYB_1:69,7VZS_1:69
]:21.06
]
Let d be the
Otu--Sayood
distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{475
}{\log_{20}
475}-\frac{141}{\log_{20}141})=98.9\)
Status
Protein1
Protein2
d
d1/2
Query variables
1YIH_1
3WYB_1
126
87
Was not able to put for d Was not able to put for d1
In notation analogous to
[Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]