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 \(c\).
\(|P_{f(4ZXQ_1)}(2) \setminus P_{f(5TGZ_1)}(2)|=33\),
\(|P_{f(5TGZ_1)}(2) \setminus P_{f(4ZXQ_1)}(2)|=156\).
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 1:111011010100111101001010111001100001101110100111111010011011001010000001011000101001010100000010100011011010100101011010001011010100100010110100
Pair
\(Z_2\)
Length of longest common subsequence
4ZXQ_1,5TGZ_1
189
4
4ZXQ_1,8YOZ_1
136
3
5TGZ_1,8YOZ_1
209
3
Newick tree
[
5TGZ_1:10.13,
[
4ZXQ_1:68,8YOZ_1:68
]:40.13
]
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 \((0.85)(0.8)(\frac{596
}{\log_{20}
596}-\frac{144}{\log_{20}144})=130.\)
Status
Protein1
Protein2
d
d1/2
Query variables
4ZXQ_1
5TGZ_1
165
106.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]