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 \(c\).
\(|P_{f(6TKK_1)}(2) \setminus P_{f(3FEU_1)}(2)|=72\),
\(|P_{f(3FEU_1)}(2) \setminus P_{f(6TKK_1)}(2)|=90\).
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 1:10110010111100101000010100000001010000100100110110000001101011110110111001110000000001000010100010011010010011110100010011111110111001101011010011010101010010
Pair
\(Z_2\)
Length of longest common subsequence
6TKK_1,3FEU_1
162
3
6TKK_1,8ZZV_1
169
4
3FEU_1,8ZZV_1
165
5
Newick tree
[
8ZZV_1:84.32,
[
6TKK_1:81,3FEU_1:81
]:3.32
]
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 \((0.85)(0.8)(\frac{343
}{\log_{20}
343}-\frac{158}{\log_{20}158})=56.1\)
Status
Protein1
Protein2
d
d1/2
Query variables
6TKK_1
3FEU_1
70
65.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]