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 \(c\).
\(|P_{f(6FLK_1)}(2) \setminus P_{f(7YAQ_1)}(2)|=84\),
\(|P_{f(7YAQ_1)}(2) \setminus P_{f(6FLK_1)}(2)|=86\).
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 1:1100001010101001011011111001100001111011111001110100010111100001101101100100011011111011000010001111110101001100000011100100010000000111111010000110100010100011101
Pair
\(Z_2\)
Length of longest common subsequence
6FLK_1,7YAQ_1
170
3
6FLK_1,4LAN_1
177
4
7YAQ_1,4LAN_1
187
4
Newick tree
[
4LAN_1:92.95,
[
6FLK_1:85,7YAQ_1:85
]:7.95
]
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 \((0.85)(0.8)(\frac{317
}{\log_{20}
317}-\frac{154}{\log_{20}154})=49.8\)
Status
Protein1
Protein2
d
d1/2
Query variables
6FLK_1
7YAQ_1
66
64
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]