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 \(c\).
\(|P_{f(2LKU_1)}(2) \setminus P_{f(7FPS_1)}(2)|=62\),
\(|P_{f(7FPS_1)}(2) \setminus P_{f(2LKU_1)}(2)|=92\).
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 1:1110111000100110000111110100011001100011011001011100101100011001100000110110101010111100101010001010001001100
Pair
\(Z_2\)
Length of longest common subsequence
2LKU_1,7FPS_1
154
3
2LKU_1,1AQH_1
199
3
7FPS_1,1AQH_1
205
3
Newick tree
[
1AQH_1:10.83,
[
2LKU_1:77,7FPS_1:77
]:30.83
]
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 \((0.85)(0.8)(\frac{268
}{\log_{20}
268}-\frac{109}{\log_{20}109})=50.3\)
Status
Protein1
Protein2
d
d1/2
Query variables
2LKU_1
7FPS_1
66
54.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]