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 \(c\).
\(|P_{f(1KWZ_1)}(2) \setminus P_{f(6FWA_1)}(2)|=45\),
\(|P_{f(6FWA_1)}(2) \setminus P_{f(1KWZ_1)}(2)|=154\).
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 1:11010110101010010001010001011011000100111000001110010110001010111100100001100110001111100010010110101101000010000100110100010110011100100010001100111
Pair
\(Z_2\)
Length of longest common subsequence
1KWZ_1,6FWA_1
199
4
1KWZ_1,8GZG_1
160
4
6FWA_1,8GZG_1
177
4
Newick tree
[
6FWA_1:98.42,
[
1KWZ_1:80,8GZG_1:80
]:18.42
]
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 \((0.85)(0.8)(\frac{566
}{\log_{20}
566}-\frac{149}{\log_{20}149})=121.\)
Status
Protein1
Protein2
d
d1/2
Query variables
1KWZ_1
6FWA_1
153
103.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]