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 \(c\).
\(|P_{f(7ULN_1)}(2) \setminus P_{f(7CTV_1)}(2)|=78\),
\(|P_{f(7CTV_1)}(2) \setminus P_{f(7ULN_1)}(2)|=115\).
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 1:0000111001111001110000011010011000100100001010010001000001011111111110101010101010010000001000000111100001001101110001010000100010010101010100100100100010100010001001011111111000000001100111000110111111110001001010000100110011100100101100
Pair
\(Z_2\)
Length of longest common subsequence
7ULN_1,7CTV_1
193
3
7ULN_1,7XPK_1
179
4
7CTV_1,7XPK_1
168
4
Newick tree
[
7ULN_1:95.89,
[
7XPK_1:84,7CTV_1:84
]:11.89
]
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 \((0.85)(0.8)(\frac{574
}{\log_{20}
574}-\frac{238}{\log_{20}238})=95.4\)
Status
Protein1
Protein2
d
d1/2
Query variables
7ULN_1
7CTV_1
124
105.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]