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 \(c\).
\(|P_{f(2WLB_1)}(2) \setminus P_{f(7WJX_1)}(2)|=60\),
\(|P_{f(7WJX_1)}(2) \setminus P_{f(2WLB_1)}(2)|=56\).
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 1:1011011110101001110100100110110100101011001011000001110100001101100000011011111000001100111000101101011
Pair
\(Z_2\)
Length of longest common subsequence
2WLB_1,7WJX_1
116
2
2WLB_1,6FMY_1
197
3
7WJX_1,6FMY_1
189
3
Newick tree
[
6FMY_1:10.30,
[
2WLB_1:58,7WJX_1:58
]:48.30
]
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 \((0.85)(0.8)(\frac{198
}{\log_{20}
198}-\frac{95}{\log_{20}95})=33.7\)
Status
Protein1
Protein2
d
d1/2
Query variables
2WLB_1
7WJX_1
42
41
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]