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 \(c\).
\(|P_{f(4LPJ_1)}(2) \setminus P_{f(4XVA_1)}(2)|=55\),
\(|P_{f(4XVA_1)}(2) \setminus P_{f(4LPJ_1)}(2)|=115\).
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 1:11001000000000111001101010100000010111000011100110110000000000100110000001101110010010000110010110111010001001110100110001010100110000100010010011011001001111001110011000010000
Pair
\(Z_2\)
Length of longest common subsequence
4LPJ_1,4XVA_1
170
4
4LPJ_1,5AZP_1
170
5
4XVA_1,5AZP_1
162
4
Newick tree
[
4LPJ_1:86.29,
[
4XVA_1:81,5AZP_1:81
]:5.29
]
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 \((0.85)(0.8)(\frac{469
}{\log_{20}
469}-\frac{176}{\log_{20}176})=85.9\)
Status
Protein1
Protein2
d
d1/2
Query variables
4LPJ_1
4XVA_1
110
87
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]