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 \(c\).
\(|P_{f(6AFH_1)}(2) \setminus P_{f(5NIU_1)}(2)|=87\),
\(|P_{f(5NIU_1)}(2) \setminus P_{f(6AFH_1)}(2)|=63\).
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 1:110001111110110010011110110011101011111100110000011101010100100011001111111011100100011100110000000111111011101111001111001000111000110110000000010001111000111001011111101101001110101111100
Pair
\(Z_2\)
Length of longest common subsequence
6AFH_1,5NIU_1
150
3
6AFH_1,6EGL_1
124
3
5NIU_1,6EGL_1
98
4
Newick tree
[
6AFH_1:74.24,
[
6EGL_1:49,5NIU_1:49
]:25.24
]
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 \((0.85)(0.8)(\frac{317
}{\log_{20}
317}-\frac{128}{\log_{20}128})=58.3\)
Status
Protein1
Protein2
d
d1/2
Query variables
6AFH_1
5NIU_1
74
61.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]