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 \(c\).
\(|P_{f(4GUO_1)}(2) \setminus P_{f(4QKL_1)}(2)|=146\),
\(|P_{f(4QKL_1)}(2) \setminus P_{f(4GUO_1)}(2)|=14\).
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 1:110000000001110000011100101010000010010100011100100011000110101001111101101111000100100110001000110010010011100110101001000100110100011110011011001001100110000011110001111110101001011100010101010110000100000000
Pair
\(Z_2\)
Length of longest common subsequence
4GUO_1,4QKL_1
160
3
4GUO_1,6VRV_1
188
4
4QKL_1,6VRV_1
168
3
Newick tree
[
6VRV_1:91.98,
[
4GUO_1:80,4QKL_1:80
]:11.98
]
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 \((0.85)(0.8)(\frac{237
}{\log_{20}
237}-\frac{27}{\log_{20}27})=71.6\)
Status
Protein1
Protein2
d
d1/2
Query variables
4GUO_1
4QKL_1
92
51.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]