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 \(c\).
\(|P_{f(3ETQ_1)}(2) \setminus P_{f(6WDH_1)}(2)|=74\),
\(|P_{f(6WDH_1)}(2) \setminus P_{f(3ETQ_1)}(2)|=90\).
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 1:111010000000000000010001010011101000100000000010110000111010111000110100001110111110101011011100101011011001100101100101100111011001000101001001101011001000101010000010010100100110001110011001110010011000
Pair
\(Z_2\)
Length of longest common subsequence
3ETQ_1,6WDH_1
164
3
3ETQ_1,7XEK_1
170
3
6WDH_1,7XEK_1
168
3
Newick tree
[
7XEK_1:85.31,
[
3ETQ_1:82,6WDH_1:82
]:3.31
]
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 \((0.85)(0.8)(\frac{475
}{\log_{20}
475}-\frac{204}{\log_{20}204})=78.8\)
Status
Protein1
Protein2
d
d1/2
Query variables
3ETQ_1
6WDH_1
101
89.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]