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 \(c\).
\(|P_{f(5QEW_1)}(2) \setminus P_{f(8EMW_1)}(2)|=32\),
\(|P_{f(8EMW_1)}(2) \setminus P_{f(5QEW_1)}(2)|=136\).
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 1:101000100100010111100010001001100110110000000000101100001010000000101011010010000110011110011011011100000111110011001010010011000000111000010101100010000010010100100000001101000011011110011011011101000101010011111000111100101011000111100000100101001110100101111001001010011110110111100010001001000010111001111101100110100
Pair
\(Z_2\)
Length of longest common subsequence
5QEW_1,8EMW_1
168
4
5QEW_1,4BRX_1
174
4
8EMW_1,4BRX_1
180
4
Newick tree
[
4BRX_1:89.96,
[
5QEW_1:84,8EMW_1:84
]:5.96
]
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 \((0.85)(0.8)(\frac{1553
}{\log_{20}
1553}-\frac{321}{\log_{20}321})=317.\)
Status
Protein1
Protein2
d
d1/2
Query variables
5QEW_1
8EMW_1
401
251.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]