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 \(c\).
\(|P_{f(4TPN_1)}(2) \setminus P_{f(4OUE_1)}(2)|=60\),
\(|P_{f(4OUE_1)}(2) \setminus P_{f(4TPN_1)}(2)|=94\).
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 1:1011011101011101011010110001101100101111100100111100101010010011110111011110011001000111000111001100010111001100000011001100111011111001011111110111001101111100011111101001100001100011000111001001000100110110101110011101111001100001000100111111111110011010010111100100100111001010001001100010001010111001011110101100111111010001011001001010010010010110110101110110101011011110011110011011100010100100111010
Pair
\(Z_2\)
Length of longest common subsequence
4TPN_1,4OUE_1
154
3
4TPN_1,7RSV_1
152
4
4OUE_1,7RSV_1
150
6
Newick tree
[
4TPN_1:76.99,
[
7RSV_1:75,4OUE_1:75
]:1.99
]
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 \((0.85)(0.8)(\frac{875
}{\log_{20}
875}-\frac{406}{\log_{20}406})=125.\)
Status
Protein1
Protein2
d
d1/2
Query variables
4TPN_1
4OUE_1
161
148
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]