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 \(c\).
\(|P_{f(4UOV_1)}(2) \setminus P_{f(6TOM_1)}(2)|=109\),
\(|P_{f(6TOM_1)}(2) \setminus P_{f(4UOV_1)}(2)|=49\).
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 1:1001110111111110111111110110010111001101010011001100001101001011010111101001001001100100101111101011101001010010101100001010001101011010001010111111011000101001101110011000010101010011100000000010100110001101111001101000010010011110000110110100110
Pair
\(Z_2\)
Length of longest common subsequence
4UOV_1,6TOM_1
158
4
4UOV_1,5RHC_1
179
3
6TOM_1,5RHC_1
177
3
Newick tree
[
5RHC_1:92.09,
[
4UOV_1:79,6TOM_1:79
]:13.09
]
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 \((0.85)(0.8)(\frac{391
}{\log_{20}
391}-\frac{144}{\log_{20}144})=74.4\)
Status
Protein1
Protein2
d
d1/2
Query variables
4UOV_1
6TOM_1
95
73.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]