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 \(c\).
\(|P_{f(9FFO_1)}(2) \setminus P_{f(4DXR_1)}(2)|=124\),
\(|P_{f(4DXR_1)}(2) \setminus P_{f(9FFO_1)}(2)|=41\).
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 1:100000110110110111101001010100010100010001100001010101010000100000110011001101000010111100100100011100111100001000101110001000010101110110100111001101001100100011001011001101000101100101010100110100111010101101100100010110010001100111100100100001110010011100001001110001010001101110001101101110010111000011100111100110100101010001101101011011111001111011101101001000011011010010011111111110110110010001010110100
Pair
\(Z_2\)
Length of longest common subsequence
9FFO_1,4DXR_1
165
4
9FFO_1,9IIZ_1
142
4
4DXR_1,9IIZ_1
191
4
Newick tree
[
4DXR_1:94.53,
[
9FFO_1:71,9IIZ_1:71
]:23.53
]
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 \((0.85)(0.8)(\frac{613
}{\log_{20}
613}-\frac{202}{\log_{20}202})=117.\)
Status
Protein1
Protein2
d
d1/2
Query variables
9FFO_1
4DXR_1
147
108
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]