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 \(c\).
\(|P_{f(8XKV_1)}(2) \setminus P_{f(3ANI_1)}(2)|=105\),
\(|P_{f(3ANI_1)}(2) \setminus P_{f(8XKV_1)}(2)|=46\).
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 1:101010001010010011010001110010101001100001001010001011100001110000000100000010000010000110010000000100100100010101000111100100011010101010010100000101001101100001001000001001011110000101110000000011100011001110110011001000110101101011110100011011110101111101011100100110101110011010001001100011101100010110100001101001110001000100110110000010001100101111011110100111011110111111110100110111111100100111000001101111001011100011101111110000111011001011010000111111000101101111001010011011110001011110101000110000000011001100000101101001100111101000100110001101100001010010000001100101011000111111100110000110000100100010100000010111000001001101011011000111100101101000101000101111001101110100000001110111010110000101110000011000001100100111001010100110100011011001101100011110110111011011101010110111101111011000110000101100110001001010111010000001011110011
Pair
\(Z_2\)
Length of longest common subsequence
8XKV_1,3ANI_1
151
4
8XKV_1,3TEI_1
140
4
3ANI_1,3TEI_1
153
4
Newick tree
[
3ANI_1:77.89,
[
8XKV_1:70,3TEI_1:70
]:7.89
]
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 \((0.85)(0.8)(\frac{1253
}{\log_{20}
1253}-\frac{398}{\log_{20}398})=222.\)
Status
Protein1
Protein2
d
d1/2
Query variables
8XKV_1
3ANI_1
282
206
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]