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 \(c\).
\(|P_{f(7SIN_1)}(2) \setminus P_{f(8ILK_1)}(2)|=178\),
\(|P_{f(8ILK_1)}(2) \setminus P_{f(7SIN_1)}(2)|=31\).
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 1:11100001111110100010110001000101111111110111110000100010010010001011011011111100100011111010110011000001001101010111000100101001000000110011111101011001110111110110100100001100000100110011000001011101100101011101110000101110010001000010101001100000000100110110000101111100110101110011000101011110011100011111001011110111110110111100110010100010011100110001000100110111110011010000100100000110110010001001001010000101000101110011011001000111011100100101001011011001001010001100101000101110001101010100101110011000101001001110000111011000111000000011100011101010001001001010000000101000010011000000001100101101001111110111111111011111111010001110100001001111011001000111110100100010011111011101001110000111110101100100011110101111110011011101111001110000000100011110000101111111110001111101111100001100100101101011111111101111010001011011011111110111110111001011110100001000000000
Pair
\(Z_2\)
Length of longest common subsequence
7SIN_1,8ILK_1
209
4
7SIN_1,2PRC_1
170
5
8ILK_1,2PRC_1
193
3
Newick tree
[
8ILK_1:10.26,
[
7SIN_1:85,2PRC_1:85
]:20.26
]
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 \((0.85)(0.8)(\frac{1102
}{\log_{20}
1102}-\frac{224}{\log_{20}224})=236.\)
Status
Protein1
Protein2
d
d1/2
Query variables
7SIN_1
8ILK_1
301
187.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]