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 \(c\).
\(|P_{f(5JBG_1)}(2) \setminus P_{f(1OGM_1)}(2)|=83\),
\(|P_{f(1OGM_1)}(2) \setminus P_{f(5JBG_1)}(2)|=57\).
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 1:11111100101001011111101000111110111000111010000101001101111100101100010001011001101011010000001110110100111001011001110100010101001011110000000001100011100100010100011011110101101100010111001101010100011101000100100011010000010000000111001001110100010110110011001000011010001100100000101101000001111000101101100100101000000010001110010000101011110000001010010011000101110001111000000100110110001110100101111010100001011000000011011001010111000110011011000111001110001111010101010001001110100001000010001111100110110111000001011010001110101001000000010010010100100011100100100101100101010101000100101010001001011001100000001110110001011110100111101000000001001011100100100000000001
Pair
\(Z_2\)
Length of longest common subsequence
5JBG_1,1OGM_1
140
5
5JBG_1,1LCU_1
174
5
1OGM_1,1LCU_1
170
4
Newick tree
[
1LCU_1:90.71,
[
5JBG_1:70,1OGM_1:70
]:20.71
]
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 \((0.85)(0.8)(\frac{1254
}{\log_{20}
1254}-\frac{574}{\log_{20}574})=174.\)
Status
Protein1
Protein2
d
d1/2
Query variables
5JBG_1
1OGM_1
222
205.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]