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 \(c\).
\(|P_{f(1VIU_1)}(2) \setminus P_{f(8IIM_1)}(2)|=79\),
\(|P_{f(8IIM_1)}(2) \setminus P_{f(1VIU_1)}(2)|=73\).
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 1:10100010110001100001010010001000010110000010001011011100000001111001011011010001011000111100001010100011000100110100110101011110011011110000000101111100001011011100110110010100100111100100001100110000000
Pair
\(Z_2\)
Length of longest common subsequence
1VIU_1,8IIM_1
152
3
1VIU_1,1OSC_1
146
4
8IIM_1,1OSC_1
128
3
Newick tree
[
1VIU_1:77.70,
[
1OSC_1:64,8IIM_1:64
]:13.70
]
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 \((0.85)(0.8)(\frac{411
}{\log_{20}
411}-\frac{203}{\log_{20}203})=61.2\)
Status
Protein1
Protein2
d
d1/2
Query variables
1VIU_1
8IIM_1
75
74.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]