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Parikh vectors
7OFO_1 2KWK_1 5BPF_1 Letter Amino acid
7 0 26 V Valine
0 3 28 A Alanine
0 0 19 D Aspartic acid
4 2 27 G Glycine
0 3 12 T Threonine
0 0 10 Y Tyrosine
1 1 14 P Proline
2 1 25 S Serine
0 0 3 W Tryptophan
2 3 9 R Arginine
2 0 2 N Asparagine
0 0 19 E Glutamic acid
1 0 9 I Isoleucine
3 0 11 F Phenylalanine
0 0 7 H Histidine
0 0 3 C Cysteine
1 2 18 Q Glutamine
6 1 42 L Leucine
1 4 14 K Lycine
0 0 8 M Methionine 

7OFO_1|Chain A|Bcl-2 homologous antagonist/killer|Homo sapiens (9606)
>2KWK_1|Chain A[auth B]|Histone peptide|null
>5BPF_1|Chains A, B, C, D|D-alanine-D-alanine ligase|Yersinia pestis (632)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
7OFO , Knot 17 30 0.64 22 23 27 
SLGNGPILNVLVVLGVVLLGQFVVRRFFKS
2KWK , Knot 13 20 0.65 18 16 18 
ARTKQTARKSTGGKAPRKQL
5BPF , Knot 129 306 0.80 40 181 296 
MAEKVAVLLGGTSAEREVSLLSGQAVLAGLKEAGIDAYGVDTKDFPVTQLKEQGFDKVFIALHGRGGEDGTLQGVLEFLQLPYTGSGVMASALTMDKLRTKLVWQALGLPISPYVALNRQQFETLSPEELVACVAKLGLPLIVKPSHEGSSVGMSKVDHASELQKALVEAFQHDSDVLIEKWLSGPEFTVAILGDEVLPSIRIQPPGVFYDYDAKYLSDKTQYFCPSGLSDESEQQLAALALQAYHALDCSGWGRVDVMQDRDGHFYLLEVNTSPGMTSHSLVPMAARQYGLSFSQLVARILMLAD

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 \(c\).

\(|P_{f(7OFO_1)}(2) \setminus P_{f(2KWK_1)}(2)|=22\), \(|P_{f(2KWK_1)}(2) \setminus P_{f(7OFO_1)}(2)|=15\). 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 1:011011110111111111110111001100
Pair \(Z_2\) Length of longest common subsequence
7OFO_1,2KWK_1 37 2
7OFO_1,5BPF_1 180 4
2KWK_1,5BPF_1 183 3

Newick tree

 
[
	5BPF_1:10.24,
	[
		7OFO_1:18.5,2KWK_1:18.5
	]:85.74
]

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 \((0.85)(0.8)(\frac{50 }{\log_{20} 50}-\frac{20}{\log_{20}20})=12.4\)
Status Protein1 Protein2 d d1/2
Query variables 7OFO_1 2KWK_1 15 12
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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