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Parikh vectors
5NUB_1 6KQP_1 5NIN_1 Letter Amino acid
8 8 2 Y Tyrosine
13 10 7 V Valine
6 16 0 C Cysteine
10 15 8 K Lycine
4 2 0 W Tryptophan
8 8 6 Q Glutamine
10 16 21 E Glutamic acid
11 18 11 G Glycine
2 4 1 H Histidine
4 14 8 I Isoleucine
14 15 6 R Arginine
9 15 6 N Asparagine
18 12 17 D Aspartic acid
5 12 2 P Proline
11 6 4 S Serine
9 13 12 T Threonine
13 10 11 A Alanine
13 14 9 L Leucine
4 3 10 M Methionine
10 9 8 F Phenylalanine 

5NUB_1|Chain A|Retinol-binding protein 4|Homo sapiens (9606)
>6KQP_1|Chain A|Histone-lysine N-methyltransferase, H3 lysine-36 and H4 lysine-20 specific|Homo sapiens (9606)
>5NIN_1|Chains A, B|Calmodulin|Homo sapiens (9606)
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 \(w=f(c)\)
5NUB , Knot 86 182 0.82 40 135 176 
ERDCRVSSFRVKENFDKARFSGTWYAMAKKDPEGLFLQDNIVAEFSVDETGQMSATAKGRVRLLNNWDVCADMVGTFTDTEDPAKFKMKYWGVASFLQKGNDDHWIVDTDYDTYAVQYSCRLLNLDGTCADSYSFVFSRDPNGLPPEAQKIVRQRQEELCLARQYRLIVHNGYCDGRSERNL
6KQP , Knot 104 220 0.85 40 166 215 
DKKPPPYKHIKVNRPIGRVQIFTADLSEIPRCNCKATDENPCGIDSECINRMLLYECHPTVCPAGGRCQNQCFSKRQYPEVEIFRTLQRGWGLRTKTDIKKGEFVNEYVGELIDEEECRARIRYAQEHDITNFYMLTLDKDRIIDAGPKGNYARFMNHCCQPNCETQKWSVNGDTRVGLFALSDIKAGTELTFNYNLECLGNGKTVCKCGAPNCSGFLGV
5NIN , Knot 67 149 0.75 36 97 134 
MADQLTEEQIAEFKEAFSLFDKDGDGTITTKELGTVMRSLGQNPTEAELQDMINEVDADGNGTIDFPEFLTMMARKMKDTDSEEEIREAFRVFDKDGNGYISAAELRHVMTNLGEKLTDEEVDEMIREADIDGDGQVNYEEFVQMMTAK

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 \(c\).

\(|P_{f(5NUB_1)}(2) \setminus P_{f(6KQP_1)}(2)|=75\), \(|P_{f(6KQP_1)}(2) \setminus P_{f(5NUB_1)}(2)|=106\). 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 1:00000100101000100101010101110001011110001110101000101010101010110010101011101000001101010011110110010000111000000011000001101010010000111000101111010011000000101100001110010001000001
Pair \(Z_2\) Length of longest common subsequence
5NUB_1,6KQP_1 181 3
5NUB_1,5NIN_1 140 3
6KQP_1,5NIN_1 161 3

Newick tree

 
[
	6KQP_1:90.26,
	[
		5NUB_1:70,5NIN_1:70
	]: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 \((0.85)(0.8)(\frac{402 }{\log_{20} 402}-\frac{182}{\log_{20}182})=65.3\)
Status Protein1 Protein2 d d1/2
Query variables 5NUB_1 6KQP_1 85 78
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]