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Parikh vectors
4KYH_1 3HQX_1 6BYN_1 Letter Amino acid
6 3 17 N Asparagine
3 3 10 C Cysteine
11 8 21 G Glycine
17 8 27 L Leucine
4 2 0 M Methionine
13 9 20 T Threonine
3 0 10 W Tryptophan
11 4 18 A Alanine
18 5 22 D Aspartic acid
6 4 5 Q Glutamine
3 4 8 H Histidine
12 4 9 P Proline
5 3 3 R Arginine
5 10 21 V Valine
10 11 8 E Glutamic acid
18 6 32 K Lycine
11 7 11 F Phenylalanine
12 10 36 S Serine
6 3 9 Y Tyrosine
10 7 19 I Isoleucine 

4KYH_1|Chains A, B|Lactoylglutathione lyase|Mus musculus (10090)
>3HQX_1|Chain A|UPF0345 protein ACIAD0356|Acinetobacter sp. ADP1 (62977)
>6BYN_1|Chain A[auth W]|WD repeat-containing protein 5|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 \(w=f(c)\)
4KYH , Knot 86 184 0.81 40 138 176 
MAEPQPASSGLTDETAFSCCSDPDPSTKDFLLQQTMLRIKDPKKSLDFYTRVLGLTLLQKLDFPAMKFSLYFLAYEDKNDIPKDKSEKTAWTFSRKATLELTHNWGTEDDETQSYHNGNSDPRGFGHIGIAVPDVYSACKRFEELGVKFVKKPDDGKMKGLAFIQDPDGYWIEILNPNKIATII
3HQX , Knot 59 111 0.83 38 96 109 
SNAMSSAQFDHVTVIKKSNVYFGGLCISHTVQFEDGTKKTLGVILPTEQPLTFETHVPERMEIISGECRVKIADSTESELFRAGQSFYVPGNSLFKIETDEVLDYVCHLEG
6BYN , Knot 120 306 0.74 38 162 262 
GTVKPNYALKFTLAGHTKAVSSVKFSPNGEWLASSSADKLIKIWGAYDGKFEKTISGHKLGISDVAWSSDSNLLVSASDDKTLKIWDVSSGKCLKTLKGHSNYVFCCNFNPQSNLIVSGSFDESVRIWDVKTGKCLKTLPAHSDPVSAVHFNRDGSLIVSSSYDGLCRIWDTASGQCLKTLIDDDNPPVSFVKFSPNGKYILAATLDNTLKLWDYSKGKCLKTYTGHKNEKYCIFANFSVTGGKWIVSGSEDNLVYIWNLQTKEIVQKLQGHTDVVISTACHPTENIIASAALENDKTIKLWKSDC

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 \(c\).

\(|P_{f(4KYH_1)}(2) \setminus P_{f(3HQX_1)}(2)|=90\), \(|P_{f(3HQX_1)}(2) \setminus P_{f(4KYH_1)}(2)|=48\). 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 1:1101011001100001100000101000011100011010010001010001111011001011110101011100000011000000011010001010100011000000000001000101110111111010010001001110110010010101111100101011011010011011
Pair \(Z_2\) Length of longest common subsequence
4KYH_1,3HQX_1 138 3
4KYH_1,6BYN_1 154 3
3HQX_1,6BYN_1 150 3

Newick tree

 
[
	6BYN_1:78.20,
	[
		4KYH_1:69,3HQX_1:69
	]:9.20
]

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 \((0.85)(0.8)(\frac{295 }{\log_{20} 295}-\frac{111}{\log_{20}111})=57.6\)
Status Protein1 Protein2 d d1/2
Query variables 4KYH_1 3HQX_1 71 57
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]