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Parikh vectors
8JKC_1 1DNU_1 7ZHX_1 Letter Amino acid
6 3 7 M Methionine
13 6 24 F Phenylalanine
10 10 24 P Proline
7 5 26 S Serine
8 7 37 A Alanine
7 9 27 R Arginine
20 9 41 L Leucine
20 2 35 K Lycine
16 5 32 V Valine
2 2 10 C Cysteine
7 5 18 Q Glutamine
12 2 36 I Isoleucine
1 3 6 W Tryptophan
12 5 22 N Asparagine
19 7 30 D Aspartic acid
11 3 19 Y Tyrosine
16 6 36 E Glutamic acid
23 7 35 G Glycine
10 1 12 H Histidine
18 7 26 T Threonine 

8JKC_1|Chains A, B, C, D|Green fluorescent protein|Aequorea victoria (6100)
>1DNU_1|Chains A, C[auth B]|MYELOPEROXIDASE|Homo sapiens (9606)
>7ZHX_1|Chain A|Glucose-6-phosphate 1-dehydrogenase|Leishmania donovani (5661)
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 \(w=f(c)\)
8JKC , Knot 108 238 0.82 40 158 223 
MSKGEELFTGVVPILVELDGDVNGHKFSVSGEGEGDATYGKLTLKFICTTGKLPVPWPTLVTTLTYGVQCFSRYPDHMKRHDFFKSAMPEGYVQERTIFFKDDGNYKTRAEVKFEGDTLVNRIELKGIDFKEDGNILGHKLEYNYNDHQVYIMADKQKNGIKANFKIRHNIEDGGVQLADHYQQNTPIGDGPVLLPDNHYLFTTSTLSKDPNEKRDHMVLLEFVTAAGITHGMDELYK
1DNU , Knot 55 104 0.81 40 86 102 
CPEQDKYRTITGMCNNRRSPTLGASNRAFVRWLPAEYEDGFSLPYGWTPGVKRNGFPVALARAVSNEIVRFPTDQLTPDQERSLMFMQWGQLLDHDLDFTPEPA
7ZHX , Knot 211 503 0.87 40 266 482 
NCKVKDEQKSRALTIIVFGASGDLAKKKTFPALFDLYCGGLLPPEVNIIGYARTKVDDVEKWKHETLMKYFSNLSERGCHAEDFLKHISYFCGAYDSVDDFKRLDAVIREKENAFKGPEKGGNRLFYLALPPSVFASVCESIHKGAMPQEVGGWVRVIIEKPFGRDTKSSAELSQALEPFFDESQLYRIDHYLGKEMVQNIITTRFANRIFSAVWNASNIACVQITFKETIGTEGRGGYFDNIGIIRDVMQNHLTQILALLAMEKPRSLDAECIRDEKVSVLKCIEPITKENCVLGQYTASADGSIPGYLEDVTVPEGSTCPTFAVMRLNINNDRWAGVPFILKAGKAVEQKYVAIRIQFRDEVHPYGEATQRNELVIRAQPSEAMYVKITTKVPGLSGDLRQTHQTELDLTYHTRYDVRLPDAYESLINDALLGNSTNFVRKDELDVAWRIFTPLLHQIDSGEIKPIPYQAGTRGPKEADEFIANNGFKHQKGYHWLPSNKL

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 \(c\).

\(|P_{f(8JKC_1)}(2) \setminus P_{f(1DNU_1)}(2)|=118\), \(|P_{f(1DNU_1)}(2) \setminus P_{f(8JKC_1)}(2)|=46\). 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 1:1001001101111111010101010010101010101001010101100010111111011001001100100010010000110011101010000111000100000101010100110010101101000101110010000000010111000001101010100010011101100000001110111111000011000010001000000111101101111001100100
Pair \(Z_2\) Length of longest common subsequence
8JKC_1,1DNU_1 164 3
8JKC_1,7ZHX_1 190 3
1DNU_1,7ZHX_1 216 3

Newick tree

 
[
	7ZHX_1:10.47,
	[
		8JKC_1:82,1DNU_1:82
	]:25.47
]

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 \((0.85)(0.8)(\frac{342 }{\log_{20} 342}-\frac{104}{\log_{20}104})=73.7\)
Status Protein1 Protein2 d d1/2
Query variables 8JKC_1 1DNU_1 92 66
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]