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Parikh vectors
5BZQ_1 2REO_1 2UYA_1 Letter Amino acid
10 13 24 A Alanine
9 20 25 D Aspartic acid
8 8 15 R Arginine
8 25 33 E Glutamic acid
10 21 20 I Isoleucine
4 7 11 Q Glutamine
9 11 28 G Glycine
6 17 19 F Phenylalanine
7 19 23 P Proline
10 16 23 S Serine
0 8 7 W Tryptophan
8 11 14 Y Tyrosine
11 12 16 N Asparagine
6 3 1 C Cysteine
5 11 14 H Histidine
7 24 28 L Leucine
5 34 26 K Lycine
3 16 9 M Methionine
15 16 19 T Threonine
10 13 28 V Valine 

5BZQ_1|Chain A|CD44 antigen|Mus musculus (10090)
>2REO_1|Chain A|Putative sulfotransferase 1C3|Homo sapiens (9606)
>2UYA_1|Chain A|OXALATE DECARBOXYLASE OXDC|BACILLUS SUBTILIS (1423)
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 \(w=f(c)\)
5BZQ , Knot 75 151 0.83 38 122 146 
MNQIDLNVTCRYAGVFHVEKNGRYSISRTEAADLCQAFNSTLPTMDQMKLALSKGFETCRYGFIEGNVVIPRIHPNAICAANHTGVYILVTSNTSHYDTYCFNASAPPEEDCTSVTDLPNSFDGPVTITIVNRDGTRYSKKGEYRTHQEDI
2REO , Knot 136 305 0.85 40 192 294 
GMAKIEKNAPTMEKKPELFNIMEVDGVPTLILSKEWWEKVANFQAKPDDLILATYPKSGTTWMHEILDMILNDGDVEKCKRAQTLDRHAFLELKFPHKEKPDLEFVLEMSSPQLIKTHLPSHLIPPSIWKENCKIVYVARNPKDCLVSYYHFHRMASFMPDPQNLEEFYEKFMSGKVVGGSWFDHVKGWWAAKDMHRILYLFYEDIKKDPKREIEKILKFLEKDISEEILNKIIYHTSFDVMKQNPMTNYTTLPTSIMDHSISPFMRKGMPGDWKNYFTVAQNEEFDKDYQKKMAGSTLTFRTEI
2UYA , Knot 161 383 0.83 40 218 365 
MKKQNDIPQPIRGDKGATVKIPRNIERDRQNPDMLVPPETDHGTVSNMKFSFSDTHNRLEKGGYAREVTVRELPISENLASVNMRLKPGAIRELHWHKEAEWAYMIYGSARVTIVDEKGRSFIDDVGEGDLWYFPSGLPHSIQALEEGAEFLLVFDDGSFSSTFQLTDWLAHTPKEVIAANFGVTKEEISNLPGKEKYIFENQLPGSLKDDIVEGPNGEVPYPFTYRLLEQEPIESEGGKVYIADSTNFKVSKTIASALVTVEPGAMRELHWHPNTHEWQYYISGKARMTVFASDGHARTFNYQAGDVGYVPFAMGHYVENIGDEPLVFLEIFKDDHYADVSLNQWLAMLPETFVQAHLDLGKDFTDVLSKEKHPVVKKKCSK

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 \(c\).

\(|P_{f(5BZQ_1)}(2) \setminus P_{f(2REO_1)}(2)|=66\), \(|P_{f(2REO_1)}(2) \setminus P_{f(5BZQ_1)}(2)|=136\). 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 1:1001010100001111010001000100001101001100011010010111001100000111010111101010110110001101110000000000010101110000001001100101110101100010000001000000001
Pair \(Z_2\) Length of longest common subsequence
5BZQ_1,2REO_1 202 4
5BZQ_1,2UYA_1 188 3
2REO_1,2UYA_1 168 4

Newick tree

 
[
	5BZQ_1:10.68,
	[
		2UYA_1:84,2REO_1:84
	]:17.68
]

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 \((0.85)(0.8)(\frac{456 }{\log_{20} 456}-\frac{151}{\log_{20}151})=90.4\)
Status Protein1 Protein2 d d1/2
Query variables 5BZQ_1 2REO_1 116 86
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]