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Parikh vectors
5OWX_1 3PKG_1 4EUU_1 Letter Amino acid
17 20 30 L Leucine
14 26 15 T Threonine
9 10 9 Y Tyrosine
4 20 24 E Glutamic acid
10 18 18 I Isoleucine
3 4 8 M Methionine
5 10 16 F Phenylalanine
12 23 20 S Serine
12 26 23 V Valine
8 14 15 N Asparagine
8 11 16 H Histidine
9 15 27 G Glycine
12 12 17 R Arginine
9 17 16 D Aspartic acid
6 12 10 Q Glutamine
8 25 18 K Lycine
11 9 13 P Proline
1 7 4 W Tryptophan
23 19 16 A Alanine
1 3 4 C Cysteine 

5OWX_1|Chain A[auth 1]|Genome polyprotein|Foot-and-mouth disease virus (strain A10-61) (12112)
>3PKG_1|Chain A|Uricase|Aspergillus flavus (5059)
>4EUU_1|Chains A, B|Serine/threonine-protein kinase TBK1|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 \(w=f(c)\)
5OWX , Knot 88 182 0.83 40 139 180 
RHHTDVGFIMDRFVKINSLSPTHVIDLMQTHKHGIVGALLRAATYYFSDLEIVVRHDGNLTWVPNGAPEAALSNTSNPTAYNKAPFTRLALPYTAPHRVLATVYDGTNKYSASDSRSGDLGSTAARVATQLPASFNYGAIQAQAIHELLVRMKRAELYCPRPLLAIKVTSQDRYKQKIIAPA
3PKG , Knot 136 302 0.85 42 204 291 
XSAVKAARYGKDNVRVYKVHKDEKTGVQTVYEMTVCVLLEGEIETSYTKADNSVIVATDSIKNTIYITAKQNPVTPPELFGSILGTHFIEKYNHIHAAHVNIVCHRWTRMDIDGKPHPHSFIRDSEEKRNVQVDVVEGKGIDIKSSLSGLTVLKSTNSQFWGFLRDEYTTLKETWDRILSTDVDATWQWKNFSGLQEVRSHVPKFDATWATAREVTLKTFAEDNSASVQATMYKMAEQILARQQLIETVEYSLPNKHYFEIDLSWHKGLQNTGKNAEVFAPQSDPNGLIKCTVGRSSLKSKL
4EUU , Knot 139 319 0.83 40 196 303 
MGSQSTSNHLWLLSDILGQGATANVFRGRHKKTGDLFAIKVFNNISFLRPVDVQMREFEVLKKLNHKNIVKLFAIEEETTTRHKVLIMEFCPCGSLYTVLEEPSNAYGLPESEFLIVLRDVVGGMNHLRENGIVHRNIKPGNIMRVIGEDGQSVYKLTDFGAARELEDDEQFVSLYGTEEYLHPDMYERAVLRKDHQKKYGATVDLWSIGVTFYHAATGSLPFRPFEGPRRNKEVMYKIITGKPSGAISGVQKAENGPIDWSGDMPVSCSLSRGLQVLLTPVLANILEADQEKCWGFDQFFAETSDILHRGNSHHHHHH

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 \(c\).

\(|P_{f(5OWX_1)}(2) \setminus P_{f(3PKG_1)}(2)|=52\), \(|P_{f(3PKG_1)}(2) \setminus P_{f(5OWX_1)}(2)|=117\). 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 1:00000111110011010010100110110000011111110110001001011100010101110111011100000101000111001111001100111010010000010000010110011011001110100111010110011101001010010111110100000000011111
Pair \(Z_2\) Length of longest common subsequence
5OWX_1,3PKG_1 169 3
5OWX_1,4EUU_1 183 4
3PKG_1,4EUU_1 162 3

Newick tree

 
[
	5OWX_1:90.30,
	[
		3PKG_1:81,4EUU_1:81
	]:9.30
]

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 \((0.85)(0.8)(\frac{484 }{\log_{20} 484}-\frac{182}{\log_{20}182})=88.2\)
Status Protein1 Protein2 d d1/2
Query variables 5OWX_1 3PKG_1 113 88.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]