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Parikh vectors
5AOU_1 3IAR_1 8GDJ_1 Letter Amino acid
30 37 20 L Leucine
20 17 18 S Serine
1 4 7 W Tryptophan
12 21 14 D Aspartic acid
28 17 30 I Isoleucine
13 25 21 V Valine
28 31 21 E Glutamic acid
6 16 30 T Threonine
4 15 14 Q Glutamine
9 9 8 M Methionine
12 17 8 Y Tyrosine
12 31 14 A Alanine
0 5 16 C Cysteine
11 24 28 G Glycine
7 11 9 H Histidine
14 24 25 K Lycine
8 11 11 F Phenylalanine
12 21 18 P Proline
17 17 8 R Arginine
14 14 35 N Asparagine 

5AOU_1|Chain A|INDOLE-3-GLYCEROL PHOSPHATE SYNTHASE|SULFOLOBUS SOLFATARICUS (2287)
>3IAR_1|Chain A|Adenosine deaminase|Homo sapiens (9606)
>8GDJ_1|Chain A|HIV-1 LM/HT Clade A/E CRF01 gp120 core|Human immunodeficiency virus 1 (11676)
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 \(w=f(c)\)
5AOU , Knot 114 258 0.81 38 156 248 
MPRYLKGWLEDVVQLSLRRPSVHASRQRPIISLNERILEFNKRNITAIIAYYLRKSPSGLDVERDPIEYAKYMEPYAVGLSIKTEEKYFDGSYEMLRKIASSVSIPILMNDFIVKESQIDDAYNLGADTVLLIVEILTERELESLLEYARGYGMEPLILINDENDLDIALRIGARFITIYSMNFETGEINKENQRKLISMIPSNVVKVPLLDFFEPNEIEELRKLGVNAFMISSSLMRNPEKIKELIEGSLEHHHHHH
3IAR , Knot 158 367 0.84 40 208 354 
MPAFDKPKVELHVHLDGSIKPETILYYGRRRGIALPANTAEGLLNVIGMDKPLTLPDFLAKFDYYMPAIAGCREAIKRIAYEFVEMKAKEGVVYVEVRYSPHLLANSKVEPIPWNQAEGDLTPDEVVALVGQGLQEGERDFGVKARSILCCMRHQPNWSPKVVELCKKYQQQTVVAIDLAGDETIPGSSLLPGHVQAYQEAVKSGIHRTVHAGEVGSAEVVKEAVDILKTERLGHGYHTLEDQALYNRLRQENMHFEICPWSSYLTGAWKPDTEHAVIRLKNDQANYSLNTDDPLIFKSTLDTDYQMTKRDMGFTEEEFKRLNINAAKSSFLPEDEKRELLDLLYKAYGMPPSASAGQNLAENLYFQ
8GDJ , Knot 148 355 0.81 40 203 330 
VPVWKDADTTLFCASDAKAYETEVHNVWATHACVPTDPNPQEIHLENVTENFNMWKNNMVEQMHEDIISLWDQSLQPCVKLTGGSVIKQACPKISFDPIPIHYCTPAGYVILKCNDKNFNGTGPCKNVSSVQCTHGIKPVVSTQLLLNGSLAEEEIIIRSENLTNNAKTIIVHLNKSVEINCTRPSNGGSGSGGDIRKAYCEINGTKWNKVLKQVTEKLKEHFNNKTIIFQPPSGGDLEITMHTFNCRGEFFYCNTTQLFNNTCIGNETMKGCNGTITLPCKIKQIINMWQGTGQAMYAPPIDGKINCVSNITGILLTRDGGANNTSNETFRPGGGDMRDNWRSELYKYKVVQIE

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 \(c\).

\(|P_{f(5AOU_1)}(2) \setminus P_{f(3IAR_1)}(2)|=50\), \(|P_{f(3IAR_1)}(2) \setminus P_{f(5AOU_1)}(2)|=102\). 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 1:110010111001101010010101000011101000110100001011110010001011010001100100101011110100000010100011001100101111100111000010010011100111110110000100110010101101111100000101110111011010010100101000000011011100110111101101001001001110111100011001001001101010000000
Pair \(Z_2\) Length of longest common subsequence
5AOU_1,3IAR_1 152 3
5AOU_1,8GDJ_1 183 4
3IAR_1,8GDJ_1 177 3

Newick tree

 
[
	8GDJ_1:94.22,
	[
		5AOU_1:76,3IAR_1:76
	]:18.22
]

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 \((0.85)(0.8)(\frac{625 }{\log_{20} 625}-\frac{258}{\log_{20}258})=103.\)
Status Protein1 Protein2 d d1/2
Query variables 5AOU_1 3IAR_1 131 110
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]