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Parikh vectors
1ICS_1 5HHK_1 5XFI_1 Letter Amino acid
26 14 6 A Alanine
24 21 4 R Arginine
18 17 14 N Asparagine
15 10 7 F Phenylalanine
17 12 7 Y Tyrosine
26 21 11 V Valine
11 13 2 Q Glutamine
24 23 6 L Leucine
9 14 3 M Methionine
21 11 7 P Proline
16 15 12 S Serine
20 13 15 T Threonine
4 1 2 W Tryptophan
7 2 0 C Cysteine
10 4 2 H Histidine
19 24 6 K Lycine
24 23 7 D Aspartic acid
27 20 3 E Glutamic acid
30 22 26 G Glycine
28 25 13 I Isoleucine 

1ICS_1|Chains A, B|12-OXOPHYTODIENOATE REDUCTASE 1|Solanum lycopersicum (4081)
>5HHK_1|Chains A, B|Retron-type reverse transcriptase|Roseburia intestinalis XB6B4 (718255)
>5XFI_1|Chains A, B|Lectin|Calystegia sepium (47519)
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 \(w=f(c)\)
1ICS , Knot 160 376 0.84 40 213 361 
MENKVVEEKQVDKIPLMSPCKMGKFELCHRVVLAPLTRQRSYGYIPQPHAILHYSQRSTNGGLLIGEATVISETGIGYKDVPGIWTKEQVEAWKPIVDAVHAKGGIFFCQIWHVGRVSNKDFQPNGEDPISCTDRGLTPQIMSNGIDIAHFTRPRRLTTDEIPQIVNEFRVAARNAIEAGFDGVEIHGAHGYLIDQFMKDQVNDRSDKYGGSLENRCRFALEIVEAVANEIGSDRVGIRISPFAHYNEAGDTNPTALGLYMVESLNKYDLAYCHVVEPRMKTAWEKIECTESLVPMRKAYKGTFIVAGGYDREDGNRALIEDRADLVAYGRLFISNPDLPKRFELNAPLNKYNRDTFYTSDPIVGYTDYPFLETMT
5HHK , Knot 137 305 0.85 40 192 294 
MVKSSGTERKERMDTSSLMEQILSNDNLNRAYLQVVRNKGAEGVDGMKYTELKEYLAKNGEIIKEQLRIRKYKPQPVRRVEIPKPDGGVRNLGVPTVTDRFIQQAIAQVLTPIYEEQFHDHSYGFRPNRCAQQAILTALDMMNDGNDWIVDIDLEKFFDTVNHDKLMTIIGRTIKDGDVISIVRKYLVSGIMIDDEYEDSIVGTPQGGNLSPLLANIMLNELDKEMEKRGLNFVRYADDCIIMVGSEMSANRVMRNISRFIEEKLGLKVNMTKSKVDRPRGIKYLGFGFYYDTSAQQFKAKPHAK
5XFI , Knot 68 153 0.74 38 108 141 
MAVPMDTISGPWGNNGGNFWSFRPVNKINQIVISYGGGGNNPIALTFSSTKADGSKDTITVGGGGPDSITGTEMVNIGTDEYLTGISGTFGIYLDNNVLRSITFTTNLKAHGPYGQKVGTPFSSANVVGNEIVGFLGRSGYYVDAIGTYNRHK

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 \(c\).

\(|P_{f(1ICS_1)}(2) \setminus P_{f(5HHK_1)}(2)|=87\), \(|P_{f(5HHK_1)}(2) \setminus P_{f(1ICS_1)}(2)|=66\). 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 1:1000110000100111101001101010001111110000001011010111000000001111110101100011100011111000010110111011010111110011011010000101010011000001101011001101101001001000011011001011100110111011010110101100110001000000011010000011101101110011000111010111000011000101111011001000011000110101001100100000111100100101111110000010011100010111010111001011001010111000000010000111100001110010
Pair \(Z_2\) Length of longest common subsequence
1ICS_1,5HHK_1 153 3
1ICS_1,5XFI_1 187 3
5HHK_1,5XFI_1 160 4

Newick tree

 
[
	5XFI_1:90.24,
	[
		1ICS_1:76.5,5HHK_1:76.5
	]:13.74
]

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 \((0.85)(0.8)(\frac{681 }{\log_{20} 681}-\frac{305}{\log_{20}305})=104.\)
Status Protein1 Protein2 d d1/2
Query variables 1ICS_1 5HHK_1 132 118
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]