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Parikh vectors
6URL_1 7BTO_1 4GIU_1 Letter Amino acid
7 41 22 D Aspartic acid
6 52 71 A Alanine
3 32 21 S Serine
1 20 1 Y Tyrosine
6 32 35 V Valine
2 13 19 P Proline
3 24 19 T Threonine
2 3 6 W Tryptophan
5 15 27 R Arginine
1 36 9 N Asparagine
8 29 14 E Glutamic acid
9 29 42 G Glycine
1 42 12 I Isoleucine
2 6 8 M Methionine
6 32 11 F Phenylalanine
4 4 2 C Cysteine
3 20 9 Q Glutamine
1 13 13 H Histidine
10 41 33 L Leucine
9 36 4 K Lycine 

6URL_1|Chains A, B|Barrier-to-autointegration factor|Homo sapiens (9606)
>7BTO_1|Chains A, B|Type I restriction enzyme EcoR124II M protein|Escherichia coli (562)
>4GIU_1|Chains A, B|Anthranilate phosphoribosyltransferase|Mycobacterium tuberculosis (1773)
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 \(w=f(c)\)
6URL , Knot 49 89 0.82 40 77 85 
MTTSQKHRDFVAEPMGEKPVGSLAGIGEVLGKKLEERGFDKAYVVLGQFLVLKKDEDLFREWLKDTCGANAKQSRDCFGCLREWCDAFL
7BTO , Knot 213 520 0.85 40 251 492 
MKMTSIQQRAELHRQIWQIANDVRGSVDGWDFKQYVLGALFYRFISENFSSYIEAGDDSICYAKLDDSVITDDIKDDAIKTKGYFIYPSQLFCNVAAKANTNDRLNADLNSIFVAIESSAYGYPSEADIKGLFADFDTTSNRLGNTVKDKNARLAAVLKGVEGLKLGDFNEHQIDLFGDAYEFLISNYAANAGKSGGEFFTPQHVSKLIAQLAMHGQTHVNKIYDPAAGSGSLLLQAKKQFDNHIIEEGFFGQEINHTTYNLARMNMFLHNINYDKFDIKLGNTLTEPHFRDEKPFDAIVSNPPYSVKWIGSDDPTLINDERFAPAGVLAPKSKADFAFVLHALNYLSAKGRAAIVCFPGIFYRGGAEQKIRQYLVDNNYVETVISLAPNLFFGTTIAVNILVLSKHKTDTNVQFIDASELFKKETNNNILTDAHIEQIMQVFASKEDVAHLAKSVAFETVVANDYNLSVSSYVEAKDNREIIDIAELNAELKTTVSKIDQLRKDIDAIVAEIEGCEVQK
4GIU , Knot 149 378 0.78 40 162 338 
MALSAEGSSGGSRGGSPKAEAASVPSWPQILGRLTDNRDLARGQAAWAMDQIMTGNARPAQIAAFAVAMTMKAPTADEVGELAGVMLSHAHPLPADTVPDDAVDVVGTGGDGVNTVNLSTMAAIVVAAAGVPVVKHGNRAASSLSGGADTLEALGVRIDLGPDLVARSLAEVGIGFCFAPRFHPSYRHAAAVRREIGVPTVFNLLGPLTNPARPRAGLIGCAFADLAEVMAGVFAARRSSVLVVHGDDGLDELTTTTTSTIWRVAAGSVDKLTFDPAGFGFARAQLDQLAGGDAQANAAAVRAVLGGARGPVRDAVVLNAAGAIVAHAGLSSRAEWLPAWEEGLRRASAAIDTGAAEQLLARWVRFGRQILEHHHHHH

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 \(c\).

\(|P_{f(6URL_1)}(2) \setminus P_{f(7BTO_1)}(2)|=15\), \(|P_{f(7BTO_1)}(2) \setminus P_{f(6URL_1)}(2)|=189\). 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 1:10000000011101110011101111101110010001100101111011110000011001100001101000000110100100111
Pair \(Z_2\) Length of longest common subsequence
6URL_1,7BTO_1 204 3
6URL_1,4GIU_1 147 3
7BTO_1,4GIU_1 173 4

Newick tree

 
[
	7BTO_1:10.61,
	[
		6URL_1:73.5,4GIU_1:73.5
	]:27.11
]

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 \((0.85)(0.8)(\frac{609 }{\log_{20} 609}-\frac{89}{\log_{20}89})=153.\)
Status Protein1 Protein2 d d1/2
Query variables 6URL_1 7BTO_1 194 112.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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