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Parikh vectors
3BBB_1 6OKU_1 7SFQ_1 Letter Amino acid
8 37 9 A Alanine
5 75 2 N Asparagine
13 49 12 G Glycine
4 5 1 H Histidine
8 71 1 D Aspartic acid
12 58 2 E Glutamic acid
13 81 1 K Lycine
8 26 5 F Phenylalanine
7 36 5 P Proline
3 7 4 W Tryptophan
4 49 5 Y Tyrosine
9 20 3 R Arginine
4 26 2 Q Glutamine
10 63 15 I Isoleucine
8 70 7 S Serine
4 78 7 T Threonine
14 39 6 V Valine
2 1 3 C Cysteine
11 68 16 L Leucine
4 17 4 M Methionine 

3BBB_1|Chains A, B, C, D, E, F|Nucleoside diphosphate kinase B|Homo sapiens (9606)
>6OKU_1|Chains A, B, C, D, E, F, G|ADP-ribosyltransferase binding component|Clostridioides difficile (1496)
>7SFQ_1|Chains A, B|Multidrug transporter EmrE|Escherichia coli (562)
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 \(w=f(c)\)
3BBB , Knot 81 151 0.89 40 126 147 
ANLERTFIAIKPDGVQRGLVGEIIKRFEQKGFRLVAMKFLRASEEHLKQHYIDLKDRPFFPGLVKYMNSGPVVAMVWEGLNVVKTGRVMLGETNPADSKPGTIRGDFCIQVGRNIIHGSDSVKSAEKEISLWFKPEELVDYKSCAHDWVYE
6OKU , Knot 322 876 0.83 40 270 770 
MKIQMRNKKVLSFLTLTAIVSQALVYPVYAQTSTSNHSNKKKEIVNEDILPNNGLMGYYFTDEHFKDLKLMAPIKDGNLKFEEKKVDKLLDKDKSDVKSIRWTGRIIPSKDGEYTLSTDRDDVLMQVNTESTISNTLKVNMKKGKEYKVRIELQDKNLGSIDNLSSPNLYWELDGMKKIIPEENLFLRDYSNIEKDDPFIPNNNFFDPKLMSDWEDEDLDTDNDNIPDSYERNGYTIKDLIAVKWEDSFAEQGYKKYVSNYLESNTAGDPYTDYEKASGSFDKAIKTEARDPLVAAYPIVGVGMEKLIISTNEHASTDQGKTVSRATTNSKTESNTAGVSVNVGYQNGFTANVTTNYSHTTDNSTAVQDSNGESWNTGLSINKGESAYINANVRYYNTGTAPMYKVTPTTNLVLDGDTLSTIKAQENQIGNNLSPGDTYPKKGLSPLALNTMDQFSSRLIPINYDQLKKLDAGKQIKLETTQVSGNFGTKNSSGQIVTEGNSWSDYISQIDSISASIILDTENESYERRVTAKNLQDPEDKTPELTIGEAIEKAFGATKKDGLLYFNDIPIDESCVELIFDDNTANKIKDSLKTLSDKKIYNVKLERGMNILIKTPTYFTNFDDYNNYPSTWSNVNTTNQDGLQGSANKLNGETKIKIPMSELKPYKRYVFSGYSKDPLTSNSIIVKIKAKEEKTDYLVPEQGYTKFSYEFETTEKDSSNIEITLIGSGTTYLDNLSITELNSTPEILDEPEVKIPTDQEIMDAHKIYFADLNFNPSTGNTYINGMYFAPTQTNKEALDYIQKYRVEATLQYSGFKDIGTKDKEMRNYLGDPNQPKTNYVNLRSYFTGGENIMTYKKLRIYAITPDDRELLVLSVD
7SFQ , Knot 55 110 0.78 40 84 106 
MNPYIYLGGAILAEVIGTTLMKFSEGFTRLWPSVGTIICYCASFWLLAQTLAYIPTGIAYAIWVGVGIVLISLLSWGFFGQRLDLPAIIGMMLICAGVLIINLLSRSTPH

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 \(c\).

\(|P_{f(3BBB_1)}(2) \setminus P_{f(6OKU_1)}(2)|=29\), \(|P_{f(6OKU_1)}(2) \setminus P_{f(3BBB_1)}(2)|=173\). 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 1:1010001111010110011110110010001101111011010000100001010001111111001001111111101101100101111000110001101010101011001101000100100010111010011000001001100
Pair \(Z_2\) Length of longest common subsequence
3BBB_1,6OKU_1 202 3
3BBB_1,7SFQ_1 146 3
6OKU_1,7SFQ_1 224 4

Newick tree

 
[
	6OKU_1:11.70,
	[
		3BBB_1:73,7SFQ_1:73
	]:42.70
]

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 \((0.85)(0.8)(\frac{1027 }{\log_{20} 1027}-\frac{151}{\log_{20}151})=240.\)
Status Protein1 Protein2 d d1/2
Query variables 3BBB_1 6OKU_1 299 176
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]