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Parikh vectors
4ZKY_1 7AQA_1 9MLA_1 Letter Amino acid
7 23 9 H Histidine
3 33 25 I Isoleucine
3 46 17 K Lycine
6 32 36 P Proline
9 34 30 S Serine
5 10 13 W Tryptophan
21 50 16 A Alanine
12 54 14 D Aspartic acid
4 19 13 Q Glutamine
11 38 26 L Leucine
3 29 9 F Phenylalanine
18 53 30 V Valine
3 28 17 N Asparagine
0 9 12 C Cysteine
7 52 17 G Glycine
9 35 19 T Threonine
5 18 13 Y Tyrosine
11 26 24 R Arginine
5 32 21 E Glutamic acid
1 25 8 M Methionine 

4ZKY_1|Chains A, B|Pyridoxamine 5-phosphate oxidase|Mycobacterium smegmatis (1772)
>7AQA_1|Chains A, B|Nitrous-oxide reductase|Pseudomonas stutzeri (316)
>9MLA_1|Chains A, B, C|Surface protein|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 \(w=f(c)\)
4ZKY , Knot 70 143 0.81 38 109 137 
GMAEFDAVTAFADAPAAVLSTLNADGAPHLVPVVFAVHVPHVEGQPARIYTAVDAKRKTTRNLRRLANIDRDSRVSLLVDHYSDDWTQLWWVRADGVATTHHSGDEVATGYALLRAKYHQYERVSLDGPVISVEVSRWASWQA
7AQA , Knot 254 646 0.84 40 280 602 
MSDKDSKNTPQVPEKLGLSRRGFLGASAVTGAAVAATALGGAVMTRESWAQAVKESKQKIHVGPGELDDYYGFWSGGHQGEVRVLGVPSMRELMRIPVFNVDSATGWGLTNESRHIMGDSAKFLNGDCHHPHISMTDGKYDGKYLFINDKANSRVARIRLDIMKCDKMITVPNVQAIHGLRLQKVPHTKYVFANAEFIIPHPNDGKVFDLQDENSYTMYNAIDAETMEMAFQVIVDGNLDNTDADYTGRFAAATCYNSEKAFDLGGMMRNERDWVVVFDIHAVEAAVKAGDFITLGDSKTPVLDGRKKDGKDSKFTRYVPVPKNPHGCNTSSDGKYFIAAGKLSPTCSMIAIDKLPDLFAGKLADPRDVIVGEPELGLGPLATTFDGRGNAYTTLFIDSQVVKWNMEEAVRAYKGEKVNYIKQKLDVHYQPGHLHASLCETNEADGKWLVALSKFSKDRFLPVGPLHPENDQLIDISGDEMKLVHDGPTFAEPHDCIMARRDQIKTKKIWDRNDPFFAPTVEMAKKDGINLDTDNKVIRDGNKVRVYMTSMAPAFGVQEFTVKQGDEVTVTITNIDQIEDVSHGFVVVNHGVSMEISPQQTSSITFVADKPGLHWYYCSWFCHALHMEMVGRMMVEPAWSHPQFEK
9MLA , Knot 161 369 0.86 40 221 353 
AAANYTYWAYVPFPPLIRAVTWMDNPIEVYVNDSVWVPGPIDDRCPAKPEEEGMMINISIGYRYPPICLGRAPGCLMPAVQNWLVEVPTVSPISRFTYHMVSGMSLRPRVNYLQDFSYQRSLKFRPKGKPCPKEIPKESKNTEVLVWEECVANSAVILQNNEFGTIIDWAPRGQFYHNCSGQTQSCPSAQVSPAVDSDLTESLDKHKHKKLQSFYPWEWGEKGISTPRPKIISPVSGPEHPELWRLTVASHHIRIWSGNQTLETRDRKPFYTVDLNSSLTVPLQSCVKPPYMLVVGNIVIKPDSQTITCENCRLLTCIDSTFNWQHRILLVRAREGVWIPCSMDRPWEASPSIHILTEVLKGVLNRRRR

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 \(c\).

\(|P_{f(4ZKY_1)}(2) \setminus P_{f(7AQA_1)}(2)|=22\), \(|P_{f(7AQA_1)}(2) \setminus P_{f(4ZKY_1)}(2)|=193\). 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 1:11101011011101111110010101110111111110110101011010011010000000100110100000101110000001001111010111000001001101011101000000010101111010100110101
Pair \(Z_2\) Length of longest common subsequence
4ZKY_1,7AQA_1 215 4
4ZKY_1,9MLA_1 206 3
7AQA_1,9MLA_1 185 4

Newick tree

 
[
	4ZKY_1:10.20,
	[
		9MLA_1:92.5,7AQA_1:92.5
	]:16.70
]

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 \((0.85)(0.8)(\frac{789 }{\log_{20} 789}-\frac{143}{\log_{20}143})=182.\)
Status Protein1 Protein2 d d1/2
Query variables 4ZKY_1 7AQA_1 226 137
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]