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Parikh vectors
1OOW_1 4IUG_1 6UBR_1 Letter Amino acid
7 73 58 A Alanine
3 47 39 I Isoleucine
6 50 38 K Lycine
7 58 55 D Aspartic acid
7 87 68 S Serine
5 66 38 T Threonine
11 59 50 V Valine
1 5 6 C Cysteine
10 50 45 E Glutamic acid
6 91 80 L Leucine
12 103 47 G Glycine
2 20 16 H Histidine
2 7 15 M Methionine
6 45 30 F Phenylalanine
5 59 46 P Proline
0 33 28 R Arginine
5 51 59 N Asparagine
1 26 52 Q Glutamine
0 20 18 W Tryptophan
3 55 28 Y Tyrosine 

1OOW_1|Chain A|Plastocyanin, chloroplast|Spinacia oleracea (3562)
>4IUG_1|Chain A|Beta-galactosidase A|Aspergillus oryzae (5062)
>6UBR_1|Chains A, B, C, D|Uncharacterized protein|Pseudoalteromonas luteoviolacea DSM 6061 (1365250)
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 \(w=f(c)\)
1OOW , Knot 49 99 0.75 36 80 95 
VEVLLGGDDGSEAFLPGDFSVASGEEIVFKNNAGFPHNVVFDEDEIPSGVDAAKISMSEEDLLNAPGETYKVTLTEKGTYKFYCSPHQGAGMVGKVTVN
4IUG , Knot 371 1005 0.85 40 308 887 
MKLLSVAAVALLAAQAAGASIKHRLNGFTILEHPDPAKRDLLQDIVTWDDKSLFINGERIMLFSGEVHPFRLPVPSLWLDIFHKIRALGFNCVSFYIDWALLEGKPGDYRAEGIFALEPFFDAAKEAGIYLIARPGSYINAEVSGGGFPGWLQRVNGTLRSSDEPFLKATDNYIANAAAAVAKAQITNGGPVILYQPENEYSGGCCGVKYPDADYMQYVMDQARKADIVVPFISNDASPSGHNAPGSGTGAVDIYGHDSYPLGFDCANPSVWPEGKLPDNFRTLHLEQSPSTPYSLLEFQAGAFDPWGGPGFEKCYALVNHEFSRVFYRNDLSFGVSTFNLYMTFGGTNWGNLGHPGGYTSYDYGSPITETRNVTREKYSDIKLLANFVKASPSYLTATPRNLTTGVYTDTSDLAVTPLIGDSPGSFFVVRHTDYSSQESTSYKLKLPTSAGNLTIPQLEGTLSLNGRDSKIHVVDYNVSGTNIIYSTAEVFTWKKFDGNKVLVLYGGPKEHHELAIASKSNVTIIEGSDSGIVSTRKGSSVIIGWDVSSTRRIVQVGDLRVFLLDRNSAYNYWVPELPTEGTSPGFSTSKTTASSIIVKAGYLLRGAHLDGADLHLTADFNATTPIEVIGAPTGAKNLFVNGEKASHTVDKNGIWSSEVKYAAPEIKLPGLKDLDWKYLDTLPEIKSSYDDSAWVSADLPKTKNTHRPLDTPTSLYSSDYGFHTGYLIYRGHFVANGKESEFFIRTQGGSAFGSSVWLNETYLGSWTGADYAMDGNSTYKLSQLESGKNYVITVVIDNLGLDENWTVGEETMKNPRGILSYKLSGQDASAITWKLTGNLGGEDYQDKVRGPLNEGGLYAERQGFHQPQPPSESWESGSPLEGLSKPGIGFYTAQFDLDLPKGWDVPLYFNFGNNTQAARAQLYVNGYQYGKFTGNVGPQTSFPVPEGILNYRGTNYVALSLWALESDGAKLGSFELSYTTPVLTGYGNVESPEQPKYEQRKGAY
6UBR , Knot 312 816 0.85 40 295 740 
MSNCQYKIYPPLGIARVGNGPAIKPLSLSTPEVPWAHLYDTNVQYLVTQQELEQLLEEAFGGNVINEISQIKTKLDERKAEKFKQEEIETITGLLGLSHLVPQQQLSRSLDNLELKSTKDSDDIVQQIKGALLKVLSDHYLHAVKKQAQNFYIYKCDEQGNPVEKLKLTDGDKVTWRVEVANKKSFWYDYNNALDLSLHTQGSGNLSKNVSKHRLAPAMTAKRRNPNVITNSLRKQLVISSQGSVSSDNNTQVPLRGKFPANEPDTNNRLSDLLNLQERHNVLQGSIECDNEGVLRFYAGNGISQALSPSSLNTDFADNSNWFDDICDGRVTAVVELKNGDTFEIQDEQSSAWVATTPPDYAPQIEPIVTMYDMVSGAALKEQDLDNLTTQFSDVFPILYRLYRMQWVNQADFTDNAVNTQIRELNSELGFAQLLDNSASAKSLREGIFNQFRNPLFDQDIDVDDPGQSSNEWVSNSRIIPSKDETNIAAKPATSSLKLPFYPNDGIDYPGSPVQWFAIPPFMYQHLQNWAAGDFSVTQVEKESANTIEELGLFYSEQFKNSPNSALLCARGALDALYGGGFHPGVELTWPMRHNLIYSQNDYVSSVTPEINLLGLREFRLKQDLQGLNSPNMYQDFGHVIAVDNVTASIDPNSDAAWLWRSTPGDLTKWMGIPWQSAAASCQAVYTPEDFPIPSWWAANLPVHVLPLARYNKFKDSQSADLPEINGMTHSIAQGMSEETFEHLRLEQFSQRLDWLHTADLGFVGYHAEGGYTNGLIQMVSQWKNMAMVMARPVENPGSSGIPNVVYVAYSQADKD

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 \(c\).

\(|P_{f(1OOW_1)}(2) \setminus P_{f(4IUG_1)}(2)|=8\), \(|P_{f(4IUG_1)}(2) \setminus P_{f(1OOW_1)}(2)|=236\). 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 1:101111100100111110101101001110001111001110000110110110101000011011100001010001000100010011111101010
Pair \(Z_2\) Length of longest common subsequence
1OOW_1,4IUG_1 244 4
1OOW_1,6UBR_1 239 5
4IUG_1,6UBR_1 103 5

Newick tree

 
[
	1OOW_1:13.23,
	[
		6UBR_1:51.5,4IUG_1:51.5
	]:84.73
]

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 \((0.85)(0.8)(\frac{1104 }{\log_{20} 1104}-\frac{99}{\log_{20}99})=277.\)
Status Protein1 Protein2 d d1/2
Query variables 1OOW_1 4IUG_1 348 189.5
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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