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Parikh vectors
1ZDA_1 1OIL_1 8WLZ_1 Letter Amino acid
4 18 86 N Asparagine
3 7 46 E Glutamic acid
2 11 73 I Isoleucine
1 30 90 V Valine
1 13 67 P Proline
3 15 71 D Aspartic acid
0 2 30 C Cysteine
0 36 88 G Glycine
1 6 24 H Histidine
0 3 11 W Tryptophan
1 14 49 Y Tyrosine
4 39 83 A Alanine
4 9 45 R Arginine
5 15 58 Q Glutamine
2 31 100 L Leucine
2 7 57 K Lycine
1 1 15 M Methionine
2 7 85 F Phenylalanine
2 22 102 S Serine
0 34 91 T Threonine 

1ZDA_1|Chain A|MINI PROTEIN A DOMAIN, Z38|synthetic construct (32630)
>1OIL_1|Chains A, B|LIPASE|Burkholderia cepacia (292)
>8WLZ_1|Chains A, B, C|Spike glycoprotein,Fibritin|Bat SARS-like coronavirus WIV1 (1415852)
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 \(w=f(c)\)
1ZDA , Knot 25 38 0.79 32 35 36 
AVAQSFNMQQQRRFYEALHDPNLNEEQRNAKIKSIRDD
1OIL , Knot 134 320 0.80 40 177 297 
ADNYAATRYPIILVHGLTGTDKYAGVLEYWYGIQEDLQQRGATVYVANLSGFQSDDGPNGRGEQLLAYVKTVLAATGATKVNLVGHSQGGLTSRYVAAVAPDLVASVTTIGTPHRGSEFADFVQGVLAYDPTGLSSTVIAAFVNVFGILTSSSNNTNQDALAALKTLTTAQAATYNQNYPSAGLGAPGSCQTGAPTETVGGNTHLLYSWAGTAIQPTISVFGVTGATDTSTIPLVDPANALDPSTLALFGTGTVMVNRGSGQNDGVVSKCSALYGQVLSTSYKWNHLDEINQLLGVRGANAEDPVAVIRTHANRLKLAGV
8WLZ , Knot 446 1271 0.83 40 331 1093 
MKLLVLVFATLVSSYTIEKCLDFDDRTPPANTQFLSSHRGVYYPDDIFRSNVLHLVQDHFLPFDSNVTRFITFGLNFDNPIIPFKDGIYFAATEKSNVIRGWVFGSTMNNKSQSVIIMNNSTNLVIRACNFELCDNPFFVVLKSNNTQIPSYIFNNAFNCTFEYVSKDFNLDLGEKPGNFKDLREFVFRNKDGFLHVYSGYQPISAASGLPTGFNALKPIFKLPLGINITNFRTLLTAFPPRPDYWGTSAAAYFVGYLKPTTFMLKYDENGTITDAVDCSQNPLAELKCSVKSFEIDKGIYQTSNFRVAPSKEVVRFPNITNLCPFGEVFNATTFPSVYAWERKRISNCVADYSVLYNSTSFSTFKCYGVSATKLNDLCFSNVYADSFVVKGDDVRQIAPGQTGVIADYNYKLPDDFTGCVLAWNTRNIDATQTGNYNYKYRSLRHGKLRPFERDISNVPFSPDGKPCTPPAFNCYWPLNDYGFYITNGIGYQPYRVVVLSFELLNAPATVCGPKLSTDLIKNQCVNFNFNGLTGTGVLTPSSKRFQPFQQFGRDVSDFTDSVRDPKTSEILDISPCSFGGVSVITPGTNTSSEVAVLYQDVNCTDVPVAIHADQLTPSWRVHSTGNNVFQTQAGCLIGAEHVDTSYECDIPIGAGICASYHTVSSLRSTSQKSIVAYTMSLGADSSIAYSNNTIAIPTNFSISITTEVMPVSMAKTSVDCNMYICGDSTECANLLLQYGSFCTQLNRALSGIAVEQDRNTREVFAQVKQMYKTPTLKDFGGFNFSQILPDPLKPTKRSFIEDLLFNKVTLADAGFMKQYGECLGDINARDLICAQKFNGLTVLPPLLTDDMIAAYTAALVSGTATAGWTFGAGAALQIPFAMQMAYRFNGIGVTQNVLYENQKQIANQFNKAISQIQESLTTTSTALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDILSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQAAPHGVVFLHVTYVPSQERNFTTAPAICHEGKAYFPREGVFVFNGTSWFITQRNFFSPQIITTDNTFVSGSCDVVIGIINNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGRSLEVLFQGPGHHHHHHHHSAWSHPQFEKGGGSGGGGSGGSAWSHPQFEK

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 \(c\).

\(|P_{f(1ZDA_1)}(2) \setminus P_{f(1OIL_1)}(2)|=21\), \(|P_{f(1OIL_1)}(2) \setminus P_{f(1ZDA_1)}(2)|=163\). 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 1:11100101000001001100101000000101001000
Pair \(Z_2\) Length of longest common subsequence
1ZDA_1,1OIL_1 184 3
1ZDA_1,8WLZ_1 302 4
1OIL_1,8WLZ_1 170 5

Newick tree

 
[
	1ZDA_1:13.77,
	[
		1OIL_1:85,8WLZ_1:85
	]:50.77
]

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 \((0.85)(0.8)(\frac{358 }{\log_{20} 358}-\frac{38}{\log_{20}38})=102.\)
Status Protein1 Protein2 d d1/2
Query variables 1ZDA_1 1OIL_1 127 72
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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