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Parikh vectors
4ZTY_1 6UUQ_1 2QTV_1 Letter Amino acid
10 8 16 M Methionine
10 2 6 W Tryptophan
44 20 46 D Aspartic acid
69 36 79 L Leucine
22 17 46 N Asparagine
34 10 37 Q Glutamine
35 16 44 I Isoleucine
33 19 39 F Phenylalanine
30 14 52 T Threonine
76 19 63 A Alanine
37 17 30 R Arginine
38 18 40 G Glycine
25 17 28 Y Tyrosine
70 15 40 V Valine
5 11 11 C Cysteine
61 20 35 E Glutamic acid
45 14 48 P Proline
44 15 63 S Serine
21 11 18 H Histidine
60 17 31 K Lycine 

4ZTY_1|Chain A|Cobalamin-Independent Methionine synthase|Neurospora crassa (5141)
>6UUQ_1|Chain A|Serine/threonine-protein phosphatase 2B catalytic subunit alpha isoform|Homo sapiens (9606)
>2QTV_1|Chain A|Protein transport protein SEC23|Saccharomyces cerevisiae (4932)
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 \(w=f(c)\)
4ZTY , Knot 297 769 0.85 40 287 695 
MVQSSVLGFPRMGVLRDLKKANEAYWADKISQEALLAEGKRLRLAHWKIQKDAGVDIIPSNDFAHYDHVLDHIQLFNAVPERYTSQKLSPLDEYFAMGRGHQKGGVDVPALEMVKWFDSNYHYVKPTLQDNQTFSLAKDPKPVREFLEAKEAGFQTRPVLVGPVSFLALGKADRGSSVDPITLLDKLVPVYVELLKQLKAAGAESVQIDEPVLVFDLRPEVKAAFKPAYEAIAAAGDAVPKVVVATYFGDIVHNFDVLPAFSGAAGLHVDLVRNPEQLEPVLKQLGPNQILSAGVVDGRNIWKNDFAKSLEILQTAVKALGSERVIVATSSSLIHTPHTLASEKKLPSDVYEWFSFAVEKVKEVATLAKAVTEPEAVKAELEANAAAIKARTDSKRTNDPAVKERQAQVTPEQHNRKAPFNTRYAEQKKHLSLPLFPTTTIGSFPQTSEIRVQRNKFTKGEISAEEYERFIEKEIELAVKIQDELDLDVYVHGEPERNDMVQYFGERLNGYVFTTHAWVQSYGSRCVRPPIIVGDISRPAPMTVKESKYAASISKKPMKGMLTGPVTCLRWSFPRVDVHQSVQCQQLALALRDEVVDLEKNGIYVIQVDEPALREGLPLRKGQEREAYLKWAVDSFKLATAGVENSTQIHSHFCYSEFQDFFHAIAALDADVLSIENSKSDAKLLKVFIDEEYPRHIGPGVYDIHSPRVPTLEEFKQRIEEMLAYLKPEQLWINPDCGLKTRKWDEVKGALSHMVEAAKYFREKYANKA
6UUQ , Knot 140 316 0.85 40 206 308 
MGSHRLTAKEVFDNDGKPRVDILKAHLMKEGRLEESVALRIITEGASILRQEKNLLDIDAPVTVCGDIHGQFFDLMKLFEVGGSPANTRYLFLGDYVDRGYFSIECVLYLWALKILYPKTLFLLRGNHECRHLTEYFTFKQECKIKYSERVYDACMDAFDCLPLAALMNQQFLCVHGGLSPEINTLDDIRKLDRFKEPPAYGPMCDILWSDPLEDFGNEKTQEHFTHNTVRGCSYFYSYPAVCEFLQHNNLLSILRAHEAQDAGYRMYRKSQTTGFPSLITIFSAPNYLDVYNNKAAVLKYENNVMNIRQFNCSPH
2QTV , Knot 296 772 0.85 40 302 701 
GAMGSDFETNEDINGVRFTWNVFPSTRSDANSNVVPVGCLYTPLKEYDELNVAPYNPVVCSGPHCKSILNPYCVIDPRNSSWSCPICNSRNHLPPQYTNLSQENMPLELQSTTIEYITNKPVTVPPIFFFVVDLTSETENLDSLKESIITSLSLLPPNALIGLITYGNVVQLHDLSSETIDRCNVFRGDREYQLEALTEMLTGQKPTGPGGAASHLPNAMNKVTPFSLNRFFLPLEQVEFKLNQLLENLSPDQWSVPAGHRPLRATGSALNIASLLLQGCYKNIPARIILFASGPGTVAPGLIVNSELKDPLRSHHDIDSDHAQHYKKACKFYNQIAQRVAANGHTVDIFAGCYDQIGMSEMKQLTDSTGGVLLLTDAFSTAIFKQSYLRLFAKDEEGYLKMAFNGNMAVKTSKDLKVQGLIGHASAVKKTDANNISESEIGIGATSTWKMASLSPYHSYAIFFEIANTAANSNPMMSAPGSADRPHLAYTQFITTYQHSSGTNRIRVTTVANQLLPFGTPAIAASFDQEAAAVLMARIAVHKAETDDGADVIRWLDRTLIKLCQKYADYNKDDPQSFRLAPNFSLYPQFTYYLRRSQFLSVFNNSPDETAFYRHIFTREDTTNSLIMIQPTLTSFSMEDDPQPVLLDSISVKPNTILLLDTFFFILIYHGEQIAQWRKAGYQDDPQYADFKALLEEPKLEAAELLVDRFPLPRFIDTEAGGSQARFLLSKLNPSDNYQDMARGGSTIVLTDDVSLQNFMTHLQQVAVSGQA

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 \(c\).

\(|P_{f(4ZTY_1)}(2) \setminus P_{f(6UUQ_1)}(2)|=118\), \(|P_{f(6UUQ_1)}(2) \setminus P_{f(4ZTY_1)}(2)|=37\). 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 1:1100011111011110010010010110010001111010010110101000111011100011000011001011011100000001011000111101000111011110110110000001010100000101100101100110100111000111111101111101001001011011001111010110010111100101001111101010101110110011111101110111100110110010111110111110101100100101110011100110111101001100011001011001101110001111000011001001100001100100110111001001101101100101101010101111010000000001110000101010000001110000100000101111100011011000010100001001010100000110001011101000101010101010000110011001010110001110001000101111110100111101000001101000110111011100101011010100010000111110001101000110110100111001111001000010101110010110111000001000100001001101111101011010000001011011100001001111100100101101001000100111010100111010011000010010111001101100100001001
Pair \(Z_2\) Length of longest common subsequence
4ZTY_1,6UUQ_1 155 4
4ZTY_1,2QTV_1 119 4
6UUQ_1,2QTV_1 174 4

Newick tree

 
[
	6UUQ_1:88.71,
	[
		4ZTY_1:59.5,2QTV_1:59.5
	]:29.21
]

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 \((0.85)(0.8)(\frac{1085 }{\log_{20} 1085}-\frac{316}{\log_{20}316})=204.\)
Status Protein1 Protein2 d d1/2
Query variables 4ZTY_1 6UUQ_1 258 180.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]