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Parikh vectors
8TZS_1 3RTV_1 5NBD_1 Letter Amino acid
18 20 17 D Aspartic acid
37 36 28 G Glycine
52 17 61 I Isoleucine
43 17 43 F Phenylalanine
38 36 32 V Valine
12 0 4 C Cysteine
17 10 7 H Histidine
45 75 76 L Leucine
27 19 54 K Lycine
26 13 10 M Methionine
29 21 19 T Threonine
16 14 26 Y Tyrosine
31 63 31 A Alanine
19 10 37 N Asparagine
16 12 18 Q Glutamine
31 61 36 E Glutamic acid
22 32 9 P Proline
25 21 41 S Serine
16 9 0 W Tryptophan
21 54 15 R Arginine 

8TZS_1|Chain A|Protein wntless homolog|Homo sapiens (9606)
>3RTV_1|Chain A|DNA polymerase I, thermostable|Thermus aquaticus (271)
>5NBD_1|Chains A, B|WlaB protein|Campylobacter jejuni (197)
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 \(w=f(c)\)
8TZS , Knot 217 541 0.84 40 272 522 
MAGAIIENMSTKKLCIVGGILLVFQIIAFLVGGLIAPGPTTAVSYMSVKCVDARKNHHKTKWFVPWGPNHCDKIRDIEEAIPREIEANDIVFSVHIPLPHMEMSPWFQFMLFILQLDIAFKLNNQIRENAEVSMDVSLAYRDDAFAEWTEMAHERVPRKLKCTFTSPKTPEHEGRYYECDVLPFMEIGSVAHKFYLLNIRLPVNEKKKINVGIGEIKDIRLVGIHQNGGFTKVWFAMKTFLTPSIFIIMVWYWRRITMMSRPPVLLEKVIFALGISMTFINIPVEWFSIGFDWTWMLLFGDIRQGIFYAMLLSFWIIFCGEHMMDQHERNHIAGYWKQVGPIAVGSFCLFIFDMCERGVQLTNPFYSIWTTDIGTELAMAFIIVAGICLCLYFLFLCFMVFQVFRNISGKQSSLPAMSKVRRLHYEGLIFRFKFLMLITLACAAMTVIFFIVSQVTEGHWKWGGVTVQVNSAFFTGIYGMWNLYVFALMFLYAPSHKNYGEDQSNGDLGVHSGEELQLTTTITHVDGPTEIYKLTRKEAQE
3RTV , Knot 207 540 0.80 38 225 472 
ALEEAPWPPPEGAFVGFVLSRKEPMWADLLALAAARGGRVHRAPEPYKALRDLKEARGLLAKDLSVLALREGLGLPPGDDPMLLAYLLDPSNTTPEGVARRYGGEWTEEAGERAALSERLFANLWGRLEGEERLLWLYREVERPLSAVLAHMEATGVRLDVAYLRALSLEVAEEIARLEAEVFRLAGHPFNLNSRDQLERVLFDELGLPAIGKTEKTGKRSTSAAVLEALREAHPIVEKILQYRELTKLKSTYIDPLPDLIHPRTGRLHTRFNQTATATGRLSSSDPNLQNIPVRTPLGQRIRRAFIAEEGWLLVALDYSQIELRVLAHLSGDENLIRVFQEGRDIHTETASWMFGVPREAVDPLMRRAAKTINFGVLYGMSAHRLSQELAIPYEEAQAFIERYFQSFPKVRAWIEKTLEEGRRRGYVETLFGRRRYVPDLEARVKSVREAAERMAFNMPVQGTAADLMKLAMVKLFPRLEEMGARMLLQVHDELVLEAPKERAEAVARLAKEVMEGVYPLAVPLEVEVGIGEDWLSAKE
5NBD , Knot 218 564 0.81 38 222 497 
MVKKLFFILSKEDKNFLFFLLVFSVFVSFIETFAISLVMPFITLASDFSYFDRNKYLISLKEYLNIPVFEIIVYFGVGLIVFYVFRALLNAYYFHLLARFSKGRKHAIAYKVFSKFLNINYEKFTQKNQSEILKSITGEVYNLSTMISSFLLLMSEIFVVLLLYALMLLINYKITLFLSIFMVLNAFILVKILSPIIKKAGLRREEAMKNFFEILNTNLNNFKFIKLKTKEDGVLSLFKAQSEAFSKANITNESVAAVPRIYLEGIGFCVLVFIVVFLVLKNESDISGILSTISIFVLALYRLMPSANRIITSYHDLLYYHSSLNIIYQNLRQEEENLGEGKLSFNQELKICNLSFGYEGKKYLFKNLNLNIKKGEKIAFIGESGCGKSTLVDLIIGLLKPKEGQILIDKQELNASNAKNYRQKIGYIPQNIYLFNDSIAKNITFGDAVDEEKLNKVIKQANLEHFIKNLPQGVQTKVGDGGSNLSGGQKQRIAIARALYLEPEILVLDQATSALDTQSEAKIMDEIYKISKDKTMIIIAHRLSTITQCDKVYRLEHGKLKEEK

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 \(c\).

\(|P_{f(8TZS_1)}(2) \setminus P_{f(3RTV_1)}(2)|=99\), \(|P_{f(3RTV_1)}(2) \setminus P_{f(8TZS_1)}(2)|=52\). 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 1:1111110010000101111111110111111111111110011001010010100000000111111100000100100111001010011101011110101011101111110101110100010001010101011000011101001100011001000100100100010000001111101101100101101011100000101111010010111100011100111110011010111111101001011001111100111111101011011101101110101111110100111011110111110100110000000111010011111110101111010001101001100110001100111111111110101011110111101100101000011110010010001111010111110110111011111100100101011110101001110110111010111111101100000100000101110010010100010010110010010000100
Pair \(Z_2\) Length of longest common subsequence
8TZS_1,3RTV_1 151 5
8TZS_1,5NBD_1 160 4
3RTV_1,5NBD_1 155 5

Newick tree

 
[
	5NBD_1:79.81,
	[
		8TZS_1:75.5,3RTV_1:75.5
	]:4.31
]

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 \((0.85)(0.8)(\frac{1081 }{\log_{20} 1081}-\frac{540}{\log_{20}540})=140.\)
Status Protein1 Protein2 d d1/2
Query variables 8TZS_1 3RTV_1 184 178.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]