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Parikh vectors
7ZPI_1 1UXZ_1 4AYG_1 Letter Amino acid
82 8 97 N Asparagine
56 5 20 H Histidine
74 2 30 F Phenylalanine
18 5 19 W Tryptophan
114 6 57 V Valine
127 12 74 A Alanine
103 20 82 G Glycine
203 5 75 L Leucine
132 3 56 K Lycine
88 4 67 Y Tyrosine
97 2 35 R Arginine
157 5 44 E Glutamic acid
89 11 45 I Isoleucine
160 10 61 S Serine
127 4 89 D Aspartic acid
48 0 0 C Cysteine
70 9 49 Q Glutamine
41 1 22 M Methionine
123 4 33 P Proline
95 15 84 T Threonine 

7ZPI_1|Chain A|Endoribonuclease Dicer|Mus musculus (10090)
>1UXZ_1|Chains A, B|CELLULASE B|CELLVIBRIO MIXTUS (39650)
>4AYG_1|Chains A, B|GLUCANSUCRASE|LACTOBACILLUS REUTERI (1598)
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 \(w=f(c)\)
7ZPI , Knot 677 2004 0.85 40 377 1641 
MVWSHPQFEKGGGSGGGSGGSAWSHPQFEKGDYPYDVPDYAGTENLYFQGLVDMKSPALQPLSMAGLQLMTPASSPMGPFFGLPWQQEAIHDNIYTPRKYQVELLEAALDHNTIVCLNTGSGKTFIAVLLTKELAHQIRGDLNPHAKRTVFLVNSANQVAQQVSAVRTHSDLKVGEYSDLEVNASWTKERWSQEFTKHQVLIMTCYVALTVLKNGYLSLSDINLLVFDECHLAILDHPYREIMKLCESCPSCPRILGLTASILNGKCDPEELEEKIQKLERILRSDAETATDLVVLDRYTSQPCEIVVDCGPFTDRSGLYERLLMELEAALDFINDCNVAVHSKERDSTLISKQILSDCRAVLVVLGPWCADKVAGMMVRELQKYIKHEQEELHRKFLLFTDTLLRKIHALCEEYFSPASLDLKYVTPKVMKLLEILRKYKPYERQQFESVEWYNNRNQDNYVSWSDSEDDDDDEEIEEKEKPETNFPSPFTNILCGIIFVERRYTAVVLNRLIKEAGKQDPELAYISSNFITGHGIGKNQPRSKQMEAEFRKQEEVLRKFRAHETNLLIATSVVEEGVDIPKCNLVVRFDLPTEYRSYVQSKGRARAPISNYVMLADTDKIKSFEEDLKTYKAIEKILRNKCSKSADGAEADVHAGVDDEDAFPPYVLRPDDGGPRVTINTAIGHINRYCARLPSDPFTHLAPKCRTRELPDGTFYSTLYLPINSPLRASIVGPPMDSVRLAERVVALICCEKLHKIGELDEHLMPVGKETVKYEEELDLHDEEETSVPGRPGSTKRRQCYPKAIPECLRESYPKPDQPCYLYVIGMVLTTPLPDELNFRRRKLYPPEDTTRCFGILTAKPIPQIPHFPVYTRSGEVTISIELKKSGFTLSQQMLELITRLHQYIFSHILRLEKPALEFKPTGAESAYCVLPLNVVNDSGTLDIDFKFMEDIEKSEARIGIPSTKYSKETPFVFKLEDYQDAVIIPRYRNFDQPHRFYVADVYTDLTPLSKFPSPEYETFAEYYKTKYNLDLTNLNQPLLDVDHTSSRLNLLTPRHLNQKGKALPLSSAEKRKAKWESLQNKQILVPELCAIHPIPASLWRKAVCLPSILYRLHCLLTAEELRAQTASDAGVGVRSLPVDFRYPNLDFGWKKSIDSKSFISSCNSSLAESDNYCKHSTTVVPEHAAHQGATRPSLENHDQMSVNCKRLPAESPAKLQSEVSTDLTAINGLSYNKNLANGSYDLVNRDFCQGNQLNYFKQEIPVQPTTSYPIQNLYNYENQPKPSNECPLLSNTYLDGNANTSTSDGSPAVSTMPAMMNAVKALKDRMDSEQSPSVGYSSRTLGPNPGLILQALTLSNASDGFNLERLEMLGDSFLKHAITTYLFCTYPDAHEGRLSYMRSKKVSNCNLYRLGKKKGLPSRMVVSIFDPPVNWLPPGYVVNQDKSNSEKWEKDEMTKDCLLANGKLGEACEEEEDLTWRAPKEEAEDEDDFLEYDQEHIQFIDSMLMGSGAFVRKISLSPFSASDSAYEWKMPKKASLGSMPFASGLEDFDYSSWDAMCYLDPSKAVEEDDFVVGFWNPSEENCGVDTGKQSISYDLHTEQCIADKSIADCVEALLGCYLTSCGERAAQLFLCSLGLKVLPVIKRTSREKALDPAQENGSSQQKSLSGSCASPVGPRSSAGKDLEYGCLKIPPRCMFDHPDAEKTLNHLISGFETFEKKINYRFKNKAYLLQAFTHASYHYNTITDCYQRLEFLGDAILDYLITKHLYEDPRQHSPGVLTDLRSALVNNTIFASLAVKYDYHKYFKAVSPELFHVIDDFVKFQLEKNEMQGMDSELRRSEEDEEKEEDIEVPKAMGDIFESLAGAIYMDSGMSLEVVWQVYYPMMQPLIEKFSANVPRSPVRELLEMEPETAKFSPAERTYDGKVRVTVEVVGKGKFKGVGRSYRIAKSAAARRALRSLKANQPQVPNSGRGENLYFQGASDYKDHDGDYKDHDGSHHHHHHHH
1UXZ , Knot 65 131 0.80 38 98 124 
MVIATIQAEDHSQQSGTQQETTTDTGGGKNVGYIDAGDWLSYAGTPVNIPSSGSYLIEYRVASQNGGGSLTFEEAGGAPVHGTIAIPATGGWQTWTTIQHTVNLSAGSHQFGIKANAGGWNLNWIRINKTH
4AYG , Knot 378 1039 0.84 38 300 922 
MGINGQQYYIDPTTGQPRKNFLLQNGNDWIYFDKDTGAGTNALKLQFDKGTISADEQYRRGNEAYSYDDKSIENVNGYLTADTWYRPKQILKDGTTWTDSKETDMRPILMVWWPNTVTQAYYLNYMKQYGNLLPASLPSFSTDADSAELNHYSELVQQNIEKRISETGSTDWLRTLMHEFVTKNSMWNKDSENVDYGGLQLQGGFLKYVNSDLTKYANSDWRLMNRTATNIDGKNYGGAEFLLANDIDNSNPVVQAEELNWLYYLMNFGTITGNNPEANFDGIRVDAVDNVDVDLLSIARDYFNAAYNMEQSDASANKHINILEDWGWDDPAYVNKIGNPQLTMDDRLRNAIMDTLSGAPDKNQALNKLITQSLVNRANDNTENAVIPSYNFVRAHDSNAQDQIRQAIQAATGKPYGEFNLDDEKKGMEAYINDQNSTNKKWNLYNMPSAYTILLTNKDSVPRVYYGDLYQDGGQYMEHKTRYFDTITNLLKTRVKYVAGGQTMSVDKNGILTNVRFGKGAMNATDTGTDETRTEGIGVVISNNTNLKLNDGESVVLHMGAAHKNQKYRAVILTTEDGVKNYTNDTDAPVAYTDANGDLHFTNTNLDGQQYTAVRGYANPDVTGYLAVWVPAGAADDQDARTAPSDEAHTTKTAYRSNAALDSNVIYEGFSNFIYWPTTESERTNVRIAQNADLFKSWGITTFELAPQYNSSKDGTFLDSIIDNGYAFTDRYDLGMSTPNKYGSDEDLRNALQALHKAGLQAIADWVPDQIYNLPGKEAVTVTRSDDHGTTWEVSPIKNVVYITNTIGGGEYQKKYGGEFLDTLQKEYPQLFSQVYPVTQTTIDPSVKIKEWSAKYFNGTNILHRGAGYVLRSNDGKYYNLGTSTQQFLPSQLSVQDNEGYGFVKEGNNYHYYDENKQMVKDAFIQDSVGNWYYLDKNGNMVANQSPVEISSNGASGTYLFLNNGTSFRSGLVKTDAGTYYYDGDGRMVRNQTVSDGAMTYVLDENGKLVSESFDSSATEAHPLKPGDLNGQKHHHHHH

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 \(c\).

\(|P_{f(7ZPI_1)}(2) \setminus P_{f(1UXZ_1)}(2)|=282\), \(|P_{f(1UXZ_1)}(2) \setminus P_{f(7ZPI_1)}(2)|=3\). 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 1:111001010011101110110110010100100100110011000101011101001110110111101101100111111111100011000100100001011011100001101001010011111100011001010101010001111001001100101100000101100001010101000010001000011110001110110010101001011110000111100100011010000100101111010110100010010001001001100010010011110000001001110011100001100011101011101100001110000000011000110000111111111010011111100100010000001000111100011001011000010110101001010110110110000100000100101000000000101000000000001000001000110110011011111000001111001100110001011010001101011100010000101010000011001010000111100110011011000111010110000001000101011100011110000100100010000110011000000010110101011100001111011010011101010011101000010110011001110000001101010001011100110101111110010110011111000010011010001111100010000010100000001110110000000010111001000010100100101111110011100101000010110000001111010111011011100001010101010001101000110110010001100110100111010101100100111101100010101010110010000101111000000001111010000011111000010010010110100010110011010000110000000010100100111010000001011010010001011110010000101001000011110101101111011001101101100100110100101001001111100111010010101110001000011000000110000000000011100110011001010000010100001110011010001000101101100000110100011000100100100100011101000011001000000101000011100001010100000010111001111101101100010000010110000011101111101101001001101001011100110011000110001010010100100001000010011000111001110110111011111011000000000100001000011101011010000001010110001000001100000010110011110111100101011010001001011001011011110110010000101100101001100001111110100000110010001000100000110001100101111001000100110111001110111110000000110110001000000101001011110001100100101011100110010100010011011001000100010001011011001000000100000010111011100110001000100001111001001110001110111000000010110101101100110101000010110001000000000000101101110110011111010011010111010011101110010101100110011010100101011000001010101011101010111000011001110011001010010110010100101011000000010000001000000000
Pair \(Z_2\) Length of longest common subsequence
7ZPI_1,1UXZ_1 285 4
7ZPI_1,4AYG_1 103 6
1UXZ_1,4AYG_1 218 4

Newick tree

 
[
	1UXZ_1:14.43,
	[
		7ZPI_1:51.5,4AYG_1:51.5
	]:91.93
]

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 \((0.85)(0.8)(\frac{2135 }{\log_{20} 2135}-\frac{131}{\log_{20}131})=512.\)
Status Protein1 Protein2 d d1/2
Query variables 7ZPI_1 1UXZ_1 651 345.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]