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Parikh vectors
8EXO_1 1UOR_1 7ENB_1 Letter Amino acid
70 8 8 I Isoleucine
39 6 3 M Methionine
45 31 16 F Phenylalanine
18 1 6 W Tryptophan
49 20 10 Q Glutamine
78 62 18 E Glutamic acid
52 12 15 G Glycine
34 16 18 H Histidine
44 18 8 Y Tyrosine
32 35 6 C Cysteine
127 61 18 L Leucine
70 59 13 K Lycine
49 24 23 P Proline
58 24 23 S Serine
45 62 22 A Alanine
58 24 23 R Arginine
57 17 5 N Asparagine
59 36 20 D Aspartic acid
40 28 16 T Threonine
56 41 27 V Valine 

8EXO_1|Chain A|Phosphatidylinositol 4,5-bisphosphate 3-kinase catalytic subunit alpha isoform|Homo sapiens (9606)
>1UOR_1|Chain A|SERUM ALBUMIN|Homo sapiens (9606)
>7ENB_1|Chains A, B|NvfI|Aspergillus novofumigatus IBT 16806 (1392255)
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 \(w=f(c)\)
8EXO , Knot 399 1080 0.86 40 344 969 
MSYYHHHHHHDYDIPTTENLYFQGAMGSGELWGIHLMPPRILVECLLPNGMIVTLECLREATLITIKHELFKEARKYPLHQLLQDESSYIFVSVTQEAEREEFFDETRRLCDLRLFQPFLKVIEPVENLYFQGGNREEKILNREIGFAIGMPVCEFDMVKDPEVQDFRRNILNVCKEAVDLRDLNSPHSRAMYVYPPNVESSPELPKHIYNKLDKGQIIVVIWVIVSPNNDKQKYTLKINHDCVPEQVIAEAIRKKTRSMLLSSEQLKLCVLEYQGKYILKVCGCDEYFLEKYPLSQYKYIRSCIMLGRMPNLMLMAKESLYSQLPMDCFTMPSYSRRISTATPYMNGETSTKSLWVINSALRIKILCATYVNVNIRDIDKIYVRTGIYHGGEPLCDNVNTQRVPCSNPRWNEWLNYDIYIPDLPRAARLCLSICSVKGRKGAKEEHCPLAWGNINLFDYTDTLVSGKMALNLWPVPHGLEDLLNPIGVTGSNPNKETPCLELEFDWFSSVVKFPDMSVIEEHANWSVSREAGFSYSHAGLSNRLARDNELRENDKEQLKAISTRDPLSEITEQEKDFLWSHRHYCVTIPEILPKLLLSVKWNSRDEVAQMYCLVKDWPPIKPEQAMELLDCNYPDPMVRGFAVRCLEKYLTDDKLSQYLIQLVQVLKYEQYLDNLLVRFLLKKALTNQRIGHFFFWHLKSEMHNKTVSQRFGLLLESYCRACGMYLKHLNRQVEAMEKLINLTDILKQEKKDETQKVQMKFLVEQMRRPDFMDALQGFLSPLNPAHQLGNLRLEECRIMSSAKRPLWLNWENPDIMSELLFQNNEIIFKNGDDLRQDMLTLQIIRIMENIWQNQGLDLRMLPYGCLSIGDCVGLIEVVRNSHTIMQIQCKGGLKGALQFNSHTLHQWLKDKNKGEIYDAAIDLFTRSCAGYCVATFILGIGDRHNSNIMVKDDGQLFHIDFGHFLDHKKKKFGYKRERVPFVLTQDFLIVISKGAQECTKTREFERFQEMCYKAYLAIRQHANLFINLFSMMLGSGMPELQSFDDIAYIRKTLALDKTEQEALEYFMKQMNDAHHGGWT
1UOR , Knot 229 585 0.83 40 238 530 
DAHKSEVAHRFKDLGEENFKALVLIAFAQYLQQCPFEDHVKLVNEVTEFAKTCVADESAENCDKSLHTLFGDKLCTVATLRETYGEMADCCAKQEPERNECFLQHKDDNPNLPRLVRPEVDVMCTAFHDNEETFLKKYLYEIARRHPYFYAPELLFFAKRYKAAFTECCQAADKAACLLPKLDELRDEGKASSAKQRLKCASLQKFGERAFKAWAVARLSQRFPKAEFAEVSKLVTDLTKVHTECCHGDLLECADDRADLAKYICENQDSISSKLKECCEKPLLEKSHCIAEVENDEMPADLPSLAADFVESKDVCKNYAEAKDVFLGMFLYEYARRHPDYSVVLLLRLAKTYETTLEKCCAAADPHECYAKVFDEFKPLVEEPQNLIKQNCELFEQLGEYKFQNALLVRYTKKVPQVSTPTLVEVSRNLGKVGSKCCKHPEAKRMPCAEDYLSVVLNQLCVLHEKTPVSDRVTKCCTESLVNRRPCFSALEVDETYVPKEFNAETFTFHADICTLSEKERQIKKQTALVELVKHKPKATKEQLKAVMDDFAAFVEKCCKADDKETCFAEEGKKLVAASQAALGL
7ENB , Knot 134 298 0.85 40 201 291 
MGSSHHHHHHSSGLVPRGSHMVGSRTWCESEMLFVQPDAGTKEELYYRVTPKPGQTQANFNWTPHKVRFHDARPQRDSFDLNTHGFTFVEDAISPQLIERIRADDTAAVEGDYFASVAALVKRVTGADHVVCFSPYTRKENSEKGIFGQPARTVHCDHTPAAAIELTHKLCGEDAVRLLQSRFRAFSVWRPLVEPVLDWPLAVVDGRTIAPDDLHPVHWLRYEKKDTEPPFQLSFSETQKWYYLSRQRSDEVSIVKNYDSEVVPSPRSAHCAFKHPFVPKDAPPRESIDVRCLVFGGR

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 \(c\).

\(|P_{f(8EXO_1)}(2) \setminus P_{f(1UOR_1)}(2)|=135\), \(|P_{f(1UOR_1)}(2) \setminus P_{f(8EXO_1)}(2)|=29\). 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 1:100000000000011000010101111010111101111011100111011110100100101101000110010001100110000001110100010000110000010010110111011011001010110000011000111111111001011001010010001101000110100100100011010110100010110010001001011111111101000000001010000110011101100000011100001010110001001101010000110001100000100011110110111110001000111001011000001001010101000000111100110101101001010100100101001100110110001000011000101001100010110110110101010010100110000011111010110000011010111011111011001101111010010000101010101100110110101100010101000111000011100011000010000000101100001100100000011100000010110111011101010000011010011001111010011011000010111011110010001000010001101101100000100111011100110000110111101000100001000111110000010110100100010110011010011000000000010101110010010110110111011011001101010000110010011110100101100111000011100100100011010110110011000110101110101011001111011000001101000111011101000010011000001010011101100001100110111111000000111000101101011011000000110000011111000111110011000000001001001000101110001011101101111011101001001101000111000000110011001001001110
Pair \(Z_2\) Length of longest common subsequence
8EXO_1,1UOR_1 164 6
8EXO_1,7ENB_1 197 6
1UOR_1,7ENB_1 161 4

Newick tree

 
[
	8EXO_1:93.75,
	[
		1UOR_1:80.5,7ENB_1:80.5
	]:13.25
]

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 \((0.85)(0.8)(\frac{1665 }{\log_{20} 1665}-\frac{585}{\log_{20}585})=270.\)
Status Protein1 Protein2 d d1/2
Query variables 8EXO_1 1UOR_1 355 268
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]