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Parikh vectors
6YWS_1 2PIQ_1 4OVU_1 Letter Amino acid
7 11 57 N Asparagine
23 9 51 G Glycine
8 17 45 F Phenylalanine
17 17 60 S Serine
19 6 42 T Threonine
26 13 46 A Alanine
2 4 19 W Tryptophan
5 12 77 E Glutamic acid
11 15 72 I Isoleucine
7 11 60 D Aspartic acid
3 6 32 C Cysteine
9 15 49 Q Glutamine
6 8 36 H Histidine
20 28 127 L Leucine
27 14 72 K Lycine
10 13 41 M Methionine
16 12 52 P Proline
21 14 59 R Arginine
16 19 56 V Valine
5 7 43 Y Tyrosine 

6YWS_15|Chain O[auth M]|Mitochondrial ribosomal protein|Neurospora crassa OR74A (367110)
>2PIQ_1|Chain A|Androgen receptor|Homo sapiens (9606)
>4OVU_1|Chain A|Phosphatidylinositol 4,5-bisphosphate 3-kinase catalytic subunit alpha isoform|Homo sapiens (9606)
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 \(w=f(c)\)
6YWS , Knot 113 258 0.81 40 159 247 
MNVTASLSRRPLGCLKAGLCQSLIRGYATAVAASTTQSTGALPDHNFFRIADKHTKKTRTAFAVFTPPKSPTTLPPSSVVTKSKAFKIANSKLPPLMTKPPSDPMPLLTQQQIARMDPTGARTALFSKARHAARVGDVLMVTHRRGGEPFAGVCLAIRRSGIDTAILLRNHLGKVGVEMWYKIYNKNVAGIEIIKRRAKRARRAKLMYMRQPKHDMGSVEQYVFAWKKMRKVLSSKGLTGGVGGGGGKQKGQESKKKN
2PIQ , Knot 120 251 0.88 40 185 243 
CQPIFLNVLEAIEPGVVCAGHDNNQPDSFAALLSSLNELGERQLVHVVKWAKALPGFRNLHVDDQMAVIQYSWMGLMVFAMGWRSFTNVNSRMLYFAPDLVFNEYRMHKSRMYSQCVRMRHLSQEFGWLQITPQEFLCMKALLLFSIIPVDGLKNQKFFDELRMNYIKELDRIIACKRKNPTSCSRRFYQLTKLLDSVQPIARELHQFTFDLLIKSHMVSVDFPEMMAEIISVQVPKILSGKVKPIYFHTQ
4OVU , Knot 404 1096 0.86 40 345 985 
MSYYHHHHHHDYDIPTTENLYFQGAMGSMPPRPSSGELWGIHLMPPRILVECLLPNGMIVTLECLREATLITIKHELFKEARKYPLHQLLQDESSYIFVSVTQEAEREEFFDETRRLCDLRLFQPFLKVIEPVGNREEKILNREIGFAIGMPVCEFDMVKDPEVQDFRRNILNVCKEAVDLRDLNSPHSRAMYVYPPNVESSPELPKHIYNKLDKGQIIVVIWVIVSPNNDKQKYTLKINHDCVPEQVIAEAIRKKTRSMLLSSEQLKLCVLEYQGKYILKVCGCDEYFLEKYPLSQYKYIRSCIMLGRMPNLMLMAKESLYSQLPMDCFTMPSYSRRISTATPYMNGETSTKSLWVINSALRIKILCATYVNVNIRDIDKIYVRTGIYHGGEPLCDNVNTQRVPCSNPRWNEWLNYDIYIPDLPRAARLCLSICSVKGRKGAKEEHCPLAWGNINLFDYTDTLVSGKMALNLWPVPHGLEDLLNPIGVTGSNPNKETPCLELEFDWFSSVVKFPDMSVIEEHANWSVSREAGFSYSHAGLSNRLARDNELRENDKEQLKAISTRDPLSEITEQEKDFLWSHRHYCVTIPEILPKLLLSVKWNSRDEVAQMYCLVKDWPPIKPEQAMELLDCNYPDPMVRGFAVRCLEKYLTDDKLSQYLIQLVQVLKYEQYLDNLLVRFLLKKALTNQRIGHFFFWHLKSEMHNKTVSQRFGLLLESYCRACGMYLKHLNRQVEAMEKLINLTDILKQEKKDETQKVQMKFLVEQMRRPDFMDALQGFLSPLNPAHQLGNLRLEECRIMSSAKRPLWLNWENPDIMSELLFQNNEIIFKNGDDLRQDMLTLQIIRIMENIWQNQGLDLRMLPYGCLSIGDCVGLIEVVRNSHTIMQIQCKGGLKGALQFNSHTLHQWLKDKNKGEIYDAAIDLFTRSCAGYCVATFILGIGDRHNSNIMVKDDGQLFHIDFGHFLDHKKKKFGYKRERVPFVLTQDFLIVISKGAQECTKTREFERFQEMCYKAYLAIRQHANLFINLFSMMLGSGMPELQSFDDIAYIRKTLALDKTEQEALEYFMKQMNDAHHGGWTTKMDWIFHTIKQHALN

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 \(c\).

\(|P_{f(6YWS_1)}(2) \setminus P_{f(2PIQ_1)}(2)|=76\), \(|P_{f(2PIQ_1)}(2) \setminus P_{f(6YWS_1)}(2)|=102\). 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 1:101010100011101011100011010101111000000111100011011000000000111110110010011100110000110110001111100110011111000011010101100111001001101101111000011011111011100011001111000110111011001000011110110001001001011010010001101000111100100110001101111111100010000000
Pair \(Z_2\) Length of longest common subsequence
6YWS_1,2PIQ_1 178 3
6YWS_1,4OVU_1 216 4
2PIQ_1,4OVU_1 194 4

Newick tree

 
[
	4OVU_1:10.80,
	[
		6YWS_1:89,2PIQ_1:89
	]:17.80
]

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 \((0.85)(0.8)(\frac{509 }{\log_{20} 509}-\frac{251}{\log_{20}251})=73.8\)
Status Protein1 Protein2 d d1/2
Query variables 6YWS_1 2PIQ_1 96 93
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]