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Parikh vectors
9BAQ_1 9GOA_1 5SWO_1 Letter Amino acid
83 95 46 A Alanine
86 79 60 D Aspartic acid
71 74 77 E Glutamic acid
43 89 72 I Isoleucine
83 85 72 K Lycine
100 120 127 L Leucine
44 44 43 Y Tyrosine
90 71 56 V Valine
19 5 19 W Tryptophan
52 84 57 N Asparagine
18 9 32 C Cysteine
88 65 51 G Glycine
34 29 36 H Histidine
69 65 42 T Threonine
71 71 60 S Serine
70 41 59 R Arginine
51 41 49 Q Glutamine
22 20 41 M Methionine
55 54 45 F Phenylalanine
95 34 52 P Proline 

9BAQ_1|Chain A|DNA (cytosine-5-)-methyltransferase|Neurospora crassa (5141)
>9GOA_1|Chains A, B, C, D|Alpha-latrotoxin-Lt1a|Latrodectus tredecimguttatus (6925)
>5SWO_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 \(w=f(c)\)
9BAQ , Knot 440 1244 0.84 40 344 1088 
GSMDSPDRSHGGMFIDVPAETMGFQEDYLDMFASVLSQGLAKEGDYAHHQPLPAGKEECLEPIAVATTITPSPDDPQLQLQLELEQQFQTESGLNGVDPAPAPESEDEADLPDGFSDESPDDDFVVQRSKHITVDLPVSTLINPRSTFQRIDENDNLVPPPQSTPERVAVEDLLKAAKAAGKNKEDYIEFELHDFNFYVNYAYHPQEMRPIQLVATKVLHDKYYFDGVLKYGNTKHYVTGMQVLELPVGNYGASLHSVKGQIWVRSKHNAKKEIYYLLKKPAFEYQRYYQPFLWIADLGKHVVDYCTRMVERKREVTLGCFKSDFIQWASKAHGKSKAFQNWRAQHPSDDFRTSVAANIGYIWKEINGVAGAKRAAGDQLFRELMIVKPGQYFRQEVPPGPVVTEGDRTVAATIVTPYIKECFGHMILGKVLRLAGEDAEKEKEVKLAKRLKIENKNATKADTKDDMKNDTATESLPTPLRSLPVQVLEATPIESDIVSIVSSDLPPSENNPPPLTNGSVKPKAKANPKPKPSTQPLHAAHVKYLSQELVNKIKVGDVISTPRDDSSNTDTKWKPTDTDDHRWFGLVQRVHTAKTKSSGRGLNSKSFDVIWFYRPEDTPCCAMKYKWRNELFLSNHCTCQEGHHARVKGNEVLAVHPVDWFGTPESNKGEFFVRQLYESEQRRWITLQKDHLTCYHNQPPKPPTAPYKPGDTVLATLSPSDKFSDPYEVVEYFTQGEKETAFVRLRKLLRRRKVDRQDAPANELVYTEDLVDVRAERIVGKCIMRCFRPDERVPSPYDRGGTGNMFFITHRQDHGRCVPLDTLPPTLRQGFNPLGNLGKPKLRGMDLYCGGGNFGRGLEEGGVVEMRWANDIWDKAIHTYMANTPDPNKTNPFLGSVDDLLRLALEGKFSDNVPRPGEVDFIAAGSPCPGFSLLTQDKKVLNQVKNQSLVASFASFVDFYRPKYGVLENVSGIVQTFVNRKQDVLSQLFCALVGMGYQAQLILGDAWAHGAPQSRERVFLYFAAPGLPLPDPPLPSHSHYRVKNRNIGFLCNGESYVQRSFIPTAFKFVSAGEGTADLPKIGDGKPDACVRFPDHRLASGITPYIRAQYACIPTHPYGMNFIKAWNNGNGVMSKSDRDLFPSEGKTRTSDASVGWKRLNPKTLFPTVTTTSNPSDARMGPGLHWDEDRPYTVQEMRRAQGYLDEEVLVGRTTDQWKLVGNSVSRHMALAIGLKFREAWLGTLYD
9GOA , Knot 413 1175 0.82 40 303 968 
EGEDLTLEEKAEICSELELQQKYVDIASNIIGDLSSLPIAGKIAGTIAAAAMTATHVASGRLDIEQTLLGCSDLPFDQIKEVLENRFNEIDRKLDSHSAALEEITKLVEKSISVVEKTRKQMNKRFDEVMKSIQDAKVSPIISKINNFARYFDTEKERIRGLKLNDYILKLEEPNGILLHFKESRTPTDDSLQAPLFSIIEEGYAVPKSIDDELAFKVLYALLYGTQTYVSVMFFLLEQYSFLANHYYEKGYLEKYDEYFNSLNNVFLDFKSSLVGTGTSNNEGLLDRVLQVLMTVKNSEFLGLEKNGVDEMLNEKINLFNKIKEEIEGKQKMTLSETPENFAQISFDKDITTPIGDWRDGREVRYAVQYASETLFSKISHWSDPVSVREKACPTLRMPVDQTRRNVLVFRKFDSSKPQLVGEITPYLSNFIDIDRDLYNAASNPDSAVGFKEFTKLNYDGANIRATFDHGRTVFHAAAKSGNDKIMFGLTFLAKSTELNQPDKKGYTPIHVAADSGNAGIVNLLIQRGVSINSKTYHFLQTPLHLAAQRGFVTTFQRLMESPEININERDKDGFTPLHYAIRGGERILEAFLNQISIDVNAKSNTGLTPFHLAIIKNDWPVASTLLGSKKVDINAVDENNITALHYAAILGYLETTKQLINLKEINANVVSSPGLLSALHYAILYKHDDVASFLMRSSNVNVNLKALGGITPLHLAVIQGRKQILSLMFDIGVNIEQKTDEKYTPLHLAAMSKYPELIQILLDQGSNFEAKTNSGATPLHLATFKGKSQAALILLNNEVNWRDTDENGQMPIHGAAMTGLLDVAQAIISIDATVVDIEDKNSDTPLNLAAQNSHIDVIKYFIDQGADINTRNKKGLAPLLAFSKKGNLDMVKYLFDKNANVYIADNDGMNFFYYAVQNGHLNIVKYAMSEKDKFEWSNTDNNRRDECPNEECAISHFAVCDAVQFDRIEIVKYFVGTLGNFAICGPLHQAARYGHLDIVKYLVEEEFLSVDGSKTDTPLCYASENGHFTVVQYLVSNGAKVNHDCGNGMTAIDKAITKNHLQVVQFLAANGVDFRRKNSRGTTPFLTAVAENALHIAEYLIREKRQDININEQNVDKDTALHLAVYYKNLQMIKLLIKYGIDVTIRNAYDKTALDIAIDAKFSNIVEYLKTKSG
5SWO , 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 \(c\).

\(|P_{f(9BAQ_1)}(2) \setminus P_{f(9GOA_1)}(2)|=61\), \(|P_{f(9GOA_1)}(2) \setminus P_{f(9BAQ_1)}(2)|=20\). 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 1:10100100001111101110011100001011101100111001001000111110000101111100101010010101010100010000110110111110000010110110000100011100000101011100110100010010000011111000100111001101101110000001010100101010010010010110111001100000101110010000010110110111100110100101011100000100010011001110000000111111011001100000110000010110100011011001010001100101001000100011101101100101111100111001100111101100100011111110010001110110101000110111101101110010000010110010100001001000001000010001101100111011010110001101100011100001111001010101010101010001101101001000110010110110010000000000101000000011111001001000001011000010111100100010011000100011100000000100101010011110110111010000101110010000000110100001000000110110110011001110101000100100110010010000111010011000010000111001100001101010011100110010100011010001101011110000001001110011101001101110110101011010011101101100111101011001100110001100101000011110100110111010100011011010111110101110110000011001000011101101101001001110010111001100000110011011111100101111011101110000011101111111110111100000010000111100100010001110110110110101011011010101010110001101101010100101100101101101100101110000001110010000001011100101001110100000100101111101000010010010010101000111100000101110010001111111010011110100
Pair \(Z_2\) Length of longest common subsequence
9BAQ_1,9GOA_1 81 4
9BAQ_1,5SWO_1 81 4
9GOA_1,5SWO_1 98 4

Newick tree

 
[
	5SWO_1:46.34,
	[
		9BAQ_1:40.5,9GOA_1:40.5
	]:5.84
]

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 \((0.85)(0.8)(\frac{2419 }{\log_{20} 2419}-\frac{1175}{\log_{20}1175})=293.\)
Status Protein1 Protein2 d d1/2
Query variables 9BAQ_1 9GOA_1 386 368.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]