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Parikh vectors
1MKW_1 8AYK_1 6WOT_1 Letter Amino acid
2 17 297 R Arginine
1 4 101 C Cysteine
4 23 367 G Glycine
1 25 197 I Isoleucine
2 19 282 S Serine
1 16 319 V Valine
2 13 394 A Alanine
4 6 200 Q Glutamine
0 7 146 M Methionine
6 10 207 F Phenylalanine
1 6 170 N Asparagine
3 30 561 L Leucine
4 14 227 K Lycine
1 9 142 Y Tyrosine
1 9 132 H Histidine
8 25 482 E Glutamic acid
2 16 271 P Proline
3 19 224 T Threonine
0 1 64 W Tryptophan
3 8 254 D Aspartic acid 

1MKW_1|Chain A[auth L]|ALPHA-THROMBIN|Bos taurus (9913)
>8AYK_1|Chains A, B|Aminotransferase class IV|Aminobacterium colombiense (81468)
>6WOT_1|Chains A, B, C, D|Ryanodine receptor 1|Oryctolagus cuniculus (9986)
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 \(w=f(c)\)
1MKW , Knot 31 49 0.82 36 44 46 
TSEDHFQPFFNEKTFGAGEADCGLRPLFEKKQVQDQTEKELFESYIEGR
8AYK , Knot 125 277 0.84 40 174 267 
GHMNLCYIDGKFLPLEEAKLPVTDLIIQRGVGVFETISTHSRRPLMLTPHLKRLEGSATASSIVMPATLDEMARIIREGIKKMGCETMVRPYITGGDSFGKDHLFSSSRYFVIFEEIRKPDPILYEKGVALHPINAERYLPSTKSINYMLSFTGQRDSKGAYEILYCPEGEIVEGSHSTFFLIKNGHLITAPTSRALSGTTRQIVLELARRGNIQVEERCPLLTELPEAEEAFITGTVKELLPVVRIGDQIIGNGVPGKLTKHLHQVYLSSIVEWLE
6WOT , Knot 1527 5037 0.86 40 399 3133 
MGDGGEGEDEVQFLRTDDEVVLQCSATVLKEQLKLCLAAEGFGNRLCFLEPTSNAQNVPPDLAICCFTLEQSLSVRALQEMLANTVEAGVESSQGGGHRTLLYGHAILLRHAHSRMYLSCLTTSRSMTDKLAFDVGLQEDATGEACWWTMHPASKQRSEGEKVCVGDDLILVSVSSERYLHLSTASGELQVDASFMQTLWNMNPICSCCEEGYVTGGHVLRLFHGHMDECLTISAADSDDQRRLVYYEGGAVCTHARSLWRLEPLRISWSGSHLRWGQPLRIRHVTTGRYLALTEDQGLVVVDACKAHTKATSFCFRVSKEKLDTAPKRDVEGMGPPEIKYGESLCFVQHVASGLWLTYAAPDPKALRLGVLKKKAILHQEGHMDDALFLTRCQQEESQAARMIHSTAGLYNQFIKGLDSFSGKPRGSGPPAGPALPIEAVILSLQDLIGYFEPPSEELQHEEKQSKLRSLRNRQSLFQEEGMLSLVLNCIDRLNVYTTAAHFAEYAGEEAAESWKEIVNLLYELLASLIRGNRANCALFSTNLDWVVSKLDRLEASSGILEVLYCVLIESPEVLNIIQENHIKSIISLLDKHGRNHKVLDVLCSLCVCNGVAVRSNQDLITENLLPGRELLLQTNLINYVTSIRPNIFVGRAEGSTQYGKWYFEVMVDEVVPFLTAQATHLRVGWALTEGYSPYPGGGEGWGGNGVGDDLYSYGFDGLHLWTGHVARPVTSPGQHLLAPEDVVSCCLDLSVPSISFRINGCPVQGVFEAFNLDGLFFPVVSFSAGVKVRFLLGGRHGEFKFLPPPGYAPCHEAVLPRERLRLEPIKEYRREGPRGPHLVGPSRCLSHTDFVPCPVDTVQIVLPPHLERIREKLAENIHELWALTRIEQGWTYGPVRDDNKRLHPCLVNFHSLPEPERNYNLQMSGETLKTLLALGCHVGMADEKAEDNLKKTKLPKTYMMSNGYKPAPLDLSHVRLTPAQTTLVDRLAENGHNVWARDRVAQGWSYSAVQDIPARRNPRLVPYRLLDEATKRSNRDSLCQAVRTLLGYGYNIEPPDQEPSQVENQSRWDRVRIFRAEKSYTVQSGRWYFEFEAVTTGEMRVGWARPELRPDVELGADELAYVFNGHRGQRWHLGSEPFGRPWQSGDVVGCMIDLTENTIIFTLNGEVLMSDSGSETAFREIEIGDGFLPVCSLGPGQVGHLNLGQDVSSLRFFAICGLQEGFEPFAINMQRPVTTWFSKSLPQFEPVPPEHPHYEVARMDGTVDTPPCLRLAHRTWGSQNSLVEMLFLRLSLPVQFHQHFRCTAGATPLAPPGLQPPAEDEARAAEPDPDYENLRRSAGGWGEAEGGKEGTAKEGTPGGTPQPGVEAQPVRAENEKDATTEKNKKRGFLFKAKKAAMMTQPPATPALPRLPHDVVPADNRDDPEIILNTTTYYYSVRVFAGQEPSCVWVGWVTPDYHQHDMNFDLSKVRAVTVTMGDEQGNVHSSLKCSNCYMVWGGDFVSPGQQGRISHTDLVIGCLVDLATGLMTFTANGKESNTFFQVEPNTKLFPAVFVLPTHQNVIQFELGKQKNIMPLSAAMFLSERKNPAPQCPPRLEVQMLMPVSWSRMPNHFLQVETRRAGERLGWAVQCQDPLTMMALHIPEENRCMDILELSERLDLQRFHSHTLRLYRAVCALGNNRVAHALCSHVDQAQLLHALEDAHLPGPLRAGYYDLLISIHLESACRSRRSMLSEYIVPLTPETRAITLFPPGRKGGNARRHGLPGVGVTTSLRPPHHFSPPCFVAALPAAGVAEAPARLSPAIPLEALRDKALRMLGEAVRDGGQHARDPVGGSVEFQFVPVLKLVSTLLVMGIFGDEDVKQILKMIEPEVFTEEEEEEEEEEEEEEEEEEDEEEKEEDEEEEEKEDAEKEEEEAPEGEKEDLEEGLLQMKLPESVKLQMCNLLEYFCDQELQHRVESLAAFAERYVDKLQANQRSRYALLMRAFTMSAAETARRTREFRSPPQEQINMLLHFKDEADEEDCPLPEDIRQDLQDFHQDLLAHCGIQLEGEEEEPEEETSLSSRLRSLLETVRLVKKKEEKPEEELPAEEKKPQSLQELVSHMVVRWAQEDYVQSPELVRAMFSLLHRQYDGLGELLRALPRAYTISPSSVEDTMSLLECLGQIRSLLIVQMGPQEENLMIQSIGNIMNNKVFYQHPNLMRALGMHETVMEVMVNVLGGGETKEIRFPKMVTSCCRFLCYFCRISRQNQRSMFDHLSYLLENSGIGLGMQGSTPLDVAAASVIDNNELALALQEQDLEKVVSYLAGCGLQSCPMLLAKGYPDIGWNPCGGERYLDFLRFAVFVNGESVEENANVVVRLLIRKPECFGPALRGEGGSGLLAAIEEAIRISEDPARDGPGVRRDRRREHFGEEPPEENRVHLGHAIMSFYAALIDLLGRCAPEMHLIQAGKGEALRIRAILRSLVPLDDLVGIISLPLQIPTLGKDGALVQPKMSASFVPDHKASMVLFLDRVYGIENQDFLLHVLDVGFLPDMRAAASLDTATFSTTEMALALNRYLCLAVLPLITKCAPLFAGTEHRAIMVDSMLHTVYRLSRGRSLTKAQRDVIEDCLMALCRYIRPSMLQHLLRRLVFDVPILNEFAKMPLKLLTNHYERCWKYYCLPTGWANFGVTSEEELHLTRKLFWGIFDSLAHKKYDQELYRMAMPCLCAIAGALPPDYVDASYSSKAEKKATVDAEGNFDPRPVETLNVIIPEKLDSFINKFAEYTHEKWAFDKIQNNWSYGENVDEELKTHPMLRPYKTFSEKDKEIYRWPIKESLKAMIAWEWTIEKAREGEEERTEKKKTRKISQTAQTYDPREGYNPQPPDLSGVTLSRELQAMAEQLAENYHNTWGRKKKQELEAKGGGTHPLLVPYDTLTAKEKARDREKAQELLKFLQMNGYAVTRGLKDMELDTSSIEKRFAFGFLQQLLRWMDISQPFIAHLEAVVSSGRVEKSPHEQEIKFFAKILLPLINQYFTNHCLYFLSTPAKVLGSGGHASNKEKEMITSLFCKLAALVRHRVSLFGTDAPAVVNCLHILARSLDARTVMKSGPEIVKAGLRSFFESASEDIEKMVENLRLGKVSQARTQVKGVGQNLTYTTVALLPVLTTLFQHIAQHQFGDDVILDDVQVSCYRTLCSIYSLGTTKNTYVEKLRPALGECLARLAAAMPVAFLEPQLNEYNACSVYTTKSPRERAILGLPNSVEEMCPDIPVLDRLMADIGGLAESGARYTEMPHVIEITLPMLCSYLPRWWERGPEAPPPALPAGAPPPCTAVTSDHLNSLLGNILRIIVNNLGIDEATWMKRLAVFAQPIVSRARPELLHSHFIPTIGRLRKRAGKVVAEEEQLRLEAKAEAEEGELLVRDEFSVLCRDLYALYPLLIRYVDNNRAHWLTEPNANAEELFRMVGEIFIYWSKSHNFKREEQNFVVQNEINNMSFLTADSKSKMAKAGDAQSGGSDQERTKKKRRGDRYSVQTSLIVATLKKMLPIGLNMCAPTDQDLIMLAKTRYALKDTDEEVREFLQNNLHLQGKVEGSPSLRWQMALYRGLPGREEDADDPEKIVRRVQEVSAVLYHLEQTEHPYKSKKAVWHKLLSKQRRRAVVACFRMTPLYNLPTHRACNMFLESYKAAWILTEDHSFEDRMIDDLSKAGEQEEEEEEVEEKKPDPLHQLVLHFSRTALTEKSKLDEDYLYMAYADIMAKSCHLEEGGENGEAEEEEVEVSFEEKEMEKQRLLYQQSRLHTRGAAEMVLQMISACKGETGAMVSSTLKLGISILNGGNAEVQQKMLDYLKDKKEVGFFQSIQALMQTCSVLDLNAFERQNKAEGLGMVNEDGTVINRQNGEKVMADDEFTQDLFRFLQLLCEGHNNDFQNYLRTQTGNTTTINIIICTVDYLLRLQESISDFYWYYSGKDVIEEQGKRNFSKAMSVAKQVFNSLTEYIQGPCTGNQQSLAHSRLWDAVVGFLHVFAHMMMKLAQDSSQIELLKELLDLQKDMVVMLLSLLEGNVVNGMIARQMVDMLVESSSNVEMILKFFDMFLKLKDIVGSEAFQDYVTDPRGLISKKDFQKAMDSQKQFTGPEIQFLLSCSEADENEMINFEEFANRFQEPARDIGFNVAVLLTNLSEHVPHDPRLRNFLELAESILEYFRPYLGRIEIMGASRRIERIYFEISETNRAQWEMPQVKESKRQFIFDVVNEGGEAEKMELFVSFCEDTIFEMQIAAQISEPEGEPEADEDEGMGEAAAEGAEEGAAGAEGAAGTVAAGATARLAAAAARALRGLSYRSLRRRVRRLRRLTAREAATALAALLWAVVARAGAAGAGAAAGALRLLWGSLFGGGLVEGAKKVTVTELLAGMPDPTSDEVHGEQPAGPGGDADGAGEGEGEGDAAEGDGDEEVAGHEAGPGGAEGVVAVADGGPFRPEGAGGLGDMGDTTPAEPPTPEGSPILKRKLGVDGEEEELVPEPEPEPEPEPEKADEENGEKEEVPEAPPEPPKKAPPSPPAKKEEAGGAGMEFTGELEVQRVKFLNYLSRNFYTLRFLALFLAFAINFILLFYKVSDSPPGEDDMEGSAAGDLAGAGSGGGSGWGSGAGEEAEGDEDENMVYYFLEESTGYMEPALWCLSLLHTLVAFLCIIGYNCLKVPLVIFKREKELARKLEFDGLYITEQPGDDDVKGQWDRLVLNTPSFPSNYWDKFVKRKVLDKHGDIFGRERIAELLGMDLASLEITAHNERKPDPPPGLLTWLMSIDVKYQIWKFGVIFTDNSFLYLGWYMVMSLLGHYNNFFFAAHLLDIAMGVKTLRTILSSVTHNGKQLVMTVGLLAVVVYLYTVVAFNFFRKFYNKSEDEDEPDMKCDDMMTCYLFHMYVGVRAGGGIGDEIEDPAGDEYELYRVVFDITFFFFVIVILLAIIQGLIIDAFGELRDQQEQVKEDMETKCFICGIGSDYFDTTPHGFETHTLEEHNLANYMFFLMYLINKDETEHTGQESYVWKMYQERCWDFFPAGDCFRKQYEDQLS

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 \(c\).

\(|P_{f(1MKW_1)}(2) \setminus P_{f(8AYK_1)}(2)|=23\), \(|P_{f(8AYK_1)}(2) \setminus P_{f(1MKW_1)}(2)|=153\). 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 1:0000010111000011110100110111000010000000110001010
Pair \(Z_2\) Length of longest common subsequence
1MKW_1,8AYK_1 176 3
1MKW_1,6WOT_1 355 4
8AYK_1,6WOT_1 225 5

Newick tree

 
[
	6WOT_1:16.89,
	[
		1MKW_1:88,8AYK_1:88
	]:75.89
]

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 \((0.85)(0.8)(\frac{326 }{\log_{20} 326}-\frac{49}{\log_{20}49})=89.1\)
Status Protein1 Protein2 d d1/2
Query variables 1MKW_1 8AYK_1 111 64
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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