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Parikh vectors
5URK_1 3NNR_1 4WKA_1 Letter Amino acid
79 11 15 P Proline
100 12 24 S Serine
94 13 26 T Threonine
88 16 15 R Arginine
91 14 22 D Aspartic acid
47 5 7 M Methionine
52 5 10 H Histidine
19 3 9 W Tryptophan
61 14 16 Y Tyrosine
123 8 20 V Valine
103 17 33 A Alanine
84 7 25 Q Glutamine
84 8 32 G Glycine
70 9 16 N Asparagine
25 1 4 C Cysteine
127 20 16 E Glutamic acid
65 10 20 F Phenylalanine
112 16 8 I Isoleucine
206 30 36 L Leucine
108 9 23 K Lycine 

5URK_1|Chain A|U5 small nuclear ribonucleoprotein 200 kDa helicase|Homo sapiens (9606)
>3NNR_1|Chain A|Transcriptional regulator, TetR family|Marinobacter aquaeolei (351348)
>4WKA_1|Chain A|Chitotriosidase-1|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 \(w=f(c)\)
5URK , Knot 595 1738 0.85 40 363 1424 
GGSDLDQGGEALAPRQVLDLEDLVFTQGSHFMANKRCQLPDGSFRRQRKGYEEVHVPALKPKPFGSEEQLLPVEKLPKYAQAGFEGFKTLNRIQSKLYRAALETDENLLLCAPTGAGKTNVALMCMLREIGKHINMDGTINVDDFKIIYIAPMRSLVQEMVGSFGKRLATYGITVAELTGDHQLCKEEISATQIIVCTPEKWDIITRKGGERTYTQLVRLIILDEIHLLHDDRGPVLEALVARAIRNIEMTQEDVRLIGLSATLPNYEDVATFLRVDPAKGLFYFDNSFRPVPLEQTYVGITEKKAIKRFQIMNEIVYEKIMEHAGKNQVLVFVHSRKETGKTARAIRDMCLEKDTLGLFLREGSASTEVLRTEAEQCKNLELKDLLPYGFAIHHAGMTRVDRTLVEDLFADKHIQVLVSTATLAWGVNLPAHTVIIKGTQVYSPEKGRWTELGALDILQMLGRAGRPQYDTKGEGILITSHGELQYYLSLLNQQLPIESQMVSKLPDMLNAEIVLGNVQNAKDAVNWLGYAYLYIRMLRSPTLYGISHDDLKGDPLLDQRRLDLVHTAALMLDKNNLVKYDKKTGNFQVTELGRIASHYYITNDTVQTYNQLLKPTLSEIELFRVFSLSSEFKNITVREEEKLELQKLLERVPIPVKESIEEPSAKINVLLQAFISQLKLEGFALMADMVYVTQSAGRLMRAIFEIVLNRGWAQLTDKTLNLCKMIDKRMWQSMCPLRQFRKLPEEVVKKIEKKNFPFERLYDLNHNEIGELIRMPKMGKTIHKYVHLFPKLELSVHLQPITRSTLKVELTITPDFQWDEKVHGSSEAFWILVEDVDSEVILHHEYFLLKAKYAQDEHLITFFVPVFEPLPPQYFIRVVSDRWLSCETQLPVSFRHLILPEKYPPPTELLDLQPLPVSALRNSAFESLYQDKFPFFNPIQTQVFNTVYNSDDNVFVGAPTGSGKTICAEFAILRMLLQSSEGRCVYITPMEALAEQVYMDWYEKFQDRLNKKVVLLTGETSTDLKLLGKGNIIISTPEKWDILSRRWKQRKNVQNINLFVVDEVHLIGGENGPVLEVICSRMRYISSQIERPIRIVALSSSLSNAKDVAHWLGCSATSTFNFHPNVRPVPLELHIQGFNISHTQTRLLSMAKPVYHAITKHSPKKPVIVFVPSRKQTRLTAIDILTTCAADIQRQRFLHCTEKDLIPYLEKLSDSTLKETLLNGVGYLHEGLSPMERRLVEQLFSSGAIQVVVASRSLCWGMNVAAHLVIIMDTQYYNGKIHAYVDYPIYDVLQMVGHANRPLQDDEGRCVIMCQGSKKDFFKKFLYEPLPVESHLDHCMHDHFNAEIVTKTIENKQDAVDYLTWTFLYRRMTQNPNYYNLQGISHRHLSDHLSELVEQTLSDLEQSKCISIEDEMDVAPLNLGMIAAYYYINYTTIELFSMSLNAKTKVRGLIEIISNAAEYENIPIRHHEDNLLRQLAQKVPHKLNNPKFNDPHVKTNLLLQAHLSRMQLSAELQSDTEEILSKAIRLIQACVDVLSSNGWLSPALAAMELAQMVTQAMWSKDSYLKQLPHFTSEHIKRCTDKGVESVFDIMEMEDEERNALLQLTDSQIADVARFCNRYPNIELSYEVVDKDSIRSGGPVVVLVQLEREEEVTGPVIAPLFPQKREEGWWVVIGDAKSNSLISIKRLTLQQKAKVKLDFVAPATGAHNYTLYFMSDAYMGCDQEYKFSVDVKEA
3NNR , Knot 104 228 0.82 40 156 222 
GMTMKTRDKILLSSLELFNDKGERNITTNHIAAHLAISPGNLYYHFRNKSDIIYEIFQEYEKLVDYYLDIPEDRPITLEDMTFYLESVFDGLWSYRFFHRDLEYLLDSDPRLRQDYREFTNRCLAAINRIFAKLADAGIIQPQPEDLRSAMSLNVWLVITNWMAFLKTAHAAEEPASLSLTELKQGIYQVLTLEVPYLTPEYRERVLALREKYRPTLPEAQGISGVEA
4WKA , Knot 160 377 0.84 40 221 358 
AKLVCYFTNWAQYRQGEARFLPKDLDPSLCTHLIYAFAGMTNHQLSTTEWNDETLYQEFNGLKKMNPKLKTLLAIGGWNFGTQKFTDMVATANNRQTFVNSAIRFLRKYSFDGLDLDWEYPGSQGSPAVDKERFTTLVQDLANAFQQEAQTSGKERLLLSAAVPAGQTYVDAGYEVDKIAQNLDFVNLMAYDFHGSWEKVTGHNSPLYKRQEESGAAASLNVDAAVQQWLQKGTPASKLILGMPTYGRSFTLASSSDTRVGAPATGSGTPGPFTKEGGMLAYYEVCSWKGATKQRIQDQKVPYIFRDNQWVGFDDVESFKTKVSYLKQKGLGGAMVWALDLDDFAGFSCNQGRYPLIQTLRQELSLVPRGSHHHHHH

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 \(c\).

\(|P_{f(5URK_1)}(2) \setminus P_{f(3NNR_1)}(2)|=213\), \(|P_{f(3NNR_1)}(2) \setminus P_{f(5URK_1)}(2)|=6\). 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 1:1100100110111100110100111001001110000011010100000100010111101011100001111001100101110110010010001001110000011101101110001111011001100101010101001011011110011001110110011001101101010001000010100111001001011000110000001101111001011000011110111101100101000010111101011000011011010110111010001011110000111000011001011001100011001100011111000000100101100101000011111001010001100010000010100111011110011100100011001110001011100101111101110011101001001001010011110110111011010000010111100010100010110001110001100110110101111010010011011101010101100101011000010101110000101100111110000110000001010100110110000100001000001101010010110110100010010100000101001100111110001001010101110111001010111111011010001101101110111001110100001010011000110010110010011001100100001110010010000110110110110010001011101010101011000010101010101010001010001111110010001110000111010010000110111111011110011011000110000011101001111000111001101011110110001100100001111011000110010000001111110101001010111101110000100101011011100101010001000100011110100000101110101110010010110001000001001011110010111100111101100010010001001101111000100100110111001000101010101111010101101000000110110110011000010011111110000001011011000110100001100000011101001000010001101110100110110001100110011101111000101110111011111000000101010100110011011101001100001001110010000110011001111000100010001010110001000001100101011000100010000101100001000100110001001000001010001011110111111000100001011010101000101110110011000011100000011001100110010010100101000111010100101010100000011001101101010110001110111111011011001110000010011010000100000011001101101000000111010000110110100001010100011000010011111111010000010111111111000001111111010000110100101000101010111110110000101100101100000010101001
Pair \(Z_2\) Length of longest common subsequence
5URK_1,3NNR_1 219 5
5URK_1,4WKA_1 160 5
3NNR_1,4WKA_1 167 4

Newick tree

 
[
	3NNR_1:10.51,
	[
		5URK_1:80,4WKA_1:80
	]:22.51
]

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 \((0.85)(0.8)(\frac{1966 }{\log_{20} 1966}-\frac{228}{\log_{20}228})=442.\)
Status Protein1 Protein2 d d1/2
Query variables 5URK_1 3NNR_1 560 314
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]