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Parikh vectors
4BCC_1 9KBJ_1 2HRZ_1 Letter Amino acid
36 21 36 A Alanine
57 18 35 G Glycine
38 24 11 T Threonine
12 10 2 W Tryptophan
41 7 7 Y Tyrosine
25 5 24 R Arginine
27 21 7 N Asparagine
27 13 7 Q Glutamine
39 15 26 I Isoleucine
12 7 7 M Methionine
23 6 10 H Histidine
47 24 15 K Lycine
36 14 18 F Phenylalanine
31 13 22 P Proline
38 17 21 S Serine
52 16 15 D Aspartic acid
16 6 3 C Cysteine
42 10 27 E Glutamic acid
63 26 28 L Leucine
48 29 21 V Valine 

4BCC_1|Chain A|PROLYL ENDOPEPTIDASE|SUS SCROFA (9823)
>9KBJ_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|Non-structural protein 1|Human parvovirus B19 (10798)
>2HRZ_1|Chain A|Nucleoside-diphosphate-sugar epimerase|Agrobacterium tumefaciens (176299)
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 \(w=f(c)\)
4BCC , Knot 278 710 0.85 40 302 663 
MLSFQYPDVYRDETAIQDYHGHKVCDPYAWLEDPDSEQTKAFVEAQNKITVPFLEQCPIRGLYKERMTELYDYPKYSCHFKKGKRYFYFYNTGLQNQRVLYVQDSLEGEARVFLDPNILSDDGTVALRGYAFSEDGEYFAYGLSASGSDWVTIKFMKVDGAKELPDVLERVKFSCMAWTHDGKGMFYNAYPQQDGKSDGTETSTNLHQKLYYHVLGTDQSEDILCAEFPDEPKWMGGAELSDDGRYVLLSIREGCDPVNRLWYCDLQQESNGITGILKWVKLIDNFEGEYDYVTNEGTVFTFKTNRHSPNYRLINIDFTDPEESKWKVLVPEHEKDVLEWVACVRSNFLVLCYLHDVKNTLQLHDLATGALLKIFPLEVGSVVGYSGQKKDTEIFYQFTSFLSPGIIYHCDLTKEELEPRVFREVTVKGIDASDYQTVQIFYPSKDGTKIPMFIVHKKGIKLDGSHPAFLYGYGGFNISITPNYSVSRLIFVRHMGGVLAVANIRGGGEYGETWHKGGILANKQNCFDDFQCAAEYLIKEGYTSPKRLTINGGSNGGLLVATCANQRPDLFGCVIAQVGVMDMLKFHKYTIGHAWTTDYGCSDSKQHFEWLIKYSPLHNVKLPEADDIQYPSMLLLTADHDDRVVPLHSLKFIATLQYIVGRSRKQNNPLLIHVDTKAGHGAGKPTAKVIEEVSDMFAFIARCLNIDWIP
9KBJ , Knot 135 302 0.85 40 196 291 
AEVVPFNGKGTKASIKFQTMVNWLCENRVFTEDKWKLVDFNQYTLLSSSHSGSFQIQSALKLAIYKATNLVPTSTFLLHADFEQVMCIKDNKIVKLLLCQNYDPLLVGQHVLKWIDKKCGKKNTLWFYGPPSTGKTNLAMAIAKSVPVYGMVNWNNENFPFNDVAGKSLVVWDEGIIKSTIVEAAKAILGGQPTRVDQKMRGSVAVPGVPVVITSNGDITFVVSGNTTTTVHAKALKERMVKLNFTVRCSPDMGLLTEADVQQWLTWCNAQSWDHYENWAINYTFDFPGINADALHPDLQTT
2HRZ , Knot 141 342 0.80 40 190 327 
HHHSSGRENLYFQGMHIAIIGAAGMVGRKLTQRLVKDGSLGGKPVEKFTLIDVFQPEAPAGFSGAVDARAADLSAPGEAEKLVEARPDVIFHLAAIVSGEAELDFDKGYRINLDGTRYLFDAIRIANGKDGYKPRVVFTSSIAVFGAPLPYPIPDEFHTTPLTSYGTQKAICELLLSDYSRRGFFDGIGIRLPTICIRPGKPNAAASGFFSNILREPLVGQEAVLPVPESIRHWHASPRSAVGFLIHGAMIDVEKVGPRRNLSMPGLSATVGEQIEALRKVAGEKAVALIRREPNEMIMRMCEGWAPGFEAKRARELGFTAESSFEEIIQVHIEDELGGSLK

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 \(c\).

\(|P_{f(4BCC_1)}(2) \setminus P_{f(9KBJ_1)}(2)|=139\), \(|P_{f(9KBJ_1)}(2) \setminus P_{f(4BCC_1)}(2)|=33\). 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 1:11010010100000110000100100101110010000001110100010111100011011000010010001000001001000101000110000110100010101011101011000101110101100010011011010100110101101011001101100101001110001011100101000100010000001000100011100000011010110010111110100010011101001001100110001000001101110110110010100001000101101000000100011010100100001011110000011011101000111100100100010100110111101111011011100100000011001001101111000010000101011001010110100000101101000100111111000110101001111010111010101000100111100111111110101110010010011111000001001001100110010001001010110011111100100010111011101111011010000110110000100000001011100011001011010010010111101000001111001011101001110000000111101000110111010101100100111111001010111
Pair \(Z_2\) Length of longest common subsequence
4BCC_1,9KBJ_1 172 4
4BCC_1,2HRZ_1 180 4
9KBJ_1,2HRZ_1 172 4

Newick tree

 
[
	2HRZ_1:88.68,
	[
		4BCC_1:86,9KBJ_1:86
	]:2.68
]

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 \((0.85)(0.8)(\frac{1012 }{\log_{20} 1012}-\frac{302}{\log_{20}302})=190.\)
Status Protein1 Protein2 d d1/2
Query variables 4BCC_1 9KBJ_1 242 171.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]