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Parikh vectors
7SFE_1 5ISX_1 1JUS_1 Letter Amino acid
57 23 9 A Alanine
60 19 2 R Arginine
49 38 3 D Aspartic acid
30 30 6 Q Glutamine
56 39 23 K Lycine
25 2 0 C Cysteine
64 29 8 G Glycine
33 47 17 I Isoleucine
19 13 3 M Methionine
61 24 10 S Serine
55 25 7 V Valine
36 34 23 N Asparagine
38 27 9 F Phenylalanine
13 8 3 W Tryptophan
36 27 13 Y Tyrosine
54 53 20 E Glutamic acid
20 26 9 H Histidine
73 62 17 L Leucine
58 15 1 P Proline
37 32 11 T Threonine

7SFE_1|Chain A|DNA (cytosine-5)-methyltransferase 1|Homo sapiens (9606)
>5ISX_1|Chains A, B|Gramicidin S synthase 1|Brevibacillus brevis (1393)
>1JUS_1|Chains A, B, C[auth D], D[auth E]|HYPOTHETICAL TRANSCRIPTIONAL REGULATOR IN QACA 5'REGION|Staphylococcus aureus (1280)
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 \(w=f(c)\)
7SFE , Knot 338 874 0.87 40 331 797
HMNRISWVGEAVKTDGKKSYYKKVCIDAETLEVGDCVSVIPDDSSKPLYLARVTALWEDSSNGQMFHAHWFCAGTDTVLGATSDPLELFLVDECEDMQLSYIHSKVKVIYKAPSENWAMEGGMDPESLLEGDDGKTYFYQLWYDQDYARFESPPKTQPTEDNKFKFCVSCARLAEMRQKEIPRVLEQLEDLDSRVLYYSATKNGILYRVGDGVYLPPEAFTFNIKLSSPVKRPRKEPVDEDLYPEHYRKYSDYIKGSNLDAPEPYRIGRIKEIFCPKKSNGRPNETDIKIRVNKFYRPENTHKSTPASYHADINLLYWSDEEAVVDFKAVQGRCTVEYGEDLPECVQVYSMGGPNRFYFLEAYNAKSKSFEDPPNHARSPGNKGKGKGKGKGKPKSQACEPSEPEIEIKLPKLRTLDVFSGCGGLSEGFHQAGISDTLWAIEMWDPAAQAFRLNNPGSTVFTEDCNILLKLVMAGETTNSRGQRLPQKGDVEMLCGGPPCQGFSGMNRFNSRTYSKFKNSLVVSFLSYCDYYRPRFFLLENVRNFVSFKRSMVLKLTLRCLVRMGYQCTFGVLQAGQYGVAQTRRRAIILAAAPGEKLPLFPEPLHVFAPRACQLSVVVDDKKFVSNITRLSSGPFRTITVRDTMSDLPEVRNGASALEISYNGEPQSWFQRQLRGAQYQPILRDHICKDMSALVAARMRHIPLAPGSDWRDLPNIEVRLSDGTMARKLRYTHHDRKNGRSSSGALRGVCSCVEAGKACDPAARQFNTLIPWCLPHTGNRHNHWAGLYGRLEWDGFFSTTVTNPEPMGKQGRVLHPEQHRVVSVRECARSQGFPDTYRLFGNILDKHRQVGNAVPPPLAKAIGLEIKLCMLAKA
5ISX , Knot 230 573 0.85 40 264 525
MAPRNEIEETLVTIWQDVLGIEKIGIKDNFYALGGDSIKAIQVAARLHSYQLKLETKDLLKYPTIDQLVHYIKDSKRRSEQGIVEGEIGLTPIQHWFFEQQFTNMHHWNQSYMLYRPNGFDKEILLRVFNKIVEHHDALRMIYKHHNGKIVQINRGLEGTLFDFYTFDLTANDNEQQVICEESARLQNSINLEVGPLVKIALFHTQNGDHLFMAIHHLVVDGISWRILFEDLATAYEQAMHQQTIALPEKTDSFKDWSIELEKYANSELFLEEAEYWHHLNYYTDNVQIKKDYVTMNNKQKNIRYVGMELTIEETEKLLKNVNKAYRTEINDILLTALGFALKEWADIDKIVINLEGHGREEILEQMNIARTVGWFTSQYPVVLDMQKSDDLSYQIKLMKENLRRIPNKGIGYEIFKYLTTEYLRPVLPFTLKPEINFNYLGQFDTDVKTELFTRSPYSMGNSLGPDGKNNLGPEGESYFVLNINGFIEEGKLHITFSYNEQQYKEDTIQQLSRSYKQHLLAIIEHCVQKEDTELTPSDFSFKELELEEMDDIFDLLADSLTDDDDKHHHHHH
1JUS , Knot 85 194 0.77 38 118 180
MNLKDKILGVAKELFIKNGYNATTTGEIVKLSESSKGNLYYHFKTKENLFLEILNIEESKWQEQWKKEQIKAKTNREKFYLYNELSLTTEYYYPLQNAIIEFYTEYYKTNSINEKMNKLENKYIDAYHVIFKEGNLNGEWSINDVNAVSKIAANAVNGIVTFTHEQNINERIKLMNKFSQIFLNGLSKHHHHHH

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 \(c\).

\(|P_{f(7SFE_1)}(2) \setminus P_{f(5ISX_1)}(2)|=99\), \(|P_{f(5ISX_1)}(2) \setminus P_{f(7SFE_1)}(2)|=32\). 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 1:0100101110110001000000010101001011001011100000110110101110000010110101101100011110001101111000001010010001011001100011101110100110100100010011000001010011000100000101010010110100001101100100100011000100011100110110111011010101001100100011000101000000000101001011010011010011010000101000010101001001000000011000101011010000111010110100010010011001010011110010110100100001001100100110010101010101000100100101010110100101101011100110011100011110110111011010011001100000111011111000000100110010101101111001101100100000001000111011000000010111100100110100011101010011011000011110110011100000111111111001111101101111010010111000011001001001110010100010011010011011010001010011000101100011100010001011111010011111100100110101010010110010000000001000011101100010110100111001001111011001000001111010101011100010010111001011010000110100010001110000111011000001101111111011110101011101
Pair \(Z_2\) Length of longest common subsequence
7SFE_1,5ISX_1 131 4
7SFE_1,1JUS_1 251 3
5ISX_1,1JUS_1 186 7

Newick tree

 
[
	1JUS_1:12.80,
	[
		7SFE_1:65.5,5ISX_1:65.5
	]:56.30
]

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 \((0.85)(0.8)(\frac{1447 }{\log_{20} 1447}-\frac{573}{\log_{20}573})=221.\)
Status Protein1 Protein2 d d1/2
Query variables 7SFE_1 5ISX_1 283 232
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]