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Parikh vectors
8QJC_1 4IWO_1 1YYV_1 Letter Amino acid
114 33 12 R Arginine
69 46 2 I Isoleucine
65 21 6 P Proline
99 58 9 E Glutamic acid
39 26 3 F Phenylalanine
83 36 11 S Serine
33 7 2 W Tryptophan
61 26 4 N Asparagine
113 32 6 D Aspartic acid
76 27 8 Q Glutamine
115 68 19 L Leucine
54 49 4 K Lycine
36 16 3 M Methionine
137 39 4 T Threonine
75 27 2 Y Tyrosine
92 36 10 A Alanine
18 10 1 C Cysteine
145 38 9 G Glycine
39 24 4 H Histidine
67 41 12 V Valine 

8QJC_1|Chain A|T6SS-associated Rhs core and toxin domain|Salmonella bongori N268-08 (1197719)
>4IWO_1|Chain A|Serine/threonine-protein kinase TBK1|Homo sapiens (9606)
>1YYV_1|Chains A, B|putative transcriptional regulator|Salmonella typhimurium (99287)
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 \(w=f(c)\)
8QJC , Knot 534 1530 0.85 40 351 1286 
MMAKNYIARQDGLWTVISLTPDVCKTPVGSSTPPVPYAVIASLGDAVQTVQTVKANGHPVLVLDQSIIPYTKGDEPGVAKGVKSGTVGDICEPLEHSRTVFAGGKPVLRHFDKFWMNARNTTGLIIGQPPVAADSVSKADPAPEPETKEEQSIWDTMLMIQTDQSRAQIEAAPVLLDITIGAAKAFWNQFPDLATILGQGAAMQSAGEMQMNGAMLSAMGMDEMGEAQYRMGEEVMAHAGDFNLDEYRLKMSNRNQEIGGMAYDYGSLVLGGYGLARGGFMGLKTMRTASKLVHEGEAGGKFAAHMMKMDNAEDAATDAEKLADDVVPEGRDGANIKGETQGNGDRGDGVNSMLCTKSGEPVDIGTGDFLQHFSVLSLPGSLPLTLSRFYRSQAKGTGIFGPKWTDEWSCMLTVHGNDMHFTNHEGVDLYYRIPQNGIFRDTANSRQAYYRLSGDIRDELTIFDRRSQHSQIFSLTDNGIYLLSAIHDRYGNRADFIRTEGLLTGIHHSDGYTLALGWQQRQLVSIDLATPQRQRLVTCHYDKNGYLAECDTIQFSHLWHEYTSEGWMTRWRDTDKTCVDIVYDTLGRTVSTLSTEGYFDDQFLYNDDEKCTTYLDAEGGETRYWYNGDGLVTRSIDPLGREETSVWENTRLRSRTDALGRTTAYDYNNEGEISRVSLPGGYSLYYDYNEHGQLTRLSAPGNQVWLWEYDGKGSMVCLTDPQGRQQQFSYSEHGDLLRQIMPNGATWRWSHDALHQVRATTAPDGGVTQTEQDILGRLLSVKDPLGYTTQFRYSKNHAGPQGSVEEIRRPDGVRELMRQNSEKLPESFTDGEGNTTRYEYGAFDLLTAVIRPDGERLACRYDKLTRLTEITNAEGEHYRLRYDKAGQLVAETDFTGRTLTYSYDAAGRCIRTTFPDGTHLNRQYSTTDQVTREEVAQGDSDRVLSCTTFIYDALSRLTEARNNDATVTYEYDDASRVTAETINGRRTEYNYDPDQDTVSQRTTAGITECFTRGLTGELTQWQIDGHTPLTLEHDLRGQEISRQSEAGFSLRQNYTPTGMLTGQQAGDLTEQNPHYWRNNTLQRQWLYDKAYNLTMISDSLRGTMVNSVTANDQISHATWTGSSDIPMREERFAYDRNLNITRRQTQVNGVPDSEAYQHQQHGRVTSREYKAWRHTTTRINPDSGMPEEGQFVRVIRDEQTTWKYDVNGRLVEKLIDKGGYRPLRWRYRWDARSQLTGLETPEGERREYKYDPFGRRISKRCTNRDRPGTDFHWNGDQLTEEIPVGTDGTPEYENAIRWIYEPGSFTPLARYEKGQLHYTITDTVGRIQELLTEDGTIVWRGKQHLWGREEGRNKDDAPSCRLRFPGQYEDEESGLYYNRFRYYDCEAGQYLCADPIGLRGGINLYAYAPNPLSWIDPLGLKCGETAQPEWTNHRYKHFPPKNKSWKDIIKSTKTGPAKYMHDIDIKTLEYDVFNTGMPATNGKPWKVKNIGKVIGASEGKESQWMRVELSGGTIHGHPISIDEFRRLTTQ
4IWO , Knot 263 660 0.86 40 286 612 
GAMGQSTSNHLWLLSDILGQGATANVFRGRHKKTGDLFAIKVFNNISFLRPVDVQMREFEVLKKLNHKNIVKLFAIEEETTTRHKVLIMEFCPCGSLYTVLEEPSNAYGLPESEFLIVLRDVVGGMNHLRENGIVHRDIKPGNIMRVIGEDGQSVYKLTDFGAARELEDDEQFVALYGTEEYLHPDMYERAVLRKDHQKKYGATVDLWSIGVTFYHAATGSLPFRPFEGPRRNKEVMYKIITGKPSGAISGVQKAENGPIDWSGDMPVSCSLSRGLQVLLTPVLANILEADQEKCWGFDQFFAETSDILHRMVIHVFSLQQMTAHKIYIHSYNTATIFHELVYKQTKIISSNQELIYEGRRLVLEPGRLAQHFPKTTEENPIFVVSREPLNTIGLIYEKISLPKVHPRYDLDGDASMAKAITGVVCYACRIASTLLLYQELMRKGIRWLIELIKDDYNETVHKKTEVVITLDFCIRNIEKTVKVYEKLMKINLEAAELGEISDIHTKLLRLSSSQGTIETSLQDIDSRLSPGGSLADAWAHQEGTHPKDRNVEKLQVLLNCMTEIYYQFKKDKAERRLAYNEEQIHKFDKQKLYYHATKAMTHFTDECVKKYEAFLNKSEEWIRKMLHLRKQLLSLTNQCFDIEEEVSKYQEYTNELQET
1YYV , Knot 68 131 0.84 40 99 128 
SNAMRAHTLSRQLREGNLFAEQCPSREVLKHVTSRWGVLILVALRDGTHRFSDLRRKMGGVSEKMLAQSLQALEQDGFLNRVSYPVVPPHVEYSLTPLGEQVSDKVAALADWIELNLPQVLAQRERLSDGG

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 \(c\).

\(|P_{f(8QJC_1)}(2) \setminus P_{f(4IWO_1)}(2)|=92\), \(|P_{f(4IWO_1)}(2) \setminus P_{f(8QJC_1)}(2)|=27\). 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 1:111000110001110110101010001110001111011110110110010010101011111000111000100111101100101101001100000111110111001001110100001111101111100100101110100000001100111100000010101111110101111011100110110111011110011010101111011110011010001100111011010100001010000001111100010111110111011111100100100110010111011101101001001100100110011101001101010001010010110011000010110110101100101101110111010010000101011111010001001101010010100001101000110011100010000100010101000101100000000110100011011011000010010110001110110000100111110000110101101000011000000010110000101001100000011100100000001011000110010010001010001100000000001010110000100101110001011100000110000100000111000100000010100101111001000000010100101110011110001010110100101000010000010110011101101010001100101001101110000001110110100111000010000001110101001001011001100000011001001010000000111011011101010011000001001001001010000100001101110001010010000011100100011010010000000001000011010000110000110011001001000010100000010010100101000000001000010000011100010011010100101010011010001010010000011101000001011101001101000010010000100011000100101100010101100101000100101010001110000110000101000000101110001000000101000000110000001010011100101101100000010001010110011001100110100010100010110010100000000111001000000000110010101001000111100101000011011001101011100001010001000110100110001011101000111000100000110001011100000001100001000000110010101111011101010110110110111100100101010000000111000010011000001110010010100100011001111001011010011011110010000110101011010101101001001000
Pair \(Z_2\) Length of longest common subsequence
8QJC_1,4IWO_1 119 6
8QJC_1,1YYV_1 266 4
4IWO_1,1YYV_1 211 4

Newick tree

 
[
	1YYV_1:13.28,
	[
		8QJC_1:59.5,4IWO_1:59.5
	]:74.78
]

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 \((0.85)(0.8)(\frac{2190 }{\log_{20} 2190}-\frac{660}{\log_{20}660})=372.\)
Status Protein1 Protein2 d d1/2
Query variables 8QJC_1 4IWO_1 481 342
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]