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Parikh vectors
5MEN_1 5UJA_1 8ZSA_1 Letter Amino acid
10 77 15 K Lycine
8 56 15 F Phenylalanine
12 52 20 P Proline
14 110 15 S Serine
20 80 21 G Glycine
13 23 16 H Histidine
24 96 22 A Alanine
2 47 8 N Asparagine
18 63 20 D Aspartic acid
4 16 9 C Cysteine
17 58 14 Q Glutamine
22 74 24 E Glutamic acid
4 84 11 I Isoleucine
5 35 9 M Methionine
23 62 15 T Threonine
10 21 5 W Tryptophan
23 61 16 R Arginine
17 157 36 L Leucine
14 41 11 Y Tyrosine
16 122 23 V Valine 

5MEN_1|Chain A|HLA class I histocompatibility antigen, A-2 alpha chain|Homo sapiens (9606)
>5UJA_1|Chain A|bovine multidrug resistance protein 1 (MRP1),Multidrug resistance-associated protein 1|Bos taurus (9913)
>8ZSA_1|Chains A, B, C, D|4VPMT2|Locusta migratoria (7004)
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 \(w=f(c)\)
5MEN , Knot 122 276 0.82 40 182 264 
GSHSMRYFFTSVSRPGRGEPRFIAVGYVDDTQFVRFDSDAASQRMEPRAPWIEQEGPEYWDGETRKVKAHSQTHRVDLGTLRGYYNQSEAGSHTVQRMYGCDVGSDWRFLRGYHQYAYDGKDYIALKEDLRSWTAADMAAQTTKHKWEAAHVAEQLRAYLEGTCVEWLRRYLENGKETLQRTDAPKTHMTHHAVSDHEATLRCWALSFYPAEITLTWQRDGEDQTQDTELVETRPAGDGTFQKWAAVVVPSGQEQRYTCHVQHEGLPKPLTLRWEP
5UJA , Knot 467 1460 0.77 42 350 1145 
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXPNPCPESSASFLSRITFWWITGMMVQGYRQPLESTDLWSLNKEDTSEQVVPVLVKNWKKECAKSRKQPVKIVYSSKDPAKPKGSSKVDVNEEAEALIVKCPQKERDPSLFKVLYKTFGPYFLMSFLFKAVHDLMMFAGPEILKLLINFVNDKKAPEWQGYFYTALLFISACLQTLVLHQYFHICFVSGMRIKTAVIGAVYRKALVITNAARKSSTVGEIVNLMSVDAQRFMDLATYINMIWSAPLQVILALYLLWLNLGPSVLAGVAVMVLMVPLNAVMAMKTKTYQVAHMKSKDNRIKLMNEILNGIKVLKLYAWELAFKDKVLAIRQEELKVLKKSAYLAAVGTFTWVCTPFLVALSTFAVYVTVDENNILDAQKAFVSLALFNILRFPLNILPMVISSIVQASVSLKRLRVFLSHEDLDPDSIQRRPIKDAGATNSITVKNATFTWARNDPPTLHGITFSVPEGSLVAVVGQVGCGKSSLLSALLAEMDKVEGHVTVKGSVAYVPQQAWIQNISLRENILFGRQLQERYYKAVVEACALLPDLEILPSGDRTEIGEKGVNLSGGQKQRVSLARAVYCDSDVYLLDDPLSAVDAHVGKHIFENVIGPKGLLKNKTRLLVTHAISYLPQMDVIIVMSGGKISEMGSYQELLARDGAFAEFLRTYASAEQEQGQPEDGLAGVGGPGKEVKQMENGMLVTDTAGKQMQRQLSSSSSYSRDVSQHHTSTAELRKPGPTEETWKLVEADKAQTGQVKLSVYWDYMKAIGLFISFLSIFLFLCNHVASLVSNYWLSLWTDDPIVNGTQEHTQVRLSVYGALGISQGITVFGYSMAVSIGGIFASRRLHLDLLHNVLRSPISFFERTPSGNLVNRFSKELDTVDSMIPQVIKMFMGSLFNVIGACIIILLATPMAAVIIPPLGLIYFFVQRFYVASSRQLKRLESVSRSPVYSHFNETLLGVSVIRAFEEQERFIRQSDLKVDENQKAYYPSIVANRWLAVRLECVGNCIVLFASLFAVISRHSLSAGLVGLSVSYSLQVTTYLNWLVRMSSEMETNIVAVERLKEYSETEKEAPWQIQDMAPPKDWPQVGRVEFRDYGLRYREDLDLVLKHINVTIDGGEKVGIVGRTGAGKSSLTLGLFRIKESAEGEIIIDDINIAKIGLHDLRFKITIIPQDPVLFSGSLRMNLDPFSQYSDEEVWTSLELAHLKGFVSALPDKLNHECAEGGENLSVGQRQLVCLARALLRKTKILVLDEATAAVDLETDDLIQSTIRTQFDDCTVLTIAHRLNTIMDYTRVIVLDKGEIQEWGSPSDLLQQRGLFYSMAKDSGLVSNSLEVLFQ
8ZSA , Knot 139 325 0.82 40 203 308 
MGSHHHHHHHHGSDYDIPTTENLYFQGSMEKQDTHVNTEKQEAYRGIDKPHLYTLGCCSLTAMASAQLDALWSQLSWPQTPLPVLDIGCGPGDIARKVLAPRLPAGTRLVACDISPEMVDFCRQHNALPGTITYELLDVVEPRLEDTVVWRNGPFGKVFCMLMLHLVPDNRLAVKNIHKLLVPGGEAVFSVIANMALCDAYEEMAKEPRWALYMQDVEKFVSPYQHSEDPVGEFRLLLESEGFQVLRCDLGPQKVFFPTAETHREFLKSVCAFLYRIPEEQKDDFFDDWLQRLEKLKGTSQGVGRNGEAEYYIHLSVITALARKF

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 \(c\).

\(|P_{f(5MEN_1)}(2) \setminus P_{f(5UJA_1)}(2)|=18\), \(|P_{f(5UJA_1)}(2) \setminus P_{f(5MEN_1)}(2)|=186\). 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 1:100010011001001101010111110100001101000110001010111100011001010000101000000101101010000001100010010100110010110100001001000111000100101101110000001011011001010101001011000100100010000110001000110000101001110101101010100010000000011000111010100111111101000000001000111011010101
Pair \(Z_2\) Length of longest common subsequence
5MEN_1,5UJA_1 204 4
5MEN_1,8ZSA_1 169 3
5UJA_1,8ZSA_1 181 4

Newick tree

 
[
	5UJA_1:10.08,
	[
		5MEN_1:84.5,8ZSA_1:84.5
	]:15.58
]

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 \((0.85)(0.8)(\frac{1736 }{\log_{20} 1736}-\frac{276}{\log_{20}276})=374.\)
Status Protein1 Protein2 d d1/2
Query variables 5MEN_1 5UJA_1 439 263.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]