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Parikh vectors
5BMV_1 7CRZ_1 8WUL_1 Letter Amino acid
21 19 6 R Arginine
13 14 4 H Histidine
23 35 28 S Serine
19 9 5 Y Tyrosine
27 11 13 D Aspartic acid
16 19 14 Q Glutamine
37 27 8 E Glutamic acid
36 52 10 G Glycine
27 43 9 I Isoleucine
20 21 10 P Proline
29 33 14 T Threonine
34 41 8 V Valine
16 17 9 N Asparagine
12 8 5 C Cysteine
20 40 9 F Phenylalanine
4 5 4 W Tryptophan
36 38 10 A Alanine
32 54 14 L Leucine
19 16 11 K Lycine
10 16 7 M Methionine 

5BMV_1|Chains A, C|Tubulin alpha-1B chain|Bos taurus (9913)
>7CRZ_1|Chain A|Solute carrier family 2, facilitated glucose transporter member 3|Homo sapiens (9606)
>8WUL_1|Chains A, C, E, G|TCR alpha chain|Mus musculus (10090)
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 \(w=f(c)\)
5BMV , Knot 192 451 0.86 40 258 426 
MRECISIHVGQAGVQIGNACWELYCLEHGIQPDGQMPSDKTIGGGDDSFNTFFSETGAGKHVPRAVFVDLEPTVIDEVRTGTYRQLFHPEQLITGKEDAANNYARGHYTIGKEIIDLVLDRIRKLADQCTGLQGFLVFHSFGGGTGSGFTSLLMERLSVDYGKKSKLEFSIYPAPQVSTAVVEPYNSILTTHTTLEHSDCAFMVDNEAIYDICRRNLDIERPTYTNLNRLISQIVSSITASLRFDGALNVDLTEFQTNLVPYPRIHFPLATYAPVISAEKAYHEQLSVAEITNACFEPANQMVKCDPRHGKYMACCLLYRGDVVPKDVNAAIATIKTKRSIQFVDWCPTGFKVGINYQPPTVVPGGDLAKVQRAVCMLSNTTAIAEAWARLDHKFDLMYAKRAFVHWYVGEGMEEGEFSEAREDMAALEKDYEEVGVDSVEGEGEEEGEEY
7CRZ , Knot 207 518 0.83 40 240 476 
MHHHHHHHHHHSGDEVDAGSGHMGTQKVTPALIFAITVATIGSFQFGYNTGVINAPEKIIKEFITKTLTDKGNAPPSEVLLTSLWSLSVAIFSVGGMIGSFSVGLFVNRFGRRNSMLIVNLLAVTGGCFMGLCKVAKSVEMLILGRLVIGLFCGLCTGFVPMYIGEISPTALRGAFGTLNQLGIVVGILVAQIFGLEFILGSEELWPLLLGFTILPAILQSAALPFCPESPRFLLINRKEEENAKQILQRLWGTQDVSQDIQEMKDESARMSQEKQVTVLELFRVSSYRQPIIISIVLQLSQQLSGINAVFYYSTGIFKDAGVQEPIYATIGAGVVNTIFTVVSLFLVERAGRRTLHMIGLGGMAFCSTLMTVSLLLKDNYNGMSFVCIGAILVFVAFFEIGPGPIPWFIVAELFSQGPRPAAMAVAGCSNWTSNFLVGLLFPSAAHYLGAYVFIIFTGFLITFLAFTFFKVPETRGRTFEDITRAFEGQAHGADRSGKDGVMEMNSIEPAKETTTNV
8WUL , Knot 93 198 0.82 40 146 191 
QQKVQQSPESLIVPEGGMASLNCTSSDRNVDYFWWYRQHSGKSPKMLMSIFSNGEKEEGRFTVHLNKASLHTSLHIRDSQPSDSALYLCAARSSGSWQLIFGSGTQLTVMPDIQNPEPAVYQLKDPRSQDSTLCLFTDFDSQINVPKTMESGTFITDKCVLDMKAMDSKSNGAIAWSNQTSFTCQDIFKETNATYPSS

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 \(c\).

\(|P_{f(5BMV_1)}(2) \setminus P_{f(7CRZ_1)}(2)|=88\), \(|P_{f(7CRZ_1)}(2) \setminus P_{f(5BMV_1)}(2)|=70\). 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 1:1000101011011101101010100100110101011000011110001001100011100110111101010110010010000110100110100011000101000110011011100100110000110111110011110101100111001010010000101010111010011101000110000010000011110001100100001010010000100110011001010101011101010010001110101011110011110100100001011010010101100110001001001100110010111001011110100000101101010110111000110111110110100110110000111011101000101101001110101101100101001000111100000011100101010001000
Pair \(Z_2\) Length of longest common subsequence
5BMV_1,7CRZ_1 158 4
5BMV_1,8WUL_1 196 3
7CRZ_1,8WUL_1 190 4

Newick tree

 
[
	8WUL_1:10.68,
	[
		5BMV_1:79,7CRZ_1:79
	]:22.68
]

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 \((0.85)(0.8)(\frac{969 }{\log_{20} 969}-\frac{451}{\log_{20}451})=136.\)
Status Protein1 Protein2 d d1/2
Query variables 5BMV_1 7CRZ_1 172 160.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]