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Parikh vectors
6ZPU_1 8TSI_1 5LXS_1 Letter Amino acid
27 16 21 R Arginine
26 9 16 N Asparagine
35 7 20 P Proline
28 20 34 V Valine
19 2 4 W Tryptophan
26 10 19 Y Tyrosine
45 19 36 A Alanine
30 19 27 D Aspartic acid
36 6 16 Q Glutamine
40 14 37 E Glutamic acid
32 8 29 T Threonine
7 2 12 C Cysteine
32 21 36 G Glycine
20 4 13 H Histidine
25 11 27 I Isoleucine
34 15 23 S Serine
63 17 32 L Leucine
30 12 19 K Lycine
17 9 10 M Methionine
25 13 20 F Phenylalanine 

6ZPU_1|Chain A|Angiotensin-converting enzyme|Homo sapiens (9606)
>8TSI_1|Chains A, L[auth B]|ABC transporter ATP-binding protein|Caldimonas thermodepolymerans (215580)
>5LXS_1|Chains A, C|Tubulin alpha-1B chain|Bos taurus (9913)
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 \(w=f(c)\)
6ZPU , Knot 245 597 0.87 40 296 562 
LVTDEAEASKFVEEYDRTSQVVWNEYAGANWNYNTNITTETSKILLQKNMQIAQHTLKYGTQARKFDVNQLQNTTIKRIIKKVQDLERAALPAQELEEYNKILLDMETTYSVATVCHPQGSCLQLEPDLTNVMATSRKYEDLLWAWEGWRDKAGRAILQFYPKYVELINQAARLNGYVDAGDSWRSMYETPSLEQDLERLFQELQPLYLNLHAYVRRALHRHYGAQHINLEGPIPAHLLGNMWAQTWSNIYDLVVPFPSAPSMDTTEAMLKQGWTPRRMFKEADDFFTSLGLLPVPPEFWQKSMLEKPTDGREVVCHASAWDFYNGKDFRIKQCTTVNLEDLVVAHHEMGHIQYFMQYKDLPVALREGANPGFHEAIGDVLALSVSTPKHLHSLNLLSSEGGSDEHDINFLMKMALDKIAFIPFSYLVDQWRWRVFDGSITKENYNQEWWSLRLKYQGLCPPVPRTQGDFDPGAKFHIPSSVPYIRYFVSFIIQFQFHEALCQAAGHTGPLHKCDIYQSKEAGQRLATAMKLGFSRPWPEAMQLITGQPQMSASAMLSYFKPLLDWLRTENELHGEKLGWPQYNWTPNSARSEGPLP
8TSI , Knot 107 234 0.83 40 165 223 
MIELRNLTKWYPTPHGRRYVFRNLNFRFPDDVSIGLIGRNGAGKSTLMRLLGGIEAPNEGEVVTDVSISWPVGLSGGFQGSLTARENVKFVCRIYGTSHEDMLRKVRFVEEFAEIGEHFDLPMKTYSSGMRSRVAFGLSMAFDFDYYLIDEAMAVGDAQFRAKSRAVFDSRVGQANMILVSHNMNDIKEYCDVVVLVDQGQATLYEDVEAGIAAYQGSLKKAAAKPDYKDDDDK
5LXS , Knot 192 451 0.86 40 258 426 
MRECISIHVGQAGVQIGNACWELYCLEHGIQPDGQMPSDKTIGGGDDSFNTFFSETGAGKHVPRAVFVDLEPTVIDEVRTGTYRQLFHPEQLITGKEDAANNYARGHYTIGKEIIDLVLDRIRKLADQCTGLQGFLVFHSFGGGTGSGFTSLLMERLSVDYGKKSKLEFSIYPAPQVSTAVVEPYNSILTTHTTLEHSDCAFMVDNEAIYDICRRNLDIERPTYTNLNRLISQIVSSITASLRFDGALNVDLTEFQTNLVPYPRIHFPLATYAPVISAEKAYHEQLSVAEITNACFEPANQMVKCDPRHGKYMACCLLYRGDVVPKDVNAAIATIKTKRSIQFVDWCPTGFKVGINYQPPTVVPGGDLAKVQRAVCMLSNTTAIAEAWARLDHKFDLMYAKRAFVHWYVGEGMEEGEFSEAREDMAALEKDYEEVGVDSVEGEGEEEGEEY

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 \(c\).

\(|P_{f(6ZPU_1)}(2) \setminus P_{f(8TSI_1)}(2)|=161\), \(|P_{f(8TSI_1)}(2) \setminus P_{f(6ZPU_1)}(2)|=30\). 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 1:110001010011000000001110001110100000100000011100010110001001001001010010000100110010010011111001000001110100000110100101001010101001110000000111110110001101110101001011001101010101100100100010100010011001011010101010011000011001010111110111011100100100111111011010000111001101001100100110011111111011000110010010011001011010010010100000101001111000110100110000111110011011100111011110100100100101100011000001011101110011111100110010101101010000000011010100011011110001010111010110011010011011101010011001110011100001000001100110110111001110110110101010101110010111011000001010011110001010010001111
Pair \(Z_2\) Length of longest common subsequence
6ZPU_1,8TSI_1 191 4
6ZPU_1,5LXS_1 148 5
8TSI_1,5LXS_1 167 4

Newick tree

 
[
	8TSI_1:94.35,
	[
		6ZPU_1:74,5LXS_1:74
	]:20.35
]

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 \((0.85)(0.8)(\frac{831 }{\log_{20} 831}-\frac{234}{\log_{20}234})=164.\)
Status Protein1 Protein2 d d1/2
Query variables 6ZPU_1 8TSI_1 214 145
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]