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Parikh vectors
8YHN_1 6NOQ_1 4HXQ_1 Letter Amino acid
53 50 15 A Alanine
23 40 36 G Glycine
48 48 33 L Leucine
11 18 8 N Asparagine
32 30 19 E Glutamic acid
16 18 23 I Isoleucine
21 18 8 H Histidine
14 30 19 S Serine
5 4 2 W Tryptophan
12 14 9 Y Tyrosine
13 21 9 F Phenylalanine
32 23 21 P Proline
30 28 13 R Arginine
30 31 19 D Aspartic acid
3 4 3 C Cysteine
6 14 5 Q Glutamine
11 16 22 K Lycine
8 10 3 M Methionine
17 20 22 T Threonine
26 31 25 V Valine 

8YHN_1|Chain A|Cytochrome P450|Streptomyces avermitilis (33903)
>6NOQ_1|Chain A|Phosphoglucomutase|Xanthomonas axonopodis pv. citri (strain 306) (190486)
>4HXQ_1|Chains A, B|Arginase-1|Homo sapiens (9606)
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 \(w=f(c)\)
8YHN , Knot 163 411 0.79 40 202 382 
MAAAFDLAFDPWDPAFLADPYPAYADLRAKGRVHYYEPTNQWLVPHHADVSALLRDRRLGRAYQHRYTHEDFGRTAPPAEHEPFHTLNDHGMLDLEPPDHTRIRRLVSKAFTPRTVEQLKPYVAKLAGELVDRLVAAGGGDLLADVAEPLPVAVIAEMLGIPESDRAPLRPWSADICGMYELNPPKDVAAKAVRASVEFSDYLRELIAERRKEPGDDLISGLIAAHDEGDRLTEQEMISTCVLLLNAGHEATVNATVNGWYALFRNPDQLAALRADHSLVPAAVEELMRYDTPLQLFERWVLDEIEIDGTTVPRGAEIAMLFGSANHDPEVFRNPEKLDLTREDNPHISFSAGIHYCIGAPLARIELAASMTALLEKAPTLGLVAEPKRKPNFVIRGLEGLSVAVHHHHHH
6NOQ , Knot 196 468 0.85 40 242 448 
MGSSHHHHHHSSENLYFQSHMTLPAFKAYDIRGRVPDELNEDLARRIGVALAAQLDQGPVVLGHDVRLASPALQEALSAGLRASGRDVIDIGLCGTEEVYFQTDYLKAAGGVMVTASHNPMDYNGMKLVREQARPISSDTGLFAIRDTVAADTAAPGEPTASEQSRTDKTAYLEHLLSYVDRSTLKPLKLVVNAGNGGAGLIVDLLAPHLPFEFVRVFHEPDGNFPNGIPNPLLPENRDATAKAVKDNGADFGIAWDGDFDRCFFFDHTGRFIEGYYLVGLLAQAILAKQPGGKVVHDPRLTWNTVEQVEEAGGIPVLCKSGHAFIKEKMRSENAVYGGEMSAHHYFREFAYADSGMIPWLLIAELVSQSGRSLADLVEARMQKFPCSGEINFKVADAKASVARVMEHYASLSPELDYTDGISADFGQWRFNLRSSNTEPLLRLNVETRGDAALLETRTQEISNLLRG
4HXQ , Knot 136 314 0.83 40 188 304 
SRTIGIIGAPFSKGQPRGGVEEGPTVLRKAGLLEKLKEQECDVKDYGDLPFADIPNDSPFQIVKNPRSVGKASEQLAGKVAEVKKNGRISLVLGGDHSLAIGSISGHARVHPDLGVIWVDAHTDINTPLTTTSGNLHGQPVSFLLKELKGKIPDVPGFSWVTPCISAKDIVYIGLRDVDPGEHYILKTLGIKYFSMTEVDRLGIGKVMEETLSYLLGRKKRPIHLSFDVDGLDPSFTPATGTPVVGGLTYREGLYITEEIYKTGLLSGLDIMEVNPSLGKTPEEVTRTVNTAVAITLACFGLAREGNHKPIDYL

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 \(c\).

\(|P_{f(8YHN_1)}(2) \setminus P_{f(6NOQ_1)}(2)|=48\), \(|P_{f(6NOQ_1)}(2) \setminus P_{f(8YHN_1)}(2)|=88\). 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 1:111110111011011111010110101010101000010001111001010111000011010000000001100111100011001000111010110000100110011010010010101101110110011111110111011011111111011111000011101101010110010110011101101010100010011100000110011011111000100100001100011110110010101010110111001001111010001111110011000011011001110010101001101101111110100010110010010100000101010111000111111010111010111001101111101000101110110110111000000
Pair \(Z_2\) Length of longest common subsequence
8YHN_1,6NOQ_1 136 6
8YHN_1,4HXQ_1 148 4
6NOQ_1,4HXQ_1 140 4

Newick tree

 
[
	4HXQ_1:73.32,
	[
		8YHN_1:68,6NOQ_1:68
	]:5.32
]

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 \((0.85)(0.8)(\frac{879 }{\log_{20} 879}-\frac{411}{\log_{20}411})=125.\)
Status Protein1 Protein2 d d1/2
Query variables 8YHN_1 6NOQ_1 157 144
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]