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Parikh vectors
6NNU_1 5WGT_1 2OGV_1 Letter Amino acid
50 59 27 A Alanine
31 36 16 D Aspartic acid
14 23 15 Q Glutamine
23 41 14 P Proline
18 27 16 N Asparagine
21 34 15 F Phenylalanine
4 17 6 W Tryptophan
31 38 20 V Valine
30 44 16 S Serine
20 36 11 T Threonine
4 16 7 C Cysteine
30 40 19 E Glutamic acid
40 49 22 G Glycine
48 61 33 L Leucine
16 33 22 K Lycine
10 10 8 M Methionine
14 30 13 Y Tyrosine
28 31 12 R Arginine
18 10 8 H Histidine
18 32 17 I Isoleucine 

6NNU_1|Chain A|Phosphoglucomutase|Xanthomonas axonopodis pv. citri (strain 306) (190486)
>5WGT_1|Chain A|Flavin-dependent halogenase|Malbranchea aurantiaca (78605)
>2OGV_1|Chain A|Macrophage colony-stimulating factor 1 receptor precursor|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 \(w=f(c)\)
6NNU , Knot 196 468 0.85 40 242 448 
MGSSHHHHHHSSENLYFQSHMTLPAFKAYDIRGRVPDELNEDLARRIGVALAAQLDQGPVVLGHDVRLASPALQEALSAGLRASGRDVIDIGLCGTEEVYFQTDYLKAAGGVMVTASHNPMDYNGMKLVREQARPISSDTGLFAIRDTVAADTAAPGEPTASEQSRTDKTAYLEHLLSYVDRSTLKPLKLVVNAGNGGAGLIVDLLAPHLPFEFVRVFHEPDGNFPNGIPNPLLPENRDATAKAVKDNGADFGIAWDGDFDRCFFFDHTGRFIEGYYLVGLLAQAILAKQPGGKVVHDPRLTWNTVEQVEEAGGIPVLCKSGHAFIKEKMRSENAVYGGEMSAHHYFREFAYADSGMIPWLLIAELVSQSGRSLADLVEARMQKFPCSGEINFKVADAKASVARVMEHYASLSPELDYTDGISADFGQWRFNLRSSNTEPLLRLNVETRGDAALLETRTQEISNLLRG
5WGT , Knot 265 667 0.86 40 301 628 
MAPTPKYTFTERAAAGNLSDAEILNSNNPTGSELPDESDVVVGGAGIHGLIYALHASKYKPNNLKISVIEKNTRPGYKIGESTLPIFYTWCKLHGISAAYLLRLFGLKDGLCFYFLDRENQGQYTDFCSVGAPGLVLASLQIERPMSELLFTILAQRNGVNVYHGREVDFKSTVVQGGGQGNKIAVSRGKYDSTPKTIDSALFVDATGRFRQFCSKKAPRHRFDGWNCNAFWGYFTAPKDESKIPFDLYEGDATNHLCFPEGWVWVIRLPSWEGSPIANLMDMVTYILECADAGVPGDELPSSEELARMFGLKFQWVTSIGFAVRNDVKYPEDLSAYGTREAEQKFNYFVQKYELLQQFMSNFELIENLYGPGTTWFIRKTLAYQSPVVSGPGWLAIGDACGFTNPLYSPGINVGMSTSTWAAQLSHPIVEIGKSAPADAAESSIRKLLVPYDDYCKSLVPALEQMNRFNYVCYRDTRLGPQVACLWQFFAGIERYLSDVNIETFAHYAIKWVWGAMVPEYQQVAQKCIEHIETVPLDERLPDAMVDELLAFSNRIKSAAVAADDFSLRWDAILRSFDRSLNFVEGKTSRDIYTRQCSGCGAWLQLRPDWKKCHSCGLLGTEPQTAVTFDPPLTAEEEALLYAAWNTAPKYDPSKELKLPTPTRPAA
2OGV , Knot 141 317 0.85 40 209 308 
KPKYQVRWKIIESYEGNSYTFIDPTQLPYNEKWEFPRNNLQFGKTLGAGAFGKVVEATAFGLGKEDAVLKVAVKMLKSTAHADEKEALMSELKIMSHLGQHENIVNLLGACTHGGPVLVITEYCCYGDLLNFLRRKAEADLDKEDGRPLELRDLLHFSSQVAQGMAFLASKNCIHRDVAARNVLLTNGHVAKIGDFGLARDIMNDSNYIVKGNARLPVKWMAPESIFDCVYTVQSDVWSYGILLWEIFSLGLNPYPGILVNSKFYKLVKDGYQMAQPAFAPKNIYSIMQACWALEPTHRPTFQQICSFLQEQAQEDR

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 \(c\).

\(|P_{f(6NNU_1)}(2) \setminus P_{f(5WGT_1)}(2)|=45\), \(|P_{f(5WGT_1)}(2) \setminus P_{f(6NNU_1)}(2)|=104\). 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 1:110000000000001010001011110100101011001000110011111110100111111001011011100110111010100110111010001010000101111111010001100011011000101100001111100011100111101010000000001010011001000010110111011011111110111101110110110010101101110111100001010110001101111101010001110001011010011111101111001110110010101001001001111111000101110001000011011010100010011010011111111101100010011011010100110010101011010101101100010101010000110101101010100000011101010001011110000001001101
Pair \(Z_2\) Length of longest common subsequence
6NNU_1,5WGT_1 149 4
6NNU_1,2OGV_1 175 4
5WGT_1,2OGV_1 160 4

Newick tree

 
[
	2OGV_1:86.72,
	[
		6NNU_1:74.5,5WGT_1:74.5
	]:12.22
]

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 \((0.85)(0.8)(\frac{1135 }{\log_{20} 1135}-\frac{468}{\log_{20}468})=173.\)
Status Protein1 Protein2 d d1/2
Query variables 6NNU_1 5WGT_1 223 186.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]