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Parikh vectors
6BRA_1 8OXG_1 1FXX_1 Letter Amino acid
1 16 20 H Histidine
6 20 28 P Proline
8 26 23 T Threonine
5 11 20 Q Glutamine
4 28 30 E Glutamic acid
15 25 21 I Isoleucine
10 26 47 L Leucine
13 30 19 G Glycine
2 12 24 F Phenylalanine
2 4 9 W Tryptophan
6 24 29 V Valine
5 23 49 A Alanine
4 18 31 R Arginine
3 25 38 D Aspartic acid
0 15 5 C Cysteine
1 16 17 Y Tyrosine
4 17 27 N Asparagine
7 25 21 K Lycine
2 9 11 M Methionine
1 14 13 S Serine 

6BRA_1|Chains A, B|Protease|Human immunodeficiency virus 1 (11676)
>8OXG_1|Chains A, B|Methionine aminopeptidase 2|Homo sapiens (9606)
>1FXX_1|Chain A|EXONUCLEASE I|Escherichia coli (562)
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 \(w=f(c)\)
6BRA , Knot 53 99 0.82 38 77 95 
PQITLWKRPLVTIKIGGQLKEALLNTGADDTVIEEMSLPGRWKPKMIGGIGGFIKVRQYDQIIIEIAGHKAIGTVLVGPTPVNIIGRNLLTQIGATLNF
8OXG , Knot 166 384 0.85 40 230 365 
MHHHHHHENLYFQGPKVQTDPPSVPICDLYPNGVFPKGQECEYPPTQDGRTAAWRTTSEEKKALDQASEEIWNDFREAAEAHRQVRKYVMSWIKPGMTMIEICEKLEDCSRKLIKENGLNAGLAFPTGCSLNNCAAHYTPNAGDTTVLQYDDICKIDFGTHISGRIIDCAFTVTFNPKYDTLLKAVKDATNTGIKCAGIDVRLCDVGEAIQEVMESYEVEIDGKTYQVKPIRNLNGHSIGQYRIHAGKTVPIVKGGEATRMEEGEVYAIETFGSTGKGVVHDDMECSHYMKNFDVGHVPIRLPRTKHLLNVINENFGTLAFCRRWLDRLGESKYLMALKNLCDLGIVDPYPPLCDIKGSYTAQFEHTILLRPTCKEVVSRGDDY
1FXX , Knot 200 482 0.85 40 242 452 
MMNDGKQQSTFLFHDYETFGTHPALDRPAQFAAIRTDSEFNVIGEPEVFYCKPADDYLPQPGAVLITGITPQEARAKGENEAAFAARIHSLFTVPKTCILGYNNVRFDDEVTRNIFYRNFYDPYAWSWQHDNSRWDLLDVMRACYALRPEGINWPENDDGLPSFRLEHLTKANGIEHSNAHDAMADVYATIAMAKLVKTRQPRLFDYLFTHRNKHKLMALIDVPQMKPLVHVSGMFGAWRGNTSWVAPLAWHPENRNAVIMVDLAGDISPLLELDSDTLRERLYTAKTDLGDNAAVPVKLVHINKCPVLAQANTLRPEDADRLGINRQHCLDNLKILRENPQVREKVVAIFAEAEPFTPSDNVDAQLYNGFFSDADRAAMKIVLETEPRNLPALDITFVDKRIEKLLFNYRARNFPGTLDYAEQQRWLEHRRQVFTPEFLQGYADELQMLVQQYADDKEKVALLKALWQYADEIVEHHHHHH

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 \(c\).

\(|P_{f(6BRA_1)}(2) \setminus P_{f(8OXG_1)}(2)|=20\), \(|P_{f(8OXG_1)}(2) \setminus P_{f(6BRA_1)}(2)|=173\). 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 1:101011001110101110100111001100011001011101010111111111010000011101110011101111101101110011001110101
Pair \(Z_2\) Length of longest common subsequence
6BRA_1,8OXG_1 193 3
6BRA_1,1FXX_1 207 3
8OXG_1,1FXX_1 156 6

Newick tree

 
[
	6BRA_1:10.40,
	[
		8OXG_1:78,1FXX_1:78
	]:28.40
]

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 \((0.85)(0.8)(\frac{483 }{\log_{20} 483}-\frac{99}{\log_{20}99})=115.\)
Status Protein1 Protein2 d d1/2
Query variables 6BRA_1 8OXG_1 151 93.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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