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Parikh vectors
1YZU_1 7RLO_1 5QLU_1 Letter Amino acid
5 18 3 H Histidine
17 31 17 K Lycine
8 30 9 T Threonine
5 35 14 Q Glutamine
13 46 13 G Glycine
17 74 12 L Leucine
1 32 7 P Proline
2 9 2 W Tryptophan
10 49 17 A Alanine
8 38 3 R Arginine
9 31 7 N Asparagine
5 13 2 C Cysteine
12 34 5 I Isoleucine
6 26 11 F Phenylalanine
10 58 8 S Serine
11 61 21 V Valine
8 44 12 D Aspartic acid
14 60 12 E Glutamic acid
2 15 6 M Methionine
7 17 8 Y Tyrosine 

1YZU_1|Chains A, B|Ras-related protein Rab-21|Homo sapiens (9606)
>7RLO_1|Chains A, B|Translation initiation factor eIF-2B subunit epsilon|Homo sapiens (9606)
>5QLU_1|Chains A, B|Thiol:disulfide interchange protein|Escherichia coli K-12 (83333)
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 \(w=f(c)\)
1YZU , Knot 82 170 0.82 40 136 167 
GSRAYSFKVVLLGEGCVGKTSLVLRYCENKFNDKHITTLQASFLTKKLNIGGKRVNLAIWDTAGQERFHALGPIYYRDSNGAILVYDITDEDSFQKVKNWVKELRKMLGNEICLCIVGNKIDLEKERHVSIQEAESYAESVGAKHYHTSAKQNKGIEELFLDLCKRMIET
7RLO , Knot 274 721 0.83 40 292 640 
MAAPVVAPPGVVVSRANKRSGAGPGGSGGGGARGAEEEPPPPLQAVLVADSFDRRFFPISKDQPRVLLPLANVALIDYTLEFLTATGVQETFVFCCWKAAQIKEHLLKSKWCRPTSLNVVRIITSELYRSLGDVLRDVDAKALVRSDFLLVYGDVISNINITRALEEHRLRRKLEKNVSVMTMIFKESSPSHPTRCHEDNVVVAVDSTTNRVLHFQKTQGLRRFAFPLSLFQGSSDGVEVRYDLLDCHISICSPQVAQLFTDNFDYQTRDDFVRGLLVNEEILGNQIHMHVTAKEYGARVSNLHMYSAVCADVIRRWVYPLTPEANFTDSTTQSCTHSRHNIYRGPEVSLGHGSILEENVLLGSGTVIGSNCFITNSVIGPGCHIGDNVVLDQTYLWQGVRVAAGAQIHQSLLCDNAEVKERVTLKPRSVLTSQVVVGPNITLPEGSVISLHPPDAEEDEDDGEFSDDSGADQEKDKVKMKGYNPAEVGAAGKGYLWKAAGMNMEEEEELQQNLWGLKINMEEESESESEQSMDSEEPDSRGGSPQMDDIKVFQNEVLGTLQRGKEENISCDNLVLEINSLKYAYNISLKEVMQVLSHVVLEFPLQQMDSPLDSSRYCALLLPLLKAWSPVFRNYIKRAADHLEALAAIEDFFLEHEALGISMAKVLMAFYQLEILAEETILSWFSQRDTTDKGQQLRKNQQLQRFIQWLKEAEEESSEDD
5QLU , Knot 86 189 0.79 40 134 185 
AQYEDGKQYTTLEKPVAGAPQVLEFFSFFCPHCYQFEEVLHISDNVKKKLPEGVKMTKYHVNFMGGDLGKDLTQAWAVAMALGVEDKVTVPLFEGVQKTQTIRSASDIRDVFINAGIKGEEYDAAWNSFVVKSLVAQQEKAAADVQLRGVPAMFVNGKYQLNPQGMDTSNMDVFVQQYADTVKYLSEKK

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 \(c\).

\(|P_{f(1YZU_1)}(2) \setminus P_{f(7RLO_1)}(2)|=27\), \(|P_{f(7RLO_1)}(2) \setminus P_{f(1YZU_1)}(2)|=183\). 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 1:10010010111110101100011100000010000100101011000101110010111100110001011111000000111110010000010010011001001110010101110010100000101001000100111000000100001100111010001100
Pair \(Z_2\) Length of longest common subsequence
1YZU_1,7RLO_1 210 4
1YZU_1,5QLU_1 166 3
7RLO_1,5QLU_1 200 4

Newick tree

 
[
	7RLO_1:10.26,
	[
		1YZU_1:83,5QLU_1:83
	]:25.26
]

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 \((0.85)(0.8)(\frac{891 }{\log_{20} 891}-\frac{170}{\log_{20}170})=199.\)
Status Protein1 Protein2 d d1/2
Query variables 1YZU_1 7RLO_1 248 152
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]