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Parikh vectors
7UWQ_1 3LHG_1 2DAA_1 Letter Amino acid
23 34 22 V Valine
49 24 12 A Alanine
6 6 3 C Cysteine
17 17 8 Q Glutamine
32 35 18 G Glycine
35 26 22 T Threonine
8 7 3 W Tryptophan
31 19 12 R Arginine
31 23 14 D Aspartic acid
30 24 25 I Isoleucine
50 29 21 L Leucine
14 15 25 K Lycine
8 8 4 M Methionine
16 20 10 F Phenylalanine
11 15 16 N Asparagine
28 28 27 E Glutamic acid
26 13 10 H Histidine
27 22 12 P Proline
27 31 8 S Serine
23 19 10 Y Tyrosine 

7UWQ_1|Chains A, B, C, D, E, F|AMP nucleosidase|Klebsiella pneumoniae (573)
>3LHG_1|Chain A|Beta-secretase 1|Homo sapiens (9606)
>2DAA_1|Chains A, B|D-AMINO ACID AMINOTRANSFERASE|Bacillus sp. (1409)
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 \(w=f(c)\)
7UWQ , Knot 203 492 0.85 40 246 458 
MAHHHHHHMNIKAASLTPEQALAELEARYEASVTALRKAIGDYIDHNTLPDTEARAEGLFVYPQLSVSWDGADHKALKTRAWGRFTHAGCYTTTITNPKLFRNYLLEQLTLLYQDYGAHISVELSQHEIPYPYVIDGSTLTLDRSMSAGLTRYFPTTELSQIGDETADGLFHPTEFYPLSHFDARRVDFSLARLRHYTGTPAEHFQPYVLFTNYTRYVDEFVSWGCSQILDPDSPYIALSCAGGIWITAETEAPEQAISDLAWKKHQMPAWHLITHDGKGITLINIGVGPANAKTICDHLAVLRPDVWLMIGHCGGLRESQAIGDYVLAHAYLRDDHVLDAVLPPDIPIPSIAEVQRALYDATKQVSGMPGEEVKQRLRTGTVVTTDDRNWELRYSASALRFNLSRAVAIDMESATIAAQGYRFRVPYGTLLCVSDKPLHGEIKLPGQANRFYEGAISEHLQIGIRAIDLLRAEGDHMHSRKLRTFNEPPFR
3LHG , Knot 175 415 0.84 40 232 393 
ETDEEPEEPGRRGSFVEMVDNLRGKSGQGYYVEMTVGSPPQTLNILVDTGSSNFAVGAAPHPFLHRYYQRQLSSTYRDLRKGVYVPYTQGKWEGELGTDLVSIPHGPNVTVRANIAAITESDKFFINGSNWEGILGLAYAEIARPDDSLEPFFDSLVKQTHVPNLFSLQLCGAGFPLNQSEVLASVGGSMIIGGIDHSLYTGSLWYTPIRREWYYEVIIVRVEINGQDLKMDCKEYNYDKSIVDSGTTNLRLPKKVFEAAVKSIKAASSTEKFPDGFWLGEQLVCWQAGTTPWNIFPVISLYLMGEVTNQSFRITILPQQYLRPVEDVATSQDDCYKFAISQSSTGTVMGAVIMEGFYVVFDRARKRIGFAVSACHVHDEFRTAAVEGPFVTLDMEDCGYNIPQTDESTHHHHHH
2DAA , Knot 126 282 0.84 40 171 274 
GYTLWNDQIVKDEEVKIDKEDRGYQFGDGVYEVVKVYNGEMFTVNEHIDRLYASAEKIRITIPYTKDKFHQLLHELVEKNELNTGHIYFQVTRGTSPRAHQFPENTVKPVIIGYTKENPRPLENLEKGVKATFVEDIRWLRCDIKSLNLLGAVLAKQEAHEKGCYEAILHRNNTVTEGSSSNVFGIKDGILYTHPANNMILKGITRDVVIACANEINMPVKEIPFTTHEALKMDELFVTSTTSEITPVIEIDGKLIRDGKVGEWTRKLQKQFETKIPKPLHI

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 \(c\).

\(|P_{f(7UWQ_1)}(2) \setminus P_{f(3LHG_1)}(2)|=85\), \(|P_{f(3LHG_1)}(2) \setminus P_{f(7UWQ_1)}(2)|=71\). 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 1:110000001010110101001110101000101011001110010000110001010111101010101011000110001110100110000010010110001100101100001101010100001101011010010100010111000110001001100010111010010110010100101011010000101100101011100000010011011000110100101110011111101000110011001110000111101100010110110111111010010001111010111111001110000111001110101000011011111011110110100110010001011110010001001011000000101000101101010011110100101110100101101011010001101010111010010011100010111011011010100100001001001110
Pair \(Z_2\) Length of longest common subsequence
7UWQ_1,3LHG_1 156 6
7UWQ_1,2DAA_1 177 4
3LHG_1,2DAA_1 165 4

Newick tree

 
[
	2DAA_1:87.92,
	[
		7UWQ_1:78,3LHG_1:78
	]:9.92
]

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 \((0.85)(0.8)(\frac{907 }{\log_{20} 907}-\frac{415}{\log_{20}415})=131.\)
Status Protein1 Protein2 d d1/2
Query variables 7UWQ_1 3LHG_1 168 154.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]