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Parikh vectors
4DIN_1 4PUN_1 5LRQ_1 Letter Amino acid
21 22 7 I Isoleucine
32 27 14 L Leucine
25 20 6 F Phenylalanine
6 4 3 W Tryptophan
15 18 8 T Threonine
23 38 6 A Alanine
15 32 2 R Arginine
19 24 7 D Aspartic acid
6 17 6 M Methionine
22 33 4 G Glycine
35 14 12 K Lycine
20 18 6 V Valine
8 10 7 H Histidine
13 17 11 P Proline
17 30 4 S Serine
14 8 8 Y Tyrosine
16 8 13 N Asparagine
2 5 2 C Cysteine
15 17 10 Q Glutamine
26 24 7 E Glutamic acid 

4DIN_1|Chain A|cAMP-dependent protein kinase catalytic subunit alpha|Mus musculus (10090)
>4PUN_1|Chain A|Queuine tRNA-ribosyltransferase|Zymomonas mobilis subsp. mobilis (264203)
>5LRQ_1|Chain A|Bromodomain-containing protein 4|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 \(w=f(c)\)
4DIN , Knot 152 350 0.84 40 209 337 
GNAAAAKKGSEQESVKEFLAKAKEDFLKKWETPSQNTAQLDQFDRIKTLGTGSFGRVMLVKHKESGNHYAMKILDKQKVVKLKQIEHTLNEKRILQAVNFPFLVKLEFSFKDNSNLYMVMEYVAGGEMFSHLRRIGRFSEPHARFYAAQIVLTFEYLHSLDLIYRDLKPENLLIDQQGYIQVTDFGFAKRVKGRTWTLCGTPEYLAPEIILSKGYNKAVDWWALGVLIYEMAAGYPPFFADQPIQIYEKIVSGKVRFPSHFSSDLKDLLRNLLQVDLTKRFGNLKNGVNDIKNHKWFATTDWIAIYQRKVEAPFIPKFKGPGDTSNFDDYEEEEIRVSINEKCGKEFTEF
4PUN , Knot 161 386 0.82 40 215 372 
MVEATAQETDRPRFSFSIAAREGKARTGTIEMKRGVIRTPAFMPVGTAATVKALKPETVRATGADIILGNTYHLMLRPGAERIAKLGGLHSFMGWDRPILTDSGGYQVMSLSSLTKQSEEGVTFKSHLDGSRHMLSPERSIEIQHLLGSDIVMAFDECTPYPATPSRAASSMERSMRWAKRSRDAFDSRKEQAENAALFGIQQGSVFENLRQQSADALAEIGFDGYAVGGLAVGEGQDEMFRVLDFSVPMLPDDKPHYLMGVGKPDDIVGAVERGIDMFDCVLPTRSGRNGQAFTWDGPINIRNARFSEDLKPLDSECHCAVCQKWSRAYIHHLIRAGEILGAMLMTEHNIAFYQQLMQKIRDSISEGRFSQFAQDFRARYFARNS
5LRQ , Knot 73 143 0.84 40 118 136 
MHHHHHHSSGVDLGTENLYFQSMNPPPPETSNPNKPKAQTNQLQYLLRVVLKTLWKHQFAWPFQQPVDAVKLNLPDYYKIIKTPMDMGTIKKRLENNYYWNAQECIQDFNTMFTNCYIYNKPGDDIVLMAEALEKLFLQKINE

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 \(c\).

\(|P_{f(4DIN_1)}(2) \setminus P_{f(4PUN_1)}(2)|=73\), \(|P_{f(4PUN_1)}(2) \setminus P_{f(4DIN_1)}(2)|=79\). 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 1:10111100100000100111010001100100100001010010010011010110111100000100011011000011010010001000011011011111010101000001011100111101100100110100101010110111010010010110001010011100010101001111001010010101010011101110010001101111111100111101111100110100011010101100100010011001101010001101001100100001110001111000010111110101110000100000001010100001001001
Pair \(Z_2\) Length of longest common subsequence
4DIN_1,4PUN_1 152 4
4DIN_1,5LRQ_1 179 3
4PUN_1,5LRQ_1 201 4

Newick tree

 
[
	5LRQ_1:10.73,
	[
		4DIN_1:76,4PUN_1:76
	]:24.73
]

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 \((0.85)(0.8)(\frac{736 }{\log_{20} 736}-\frac{350}{\log_{20}350})=105.\)
Status Protein1 Protein2 d d1/2
Query variables 4DIN_1 4PUN_1 134 128
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]