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Parikh vectors
5RWN_1 4HBY_1 6EAF_1 Letter Amino acid
23 9 21 Q Glutamine
29 2 29 G Glycine
31 7 37 I Isoleucine
43 13 56 L Leucine
17 5 28 A Alanine
17 12 43 N Asparagine
21 6 21 D Aspartic acid
6 2 17 C Cysteine
32 12 44 K Lycine
8 3 5 W Tryptophan
37 8 28 E Glutamic acid
17 1 9 H Histidine
39 8 44 T Threonine
16 3 19 R Arginine
7 5 9 M Methionine
20 12 18 P Proline
22 5 45 V Valine
24 5 20 F Phenylalanine
38 2 55 S Serine
16 7 20 Y Tyrosine 

5RWN_1|Chain A|Phosphatidylinositol 3,4,5-trisphosphate 5-phosphatase 1|Homo sapiens (9606)
>4HBY_1|Chain A|Bromodomain-containing protein 4|Homo sapiens (9606)
>6EAF_1|Chains A, B, C|Fusion glycoprotein F0|Human respiratory syncytial virus (11250)
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 \(w=f(c)\)
5RWN , Knot 195 463 0.86 40 252 447 
SMEQPEPDMITIFIGTWNMGNAPPPKKITSWFLSKGQGKTRDDSADYIPHDIYVIGTQEDPLSEKEWLEILKHSLQEITSVTFKTVAIHTLWNIRIVVLAKPEHENRISHICTDNVKTGIANTLGNKGAVGVSFMFNGTSLGFVNSHLTSGSEKKLRRNQNYMNILRFLALGDKKLSPFNITHRFTHLFWFGDLNYRVDLPTWEAETIIQKIKQQQYADLLSHDQLLTERREQKVFLHFEEEEITFAPTYRFERLTRDKYAYTKQKATGMKYNLPSWCDRVLWKSYPLVHVVCQSYGSTSDIMTSDHSPVFATFEAGVTSQFVSKNGPGTVDSQGQIEFLRCYATLKTKSQTKFYLEFHSSCLESFVKSQEGENEEGSEGELVVKFGETLPKLKPIISDPEYLLDQHILISIKSSDSDESYGEGCIALRLEATETQLPIYTPLTHHGELTGHFQGEIKLQTSQ
4HBY , Knot 66 127 0.84 40 110 123 
SMNPPPPETSNPNKPKRQTNQLQYLLRVVLKTLWKHQFAWPFQQPVDAVKLNLPDYYKIIKTPMDMGTIKKRLENNYYWNAQECIQDFNTMFTNCYIYNKPGDDIVLMAEALEKLFLQKINELPTEE
6EAF , Knot 225 568 0.83 40 254 525 
MELLILKANAITTILTAVTFCFASGQNITEEFYQSTCSAVSKGYLSALRTGWYTSVITIELSNIKENKCNGTDAKVKLIKQELDKYKNAVTELQLLMQSTPATNNRARRELPRFMNYTLNNAKKTNVTLSKKRKRRFLGFLLSVGSAIASGVAVCKVLHLEGEVNKIKSALLSTNKAVVSLSNGVSVLTFKVLDLKNYIDKQLLPILNKQSCSISNIETVIEFQQKNNRLLEITREFSVNAGVTTPVSTYMLTNSELLSLINDMPITNDQKKLMSNNVQIVRQQSYSIMCIIKEEVLAYVVQLPLYGVIDTPCWKLHTSPLCTTNTKEGSNICLTRTDRGWYCDNAGSVSFFPQAETCKVQSNRVFCDTMNSLTLPSEVNLCNVDIFNPKYDCKIMTSKTDVSSSVITSLGAIVSCYGKTKCTASNKNRGIIKTFSNGCDYVSNKGVDTVSVGNTLYYVNKQEGKSLYVKGEPIINFYDPLVFPSDEFDASISQVNEKINQSLAFIRKSDELLSAIGGYIPEAPRDGQAYVRKDGEWVLLSTFLGGLVPRGSHHHHHHSAWSHPQFEK

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 \(c\).

\(|P_{f(5RWN_1)}(2) \setminus P_{f(4HBY_1)}(2)|=167\), \(|P_{f(4HBY_1)}(2) \setminus P_{f(5RWN_1)}(2)|=25\). 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 1:0100101011011110101101111001001110010100000010011001011100001100001101100010010010100111001101011111010000010010000100111001100111110111010011110001001000010000001011011111000101101000100111110100010110101001100100000101100001100000001110100001011100010010000010000010110001101000111000111011000010000110000011110101110001100011101000101011000101000000010101000010011000010000100101110110011010111001001100011101000000000101011101010000111001100010101010101010000
Pair \(Z_2\) Length of longest common subsequence
5RWN_1,4HBY_1 192 3
5RWN_1,6EAF_1 140 4
4HBY_1,6EAF_1 212 4

Newick tree

 
[
	4HBY_1:10.55,
	[
		5RWN_1:70,6EAF_1:70
	]:39.55
]

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 \((0.85)(0.8)(\frac{590 }{\log_{20} 590}-\frac{127}{\log_{20}127})=134.\)
Status Protein1 Protein2 d d1/2
Query variables 5RWN_1 4HBY_1 175 108.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]