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Parikh vectors
1JBV_1 6EQJ_1 1QGT_1 Letter Amino acid
31 12 6 G Glycine
40 31 21 L Leucine
31 22 12 T Threonine
13 8 5 Y Tyrosine
53 16 10 A Alanine
24 9 2 Q Glutamine
13 8 4 H Histidine
6 6 4 W Tryptophan
35 22 11 V Valine
0 3 3 C Cysteine
24 19 7 D Aspartic acid
31 9 6 I Isoleucine
16 13 2 K Lycine
5 6 3 M Methionine
13 16 7 F Phenylalanine
14 10 6 N Asparagine
17 9 10 E Glutamic acid
22 13 12 P Proline
18 23 10 S Serine
22 8 8 R Arginine 

1JBV_1|Chain A|FOLYLPOLYGLUTAMATE SYNTHASE|Lactobacillus casei (1582)
>6EQJ_1|Chain A|Glycogenin-1|Homo sapiens (9606)
>1QGT_1|Chains A, B, C, D|PROTEIN (HBV CAPSID PROTEIN)|Hepatitis B virus (10407)
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 \(w=f(c)\)
1JBV , Knot 177 428 0.83 38 210 398 
MNYTETVAYIHSFPRLAKTGDHRRILTLLHALGNPQQQGRYIHVTGTNGKGSAANAIAHVLEASGLTVGLYTSPFIMRFNERIMIDHEPIPDAALVNAVAFVRAALERLQQQQADFNVTEFEFITALAYWYFRQRQVDVAVIEVGIGGDTDSTNVITPVVSVLTEVALDHQKLLGHTITAIAKHKAGIIKRGIPVVTGNLVPDAAAVVAAKVATTGSQWLRFDRDFSVPKAKLHGWGQRFTYEDQDGRISDLEVPLVGDYQQRNMAIAIQTAKVYAKQTEWPLTPQNIRQGLAASHWPARLEKISDTPLIVIDGAHNPDGINGLITALKQLFSQPITVIAGILADKDYAAMADRLTAAFSTVYLVPVPGTPRALPEAGYEALHEGRLKDSWQEALAASLNDVPDQPIVITGSLYLASAVRQTLLGGKS
6EQJ , Knot 122 263 0.86 40 180 256 
SMTDQAFVTLTTNDAYAKGALVLGSSLKQHRTTRRLVVLATPQVSDSMRKVLETVFDEVIMVDVLDSGDSAHLTLMKRPELGVTLTKLHCWSLTQYSKCVFMDADTLVLANIDDLFDREELSAAPDPGWPDCFNSGVFVYQPSVETYNQLLHLASEQGSFDGGDQGILNTFFSSWATTDIRKHLPFIYNLSSISIFSYLPAFKVFGASAKVVHFLGRVKPWNYTYDPKTKSVKSEAHDPNMTHPEFLILWWNIFTTNVLPLLQ
1QGT , Knot 77 149 0.86 40 113 146 
MDIDPYKEFGATVELLSFLPSDFFPSVRDLLDTASALYREALESPEHCSPHHTALRQAILCWGELMTLATWVGNNLEDPASRDLVVNYVNTNMGLKIRQLLWFHISCLTFGRETVLEYLVSFGVWIRTPPAYRPPNAPILSTLPETTVV

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 \(c\).

\(|P_{f(1JBV_1)}(2) \setminus P_{f(6EQJ_1)}(2)|=99\), \(|P_{f(6EQJ_1)}(2) \setminus P_{f(1JBV_1)}(2)|=69\). 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 1:10000011010011011001000011011011101000100101010010101101110110101101110001111010001110001110111101111101110010000101010010110111010100001011110111110000001101110110011100001110010111000111100111110101110111111101100100110100010110101011100100000010100101111100000011111001010100001110100100111100111010010001111101100101101110110011001101111111000011110010111001011111101011101100110010100010011110100110011110101011011000111100
Pair \(Z_2\) Length of longest common subsequence
1JBV_1,6EQJ_1 168 3
1JBV_1,1QGT_1 195 3
6EQJ_1,1QGT_1 161 3

Newick tree

 
[
	1JBV_1:94.24,
	[
		6EQJ_1:80.5,1QGT_1:80.5
	]:13.74
]

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 \((0.85)(0.8)(\frac{691 }{\log_{20} 691}-\frac{263}{\log_{20}263})=119.\)
Status Protein1 Protein2 d d1/2
Query variables 1JBV_1 6EQJ_1 148 120.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]