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Parikh vectors
2JJF_1 5NUK_1 3VAF_1 Letter Amino acid
12 3 5 M Methionine
39 9 9 V Valine
8 5 10 N Asparagine
24 16 6 E Glutamic acid
12 9 12 Q Glutamine
45 20 20 L Leucine
26 8 9 P Proline
17 8 7 S Serine
5 2 0 W Tryptophan
7 4 5 Y Tyrosine
64 17 15 A Alanine
38 3 7 R Arginine
15 8 9 I Isoleucine
32 4 18 G Glycine
10 2 2 H Histidine
9 15 9 K Lycine
24 6 12 F Phenylalanine
25 8 7 T Threonine
32 10 11 D Aspartic acid
5 5 1 C Cysteine 

2JJF_1|Chain A|L-LYSINE EPSILON AMINOTRANSFERASE|MYCOBACTERIUM TUBERCULOSIS (1773)
>5NUK_1|Chain A|Beta-lactoglobulin|Bos taurus (9913)
>3VAF_1|Chains A, B|Splicing factor U2AF 65 kDa subunit|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 \(w=f(c)\)
2JJF , Knot 179 449 0.81 40 215 407 
MAAVVKSVALAGRPTTPDRVHEVLGRSMLVDGLDIVLDLTRSGGSYLVDAITGRRYLDMFTFVASSALGMNPPALVDDREFHAELMQAALNKPSNSDVYSVAMARFVETFARVLGDPALPHLFFVEGGALAVENALKAAFDWKSRHNQAHGIDPALGTQVLHLRGAFHGRSGYTLSLTNTKPTITARFPKFDWPRIDAPYMRPGLDEPAMAALEAEALRQARAAFETRPHDIACFVAEPIQGEGGDRHFRPEFFAAMRELCDEFDALLIFDEVQTGCGLTGTAWAYQQLDVAPDIVAFGKKTQVCGVMAGRRVDEVADNVFAVPSRLASTWGGNLTDMVRARRILEVIEAEGLFERAVQHGKYLRARLDELAADFPAVVLDPRGRGLMCAFSLPTTADRDELIRQLWQRAVIVLPAGADTVRFRPPLTVSTAEIDAAIAAVRSALPVVT
5NUK , Knot 80 162 0.83 40 119 157 
ASVTQTMKGLDIQKVAGTWYSLAMAASDISLLDAQSAPARVYVEELKPTPEGDLEFLLQKWENGECAQKKIIAEKTKIPAVFKIDALNENKVLVLDTDYKKYLLFCFENSAEPEQSLACQCLVRTPEVDDEALEKFDKALKALPMHIRLSFNPTQLEEQCHI
3VAF , Knot 85 174 0.84 38 135 169 
GPLGSARRLYVGNIPFGITEEAMMDFFNAQMRLGGLTQAPGNPVLAVQINQDKNFAFLEFRSVDETTQAMAFDGIIFQGQSLKIRRPHDYQPLPGAHKLFIGGLPNYLNDDQVKELLTSFGPLKAFNLVKDSATGLSKGYAFCEYVDINVTDQAIAGLNGMQLGDKKLLVQRAS

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 \(c\).

\(|P_{f(2JJF_1)}(2) \setminus P_{f(5NUK_1)}(2)|=134\), \(|P_{f(5NUK_1)}(2) \setminus P_{f(2JJF_1)}(2)|=38\). 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 1:11111001111101001001001110011101101110100011001101101000101101110011110111110000101011011100100001001111011001101110111101111011111100110111010000001011011110011010111010010010100001010101101011010110101110011111101011001011100010011011101101011000101011111001000101111100100101101011100010111011111000010111110010011001111100110011101001101001101101011100110010010101001110111111010101110110110010000110011001111111110010101110100101011111100111110
Pair \(Z_2\) Length of longest common subsequence
2JJF_1,5NUK_1 172 4
2JJF_1,3VAF_1 178 4
5NUK_1,3VAF_1 142 3

Newick tree

 
[
	2JJF_1:92.36,
	[
		5NUK_1:71,3VAF_1:71
	]:21.36
]

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 \((0.85)(0.8)(\frac{611 }{\log_{20} 611}-\frac{162}{\log_{20}162})=129.\)
Status Protein1 Protein2 d d1/2
Query variables 2JJF_1 5NUK_1 157 106.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]