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Parikh vectors
1EVZ_1 2NLP_1 6ETG_1 Letter Amino acid
23 1 25 E Glutamic acid
26 4 23 K Lycine
19 2 19 T Threonine
31 1 13 V Valine
37 2 25 A Alanine
3 1 12 H Histidine
8 0 10 M Methionine
16 1 18 F Phenylalanine
13 1 18 P Proline
26 3 15 S Serine
5 3 18 Y Tyrosine
12 1 13 D Aspartic acid
9 6 7 C Cysteine
32 4 25 G Glycine
45 1 22 L Leucine
17 1 18 R Arginine
13 1 19 N Asparagine
10 1 14 Q Glutamine
19 2 20 I Isoleucine
2 0 8 W Tryptophan 

1EVZ_1|Chain A|GLYCEROL-3-PHOSPHATE DEHYDROGENASE|Leishmania mexicana (5665)
>2NLP_1|Chains A, B, C, D|Beta-defensin 1|Homo sapiens (9606)
>6ETG_1|Chain A|Lysine-specific demethylase 4D|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 \(w=f(c)\)
1EVZ , Knot 153 366 0.82 40 201 344 
MSTKQHSAKDELLYLNKAVVFGSGAFGTALAMVLSKKCREVCVWHMNEEEVRLVNEKRENVLFLKGVQLASNITFTSDVEKAYNGAEIILFVIPTQFLRGFFEKSGGNLIAYAKEKQVPVLVCTKGIERSTLKFPAEIIGEFLPSPLLSVLAGPSFAIEVATGVFTCVSIASADINVARRLQRIMSTGDRSFVCWATTDTVGCEVASAVKNVLAIGSGVANGLGMGLNARAALIMRGLLEIRDLTAALGGDGSAVFGLAGLGDLQLTCSSELSRNFTVGKKLGKGLPIEEIQRTSKAVAEGVATADPLMRLAKQLKVKMPLCHQIYEIVYKKKNPRDALADLLSCGLQDEGLPPLFKRSASTPSKL
2NLP , Knot 25 36 0.83 36 35 34 
DHYNCVSSGGQCLYSACPIFTKIEGTCYRGKAKCCK
6ETG , Knot 153 342 0.87 40 223 330 
METMKSKANCAQNPNCNIMIFHPTKEEFNDFDKYIAYMESQGAHRAGLAKIIPPKEWKARETYDNISEILIATPLQQVASGRAGVFTQYHKKKKAMTVGEYRHLANSKKYQTPPHQNFEDLERKYWKNRIYNSPIYGADISGSLFDENTKQWNLGHLGTIQDLLEKECGVVIEGVNTPYLYFGMWKTTFAWHTEDMDLYSINYLHLGEPKTWYVVPPEHGQRLERLARELFPGSSRGCGAFLRHKVALISPTVLKENGIPFNRITQEAGEFMVTFPYGYHAGFNHGFNCAEAINFATPRWIDYGKMASQCSCGEARVTFSMDAFVRILQPERYDLWKRGQDR

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 \(c\).

\(|P_{f(1EVZ_1)}(2) \setminus P_{f(2NLP_1)}(2)|=181\), \(|P_{f(2NLP_1)}(2) \setminus P_{f(1EVZ_1)}(2)|=15\). 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 1:100000010001101001111101111011111100000010110100001011000000111101101100101000100100110111111100110111000110111010000111110001100001011101110111011101111101110110111001011010101100100110010001101100001100110110011111011101111110101111101110100101111101011111111101010000010001011001101111001000001110111010111011001010111000100110000010011101100110001111110001001001
Pair \(Z_2\) Length of longest common subsequence
1EVZ_1,2NLP_1 196 3
1EVZ_1,6ETG_1 182 4
2NLP_1,6ETG_1 216 3

Newick tree

 
[
	2NLP_1:10.85,
	[
		1EVZ_1:91,6ETG_1:91
	]:15.85
]

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 \((0.85)(0.8)(\frac{402 }{\log_{20} 402}-\frac{36}{\log_{20}36})=116.\)
Status Protein1 Protein2 d d1/2
Query variables 1EVZ_1 2NLP_1 142 78
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]