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Parikh vectors
2MSE_1 2HAL_1 8GCJ_1 Letter Amino acid
17 12 19 A Alanine
14 7 8 R Arginine
18 15 16 K Lycine
8 13 12 T Threonine
4 12 12 N Asparagine
0 1 6 C Cysteine
12 9 8 Q Glutamine
5 8 8 F Phenylalanine
8 23 21 V Valine
10 10 18 D Aspartic acid
28 10 23 E Glutamic acid
0 12 14 I Isoleucine
33 19 29 L Leucine
3 3 1 W Tryptophan
8 22 14 G Glycine
5 4 3 H Histidine
3 8 10 M Methionine
8 7 7 P Proline
11 11 25 S Serine
5 6 7 Y Tyrosine 

2MSE_1|Chains A, B[auth C]|Apolipoprotein A-I|Homo sapiens (9606)
>2HAL_1|Chain A|Hepatitis A Protease 3C|Hepatitis A virus (12092)
>8GCJ_1|Chains A, B, C, D, E, F|Proliferating cell nuclear antigen|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 \(w=f(c)\)
2MSE , Knot 89 200 0.78 36 118 186 
GPLKLLDNWDSVTSTFSKLREQLGPVTQEFWDNLEKETEGLRQEMSKDLEEVKAKVQPYLDDFQKKWQEEMELYRQKVEPLRAELQEGARQKLHELQEKLSPLGEEMRDRARAHVDALRTHLAPYSDELRQRLAARLEALKENGGARLAEYHAKATEHLSTLSEKAKPALEDLRQGLLPVLESFKVSFLSALEEYTKKLN
2HAL , Knot 99 212 0.83 40 152 207 
STLEIAGLVRKNLVQFGVGEKNGSVRWVMNALGVKDDWLLVPSHAYKFEKDYEMMEFYFNRGGTYYSISAGNVVIQSLDVGFQDVVLMKVPTIPKFRDITQHFIKKGDVPRALNRLATLVTTVNGTPMLISEGPLKMEEKATYVHKKNDGTTVDLTVDQAWRGKGEGLPGMCGGALVSSNQSIQNAILGIHVAGGNSILVAKLVTQEMFQNI
8GCJ , Knot 117 261 0.83 40 168 249 
MFEARLVQGSILKKVLEALKDLINEACWDISSSGVNLQSMDSSHVSLVQLTLRSEGFDTYRCDRNLAMGVNLTSMSKILKCAGNEDIITLRAEDNADTLALVFEAPNQEKVSDYEMKLMDLDVEQLGIPEQEYSCVVKMPSGEFARICRDLSHIGDAVVISCAKDGVKFSASGELGNGNIKLSQTSNVDKEEEAVTIEMNEPVQLTFALRYLNFFTKATPLSSTVTLSMSADVPLVVEYKIADMGHLKYYLAPKIEDEEGS

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 \(c\).

\(|P_{f(2MSE_1)}(2) \setminus P_{f(2HAL_1)}(2)|=66\), \(|P_{f(2HAL_1)}(2) \setminus P_{f(2MSE_1)}(2)|=100\). 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 1:11101100100100010010001111000110010000011000100010010101010100100010001010000101101010011000100100010111001000101010110001110000100011101011000111011000101000100100010111001001111110010101101100000010
Pair \(Z_2\) Length of longest common subsequence
2MSE_1,2HAL_1 166 4
2MSE_1,8GCJ_1 140 5
2HAL_1,8GCJ_1 140 4

Newick tree

 
[
	2HAL_1:78.90,
	[
		2MSE_1:70,8GCJ_1:70
	]:8.90
]

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 \((0.85)(0.8)(\frac{412 }{\log_{20} 412}-\frac{200}{\log_{20}200})=62.4\)
Status Protein1 Protein2 d d1/2
Query variables 2MSE_1 2HAL_1 81 75.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]