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Parikh vectors
1SQK_1 9FRD_1 6LBD_1 Letter Amino acid
16 13 9 Y Tyrosine
21 23 6 V Valine
18 16 8 R Arginine
11 12 12 Q Glutamine
30 25 7 I Isoleucine
12 8 6 F Phenylalanine
23 17 11 S Serine
4 6 1 W Tryptophan
12 8 8 N Asparagine
28 21 9 G Glycine
9 8 4 H Histidine
19 23 12 K Lycine
26 33 18 L Leucine
17 13 7 M Methionine
27 13 5 T Threonine
29 20 14 A Alanine
22 21 10 D Aspartic acid
6 6 1 C Cysteine
28 26 18 E Glutamic acid
19 18 4 P Proline 

1SQK_1|Chain A|ACTIN, ALPHA SKELETAL MUSCLE|Oryctolagus cuniculus (9986)
>9FRD_1|Chain A|Epidermal growth factor receptor|Homo sapiens (9606)
>6LBD_1|Chain A|Ferritin|Penaeus japonicus (27405)
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 \(w=f(c)\)
1SQK , Knot 160 377 0.84 40 220 362 
MCDEDETTALVCDNGSGLVKAGFAGDDAPRAVFPSIVGRPRHQGVMVGMGQKDSYVGDEAQSKRGILTLKYPIEHGIITNWDDMEKIWHHTFYNELRVAPEEHPTLLTEAPLNPKANREKMTQIMFETFNVPAMYVAIQAVLSLYASGRTTGIVLDSGDGVTHNVPIYEGYALPHAIMRLDLAGRDLTDYLMKILTERGYSFVTTAEREIVRDIKEKLCYVALDFENEMATAASSSSLEKSYELPDGQVITIGNERFRCPETLFQPSFIGMESAGIHETTYNSIMKCDIDIRKDLYANNVMSGGTTMYPGIADRMQKEITALAPSTMKIKIIAPPERKYSVWIGGSILASLSTFQQMWITKQEYDEAGPSIVHRKCF
9FRD , Knot 146 330 0.85 40 212 324 
GSMGEAPNQALLRILKETEFKKIKVLGSGAFGTVYKGLWIPEGEKVKIPVAIKELREATSPKANKEILDEAYVMASVDNPHVCRLLGICLTSTVQLITQLMPFGCLLDYVREHKDNIGSQYLLNWCVQIAKGMNYLEDRRLVHRDLAARNVLVKTPQHVKITDFGLAKLLGAEEKEYHAEGGKVPIKWMALESILHRIYTHQSDVWSYGVTVWELMTFGSKPYDGIPASEISSILEKGERLPQPPICTIDVYMIMVKCWMIDADSRPKFRELIIEFSKMARDPQRYLVIQGDERMHLPSPTDSNFYRALMDEEDMDDVVDADEYLIPQQG
6LBD , Knot 82 170 0.82 40 134 166 
MASQVRQNYHEDCEASINKQINMELYASYVYLSMAYYFERDDVALPGFAKFFKESSDEEREHAQTFMKYQNKRGGRIVLQQIAAPSMQEWGTGLEALQAALDLEKQVNQSLLELHSTASGNNDPHLTKLLEDEYLEEQVDSIKKIGDMITKLKRAGPRRLGEYMFDKELN

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 \(c\).

\(|P_{f(1SQK_1)}(2) \setminus P_{f(9FRD_1)}(2)|=91\), \(|P_{f(9FRD_1)}(2) \setminus P_{f(1SQK_1)}(2)|=83\). 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 1:10000000111000101110111110011011110111010001111111000001100100001110100110011100100100110001000101110001011001110101000010011100101111011101110101010001111001011000111001011101110101110010001101100010011001000110010001001110100011011000010000011010110110001001001101011110011100000001100010100010100110110010111100100010111100101011111000001111101110100100111000000011101100001
Pair \(Z_2\) Length of longest common subsequence
1SQK_1,9FRD_1 174 4
1SQK_1,6LBD_1 176 4
9FRD_1,6LBD_1 180 4

Newick tree

 
[
	6LBD_1:89.66,
	[
		1SQK_1:87,9FRD_1:87
	]:2.66
]

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 \((0.85)(0.8)(\frac{707 }{\log_{20} 707}-\frac{330}{\log_{20}330})=103.\)
Status Protein1 Protein2 d d1/2
Query variables 1SQK_1 9FRD_1 131 123
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]