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Parikh vectors
6CGS_1 3EAO_1 6WLF_1 Letter Amino acid
0 14 6 C Cysteine
2 9 9 H Histidine
12 31 22 I Isoleucine
11 39 23 K Lycine
4 8 15 M Methionine
6 18 21 R Arginine
10 13 17 Q Glutamine
13 37 23 E Glutamic acid
10 17 14 Y Tyrosine
19 39 38 V Valine
8 20 33 N Asparagine
17 52 19 G Glycine
15 42 38 L Leucine
6 18 22 F Phenylalanine
15 23 31 S Serine
16 34 25 T Threonine
2 6 14 W Tryptophan
18 25 28 D Aspartic acid
11 20 16 P Proline
12 33 32 A Alanine 

6CGS_1|Chains A, B|Cadherin-7|Mus musculus (10090)
>3EAO_1|Chains A, B, C, D, E, F|Thioredoxin reductase 1, cytoplasmic|Rattus norvegicus (10116)
>6WLF_1|Chains A, B|Phosphoethanolamine N-methyltransferase 1|Bursaphelenchus xylophilus (6326)
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 \(w=f(c)\)
6CGS , Knot 96 207 0.82 38 150 197 
SWVWNQFFVLEEYMGSDPLYVGKLHSDVDKGDGSIKYILSGEGASSIFIIDENTGDIHATKRLDREEQAYYTLRAQALDRLTNKPVEPESEFVIKIQDINDNEPKFLDGPYTAGVPEMSPVGTSVVQVTATDADDPTYGNSARVVYSILQGQPYFSVEPKTGVIKTALPNMDREAKDQYLLVIQAKDMVGQNGGLSGTTSVTVTLTD
3EAO , Knot 203 499 0.84 42 258 470 
MNDSKDAPKSYDFDLIIIGGGSGGLAAAKEAAKFDKKVMVLDFVTPTPLGTRWGLGGTCVNVGCIPKKLMHQAALLGQALKDSRNYGWKLEDTVKHDWEKMTESVQNHIGSLNWGYRVALREKKVVYENAYGKFIGPHKIMATNNKGKEKVYSAERFLIATGERPRYLGIPGDKEYCISSDDLFSLPYCPGKTLVVGASYVALECAGFLAGIGLDVTVMVRSILLRGFDQDMANKIGEHMEEHGIKFIRQFVPTKIEQIEAGTPGRLKVTAKSTNSEETIEDEFNTVLLAVGRDSCTRTIGLETVGVKINEKTGKIPVTDEEQTNVPYIYAIGDILEGKLELTPVAIQAGRLLAQRLYGGSTVKCDYDNVPTTVFTPLEYGCCGLSEEKAVEKFGEENIEVYHSFFWPLEWTVPSRDNNKCYAKVICNLKDNERVVGFHVLGPNAGEVTQGFAAALKCGLTKQQLDSTIGIHPVCAEIFTTLSVTKRSGGDILQSGCUG
6WLF , Knot 190 446 0.86 40 244 431 
TYFSFWNKFSSNADINTMMLNNNADQLEPFDRADIIQDLPDFTGMNVVDVGAGIGRFTTTFAQRAKHVVSSDFIDSFIEKNKERNAAFENITYQVSDALGLQVDPQSTDLVFTNWLLMYLNEQECVRFLMKTMEWLKPGGYLHVRESCTEPSTGKSKTGSMHSDKKANPTHYRYSSVYLQLLKELRWTDARGMQWRFNVLWAKSVPTYVERVLNWRQVHWLCVKVPAQDGGISNEKALISAMGRWKIIQDHTEYLLNGKTSWATDVINNVISKVDFANNNLPLYVYSPRVVSPFVFVDSHNIAAMTGLSVWAVETNPLFYRHMLNRCVEAKDRRVQYGWNADLQSSINDWGLKKARFGGVIATELFMANSDTSLMALKEILDDNANIFTIEPALSIEDFKENVVVPENMEMIAITDVSAYVVCREQKQAFDSPNQPCWLLVHARLL

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 \(c\).

\(|P_{f(6CGS_1)}(2) \setminus P_{f(3EAO_1)}(2)|=36\), \(|P_{f(3EAO_1)}(2) \setminus P_{f(6CGS_1)}(2)|=144\). 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 1:011100111100011001101101000100101010011010110011110000101010001000001000101011001000110100011101001000010110110011110101110011010100100100100101100110101010101001110011101000100001111010011100111010001010100
Pair \(Z_2\) Length of longest common subsequence
6CGS_1,3EAO_1 180 3
6CGS_1,6WLF_1 176 4
3EAO_1,6WLF_1 168 4

Newick tree

 
[
	6CGS_1:90.61,
	[
		6WLF_1:84,3EAO_1:84
	]:6.61
]

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 \((0.85)(0.8)(\frac{706 }{\log_{20} 706}-\frac{207}{\log_{20}207})=140.\)
Status Protein1 Protein2 d d1/2
Query variables 6CGS_1 3EAO_1 178 124
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]