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Parikh vectors
7KID_1 1LJY_1 8YFM_1 Letter Amino acid
31 21 5 F Phenylalanine
14 8 2 W Tryptophan
35 24 5 D Aspartic acid
12 5 1 C Cysteine
26 11 3 Q Glutamine
18 9 3 H Histidine
14 4 1 M Methionine
36 24 7 R Arginine
33 29 6 G Glycine
33 17 4 I Isoleucine
61 34 11 L Leucine
42 32 7 A Alanine
42 26 8 S Serine
31 24 3 T Threonine
23 16 4 Y Tyrosine
41 16 10 V Valine
31 15 3 N Asparagine
47 12 6 E Glutamic acid
31 19 11 K Lycine
44 15 5 P Proline 

7KID_1|Chain A|Protein arginine N-methyltransferase 5|Homo sapiens (9606)
>1LJY_1|Chain A|MGP-40|Capra hircus (9925)
>8YFM_1|Chains A, C[auth D]|RB1-inducible coiled-coil protein 1|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 \(w=f(c)\)
7KID , Knot 252 645 0.84 40 287 604 
MDYKDDDDKAAMAVGGAGGSRVSSGRDLNCVPEIADTLGAVAKQGFDFLCMPVFHPRFKREFIQEPAKNRPGPQTRSDLLLSGRDWNTLIVGKLSPWIRPDSKVEKIRRNSEAAMLQELNFGAYLGLPAFLLPLNQEDNTNLARVLTNHIHTGHHSSMFWMRVPLVAPEDLRDDIIENAPTTHTEEYSGEEKTWMWWHNFRTLCDYSKRIAVALEIGADLPSNHVIDRWLGEPIKAAILPTSIFLTNKKGFPVLSKMHQRLIFRLLKLEVQFIITGTNHHSEKEFCSYLQYLEYLSQNRPPPNAYELFAKGYEDYLQSPLQPLMDNLESQTYEVFEKDPIKYSQYQQAIYKCLLDRVPEEEKDTNVQVLMVLGAGRGPLVNASLRAAKQADRRIKLYAVEKNPNAVVTLENWQFEEWGSQVTVVSSDMREWVAPEKADIIVSELLGSFADNELSPECLDGAQHFLKDDGVSIPGEYTSFLAPISSSKLYNEVRACREKDRDPEAQFEMPYVVRLHNFHQLSAPQPCFTFSHPNRDPMIDNNRYCTLEFPVEVNTVLHGFAGYFETVLYQDITLSIRPETHSPGMFSWFPILFPIKQPITVREGQTICVRFWRCSNSKKVWYEWAVTAPVCSAIHNPTGRSYTIGL
1LJY , Knot 155 361 0.84 40 217 343 
YKLICYYTSWSQYREGDGSCFPDAIDPFLCTHVIYSFANISNNEIDTWEWNDVTLYDTLNTLKNRNPKLKTLLSVGGWNFGPERFSAIASKTQSRRTFIKSVPPFLRTHGFDGLDLAWLYPGRRDKRHLTALVKEMKAEFAREAQAGTERLLLSAAVSAGKIAIDRGYDIAQISRHLDFISLLTYDFHGAWRQTVGHHSPLFRGNSDGSSRFSNADYAVSYMLRLGAPANKLVMGIPTFGRSFTLASSKTDGGAPISGPGIPGRFTKEKGILAYYEICDFLHGATTHRFRDQQVPYATKGNQWVAYDDQESVKNKARYLKNRQLAGAMVWALDLDDFRGTFCGQNLTFPLTSAVKDVLAEV
8YFM , Knot 57 105 0.84 40 87 101 
SRHSEKIAIRDFQVGDLVLIILDERHDNYVLFTVSPTLYFLHSESLPALDLKPGEGASGASRRPWVLGKVMEKEYCQAKKAQNRFKVPLGTKFYRVKAVSWNKKV

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 \(c\).

\(|P_{f(7KID_1)}(2) \setminus P_{f(1LJY_1)}(2)|=111\), \(|P_{f(1LJY_1)}(2) \setminus P_{f(7KID_1)}(2)|=41\). 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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
Pair \(Z_2\) Length of longest common subsequence
7KID_1,1LJY_1 152 4
7KID_1,8YFM_1 232 4
1LJY_1,8YFM_1 182 4

Newick tree

 
[
	8YFM_1:11.09,
	[
		7KID_1:76,1LJY_1:76
	]:36.09
]

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 \((0.85)(0.8)(\frac{1006 }{\log_{20} 1006}-\frac{361}{\log_{20}361})=171.\)
Status Protein1 Protein2 d d1/2
Query variables 7KID_1 1LJY_1 220 169
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]