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Parikh vectors
6BUY_1 2ITN_1 6HGY_1 Letter Amino acid
63 8 19 N Asparagine
6 6 8 C Cysteine
42 19 24 G Glycine
93 33 12 L Leucine
41 8 5 F Phenylalanine
58 13 5 T Threonine
31 16 7 R Arginine
43 25 8 I Isoleucine
48 21 9 D Aspartic acid
24 12 4 M Methionine
69 17 16 S Serine
58 23 15 V Valine
25 6 4 W Tryptophan
36 13 13 Y Tyrosine
63 20 18 A Alanine
49 12 9 Q Glutamine
53 26 10 E Glutamic acid
20 8 2 H Histidine
38 23 19 K Lycine
42 18 8 P Proline 

6BUY_1|Chain A|Aminopeptidase N|Sus scrofa (9823)
>2ITN_1|Chain A|EPIDERMAL GROWTH FACTOR RECEPTOR|HOMO SAPIENS (9606)
>6HGY_1|Chain A|Cathepsin K|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 \(w=f(c)\)
6BUY , Knot 338 902 0.85 40 317 812 
QSKPWNRYRLPTTLLPDSYNVTLRPYLTPNADGLYIFKGKSIVRFLCQEPTDVIIIHSKKLNYTTQGHMVVLRGVGDSQVPEIDRTELVELTEYLVVHLKGSLQPGHMYEMESEFQGELADDLAGFYRSEYMEGNVKKVLATTQMQSTDARKSFPCFDEPAMKATFNITLIHPNNLTALSNMPPKGSSTPLAEDPNWSVTEFETTPVMSTYLLAYIVSEFQSVNETAQNGVLIRIWARPNAIAEGHGMYALNVTGPILNFFANHYNTSYPLPKSDQIALPDFNAGAMENWGLVTYRENALLFDPQSSSISNKERVVTVIAHELAHQWFGNLVTLAWWNDLWLNEGFASYVEYLGADHAEPTWNLKDLIVPGDVYRVMAVDALASSHPLTTPAEEVNTPAQISEMFDSISYSKGASVIRMLSNFLTEDLFKEGLASYLHAFAYQNTTYLDLWEHLQKAVDAQTSIRLPDTVRAIMDRWTLQMGFPVITVDTKTGNISQKHFLLDSESNVTRSSAFDYLWIVPISSIKNGVMQDHYWLRDVSQAQNDLFKTASDDWVLLNVNVTGYFQVNYDEDNWRMIQHQLQTNLSVIPVINRAQVIYDSFNLATAHMVPVTLALDNTLFLNGEKEYMPWQAALSSLSYFSLMFDRSEVYGPMKKYLRKQVEPLFQHFETLTKNWTERPENLMDQYSEINAISTACSNGLPQCENLAKTLFDQWMSDPENNPIHPNLRSTIYCNAIAQGGQDQWDFAWGQLQQAQLVNEADKLRSALACSNEVWLLNRYLGYTLNPDLIRKQDATSTINSIASNVIGQPLAWDFVQSNWKKLFQDYGGGSFSFSNLIQGVTRRFSSEFELQQLEQFKKNNMDVGFGSGTRALEQALEKTKANIKWVKENKEVVLNWFIEHSS
2ITN , Knot 145 327 0.85 40 211 321 
GEAPNQALLRILKETEFKKIKVLSSGAFGTVYKGLWIPEGEKVKIPVAIKELREATSPKANKEILDEAYVMASVDNPHVCRLLGICLTSTVQLITQLMPFGCLLDYVREHKDNIGSQYLLNWCVQIAKGMNYLEDRRLVHRDLAARNVLVKTPQHVKITDFGLAKLLGAEEKEYHAEGGKVPIKWMALESILHRIYTHQSDVWSYGVTVWELMTFGSKPYDGIPASEISSILEKGERLPQPPICTIDVYMIMVKCWMIDADSRPKFRELIIEFSKMARDPQRYLVIQGDERMHLPSPTDSNFYRALMDEEDMDDVVDADEYLIPQQG
6HGY , Knot 99 215 0.82 40 148 209 
APDSVDYRKKGYVTPVKNQGQCGSCWAFSSVGALEGQLKKKTGKLLNLSPQNLVDCVSENDGCGGGYMTNAFQYVQKNRGIDSEDAYPYVGQEESCMYNPTGKAAKCRGYREIPEGNEKALKRAVARVGPVSVAIDASLTSFQFYSKGVYYDESCNSDNLNHAVLAVGYGIQKGNKHWIIKNSWGENWGNKGYILMARNKNNACGIANLASFPKM

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 \(c\).

\(|P_{f(6BUY_1)}(2) \setminus P_{f(2ITN_1)}(2)|=141\), \(|P_{f(2ITN_1)}(2) \setminus P_{f(6BUY_1)}(2)|=35\). 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 1:00011000011001110000101010101010110110100110110001001111000010000010111101110001101000011010001110101010110100100010101100111100000101010011100010000100011010011101010101101001011001110100011100101010010001110001110110010010001001111011101011101011011010111101110000000111000011110101111001111000001111010000100000110111001100111011011110011100111001001110010101010011111010011110111000110011001001101001100100001101101100110001100111001011100000010110010011010001011001011100101011111101000010100001110000010000110011111100100111000011001001000110010001111010101010100000010110001000101111100101100010110101111011100011101000011101110010010111000010111000100010111001001000100010011000001011001000111000011001100110010001101010001000111011000101111010010110010010011100001111000110010101100001000100110011101111011000100110001110101001101100010001010010010000101111010011001100001010110000011101110000
Pair \(Z_2\) Length of longest common subsequence
6BUY_1,2ITN_1 176 4
6BUY_1,6HGY_1 211 4
2ITN_1,6HGY_1 163 4

Newick tree

 
[
	6BUY_1:10.82,
	[
		2ITN_1:81.5,6HGY_1:81.5
	]:20.32
]

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 \((0.85)(0.8)(\frac{1229 }{\log_{20} 1229}-\frac{327}{\log_{20}327})=236.\)
Status Protein1 Protein2 d d1/2
Query variables 6BUY_1 2ITN_1 298 201.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]