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Parikh vectors
3GJY_1 2BJA_1 8IBJ_1 Letter Amino acid
20 22 20 T Threonine
4 8 3 W Tryptophan
9 20 15 Y Tyrosine
19 15 14 D Aspartic acid
3 1 4 C Cysteine
21 56 6 E Glutamic acid
8 26 7 K Lycine
17 29 17 P Proline
19 54 18 V Valine
36 62 26 A Alanine
10 14 13 N Asparagine
24 47 25 G Glycine
10 5 4 H Histidine
13 27 11 F Phenylalanine
22 37 19 R Arginine
12 10 5 Q Glutamine
17 22 13 I Isoleucine
18 39 13 L Leucine
9 6 6 M Methionine
26 16 31 S Serine 

3GJY_1|Chain A|Spermidine synthase|Corynebacterium glutamicum ATCC 13032 (196627)
>2BJA_1|Chains A, B|1-PYRROLINE-5-CARBOXYLATE DEHYDROGENASE|THERMUS THERMOPHILUS (300852)
>8IBJ_1|Chain A|PET hydrolase|Caldimonas taiwanensis (307483)
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 \(w=f(c)\)
3GJY , Knot 139 317 0.84 40 204 303 
SNAMARKKNTSDQSRSQAANTPIAGTYEGEYSVIELEADSYTTDGWLISINGVPSSHIVLGQPQALEFEYMRWIATGARAFIDAHQDASKLRITHLGGGACTMARYFADVYPQSRNTVVELDAELARLSREWFDIPRAPRVKIRVDDARMVAESFTPASRDVIIRDVFAGAITPQNFTTVEFFEHCHRGLAPGGLYVANCGDHSDLRGAKSELAGMMEVFEHVAVIADPPMLKGRRYGNIILMGSDTEFFSSNSTEASAITRELLGGGVPAQYKDESWVRKFASGAQARHDGVSTLQMPSDTPQHPAETPEHSNTQP
2BJA , Knot 201 516 0.81 40 220 465 
MTVEPFRNEPIETFQTEEARRAMREALRRVREEFGRHYPLYIGGEWVDTKERMVSLNPSAPSEVVGTTAKAGKAEAEAALEAAWKAFKTWKDWPQEDRSRLLLKAAALMRRRKRELEATLVYEVGKNWVEASADVAEAIDFIEYYARAALRYRYPAVEVVPYPGEDNESFYVPLGAGVVIAPWNFPVAIFTGMIVGPVAVGNTVIAKPAEDAVVVGAKVFEIFHEAGFPPGVVNFLPGVGEEVGAYLVEHPRIRFINFTGSLEVGLKIYEAAGRLAPGQTWFKRAYVETGGKDAIIVDETADFDLAAEGVVVSAYGFQGQKCSAASRLILTQGAYEPVLERVLKRAERLSVGPAEENPDLGPVVSAEQERKVLSYIEIGKNEGQLVLGGKRLEGEGYFIAPTVFTEVPPKARIAQEEIFGPVLSVIRVKDFAEALEVANDTPYGLTGGVYSRKREHLEWARREFHVGNLYFNRKITGALVGVQPFGGFKLSGTNAKTGALDYLRLFLEMKAVAERF
8IBJ , Knot 117 270 0.80 40 169 254 
EFNPYQKGPDPTASALERNGPFAIRSTSVSRTSVSGFGGGRLYYPTASGTYGAIAVSPGFTGTSSTMTFWGERLASHGFVVLVIDTITLYDQPDSRARQLKAALDYLATQNGRSSSPIYRKVDTSRRAVAGHAMGGGGSLLAARDNPSYKAAIPMAPWATSSTAFRTVSVPTMIFGCQDDSIAPVFSSAIPIYNAIPNSTRKNYVEIRNDDHLCVMNGGGHDATLGKLGISWMKRFVDNDTRYSPFVCGAEYNRVVSSYEVSRSYNNCPY

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 \(c\).

\(|P_{f(3GJY_1)}(2) \setminus P_{f(2BJA_1)}(2)|=71\), \(|P_{f(2BJA_1)}(2) \setminus P_{f(3GJY_1)}(2)|=87\). 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 1:00111000000000000110011110001000110101000000111101011100011110101101001011101101110100010010100111110011001101010000011010101101000110110110101010010111001011000111001111110100100101100000111111101100100001011000111110110011111011110100010111110000110000001011000111111110000001100110110100011001011000100110010000001
Pair \(Z_2\) Length of longest common subsequence
3GJY_1,2BJA_1 158 4
3GJY_1,8IBJ_1 163 3
2BJA_1,8IBJ_1 163 5

Newick tree

 
[
	8IBJ_1:82.31,
	[
		3GJY_1:79,2BJA_1:79
	]:3.31
]

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 \((0.85)(0.8)(\frac{833 }{\log_{20} 833}-\frac{317}{\log_{20}317})=140.\)
Status Protein1 Protein2 d d1/2
Query variables 3GJY_1 2BJA_1 173 141.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]