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Parikh vectors
6OXQ_1 9FXM_1 2PTN_1 Letter Amino acid
13 58 15 I Isoleucine
12 115 14 L Leucine
2 17 4 W Tryptophan
6 50 17 V Valine
4 42 6 D Aspartic acid
5 37 10 Q Glutamine
13 41 25 G Glycine
1 31 3 H Histidine
6 35 8 P Proline
4 47 16 N Asparagine
2 16 12 C Cysteine
7 58 14 K Lycine
2 21 2 M Methionine
2 49 3 F Phenylalanine
3 61 14 A Alanine
4 44 2 R Arginine
4 72 4 E Glutamic acid
0 73 34 S Serine
8 40 10 T Threonine
1 34 10 Y Tyrosine 

6OXQ_1|Chains A, B|Protease NL4-3|Human immunodeficiency virus 1 (11676)
>9FXM_1|Chains A, B, C, D|Transient receptor potential cation channel subfamily c member 4a|Danio rerio (7955)
>2PTN_1|Chain A|TRYPSIN|Bos taurus (9913)
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 \(w=f(c)\)
6OXQ , Knot 52 99 0.80 38 79 94 
PQITLWKRPLVTIKIGGQLKEALLDTGADDTVLEEMNLPGRWKPKMIGGIGGFIKVRQYDQILIEICGHKAIGTVLVGPTPVNIIGRNLLTQIGCTLNF
9FXM , Knot 353 941 0.85 40 332 852 
MGHHHHHHHHGGSENLYFQGGSQLYFRRTDNSSYRDRIPLRIVRAESELSTQEKSYLSAVEKGDYASVKLALEEAEIYFKININCIDPLGRTALLIAIENENLEIIELLLSFNVYVGDALLHAIRKEVVGAVELLLNHKKPSGEKQVPPILLDKQFSDFTPDITPIILAAHTNNYEIIKMLVQKGVSVPQPHEVRCNCVECVSSSDVDSLRHSRSRLNIYKALASPSLIALSSEDPFLTAFQLSWELQELSKVENEFKAEYEELSHQCKHFAKDLLDQTRSSRELELILNFRDDMNLLQDEANNELARLKLAIKYRQKEFVAQPNCQQLLASRWYDEFPGWRRRHWAGKLITCVFIGLMFPLLSLCYLVAPKSRYGLFIRKPFIKFICHTASYLTFLFLLLLASQHIVSNNPDRQGPKPTTVEWMILPWVLGFIWTEIKQMWDGGFQDYIHDWWNLMDFVMNSLYLATISLKIVAYVKYSGCKPRDTWEMWHPTLVAEAVFAIANIFSSLRLISLFTANSHLGPLQISLGRMLLDILKFLFIYCLVLLAFANGLNQLYFYYENSEGMTCKGIRCERQNNAFSTLFETLQSLFWSIFGLISLYVTNVKADHKFTEFVGATMFGTYNVISLVVLLNMLIAMMNNSYQHIADHADIEWKFARTKLWMSYFEEGGTLPPPFNIIPSPKSICYLITWIKVHVFKRRSKRTETFGTLGRRAAENVRLNHQYQEVLRNLVKRYVAAMIRDAKTEEGLTEENFKELKQDISSFRYEVIGMMKGNRKSTRANKSDTSASDVSHPEGSLQYSSALKQNSKLHLYDVTTALQQQNSEEAKASLGCLANGSAVVLTEPILKDKARSDFPKDFTDFGLFPKKQNPNKIYSLAEEATESDPDILDWGKEDKPLAGKVEQDVNESKCLMEEDERVLEEQEMEHIASSHEHLEVLFQ
2PTN , Knot 101 223 0.81 40 140 216 
IVGGYTCGANTVPYQVSLNSGYHFCGGSLINSQWVVSAAHCYKSGIQVRLGEDNINVVEGNEQFISASKSIVHPSYNSNTLNNDIMLIKLKSAASLNSRVASISLPTSCASAGTQCLISGWGNTKSSGTSYPDVLKCLKAPILSDSSCKSAYPGQITSNMFCAGYLEGGKDSCQGDSGGPVVCSGKLQGIVSWGSGCAQKNKPGVYTKVCNYVSWIKQTIASN

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 \(c\).

\(|P_{f(6OXQ_1)}(2) \setminus P_{f(9FXM_1)}(2)|=6\), \(|P_{f(9FXM_1)}(2) \setminus P_{f(6OXQ_1)}(2)|=259\). 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 1:101011001110101110100111001100011001011101010111111111010000011101010011101111101101110011001100101
Pair \(Z_2\) Length of longest common subsequence
6OXQ_1,9FXM_1 265 4
6OXQ_1,2PTN_1 155 3
9FXM_1,2PTN_1 232 3

Newick tree

 
[
	9FXM_1:13.64,
	[
		6OXQ_1:77.5,2PTN_1:77.5
	]:59.14
]

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 \((0.85)(0.8)(\frac{1040 }{\log_{20} 1040}-\frac{99}{\log_{20}99})=261.\)
Status Protein1 Protein2 d d1/2
Query variables 6OXQ_1 9FXM_1 337 184
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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