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Parikh vectors
3SQN_1 5ZXW_1 3TSJ_1 Letter Amino acid
16 19 24 D Aspartic acid
24 8 26 Y Tyrosine
20 7 22 R Arginine
3 1 5 C Cysteine
37 11 17 Q Glutamine
77 27 35 L Leucine
30 13 41 A Alanine
10 24 40 G Glycine
10 12 9 H Histidine
30 9 30 I Isoleucine
12 18 39 P Proline
41 19 33 S Serine
15 17 40 V Valine
21 11 21 N Asparagine
40 13 19 E Glutamic acid
19 24 33 K Lycine
6 2 10 M Methionine
32 12 22 F Phenylalanine
37 12 22 T Threonine
5 7 12 W Tryptophan 

3SQN_1|Chains A, B|Conserved domain protein|Enterococcus faecalis (1351)
>5ZXW_1|Chain A|Carbonic anhydrase 2|Homo sapiens (9606)
>3TSJ_1|Chains A, B|Pollen allergen Phl p 4|Phleum pratense (15957)
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 \(w=f(c)\)
3SQN , Knot 194 485 0.82 40 219 438 
SNAMYSMLKRVITEKDLLRQIRLLEQLLNVPQLTAKRLAAQIQTTERTVFSDLQYIRSQLPADWSIETDSSGIRLRNQGNAQTNELWSLFLPQSISIQLLKELLFTKELVTTSFLSTSGVSYETLKRHIKKMNQALRDFHLTIQLTTMTIQLIGAESNIRIFYHRLLVPFTHNNYFFDDYSIHEEHYFQFLKQVYSSELTVETEEIFGACWFFINTIRNKANCRVSQFSFDSKDVLFQLYQPSLAKLYASEGIYLQGEESFFAFFCFLESWNYDNVYGETLASALHTHYSQLRKSLQQFVTNLSTEEARPDLIQTNLLDNLLLLFIKYTESPTLSEQFQLEYQELMTEQAAEDLALSKSNQELLEILSRYTTIEEPTYFLSLASLLEKQAIYSIQAQTMTAYFLFQGEPAWKAFLQQELAAYLGTRVKLQAIEYVELSQLTLNEADIIISNFPLDHLDLPVFYLSLIPTKNELRRLAELTLHSYF
5ZXW , Knot 114 266 0.79 40 178 254 
GPGLNSMSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK
3TSJ , Knot 205 500 0.85 40 262 475 
YFPPPAAKEDFLGCLVKEIPPRLLYAKSSPAYPSVLGQTIRNSRWSSPDNVKPLYIITPTQVSHIQSAVVCGRRHSVRIRVRSGGHDYEGLSYRSLQPETFAVVDLNKMRAVWVDGKARTAWVDSGAQLGELYYAIYKASPTLAFPAGVCPTIGVGGNFAGGGFGMLLRKYGIAAENVIDVKLVDANGKLHDKKSMGDDHFWAVRGGGGESFGIVVAWQVKLLPVPPTVTIFKISKTVSEGAVDIINKWQVVAPQLPADLMIRIIAQGPKATFEAMYLGTCKTLTPLMSSKFPELGMNPSHCNEMSWIQSIPFVHLGHRDALEDDLLNRQNSFKPFAEYKSDYVYQPFPKTVWEQILNTWLVKPGAGIMIFDPYGATISATPESATPFPHRKGVLFNIQYVNYWFAPGAAAAPLSWSKDIYNYMEPYVSKNPRQAYANYRDIDLGRNEVVNDVSTYASGKVWGQKYFKGNFERLAITKGKVDPTDYFRNEQSIPPLIKKY

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 \(c\).

\(|P_{f(3SQN_1)}(2) \setminus P_{f(5ZXW_1)}(2)|=108\), \(|P_{f(5ZXW_1)}(2) \setminus P_{f(3SQN_1)}(2)|=67\). 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 1:00110011001100001100101100110110101001110100000011001001000111010100000110100010100001101111001010110011100011000110001100001000100100110010101010010101111000101100011111000001100001000001011001000010100001111011110010001000100101000011101001011010100110101000111110110010000101001101100000010001001100100001010110001100111111000001010001010000110001100111000000110110000010010011011011000110010100101011101011101110001110110010101100101001010010111001110010111101011100001001101010001
Pair \(Z_2\) Length of longest common subsequence
3SQN_1,5ZXW_1 175 3
3SQN_1,3TSJ_1 153 5
5ZXW_1,3TSJ_1 164 4

Newick tree

 
[
	5ZXW_1:87.38,
	[
		3SQN_1:76.5,3TSJ_1:76.5
	]:10.88
]

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 \((0.85)(0.8)(\frac{751 }{\log_{20} 751}-\frac{266}{\log_{20}266})=133.\)
Status Protein1 Protein2 d d1/2
Query variables 3SQN_1 5ZXW_1 167 129.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]