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Parikh vectors
5YEW_1 5NBN_1 8AAE_1 Letter Amino acid
22 27 10 N Asparagine
23 17 12 F Phenylalanine
4 16 8 Y Tyrosine
33 43 18 S Serine
17 20 7 R Arginine
24 24 19 D Aspartic acid
39 44 13 E Glutamic acid
12 6 12 H Histidine
34 37 24 K Lycine
11 5 2 M Methionine
17 35 12 T Threonine
33 25 13 A Alanine
20 30 22 G Glycine
20 30 9 I Isoleucine
42 44 26 L Leucine
32 28 17 V Valine
6 7 1 C Cysteine
17 14 11 Q Glutamine
11 28 17 P Proline
4 9 7 W Tryptophan 

5YEW_1|Chain A|Mitofusin-1,Mitofusin-1 fusion protein|Homo sapiens (9606)
>5NBN_1|Chains A, B|Actin-related protein 4|Saccharomyces cerevisiae (strain ATCC 204508 / S288c) (559292)
>8AAE_1|Chain A|Carbonic anhydrase 2|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 \(w=f(c)\)
5YEW , Knot 175 421 0.83 40 217 395 
MAEPVSPLKHFVLAKKAITAIFDQLLEFVTEGSHFVEATYKNPELDRIATEDDLVEMQGYKDKLSIIGEVLSRRHMKVAFFGRTSSGKSSVINAMLWDKVLPSGIGHITNCFLSVEGTDGDKAYLMTEGSDEKKSVKVNNQLAHALHMDKDLKAGCLVRVFWPKAKCALLRDDLVLVDSPGTDVTTELDSWIDKFCLDADVFVLVANSESTLMNTEKHFFHKVNERLSKPNIFILNNRWDASASEPEYMEDVRRQHMERCLHFLVEELKVVNALEAQNRIFFVSAKEVLSARKQKAQGMPESGVALAEGFHARLQEFQNFEQIFEECISQSAVKTKFEQHTIRAKQILATVKNIMDSVNLAAEDGSGSGSGGSEIARLPKEIDQLEKIQNNSKLLRNKAVQLENELENFTKQFLPSSNEES
5NBN , Knot 198 489 0.83 40 240 460 
MSNAALQVYGGDEVSAVVIDPGSYTTNIGYSGSDFPQSILPSVYGKYTADEGNKKIFSEQSIGIPRKDYELKPIIENGLVIDWDTAQEQWQWALQNELYLNSNSGIPALLTEPVWNSTENRKKSLEVLLEGMQFEACYLAPTSTCVSFAAGRPNCLVVDIGHDTCSVSPIVDGMTLSKSTRRNFIAGKFINHLIKKALEPKEIIPLFAIKQRKPEFIKKTFDYEVDKSLYDYANNRGFFQECKETLCHICPTKTLEETKTELSSTAKRSIESPWNEEIVFDNETRYGFAEELFLPKEDDIPANWPRSNSGVVKTWRNDYVPLKRTKPSGVNKSDKKVTPTEEKEQEAVSKSTSPAANSADTPNETGKRPLEEEKPPKENNELIGLADLVYSSIMSSDVDLRATLAHNVVLTGGTSSIPGLSDRLMTELNKILPSLKFRILTTGHTIERQYQSWLGGSILTSLGTFHQLWVGKKEYEEVGVERLLNDRFR
8AAE , Knot 112 260 0.79 40 176 249 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK

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 \(c\).

\(|P_{f(5YEW_1)}(2) \setminus P_{f(5NBN_1)}(2)|=65\), \(|P_{f(5NBN_1)}(2) \setminus P_{f(5YEW_1)}(2)|=88\). 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 1:1101101100111100110111001101100100110100001010011000011010100001011101100001011111000010001101111001110111010001101010010010110010000001010001101101000101101101111010011100011110011001000100110010101011111100000110000011001000100101111000101010010010010000100010111001011011010001111010011010000101110011111011010100100100110001000110001000010100111010011001011100101010110011011001001001000001100011010001001000111000000
Pair \(Z_2\) Length of longest common subsequence
5YEW_1,5NBN_1 153 4
5YEW_1,8AAE_1 165 4
5NBN_1,8AAE_1 162 4

Newick tree

 
[
	8AAE_1:83.43,
	[
		5YEW_1:76.5,5NBN_1:76.5
	]:6.93
]

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 \((0.85)(0.8)(\frac{910 }{\log_{20} 910}-\frac{421}{\log_{20}421})=130.\)
Status Protein1 Protein2 d d1/2
Query variables 5YEW_1 5NBN_1 161 149.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]