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Parikh vectors
2QTN_1 5RES_1 1QNE_1 Letter Amino acid
9 11 10 R Arginine
4 10 6 M Methionine
13 11 7 Y Tyrosine
6 17 8 D Aspartic acid
6 14 7 Q Glutamine
6 7 2 H Histidine
13 11 17 I Isoleucine
12 11 20 K Lycine
7 17 12 F Phenylalanine
7 17 18 A Alanine
5 21 7 N Asparagine
11 26 11 G Glycine
11 24 8 T Threonine
12 27 14 V Valine
2 12 3 C Cysteine
19 9 10 E Glutamic acid
22 29 16 L Leucine
13 13 11 P Proline
9 16 13 S Serine
2 3 0 W Tryptophan 

2QTN_1|Chains A, B|Nicotinate (Nicotinamide) nucleotide adenylyltransferase|Bacillus anthracis (1392)
>5RES_1|Chain A|3C-like proteinase|Severe acute respiratory syndrome coronavirus 2 (2697049)
>1QNE_1|Chains A, B|TRANSCRIPTION INITIATION FACTOR TFIID-1|ARABIDOPSIS THALIANA (3702)
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 \(w=f(c)\)
2QTN , Knot 93 189 0.86 40 145 183 
MRKIGIIGGTFDPPHYGHLLIANEVYHALNLEEVWFLPNQIPPHKQGRDITSVESRLQMLELATEAEEHFSICLEELSRKGPSYTYDTMLQLTKKYPDVQFHFIIGGDMVEYLPKWYNIEALLDLVTFVGVARPGYKLRTPYPITTVEIPEFAVSSSLLRERYKEKKTCKYLLPEKVQVYIERNGLYES
5RES , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ
1QNE , Knot 91 200 0.80 38 135 189 
MTDQGLEGSNPVDLSKHPSGIVPTLQNIVSTVNLDCKLDLKAIALQARNAEYNPKRFAAVIMRIREPKTTALIFASGKMVCTGAKSEDFSKMAARKYARIVQKLGFPAKFKDFKIQNIVGSCDVKFPIRLEGLAYSHAAFSSYEPELFPGLIYRMKVPKIVLLIFVSGKIVITGAKMRDETYKAFENIYPVLSEFRKIQQ

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 \(c\).

\(|P_{f(2QTN_1)}(2) \setminus P_{f(5RES_1)}(2)|=63\), \(|P_{f(5RES_1)}(2) \setminus P_{f(2QTN_1)}(2)|=114\). 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 1:100111111010110010111100100110100111110011100010010010001011011001000101010010001100000011010000101010111110110011010010111011011111011001001011001011011100011000000000000111001010100011000
Pair \(Z_2\) Length of longest common subsequence
2QTN_1,5RES_1 177 3
2QTN_1,1QNE_1 166 3
5RES_1,1QNE_1 165 4

Newick tree

 
[
	2QTN_1:86.86,
	[
		1QNE_1:82.5,5RES_1:82.5
	]:4.36
]

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 \((0.85)(0.8)(\frac{495 }{\log_{20} 495}-\frac{189}{\log_{20}189})=89.0\)
Status Protein1 Protein2 d d1/2
Query variables 2QTN_1 5RES_1 113 91.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]