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Parikh vectors
2VHW_1 1LTK_1 7JRH_1 Letter Amino acid
58 27 2 A Alanine
9 7 2 Q Glutamine
37 43 1 L Leucine
16 21 0 S Serine
28 18 2 T Threonine
11 49 0 K Lycine
18 12 0 P Proline
2 2 1 W Tryptophan
33 33 2 V Valine
9 34 0 N Asparagine
19 24 2 D Aspartic acid
19 8 0 R Arginine
5 3 0 C Cysteine
18 32 1 E Glutamic acid
37 38 1 G Glycine
16 13 0 H Histidine
15 32 2 I Isoleucine
8 9 0 M Methionine
9 16 0 F Phenylalanine
10 4 0 Y Tyrosine 

2VHW_1|Chains A, B, C, D, E, F|ALANINE DEHYDROGENASE|MYCOBACTERIUM TUBERCULOSIS (1773)
>1LTK_1|Chains A, B, C|PHOSPHOGLYCERATE KINASE|Plasmodium falciparum (5833)
>7JRH_1|Chain A|Cyclic peptide ASP-GLN-TRP-MLE-GLN-VAL-ASP-ORD-GLU-VAL-THR-GLY-ILE-ILE-THR-ORD|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 \(w=f(c)\)
2VHW , Knot 155 377 0.81 40 196 343 
MRVGIPTETKNNEFRVAITPAGVAELTRRGHEVLIQAGAGEGSAITDADFKAAGAQLVGTADQVWADADLLLKVKEPIAAEYGRLRHGQILFTFLHLAASRACTDALLDSGTTSIAYETVQTADGALPLLAPMSEVAGRLAAQVGAYHLMRTQGGRGVLMGGVPGVEPADVVVIGAGTAGYNAARIANGMGATVTVLDINIDKLRQLDAEFCGRIHTRYSSAYELEGAVKRADLVIGAVLVPGAKAPKLVSNSLVAHMKPGAVLVDIAIDQGGCFEGSRPTTYDHPTFAVHDTLFYCVANMPASVPKTSTYALTNATMPYVLELADHGWRAACRSNPALAKGLSTHEGALLSERVATDLGVPFTEPASVLAHHHHHH
1LTK , Knot 174 425 0.82 40 208 392 
GHSMHHHHHHLGNKLSISDLKDIKNKKVLVRVDFNVPIENGIIKDTNRITATLPTINHLKKEGASKIILISHCGRPDGLRNEKYTLKPVAETLKGLLGEEVLFLNDCVGKEVEDKINAAKENSVILLENLRFHIEEEGKGVDANGNKVKANKEDVEKFQNDLTKLADVFINDAFGTAHRAHSSMVGVKLNVKASGFLMKKELEYFSKALENPQRPLLAILGGAKVSDKIQLIKNLLDKVDRMIIGGGMAYTFKKVLNNMKIGTSLFDEAGSKIVGEIMEKAKAKNVQIFLPVDFKIADNFDNNANTKFVTDEEGIPDNWMGLDAGPKSIENYKDVILTSKTVIWNGPQGVFEMPNFAKGSIECLNLVVEVTKKGAITIVGGGDTASLVEQQNKKNEISHVSTGGGASLELLEGKELPGVLALSNK
7JRH , Knot 12 16 0.69 20 15 14 
DQWLQVDAEVTGIITA

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 \(c\).

\(|P_{f(2VHW_1)}(2) \setminus P_{f(1LTK_1)}(2)|=74\), \(|P_{f(1LTK_1)}(2) \setminus P_{f(2VHW_1)}(2)|=86\). 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 1:10111100000001011101111101000100111011110101100101011110111010011101011101001111001010010111011011100100011100100011000100101111111110011101110111001100011011111111110110111111101100110110111101011010100100101010101000000100101110010111111111110110110001110101111110111001101010010000010111000110011011101100000110010110110110011011000011110110000111100011001111100110111000000
Pair \(Z_2\) Length of longest common subsequence
2VHW_1,1LTK_1 160 6
2VHW_1,7JRH_1 191 3
1LTK_1,7JRH_1 203 3

Newick tree

 
[
	7JRH_1:10.99,
	[
		2VHW_1:80,1LTK_1:80
	]:23.99
]

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 \((0.85)(0.8)(\frac{802 }{\log_{20} 802}-\frac{377}{\log_{20}377})=114.\)
Status Protein1 Protein2 d d1/2
Query variables 2VHW_1 1LTK_1 144 135.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]