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Parikh vectors
6YQN_1 8XTY_1 5XNO_1 Letter Amino acid
5 37 14 F Phenylalanine
5 73 16 V Valine
12 26 9 N Asparagine
9 23 4 Q Glutamine
8 41 14 E Glutamic acid
1 19 3 H Histidine
13 94 23 L Leucine
12 31 12 K Lycine
7 32 8 Y Tyrosine
2 10 1 C Cysteine
7 35 7 I Isoleucine
5 20 4 M Methionine
12 52 16 P Proline
8 44 8 T Threonine
3 12 6 W Tryptophan
5 90 21 A Alanine
3 39 7 R Arginine
6 35 11 D Aspartic acid
2 79 29 G Glycine
2 59 19 S Serine 

6YQN_1|Chain A|Bromodomain-containing protein 4|Homo sapiens (9606)
>8XTY_1|Chain A|Vesicular acetylcholine transporter,Green fluorescent protein,antibody|Homo sapiens (9606)
>5XNO_1|Chains A[auth 1], B[auth 2]|Chlorophyll a-b binding protein 8, chloroplastic|Pisum sativum (3888)
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 \(w=f(c)\)
6YQN , Knot 66 127 0.84 40 110 123 
SMNPPPPETSNPNKPKRQTNQLQYLLRVVLKTLWKHQFAWPFQQPVDAVKLNLPDYYKIIKTPMDMGTIKKRLENNYYWNAQECIQDFNTMFTNCYIYNKPGDDIVLMAEALEKLFLQKINELPTEE
8XTY , Knot 313 851 0.82 40 296 747 
MESAEPAGQARAAATKLSEAVGAALQSKGEELFTGVVPILVELDGDVNGHKFSVRGEGEGDATNGKLTLKFICTTGKLPVPWPTLVTTLTYGVQCFSRYPDHMKRHDFFKSAMPEGYVQERTISFKDDGTYKTRAEVKFEGDTLVNRIELKGIDFKEDGNILGHKLEYNFNSHNVYITADKQKNGIKANFKIRHNVEDGSVQLADHYQQNTPIGDGPVLLPDNHYLSTQSVLSKDPNEKRDHMVLLEFVTAAGIEPRRQRRLVLVIVCVALLLDNMLYMVIVPIVPDYIAHMRGGGEGPTRTPEVWEPTLPLPTPANASAYTANTSASPTAAWPAGSALRPRYPTESEDVKIGVLFASKAILQLLVNPLSGPFIDRMSYDVPLLIGLGVMFASTVLFAFAEDYATLFAARSLQGLGSAFADTSGIAMIADKYPEEPERSRALGVALAFISFGSLVAPPFGGILYEFAGKRVPFLVLAAVSLFDALLLLAVAKPFSAAARARANLPVGTPIHRLMLDPYIAVVAGALTTCNIPLAFLEPTIATWMKHTMAASEWEMGMAWLPAFVPHVLGVYLTVRLAARYPHLQWLYGALGLAVIGASSCIVPACRSFAPLVVSLCGLCFGIALVDTALLPTLAFLVDVRHVSVYGSVYAIADISYSVAYALGPIVAGHIVHSLGFEQLSLGMGLANLLYAPVLLLLRNVGLLTGSGQVQLVESGGALVQPGGSLRLSCAASGFPVNRYSMRWYRQAPGKEREWVAGMSSAGDRSSYEDSVKGRFTISRDDARNTVYLQMNSLKPEDTAVYYCNVNVGFEYWGQGTQVTVSMGWSHPQFEKGGGSGGGSGGSAWSHPQFEK
5XNO , Knot 103 232 0.80 40 152 223 
RKSATTKKVASSGSPWYGPDRVKYLGPFSGESPSYLTGEFPGDYGWDTAGLSADPETFSKNRELEVIHSRWAMLGALGCVFPELLSRNGVKFGEAVWFKAGSQIFSEGGLDYLGNPSLVHAQSILAIWATQVILMGAVEGYRIAGGPLGEVVDPLYPGGSFDPLGLADDPEAFAELKVKELKNGRLAMFSMFGFFVQAIVTGKGPLENLADHLSDPVNNNAWSYATNFVPGK

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 \(c\).

\(|P_{f(6YQN_1)}(2) \setminus P_{f(8XTY_1)}(2)|=29\), \(|P_{f(8XTY_1)}(2) \setminus P_{f(6YQN_1)}(2)|=215\). 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 1:0101111000010010000001001101110011000111110011011010110000110011011010001000001010001001001100001000110011111011001110010011000
Pair \(Z_2\) Length of longest common subsequence
6YQN_1,8XTY_1 244 3
6YQN_1,5XNO_1 186 3
8XTY_1,5XNO_1 186 4

Newick tree

 
[
	8XTY_1:11.16,
	[
		6YQN_1:93,5XNO_1:93
	]:20.16
]

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 \((0.85)(0.8)(\frac{978 }{\log_{20} 978}-\frac{127}{\log_{20}127})=235.\)
Status Protein1 Protein2 d d1/2
Query variables 6YQN_1 8XTY_1 296 169.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]