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Parikh vectors
6YLN_1 5HHZ_1 1APU_1 Letter Amino acid
2 11 3 W Tryptophan
10 61 24 A Alanine
6 38 0 R Arginine
2 3 2 C Cysteine
8 13 25 Q Glutamine
20 13 5 K Lycine
5 5 0 M Methionine
10 29 12 P Proline
10 12 14 Y Tyrosine
12 13 15 N Asparagine
19 29 23 D Aspartic acid
12 16 13 I Isoleucine
16 18 30 T Threonine
17 44 23 V Valine
16 26 4 E Glutamic acid
23 65 40 G Glycine
14 17 3 H Histidine
21 50 21 L Leucine
13 27 20 F Phenylalanine
9 34 46 S Serine 

6YLN_1|Chain A|EGFP|Vaccinia virus (10245)
>5HHZ_1|Chain A|Cytokinin dehydrogenase 4|Zea mays (4577)
>1APU_1|Chain A[auth E]|PROTEIN (PENICILLOPEPSIN)|Penicillium janthinellum (5079)
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 \(w=f(c)\)
6YLN , Knot 111 245 0.83 40 161 229 
MVSKGEELFTGVVPILVELDGDVNGHKFSVSGEGEGDATYGKLTLKFICTTGKLPVPWPTLVTTLSWGVQCFARYPDHMKQHDFFKSAMPEGYVQERTIFFKDDGNYKTRAEVKFEGDTLVNRIELKGIDFKEDGNILGHKLEYNYFSDNVYITADKQKNGIKANFKIRHNIEDGGVQLADHYQQNTPIGDGPVLLPDNHYLSTQSALSKDPNEKRDHMVLLEFVTAAGITLGMDELYKHHHHHH
5HHZ , Knot 196 524 0.78 40 224 470 
AGHMLPVEPPAELLQLGGGDVGGGRLSVDASDIAEASRDFGGVARAEPMAVFHPRAAGDVAGLVGAAFRSARGFRVSARGHGHSISGQAQAAGGVVVDMSRGRGPGAAVARALPVHSAALGGHYVDVWGGELWVDVLNWTLSHGGLAPRSWTDYLYLSVGGTLSNAGISGQAFHHGPQISNVYELDVVTGKGEVVTCSETENPDLFFGVLGGLGQFGIITRARIALERAPKRVRWIRALYSNFSEFTADQERLISLGSGGGRRFDYVEGFVVAAEGLINNWRSSFFSPQNPVKLTSLKHHSSVLYCLEVTKNYDDETAGSVDQDVDTLLGELNFLPGTVFTTDLPYVDFLDRVHKAELKLRAKGMWEVPHPWLNLFVPASRIADFDRGVFRGVLGGRTAGAGGPVLIYPMNKHKWDPRSSAVTPDEEVFYLVAFLRSALPGAPESLEALARQNQRILDFCAGTGIGAKQYLPGHKARHEWAEHFGAARWDRFARLKAEFDPRAILAAGQGIFRPPGSPALAADS
1APU , Knot 130 323 0.77 36 158 296 
AASGVATNTPTANDEEYITPVTIGGTTLNLNFDTGSADLWVFSTELPASQQSGHSVYNPSATGKELSGYTWSISYGDGSSASGNVFTDSVTVGGVTAHGQAVQAAQQISAQFQQDTNNDGLLGLAFSSINTVQPQSQTTFFDTVKSSLAQPLFAVALKHQQPGVYDFGFIDSSKYTGSLTYTGVDNSQGFWSFNVDSYTAGSQSGDGFSGIADTGTTLLLLDDSVVSQYYSQVSGAQQDSNAGGYVFDCSTNLPDFSVSISGYTATVPGSLINYGPSGDGSTCLGGIQSNSGIGFSIFGDIFLKSQYVVFDSDGPQLGFAPQA

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 \(c\).

\(|P_{f(6YLN_1)}(2) \setminus P_{f(5HHZ_1)}(2)|=62\), \(|P_{f(5HHZ_1)}(2) \setminus P_{f(6YLN_1)}(2)|=125\). 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 1:11001001101111111010101010010101010101001010101100010111111011001011100110010010000110011101010000111000100000101010100110010101101000101110010000100010101000001101010100010011101100000001110111111000010000110001000000111101101111011100100000000
Pair \(Z_2\) Length of longest common subsequence
6YLN_1,5HHZ_1 187 4
6YLN_1,1APU_1 161 4
5HHZ_1,1APU_1 158 4

Newick tree

 
[
	6YLN_1:89.82,
	[
		1APU_1:79,5HHZ_1:79
	]:10.82
]

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 \((0.85)(0.8)(\frac{769 }{\log_{20} 769}-\frac{245}{\log_{20}245})=145.\)
Status Protein1 Protein2 d d1/2
Query variables 6YLN_1 5HHZ_1 178 131
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]