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Parikh vectors
1TUB_1 5SDA_1 4NKO_1 Letter Amino acid
31 19 16 L Leucine
19 17 15 K Lycine
10 7 6 M Methionine
21 8 8 R Arginine
16 10 8 N Asparagine
12 1 4 C Cysteine
31 16 8 E Glutamic acid
13 3 7 H Histidine
20 9 7 F Phenylalanine
20 12 12 P Proline
22 12 40 S Serine
18 6 15 Y Tyrosine
36 5 14 A Alanine
15 7 13 Q Glutamine
26 9 9 I Isoleucine
30 7 17 T Threonine
34 14 15 V Valine
27 9 13 D Aspartic acid
35 13 38 G Glycine
4 3 4 W Tryptophan 

1TUB_1|Chain A|TUBULIN|Sus scrofa (9823)
>5SDA_1|Chain A|Dihydrofolate reductase|Homo sapiens (9606)
>4NKO_1|Chains A, B, C|Engineered scFv|Mus musculus (10090)
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 \(w=f(c)\)
1TUB , Knot 189 440 0.87 40 253 420 
MRECISIHVGQAGVQIGNACWELYCLEHGIQPDGQMPSDKTIGGGDDSFNTFFSETGAGKHVPRAVFVDLEPTVIDEVRTGTYRQLFHPEQLITGKEDAANNYARGHYTIGKEIIDLVLDRIRKLADQCTGLQGFSVFHSFGGGTGSGFTSLLMERLSVDYGKKSKLEFSIYPAPQVSTAVVEPYNSILTTHTTLEHSDCAFMVDNEAIYDICRRNLDIERPTYTNLNRLIGQIVSSITASLRFDGALNVDLTEFQTNLVPYPRGHFPLATYAPVISAEKAYHEQLSVAEITNACFEPANQMVKCDPRHGKYMACCLLYRGDVVPKDVNAAIATIKTKRTIQFVDWCPTGFKVGINYEPPTVVPGGDLAKVQRAVCMLSNTTAIAEAWARLDHKFDLMYAKRAFVHWYVGEGMEEGEFSEAREDMAALEKDYEEVGVDSV
5SDA , Knot 88 187 0.82 40 138 178 
MVGSLNCIVAVSQNMGIGKNGDLPWPPLRNEFRYFQRMTTTSSVEGKQNLVIMGKKTWFSIPEKNRPLKGRINLVLSRELKEPPQGAHFLSRSLDDALKLTEQPELANKVDMVWIVGGSSVYKEAMNHPGHLKLFVTRIMQDFESDTFFPEIDLEKYKLLPEYPGVLSDVQEEKGIKYKFEVYEKND
4NKO , Knot 114 269 0.79 40 158 240 
MADYKDIVMTQTPSSLPVSLGDQASISCRSSQSIVHSNGNTYLEWYLQKPGQSPKLLIYKVSNRFSGVPDRFSGSGSGTDFTLKISRVEAEDLGIYYCFQGSLVPPTFGAGTKLELKRGGGGSGGGGSGGGGSSGGGSQVQLQQSGPEDVKPGASVKISCKASGYTFTDYYMNWVKQSPGKGLEWIGDINPNNGGTSYNQKFKGRATLTVDKSSSTAYMELRSLTSEDSSVYYCAASSPYSMRAAMDYWGQGTTVTVSASGADHHHHHH

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 \(c\).

\(|P_{f(1TUB_1)}(2) \setminus P_{f(5SDA_1)}(2)|=161\), \(|P_{f(5SDA_1)}(2) \setminus P_{f(1TUB_1)}(2)|=46\). 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 1:10001010110111011010101001001101010110000111100010011000111001101111010101100100100001101001101000110001010001100110111001001100001101101100111101011001110010100100001010101110100111010001100000100000111100011001000010100100001001110110010101010111010100100011101010111100111101001000010110100101011001100010010011001100101110010111101000001011010101101110001101111101101001101100001110111010001011010011101011011001010010001111000000111001
Pair \(Z_2\) Length of longest common subsequence
1TUB_1,5SDA_1 207 3
1TUB_1,4NKO_1 189 3
5SDA_1,4NKO_1 182 4

Newick tree

 
[
	1TUB_1:10.65,
	[
		4NKO_1:91,5SDA_1:91
	]:10.65
]

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 \((0.85)(0.8)(\frac{627 }{\log_{20} 627}-\frac{187}{\log_{20}187})=125.\)
Status Protein1 Protein2 d d1/2
Query variables 1TUB_1 5SDA_1 166 115.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]