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Parikh vectors
7YPI_1 7QQQ_1 5IFU_1 Letter Amino acid
0 11 11 C Cysteine
23 0 11 M Methionine
9 0 13 W Tryptophan
28 0 31 S Serine
71 30 25 A Alanine
57 0 19 R Arginine
14 0 31 N Asparagine
87 0 35 L Leucine
46 0 48 K Lycine
11 0 23 F Phenylalanine
50 0 20 D Aspartic acid
33 0 16 Q Glutamine
41 0 40 I Isoleucine
48 0 24 P Proline
45 0 27 T Threonine
66 0 31 V Valine
71 0 43 E Glutamic acid
57 24 21 G Glycine
14 0 15 H Histidine
22 0 22 Y Tyrosine 

7YPI_1|Chains A, B, C, D, E, F|Lon protease|Meiothermus taiwanensis (172827)
>7QQQ_1|Chain A|AAVS1 sgRNA|synthetic construct (32630)
>5IFU_1|Chains A, B|Proline--tRNA ligase|Plasmodium falciparum (isolate 3D7) (36329)
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 \(w=f(c)\)
7YPI , Knot 294 793 0.82 38 260 689 
MRLELPVIPLRNTVILPHTTTPVDVGRAKSKRAVEEAMGADRLIFLVAQRDPEVDDPAPDDLYTWGVQAVVKQAMRLPDGTLQVMVEARARAQVTDYIPGPYLRARGEVFSEIFPIDEAVVRVLVEELKEAFEKYVANHKSLRLDRYQLEAVKGTSDPAMLADTIAYHATWTVAEKQEILELTDLEARLKKVLGLLSRDLERFELDKRVAQRVKEQMDTNQRESYLREQMKAIQKELGGEDGLSDLEALRKKIEEVGMPEAVKTKALKELDRLERMQQGSPEATVARTYLDWLTEVPWSKADPEVLDINHTRQVLDEDHYGLKDVKERILEYLAVRQLTQGLDVRNKAPILVLVGPPGVGKTSLGRSIARSMNRKFHRISLGGVRDEAEIRGHRRTYIGAMPGKLIHAMKQVGVINPVILLDEIDKMSSDWRGDPASAMLEVLDPEQNNTFTDHYLDVPYDLSKVFFITTANTLQTIPRPLLDRMEVIEIPGYTNMEKQAIARQYLWPKQVRESGMEGRIEVTDAAILRVISEYTREAGVRGLERELGKIARKGAKFWLEGAWEGLRTIDASDIPTYLGIPRYRPDKAETEPQVGTAQGLAWTPVGGTLLTIEVAAVPGSGKLSLTGQLGEVMKESAQAALTYLRAHTQDYGLPEDFYNKVDLHVHVPDGATPKDGPSAGITMATAIASALSRRPARMDIAMTGEVSLRGKVMPIGGVKEKLLAAHQAGIHKIVLPKDNEAQLEELPKEVLEGLEIKLVEDVGEVLEYLLLPEPTMPPVVQPSDNRQQPGAGA
7QQQ , Knot 26 84 0.45 8 16 44 
GGGGGGCCACUAGGGACAGGAUGUUUUAGAGCUAGAAAUAGCAAGUUAAAAUAAGGCUAGUCCGUUAUCAACUUGAAAAAGUGA
5IFU , Knot 208 506 0.85 40 268 480 
MAHHHHHHSNILGITSKKIENFSDWYTQVIVKSELIEYYDISGCYILRPAAYYIWECVQAFFNKEIKKLNVENSYFPLFVTKNKLEKEKNHIEGFSPEVAWVTKYGDSNLPEEIAIRPTSETIMYSVFPKWIRSYRDLPLKLNQWNTVVRWEFKQPTPFIRTREFLWQEGHTAHKNEEEAVKLVFDILDLYRRWYEEYLAVPIIKGIKSEGEKFGGANFTSTAEAFISENGRAIQAATSHYLGTNFAKMFKIEFEDENEVKQYVHQTSWGCTTRSIGIMIMTHGDDKGLVLPPNVSKYKVVIVPIFYKTTDENAIHSYCKDIEKILKNAQINCVYDDRASYSPGYKFNHWELRGIPIRIEVGPKDLQNNSCVIVRRDNNEKCNVKKESVLLETQQMLVDIHKNLFLKAKKKLDDSIVQVTSFSEVMNALNKKKMVLAPWCEDIATEEEIKKETQRLSLNQTNSETTLSGAMKPLCIPLDQPPMPPNMKCFWSGKPAKRWCLFGRSY

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 \(c\).

\(|P_{f(7YPI_1)}(2) \setminus P_{f(7QQQ_1)}(2)|=256\), \(|P_{f(7QQQ_1)}(2) \setminus P_{f(7YPI_1)}(2)|=12\). 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 1:1010111111000111100001101101000011001111001111110001010011100100111011100110110101011101010101000111101010101100111100111011100100110001100001010000101101000111110011001010110000110100101010011111000100101000110010001000000001000101100011100110010110001001111011000110010010010010101011000101100111001010110100000110000011001000110011100100110100011111111111110001100110010001001011110001010100000111111011011001111011111001001000101011011101101000001000010110010011110010010011011100101101110001000111000111001000110101010011110110000001110110001101100110111011101100101001100111100010010001011010111101111011010111111010101010110110001011100101000001110010001010101101101001101110110111011000110101110101010101111111000111100111001111000010100110011011010110011011001111010111110100000011111
Pair \(Z_2\) Length of longest common subsequence
7YPI_1,7QQQ_1 268 3
7YPI_1,5IFU_1 142 5
7QQQ_1,5IFU_1 276 3

Newick tree

 
[
	7QQQ_1:15.61,
	[
		7YPI_1:71,5IFU_1:71
	]:80.61
]

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 \((0.85)(0.8)(\frac{877 }{\log_{20} 877}-\frac{84}{\log_{20}84})=225.\)
Status Protein1 Protein2 d d1/2
Query variables 7YPI_1 7QQQ_1 291 157.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]