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Parikh vectors
7AVI_1 4NCL_1 5YSO_1 Letter Amino acid
18 29 20 A Alanine
13 9 7 H Histidine
33 33 20 K Lycine
5 1 2 W Tryptophan
17 10 8 Y Tyrosine
15 24 17 D Aspartic acid
39 34 13 I Isoleucine
36 16 11 P Proline
24 19 8 T Threonine
19 24 12 Q Glutamine
24 19 5 F Phenylalanine
27 24 13 S Serine
9 10 12 M Methionine
27 35 17 V Valine
34 21 5 R Arginine
26 19 6 N Asparagine
5 8 1 C Cysteine
54 38 18 E Glutamic acid
18 35 7 G Glycine
44 49 36 L Leucine 

7AVI_1|Chains A, B|Son of sevenless homolog 1|Homo sapiens (9606)
>4NCL_1|Chains A, B|Eukaryotic translation initiation factor 5B-like protein|Chaetomium thermophilum var. thermophilum (759272)
>5YSO_1|Chains A, B, C|Estrogen-related receptor gamma|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 \(w=f(c)\)
7AVI , Knot 200 487 0.84 40 246 459 
GEEQMRLPSADVYRFAEPDSEENIIFEENMQPKAGIPIIKAGTVIKLIERLTYHMYADPNFVRTFLTTYRSFCKPQELLSLIIERFEIPEPEPTEADRIAIENGDQPLSAELKRFRKEYIQPVQLRVLNVCRHWVEHHFYDFERDAYLLQRMEEFIGTVRGKAMKKWVESITKIIQRKKIARDNGPGHNITFQSSPPTVEWHISRPGHIETFDLLTLHPIEIARQLTLLESDLYRAVQPSELVGSVWTKEDKEINSPNLLKMIRHTTNLTLWFEKCIVETENLEERVAVVSRIIEILQVFQELNNFNGVLEVVSAMNSSPVYRLDHTFEQIPSRQKKILEEAHELSEDHYKKYLAKLRSINPPCVPFFGIYLTNILKTEEGNPEVLKRHGKELINFSKRRKVAEITGEIQQYQNQPYCLRVESDIKRFFENLNPMGNSMEKEFTDYLFNKSLEIEPRNPKPLPRFPKKYSYPLKSPGVRPSNPRPGT
4NCL , Knot 188 457 0.84 40 231 435 
SHMNKDNLRSPICCILGHVDTGKTKLLDKIRQTNVQEGEAGGITQQIGATYFPVEAIKQKTAVVNKDGKFEFKVPGLLIIDTPGHESFSNLRSRGSSLCNIAILVVDIMHGLEPQTIESLRLLRERKTPFVVALNKIDRLYGWKKIENNGFRESFALQNKAVQNEFRNRLDQVKLQFAEQGFNSELFYENKNFARYVSLVPTSAHTGEGIPDMLKLIVQLCQERMASSLMYLSELQATVLEVKAIEGFGVTIDVILSNGILREGDRIVLCGLEGPIKTNIRALLTPAPMRELRIKGQYIHHKEVKAAQGVKISAPGLEGAIAGSRLLVVGPDDDEEELEEEVESDLQSLFSRVEKTGKGVSVQASTLGSLEALLDFLKDCKIPVANVGIGPVYKRDVMQCGIMLEKAPDYAVMLCFDVKVDKEAQQYADENGIKIFTADIIYHLFDQFTKHMQEQLE
5YSO , Knot 109 238 0.83 40 156 230 
MLNPQLVQPAKKPYNKIVSHLLVAEPEKIYAMPDPTVPDSDIKALTTLCDLADRELVVIIGWAKHIPGFSTLSLADQMSLLQSAWMEILILGVVYRSLSFEDELVYADDYIMDEDQSKLAGLLDLNNAILQLVKKYKSMKLEKEEFVTLKAIALANSDSMHIEDVEAVQKLQDVLHEALQDYEAGQHMEDPRRAGKMLMTLPLLRQTSTKAVQHFYNIKLEGKVPMHKLFLEMLEAKV

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 \(c\).

\(|P_{f(7AVI_1)}(2) \setminus P_{f(4NCL_1)}(2)|=86\), \(|P_{f(4NCL_1)}(2) \setminus P_{f(7AVI_1)}(2)|=71\). 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 1:1000101101010011010000011100010101111110110110110010001010101100110000010010011011100101101010010011100100110101001000010110101101000110001001000101100100111010101100110010011000011000111001010001101010100110100101101011011001011000100110100111011000000100101101100000101110001100001000111100110110110010010111011011000110010001001100000110010010000000011010010110111111010011000010101100010011010000011010101000000100101000100110010111001000100011000101010010111011000001100111010010110
Pair \(Z_2\) Length of longest common subsequence
7AVI_1,4NCL_1 157 4
7AVI_1,5YSO_1 168 4
4NCL_1,5YSO_1 159 4

Newick tree

 
[
	5YSO_1:82.84,
	[
		7AVI_1:78.5,4NCL_1:78.5
	]:4.34
]

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 \((0.85)(0.8)(\frac{944 }{\log_{20} 944}-\frac{457}{\log_{20}457})=128.\)
Status Protein1 Protein2 d d1/2
Query variables 7AVI_1 4NCL_1 164 158
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]