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Parikh vectors
3KQZ_1 1PMH_1 8PZP_1 Letter Amino acid
25 14 24 D Aspartic acid
7 1 6 C Cysteine
31 10 29 I Isoleucine
48 9 33 L Leucine
14 2 22 M Methionine
15 7 17 P Proline
40 11 41 S Serine
39 12 25 N Asparagine
8 8 20 Q Glutamine
36 19 40 G Glycine
46 16 19 K Lycine
27 7 15 Y Tyrosine
37 10 36 A Alanine
9 6 53 R Arginine
34 12 37 E Glutamic acid
4 4 6 W Tryptophan
48 16 23 V Valine
13 1 12 H Histidine
22 14 18 F Phenylalanine
25 6 30 T Threonine 

3KQZ_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|M17 leucyl aminopeptidase|Plasmodium falciparum (36329)
>1PMH_1|Chain A[auth X]|beta-1,4-mannanase|Caldicellulosiruptor saccharolyticus (44001)
>8PZP_1|Chain A|Nucleoprotein|Influenza A virus (A/WSN/1933(H1N1)) (382835)
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 \(w=f(c)\)
3KQZ , Knot 209 528 0.82 40 233 490 
MASEVPQVVSLDPTSIPIEYNTPIHDIKVQVYDIKGGCNVEEGLTIFLVNNPGKENGPVKISSKVNDKQVSEFLKDENMEKFNVKLGTSKHFYMFNDNKNSVAVGYVGCGSVADLSEADMKRVVLSLVTMLHDNKLSKLTVVFEINVDKNLFRFFLETLFYEYMTDERFKSTDKNVNMEYIKHLGVYINNADTYKEEVEKARVYYFGTYYASQLIAAPSNYCNPVSLSNAAVELAQKLNLEYKILGVKELEELKMGAYLSVGKGSMYPNKFIHLTYKSKGDVKKKIALVGKGITFDSGGYNLKAAPGSMIDLMKFDMSGCAAVLGCAYCVGTLKPENVEIHFLSAVCENMVSKNSYRPGDIITASNGKTIEVGNTDAEGRLTLADALVYAEKLGVDYIVDIATLTGAMLYSLGTSYAGVFGNNEELINKILQSSKTSNEPVWWLPIINEYRATLNSKYADINQISSSVKASSIVASLFLKEFVQNTAWAHIDIAGVSWNFKARKPKGFGVRLLTEFVLNDALHHHHHH
1PMH , Knot 88 185 0.82 40 136 179 
ESSVNPVVLDFEDGTVMSFGEAWGDSLKCIKKVSVSQDLQRPGNKYALRLDVEFNPNNGWDQGDLGTWIGGVVEGQFDFTGYKSVEFEMFIPYDEFSKSQGGFAYKVVINDGWKELGSEFNITANAGKKVKINGKDYTVIHKAFAIPEDFRTKKRAQLVFQFAGQNSNYKGPIYLDNVRIRPEDA
8PZP , Knot 205 506 0.84 40 243 467 
MATKGTKRSYEQMETDGERQNATEIRASVGKMIDGIGRFYIQMCTELKLSDYEGRLIQNSLTIERMVLSAFDERRNKYLEEHPSAGKDPKKTGGPIYRRVDGKWRRELILYDKEEIRRIWRQANNGDDATAGLTHMMIWHSNLNDATYQRTRALVRTGMDPRMCSLMQGSTLPRRSGAAGAAVKGVGTMVMELIRMIKRGINDRNFWRGENGRRTRIAYERMCNILKGKFQTAAQRTMVDQVRESRNPGNAEFEDLIFLARSALILRGSVAHKSCLPACVYGSAVASGYDFEREGYSLVGIDPFRLLQNSQVYSLIRPNENPAHKSQLVWMACHSAAFEDLRVSSFIRGTKVVPRGKLSTRGVQIASNENMETMESSTLELRSRYWAIRTRSGGNTNQQRASSGQISIQPTFSVQRNLPFDRPTIMAAFTGNTEGRTSDMRTEIIRLMESARPEDVSFQGRGVFELSDEKATSPIVPSFDMSNEGSYFFGDNAEEYDNLEHHHHHH

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 \(c\).

\(|P_{f(3KQZ_1)}(2) \setminus P_{f(1PMH_1)}(2)|=138\), \(|P_{f(1PMH_1)}(2) \setminus P_{f(3KQZ_1)}(2)|=41\). 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 1:110011011010100111000011001010100101100100110111100110001110100010000100110000100101011000010110000001111011010110100101001110110110000100101110101000110111001100010000100000010100100111010010000001001010011000100111110000011010011101100101000111100100101110101101010100110100000101000111110110100110010111101101101010101111101001101010010101101100011000000110110100100101100010101011011101001110011011010111100110001111100001100110000000011111111000010100001010010001010011101110011000111010111101010100101111011001110011000000
Pair \(Z_2\) Length of longest common subsequence
3KQZ_1,1PMH_1 179 3
3KQZ_1,8PZP_1 146 6
1PMH_1,8PZP_1 181 4

Newick tree

 
[
	1PMH_1:94.99,
	[
		3KQZ_1:73,8PZP_1:73
	]:21.99
]

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 \((0.85)(0.8)(\frac{713 }{\log_{20} 713}-\frac{185}{\log_{20}185})=148.\)
Status Protein1 Protein2 d d1/2
Query variables 3KQZ_1 1PMH_1 186 125
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]