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Parikh vectors
4ZHR_1 4UMW_1 7OMK_1 Letter Amino acid
17 41 7 R Arginine
9 9 3 H Histidine
41 46 2 I Isoleucine
60 25 10 K Lycine
37 30 5 P Proline
19 36 4 S Serine
37 47 1 T Threonine
19 6 2 W Tryptophan
21 6 1 Y Tyrosine
26 29 2 D Aspartic acid
48 90 6 L Leucine
17 22 6 N Asparagine
46 45 5 E Glutamic acid
34 59 8 G Glycine
14 18 9 F Phenylalanine
41 55 5 V Valine
29 120 4 A Alanine
3 9 7 C Cysteine
36 26 4 Q Glutamine
8 13 2 M Methionine 

4ZHR_1|Chain A|RT p66 subunit|Human immunodeficiency virus 1 (11676)
>4UMW_1|Chain A|ZINC-TRANSPORTING ATPASE|SHIGELLA SONNEI (624)
>7OMK_1|Chain A|Endonuclease 8-like 3|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 \(w=f(c)\)
4ZHR , Knot 221 562 0.83 40 253 516 
MVPISPIETVPVKLKPGMDGPKVKQWPLTEEKIKALVEICTEMEKEGKISKIGPENPYNTPVFAIKKKDSTKWRKLVDFRELNKRTQDFWEVQLGIPHPAGLKQKKSVTVLDVGDAYFSVPLDKDFRKYTAFTIPSINNETPGIRYQYNVLPMGWKGSPAIFQCSMTKILEPFRKQNPDIVIYQYMDDLYVGSDLEIGQHRTKIEELRQHLLRWGFTTPDKKHQKEPPFLWMGYELHPDKWTVQPIVLPEKDSWTVNDIQKLVGKLNWASQIYAGIKVRQLCKLLRGTKALTEVVPLTEEAELELAENREILKEPVHGVYYDPSKDLIAEIQKQGQGQWTYQIYQEPFKNLKTGKYARMKGAHTNDVKQLTEAVQKIATESIVIWGKTPKFKLPIQKETWEAWWTEYWQATWIPEWEFVNTPPLVKLWYQLEKEPIIGAETFYVDGAANRETKLGKAGYVTDRGRQKVVPLTDTTNQKTELQAIHLALQDSGLEVNIVTDSQYALGIIQAQPDKSESELVSQIIEQLIKKEKVYLAWVPAHKGIGGNEQVDKLVSAGIRKVL
4UMW , Knot 271 732 0.81 40 235 634 
MSTPDNHGKKAPQFAAFKPLTTVQNANDCCCDGACSSSPTLSENVSGTRYSWKVSGMDCAACARKVENAVRQLAGVNQVQVLFATEKLVVDADNDIRAQVESAVQKAGYSLRDEQAADEPQASRLKENLPLITLIVMMAISWGLEQFNHPFGQLAFIATTLVGLYPIARQALRLIKSGSYFAIETLMSVAAIGALFIGATAEAAMVLLLFLIGERLEGWAASRARQGVSALMALKPETATRLRNGEREEVAINSLRPGDVIEVAAGGRLPADGKLLSPFASFDESALTGESIPVERATGDKVPAGATSVDRLVTLEVLSEPGASAIDRILKLIEEAEERRAPIERFIDRFSRIYTPAIMAVALLVTLVPPLLFAASWQEWIYKGLTLLLIGCPCALVISTPAAITSGLAAAARRGALIKGGAALEQLGRVTQVAFDKTGTLTVGKPRVTAIHPATGISESELLTLAAAVEQGATHPLAQAIVREAQVAELAIPTAESQRALVGSGIEAQVNGERVLICAAGKHPADAFAGLINELESAGQTVVLVVRNDDVLGIIALQDTLRADAATAISELNALGVKGVILTGDNPRAAAAIAGELGLEFKAGLLPEDKVKAVTKLNQHAPLAMVGDGINDAPAMKAAAIGIAMGSGTDVALETADAALTHNHLRGLVQMIELARATHANIRQNITIALGLKGIFLVTTLLGMTGLWLAVLADTGATVLVTANALRLLRRR
7OMK , Knot 53 93 0.86 40 80 90 
SPLCKMHHRRCVLRVVRKDGENKGRQFYACSLPRGAQCGFFEWADLSFPFCKHGKRSIMKTVLKIGPNNGKNFFVCPLEKEKQCNFFQWAENG

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 \(c\).

\(|P_{f(4ZHR_1)}(2) \setminus P_{f(4UMW_1)}(2)|=85\), \(|P_{f(4UMW_1)}(2) \setminus P_{f(4ZHR_1)}(2)|=67\). 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 1:1111011001110101110110100111000010111010001000101001110010001111100000001001101001000000110101111011110000010110110101011100010000110110100001110000011111101011110001001101100001011100010010110010110000010010001101110010000000111111100101001010111110000101001001110101100101110100100110100110011110001010110000110011011000100011101000101010001000110010010010101100001001001100110001111100101011100001011100010101110101100111101100100011111001010111000001101101000100011110000000001011011100011010110000011111010100000011001100110000101111110011110001001101110011
Pair \(Z_2\) Length of longest common subsequence
4ZHR_1,4UMW_1 152 5
4ZHR_1,7OMK_1 229 4
4UMW_1,7OMK_1 223 3

Newick tree

 
[
	7OMK_1:12.89,
	[
		4ZHR_1:76,4UMW_1:76
	]:46.89
]

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 \((0.85)(0.8)(\frac{1294 }{\log_{20} 1294}-\frac{562}{\log_{20}562})=187.\)
Status Protein1 Protein2 d d1/2
Query variables 4ZHR_1 4UMW_1 231 206.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]