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Parikh vectors
4BAO_1 9UDA_1 2KCO_1 Letter Amino acid
2 16 3 F Phenylalanine
0 3 0 W Tryptophan
0 47 7 V Valine
0 13 8 N Asparagine
1 19 11 I Isoleucine
0 12 1 M Methionine
1 23 6 P Proline
2 28 13 T Threonine
1 32 10 A Alanine
3 24 6 D Aspartic acid
3 28 14 K Lycine
4 38 13 G Glycine
0 9 1 H Histidine
3 29 4 S Serine
1 11 3 Y Tyrosine
3 20 12 R Arginine
1 5 0 C Cysteine
0 16 3 Q Glutamine
6 29 7 E Glutamic acid
5 44 11 L Leucine 

4BAO_1|Chain A|THROMBIN LIGHT CHAIN|HOMO SAPIENS (9606)
>9UDA_1|Chain A|Na(+)-translocating NADH-quinone reductase subunit A|Vibrio cholerae O395 (345073)
>2KCO_1|Chain A|30S ribosomal protein S8e|Sulfolobus solfataricus P2 (273057)
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 \(w=f(c)\)
4BAO , Knot 24 36 0.79 28 34 34 
TFGSGEADCGLRPLFEKKSLEDKTERELLESYIDGR
9UDA , Knot 178 446 0.81 40 222 419 
MITIKKGLDLPIAGTPSQVISDGKAIKKVALLGEEYVGMRPTMHVRVGDEVKKAQILFEDKKNPGVKFTSPVSGKVVEINRGAKRVLQSVVIEVAGDDQVTFDKFEANQLASLNRDAIKTQLVESGLWTAFRTRPFSKVPAIDSTSEAIFVTAMDTNPLAAEPTVVINEQSEAFVAGLDVLSALTTGKVYVCKKGTSLPRSQQPNVEEHVFDGPHPAGLAGTHMHFLYPVSADHVAWSINYQDVIAVGQLFLTGELYTQRVVSLAGPVVNKPRLVRTVMGASLEQLVDSEIMPGEVRIISGSVLSGTKATGPHAYLGRYHLQVSVLREGRDKELFGWAMPGKNKFSVTRSFLGHLFKGQVYNMTTTTNGSDRSMVPIGNYEKVMPLDMEPTLLLRDLCAGDSDSAVRLGALELDEEDLALCTFVCPGKYEYGQLLRECLDKIEKEG
2KCO , Knot 68 133 0.83 36 104 130 
MGFYQGPDNRKITGGLKGKHRDKRKYEIGNPPTFTTLSAEDIRIKDRTLGGNFKVRLKYTTTANVLDPATNTAKKVKILEILETPANKELARRGIIIRGAKIRTEAGLAVVTSRPGQDGVINAVLLKNESQRS

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 \(c\).

\(|P_{f(4BAO_1)}(2) \setminus P_{f(9UDA_1)}(2)|=7\), \(|P_{f(9UDA_1)}(2) \setminus P_{f(4BAO_1)}(2)|=195\). 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 1:011010100110111000010000000110001010
Pair \(Z_2\) Length of longest common subsequence
4BAO_1,9UDA_1 202 3
4BAO_1,2KCO_1 112 2
9UDA_1,2KCO_1 166 4

Newick tree

 
[
	9UDA_1:10.72,
	[
		4BAO_1:56,2KCO_1:56
	]:45.72
]

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 \((0.85)(0.8)(\frac{482 }{\log_{20} 482}-\frac{36}{\log_{20}36})=138.\)
Status Protein1 Protein2 d d1/2
Query variables 4BAO_1 9UDA_1 172 92
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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