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Parikh vectors
1XCR_1 6GHH_1 7HMK_1 Letter Amino acid
9 5 4 C Cysteine
9 9 10 Q Glutamine
29 4 14 G Glycine
13 2 10 H Histidine
6 4 6 M Methionine
22 4 11 F Phenylalanine
9 3 9 R Arginine
14 10 10 D Aspartic acid
1 2 6 W Tryptophan
23 9 14 V Valine
11 4 7 Y Tyrosine
11 5 10 N Asparagine
27 22 12 L Leucine
24 15 7 K Lycine
16 7 15 S Serine
18 15 8 A Alanine
14 10 9 I Isoleucine
13 8 10 T Threonine
24 16 7 E Glutamic acid
23 8 9 P Proline 

1XCR_1|Chains A, B|hypothetical protein PTD012|Homo sapiens (9606)
>6GHH_1|Chain A|Beta-lactoglobulin|Bos taurus (9913)
>7HMK_1|Chain A[auth B]|E3 ubiquitin-protein ligase TRIM21|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 \(w=f(c)\)
1XCR , Knot 139 316 0.84 40 203 304 
GSACAEFSFHVPSLEELAGVMQKGLKDNFADVQVSVVDCPDLTKEPFTFPVKGICGKTRIAEVGGVPYLLPLVNQKKVYDLNKIAKEIKLPGAFILGAGAGPFQTLGFNSEFMPVIQTESEHKPPVNGSYFAHVNPADGGCLLEKYSEKCHDFQCALLANLFASEGQPGKVIEVKAKRRTGPLNFVTCMRETLEKHYGNKPIGMGGTFIIQKGKVKSHIMPAEFSSCPLNSDEEVNKWLHFYEMKAPLVCLPVFVSRDPGFDLRLEHTHFFSRHGEGGHYHYDTTPDIVEYLGYFLPAEFLYRIDQPKETHSIGRD
6GHH , Knot 79 162 0.82 40 121 157 
LIVTQTMKGLDIQKVAGTWYSLAMAASDISLLDAQSAPLRVYVEELKPTPEGDLEILLQKWENGECAQKKIIAEKTKIPAVFKIDALNENKVLVLDTDYKKYLLFCMENSAEPEQSLACQCLVRTPEVDDEALEKFDKALKALPMHIRLSFNPTQLEEQCHI
7HMK , Knot 91 188 0.84 40 148 179 
MHHHHHHMVHITLDRNTANSWLIISKDRRQVRMGDTHQNVSDNKERFSNYPMVLGAQRFSSGKMYWEVDVTQKEAWDLGVCRDSVQRKGQFSLSPENGFWTIWLWQDSYEAGTSPQTTLHIQVPPCQIGIFVDYEAGVVSFYNITDHGSLIYTFSECVFAGPLRPFFNVGFNYSGGNAAPLKLCPLKM

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 \(c\).

\(|P_{f(1XCR_1)}(2) \setminus P_{f(6GHH_1)}(2)|=127\), \(|P_{f(6GHH_1)}(2) \setminus P_{f(1XCR_1)}(2)|=45\). 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 1:1010101010110100111110011000110101011001010001101110110100011011111011111000010010011001011111111111111001110001111100000001110100110101101101100000000010011110111001011011010100001110110010001000010011111101110010100011110100011000001001101001011110111110001110101000011000101100000001011001101111011001001000001100
Pair \(Z_2\) Length of longest common subsequence
1XCR_1,6GHH_1 172 3
1XCR_1,7HMK_1 203 3
6GHH_1,7HMK_1 177 3

Newick tree

 
[
	7HMK_1:98.10,
	[
		1XCR_1:86,6GHH_1:86
	]:12.10
]

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 \((0.85)(0.8)(\frac{478 }{\log_{20} 478}-\frac{162}{\log_{20}162})=92.9\)
Status Protein1 Protein2 d d1/2
Query variables 1XCR_1 6GHH_1 119 88
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]