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Parikh vectors
6KBX_1 3RUJ_1 4XNL_1 Letter Amino acid
3 3 5 M Methionine
6 11 34 T Threonine
13 22 19 V Valine
25 12 44 A Alanine
15 13 36 R Arginine
5 18 22 N Asparagine
9 23 13 K Lycine
3 7 0 C Cysteine
19 12 26 E Glutamic acid
17 36 41 L Leucine
10 30 33 S Serine
13 15 17 I Isoleucine
12 20 23 F Phenylalanine
17 15 18 P Proline
8 5 15 W Tryptophan
14 23 46 D Aspartic acid
11 12 20 Q Glutamine
17 8 52 G Glycine
5 4 11 H Histidine
5 7 15 Y Tyrosine 

6KBX_1|Chain A[auth B]|SOS response-associated protein|Escherichia coli (562)
>3RUJ_1|Chain A|Ubiquitin-like modifier-activating enzyme ATG7|Saccharomyces cerevisiae (559292)
>4XNL_1|Chain A|Alginate production protein AlgE|Pseudomonas aeruginosa PAO1 (208964)
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 \(w=f(c)\)
6KBX , Knot 105 227 0.83 40 157 221 
CGRFAQSQTREDYLALLAEDIERDIPYDPEPIGRYNVAPGTKVLLLSERDEHLHLDPVFWGYAPGWWDKPPLINARVETAATSRMFKPLWQHGRAICFADGWFEWKKEGDKKQPFFIYRADGQPIFMAAIGSTPFERGDEAEGFLIVTAAADQGLVDIHDRRPLVLSPEAAREWMRQEISGKEASEIAASGCVPANQFSWHPVSRAVGNVKNQGAELIQPVLEVLFQ
3RUJ , Knot 128 296 0.82 40 179 284 
GSMSSERVLSYAPAFKSFLDTSFFQELSRLKLDVLKLDSTCQPLTVNLDLHNIPKSADQVPLFLTNRSFEKHNNKRTNEVPLQGSIFNFNVLDEFKNLDKQLFLHQRALECWEDGIKDINKCVSFVIISFADLKKYRFYYWLGVPCFQRPSSTVLHVRPEPSLKGLFSKCQKWFDVNYSKWVCILDADDEIVNYDKCIIRKTKVLAIRDTSTMENVPSALTKNFLSVLQYDVPDLIDFKLLIIRQNEGSFALNATFASIDPQSSSSNPDMKVSGWERNVQGKLAPRVVDLSSLLDP
4XNL , Knot 190 490 0.80 38 229 459 
MNSSRSVNPRPSFAPRALSLAIALLLGAPAFAANSGEAPKNFGLDVKITGESENDRDLGTAPGGTLNDIGIDLRPWAFGQWGDWSAYFMGQAVAATDTIETDTLQSDTDDGNNSRNDGREPDKSYLAAREFWVDYAGLTAYPGEHLRFGRQRLREDSGQWQDTNIEALNWSFETTLLNAHAGVAQRFSEYRTDLDELAPEDKDRTHVFGDISTQWAPHHRIGVRIHHADDSGHLRRPGEEVDNLDKTYTGQLTWLGIEATGDAYNYRSSMPLNYWASATWLTGDRDNLTTTTVDDRRIATGKQSGDVNAFGVDLGLRWNIDEQWKAGVGYARGSGGGKDGEEQFQQTGLESNRSNFTGTRSRVHRFGEAFRGELSNLQAATLFGSWQLREDYDASLVYHKFWRVDDDSDIGTSGINAALQPGEKDIGQELDLVVTKYFKQGLLPASMSQYVDEPSALIRFRGGLFKPGDAYGPGTDSTMHRAFVDFIWRF

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 \(c\).

\(|P_{f(6KBX_1)}(2) \setminus P_{f(3RUJ_1)}(2)|=79\), \(|P_{f(3RUJ_1)}(2) \setminus P_{f(6KBX_1)}(2)|=101\). 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 1:01011000000001111100100011001011100011110011110000001010111110111110011110101001100011011100101101101110100010000111100101011111111001100100101111101110011101000011110101100110001010010011101011100101011001110100011011011101110
Pair \(Z_2\) Length of longest common subsequence
6KBX_1,3RUJ_1 180 4
6KBX_1,4XNL_1 178 4
3RUJ_1,4XNL_1 172 4

Newick tree

 
[
	6KBX_1:90.63,
	[
		4XNL_1:86,3RUJ_1:86
	]:4.63
]

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 \((0.85)(0.8)(\frac{523 }{\log_{20} 523}-\frac{227}{\log_{20}227})=84.9\)
Status Protein1 Protein2 d d1/2
Query variables 6KBX_1 3RUJ_1 108 97
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]