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Parikh vectors
9EPH_1 5XEK_1 6NWE_1 Letter Amino acid
22 2 20 P Proline
7 1 5 W Tryptophan
4 0 18 Y Tyrosine
22 0 31 V Valine
30 4 7 R Arginine
2 8 10 C Cysteine
16 3 17 E Glutamic acid
9 2 6 H Histidine
13 2 15 S Serine
24 0 27 T Threonine
34 3 28 L Leucine
48 2 29 A Alanine
7 1 15 N Asparagine
33 4 5 D Aspartic acid
11 1 12 Q Glutamine
21 1 31 F Phenylalanine
43 1 23 G Glycine
9 4 22 I Isoleucine
15 2 11 K Lycine
3 1 16 M Methionine 

9EPH_1|Chains A, B|Deferrochelatase|Streptomyces lividans (1916)
>5XEK_1|Chain A|RING finger protein 141|Homo sapiens (9606)
>6NWE_1|Chain A|Rhodopsin|Bos taurus (9913)
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 \(w=f(c)\)
9EPH , Knot 151 373 0.80 40 186 337 
DPAGADAGSAVPFHGAHQAGIATPVQDRLHFAAFDVTTEDRAAFVALLKEWTAAARRLTAGHAVGEGAYGGLPEAPPDDTGEALGLKPSRLTLTIGFGPSLFTRFGLADLRPEALADLPKFPGDNLDRARSGGDLCVQACADDPQVAVHAIRNLARIGFGKVVVRWSQLGFGKTSSTTPDKQTPRNLLGFKDGTRNIAGTEKDRLDRFVWAAEKDGTPWMTGGSYLVARRIRMHIETWDRASLQEQEDVFGRDKGEGAPVGKAKERDEPFLKAMKPDAHVRLAHPDSNGGATLLRRGYSFTDGTDGLGRLDAGLFFLAYQRDIRTGFVPVQRNLATDALNEFIQHVGSAVFAVPPGVRDADDWWGSTLFGKEA
5XEK , Knot 25 42 0.74 34 38 40 
EEECCICMDGRADLILPCAHSFCQKCIDKWSDRHRNCPICRL
6NWE , Knot 152 348 0.85 40 215 336 
MNGTEGPNFYVPFSNKTGVVRSPFEAPQYYLAEPWQFSMLAAYMFLLIMLGFPINFLTLYVTVQHKKLRTPLNYILLNLAVADLFMVFGGFTTTLYTSLHGYFVFGPTGCNLEGFFATLGGEIALWSLVVLAIERYVVVCKPMSNFRFGENHAIMGVAFTWVMALACAAPPLVGWSRYIPEGMQCSCGIDYYTPHEETNNESFVIYMFVVHFIIPLIVIFFCYGQLVFTVKEAAAQQQESATTQKAEKEVTRMVIIMVIAFLICWLPYAGVAFYIFTHQGSDFGPIFMTIPAFFAKTSAVYNPVIYIMMNKQFRNCMVTTLCCGKNPLGDDEASTTVSKTETSQVAPA

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 \(c\).

\(|P_{f(9EPH_1)}(2) \setminus P_{f(5XEK_1)}(2)|=172\), \(|P_{f(5XEK_1)}(2) \setminus P_{f(9EPH_1)}(2)|=24\). 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 1:0111101101111011001111011000101111010000011111110010111001011011101101111011100010111101001010111110110011110101011101101110010010011010101010010111011001101111011101001111000000100001001111001000111000001001111100010111011001110010101001001010000011100010111110100000111011010101011010001110110010010010011101011111110000100111110001100110011001101111111110010011100111001
Pair \(Z_2\) Length of longest common subsequence
9EPH_1,5XEK_1 196 3
9EPH_1,6NWE_1 175 3
5XEK_1,6NWE_1 215 3

Newick tree

 
[
	5XEK_1:10.49,
	[
		9EPH_1:87.5,6NWE_1:87.5
	]:19.99
]

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 \((0.85)(0.8)(\frac{415 }{\log_{20} 415}-\frac{42}{\log_{20}42})=117.\)
Status Protein1 Protein2 d d1/2
Query variables 9EPH_1 5XEK_1 143 80
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]