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Parikh vectors
4GRH_1 9EAK_1 8PDK_1 Letter Amino acid
17 0 4 I Isoleucine
4 0 8 K Lycine
5 0 0 M Methionine
35 0 8 V Valine
8 0 2 N Asparagine
30 0 5 D Aspartic acid
23 0 1 H Histidine
18 0 2 T Threonine
25 0 8 Q Glutamine
26 0 18 E Glutamic acid
23 0 7 S Serine
50 0 26 L Leucine
31 0 8 P Proline
5 0 4 Y Tyrosine
55 33 15 A Alanine
37 0 11 R Arginine
42 24 13 G Glycine
4 15 0 C Cysteine
13 0 4 F Phenylalanine
6 0 2 W Tryptophan 

4GRH_1|Chain A|aminodeoxychorismate synthase|Stenotrophomonas maltophilia (522373)
>9EAK_1|Chain A[auth B]|sgRNA|synthetic construct (32630)
>8PDK_1|Chains A, B|c-di-GMP binding domain of the ATPase enzyme PilF|Thermus thermophilus HB27 (262724)
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 \(w=f(c)\)
4GRH , Knot 182 457 0.81 40 212 433 
GSHGTHAPLIHPLPHPIDLLALQQHDPARFPLLMESTASGTAQGRWSLLLVAQGDGLRLDADGQVRDQHDLVQPGTFLQALDRAWQHERLSHDGSHSLPFRGGWALMLDYEVASQIEPVLPARARGDGRPTALALRCPAAVLHDHHNEASFVIAEAGEQALLDALVALASAALPEAGQGWQPPQAVGEDAPQRFTDGVRRVIEYLRAGDVFQVNLSRRWNAQFAAPVSPQALYAQLRRANPAPFAGLFSAHGRHVVSSSPERLVSVHAGHAQTRPIAGTRPRFEGDDDAARIQELVGHPKERAEHVMLIDLERNDLGRICLPGTVVVDELMTVESYAHVHHIVSNVSGHLRPEVTPGEVIAATFPGGTITGCPKVRCMQIISELEQVPRGAYTGAFGWLNRDGDLDLNILIRTAEVDGHEVSFRTGAGIVVDSDPDKELDETRAKARGLLRALGQQG
9EAK , Knot 30 97 0.47 8 16 47 
CGAUAAAGAUGAGACGCGUUUUAGAGCUAGAAAUAGCAAGUUAAAAUAAGGCUAGUCCGUUAUCAACUUGAAAAAGUGGCACCGAGUCGGUGCUUUU
8PDK , Knot 67 146 0.76 36 101 137 
GSSGEGQKDLKLGELLLQKGWISREALEEALVEQEKTGDLLGRILVRKGLPEEALYRALAEQKGLEFLESTEGIVPDPSAALLLLRSDALRYGAVPIGFQNGEVEVVLSDPRHKEAVAQLLNRPARFYLALPQAWEELFRRAYPQK

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 \(c\).

\(|P_{f(4GRH_1)}(2) \setminus P_{f(9EAK_1)}(2)|=207\), \(|P_{f(9EAK_1)}(2) \setminus P_{f(4GRH_1)}(2)|=11\). 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 1:1001001111011101101111000011011111000101010101011111010110101010100000110110110110011000010001000111011111110001100101111101010101011110011111000000101111011001110111111011110110110110111001100100110011001011011010100010101111101011010100101111111101010011000100110101101000111100101010001101001110100010011110100001101011101110011010001010011001010101010110111101111010101010010110010011011001111110001010101110010101001010011111100010001000010101110111001
Pair \(Z_2\) Length of longest common subsequence
4GRH_1,9EAK_1 218 3
4GRH_1,8PDK_1 163 4
9EAK_1,8PDK_1 113 2

Newick tree

 
[
	4GRH_1:10.22,
	[
		8PDK_1:56.5,9EAK_1:56.5
	]:49.72
]

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 \((0.85)(0.8)(\frac{554 }{\log_{20} 554}-\frac{97}{\log_{20}97})=135.\)
Status Protein1 Protein2 d d1/2
Query variables 4GRH_1 9EAK_1 180 102.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]