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Parikh vectors
8QVE_1 7AWH_1 4RBL_1 Letter Amino acid
10 31 6 N Asparagine
7 8 4 M Methionine
11 37 14 F Phenylalanine
46 30 18 P Proline
37 32 24 E Glutamic acid
17 26 18 I Isoleucine
47 48 32 L Leucine
3 31 10 K Lycine
52 43 22 G Glycine
16 7 12 H Histidine
32 33 7 T Threonine
10 18 7 Y Tyrosine
57 32 9 A Alanine
34 22 21 R Arginine
17 21 19 D Aspartic acid
7 7 5 C Cysteine
40 31 22 V Valine
21 23 11 Q Glutamine
24 35 21 S Serine
7 14 7 W Tryptophan 

8QVE_1|Chain A|Pyranose oxidase|Deinococcus aerius (200253)
>7AWH_1|Chain A|Cholinesterase|Homo sapiens (9606)
>4RBL_1|Chain A|Serine/threonine-protein kinase pim-1|Homo sapiens (9606)
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 \(w=f(c)\)
8QVE , Knot 198 495 0.82 40 226 455 
QEIDVLIVGSGPIGATFARVLAERSPQTSVLLIDAGPQLTPRPGLHVKNIPDPAERERAQVRSQGPTQYVYGTPTPAQRAEAGEEAHEGAPTRPGLLARPGTFLLGSPEMPAAALSTNVGGMGAHWTCACPPPGNTERIGFIPEEEWQAAFDEANRLLGVTQQAYPETDAARAIRQALMGAFGERLSPERPVQPMPLAARTEGGERRWAGVDVVLGPLAEEGHGTGFTLHSETLCHRLTVDGDRVTGAVLEHLPTGERYEVRARVVVVAADSLRTPQLLWASGIRPPALGHYLNDQPQVLGAVQLNPALLPENAGRAAGGVSWVPFHSPSHPYHGQVMELEASPIPIPVTGTPVVGLGWFCAKEVRYEDHIAFSESETDYLGLPRMTVHHTLTPADEAVIEQAKAEVELAIRALGRPLPFLLPSGSSLHYQGTVRMGEQDDGQSVCDSHSRVWGYRNLFVGGNGVIPTSTACNPTITSVALAVRACDGIAALLAG
7AWH , Knot 218 529 0.86 40 265 503 
EDDIIIATKNGKVRGMQLTVFGGTVTAFLGIPYAQPPLGRLRFKKPQSLTKWSDIWNATKYANSCCQNIDQSFPGFHGSEMWNPNTDLSEDCLYLNVWIPAPKPKNATVLIWIYGGGFQTGTSSLHVYDGKFLARVERVIVVSMNYRVGALGFLALPGNPEAPGNMGLFDQQLALQWVQKNIAAFGGNPKSVTLFGESAGAASVSLHLLSPGSHSLFTRAILQSGSFNAPWAVTSLYEARNRTLNLAKLTGCSRENETEIIKCLRNKDPQEILLNEAFVVPYGTPLSVNFGPTVDGDFLTDMPDILLELGQFKKTQILVGVNKDEGTAFLVYGAPGFSKDNNSIITRKEFQEGLKIFFPGVSEFGKESILFHYTDWVDDQRPENYREALGDVVGDYNFICPALEFTKKFSEWGNNAFFYYFEHRSSKLPWPEWMGVMHGYEIEFVFGLPLERRDQYTKAEEILSRSIVKRWANFAKYGNPQETQNQSTSWPVFKSTEQKYLTLNTESTRIMTKLRAQQCRFWTSFFPKV
4RBL , Knot 126 289 0.82 40 183 272 
GPHMKEKEPLESQYQVGPLLGSGGFGSVYSGIRVSDNLPVAIKHVEKDRISDWGELPNGTRVPMEVVLLKKVSSGFSGVIRLLDWFERPDSFVLILERPEPVQDLFDFITERGALQEELARSFFWQVLEAVRHCHNCGVLHRDIKDENILIDLNRGELKLIDFGSGALLKDTVYTDFDGTRVYSPPEWIRYHRYHGRSAAVWSLGILLYDMVCGDIPFEHDEEIIRGQVFFRQRVSSECQHLIRWCLALRPSDRPTFEEIQNHPWMQDVLLPQETAEIHLHSLSPGPSK

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 \(c\).

\(|P_{f(8QVE_1)}(2) \setminus P_{f(7AWH_1)}(2)|=55\), \(|P_{f(7AWH_1)}(2) \setminus P_{f(8QVE_1)}(2)|=94\). 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 1:001011111011111011011100010001111011101010111010011011000010100011000101010110010110010011100111110110111101011111100011111101001011110000111110001011100100111100010100011011001111111001010011011111100011000111101111111001010110100001000101010010111100110100001010111111001001011110110111110010001011111010111110011011111011110010010010110101011111101011111111010010000011100000001111010100010110011100101010111011101111111010010001010110000100100000011100011111011110001001010011111010011111111
Pair \(Z_2\) Length of longest common subsequence
8QVE_1,7AWH_1 149 5
8QVE_1,4RBL_1 165 4
7AWH_1,4RBL_1 174 4

Newick tree

 
[
	4RBL_1:87.93,
	[
		8QVE_1:74.5,7AWH_1:74.5
	]:13.43
]

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 \((0.85)(0.8)(\frac{1024 }{\log_{20} 1024}-\frac{495}{\log_{20}495})=138.\)
Status Protein1 Protein2 d d1/2
Query variables 8QVE_1 7AWH_1 177 169
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]