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Parikh vectors
4EMV_1 8XCR_1 5HQL_1 Letter Amino acid
7 5 12 H Histidine
17 24 30 L Leucine
16 34 23 K Lycine
17 39 23 V Valine
0 1 3 C Cysteine
13 35 21 E Glutamic acid
26 37 46 G Glycine
9 9 30 F Phenylalanine
5 20 22 P Proline
21 23 25 T Threonine
13 40 59 A Alanine
8 20 23 R Arginine
15 24 31 D Aspartic acid
15 29 23 I Isoleucine
14 12 17 S Serine
12 15 20 N Asparagine
4 10 16 Q Glutamine
6 14 15 M Methionine
2 12 5 W Tryptophan
6 16 17 Y Tyrosine 

4EMV_1|Chain A|DNA topoisomerase IV, B subunit|Streptococcus pneumoniae GA47373 (760835)
>8XCR_1|Chain A|Glutamate dehydrogenase|Thermococcus profundus (49899)
>5HQL_1|Chains A, B, C, D, E, F|Ribulose bisphosphate carboxylase|Rhodopseudomonas palustris (1076)
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 \(w=f(c)\)
4EMV , Knot 101 226 0.80 38 154 218 
MSKKEININNYNDDAIQVLEGLDAVRKRPGMYIGSTDGAGLHHLVWEIVDNAVDEALSGFGDRIDVTINKDGSLTVQDHGRGMPTGMHAMGIPTVEVIFTILHAGGKFGQGGYKTSGGLHGVGSSVVNALSSWLEVEITRDGAVYKQRFENGGKPVTTLKKIGTALKSKTGTKVTFMPDATIFSTTDFKYNTISERLNESAFLLKNVTLSLTDKRTDEAIEFHYEN
8XCR , Knot 174 419 0.83 40 213 401 
MVEIDPFEMAVKQLERAAQYMDISEEALEWLKKPMRIVEVSVPIEMDDGSVKVFTGFRVQHNWARGPTKGGIRWHPAETLSTVKALATWMTWKVAVVDLPYGGGKGGIIVNPKELSEREQERLARAYIRAVYDVIGPWTDIPAPDVYTNPKIMGWMMDEYETIMRRKGPAFGVITGKPLSIGGSLGRGTATAQGAIFTIREAAKALGIDLKGKKIAVQGYGNAGYYTAKLAKEQLGMTVVAVSDSRGGIYNPDGLDPDEVLKWKREHGSVKDFPGATNITNEELLELEVDVLAPAAIEEVITEKNADNIKAKIVAEVANGPVTPEADDILREKGILQIPDFLCNAGGVTVSYFEWVQNINGYYWTEEEVREKLDKKMTKAFWEVYNTHKDKNIHMRDAAYVVAVSRVYQAMKDRGWVKK
5HQL , Knot 192 461 0.85 40 239 437 
MDQSNRYANLNLKESELIAGGRHVLCAYIMKPKAGFGNFIQTAAHFVAESSTGTNVEVSTTDDFTRGVDALVYEVDEANSLMKIAYPIELFDRNVIDGRAMIASFLTLTIGNNQGMGDVEYAKMYDFYVPPAYLKLFDGPSTTIKDLWRVLGRPVINGGFIVGTIIKPKLGLRPQPFANACYDFWLGGDFIKNDEPQGNQVFAPFKDTVRAVADAMRRAQDKTGEAKLFSFNITADDHYEMLARGEFILETFADNADHIAFLVDGYVAGPAAVTTARRAFPKQYLHYHRAGHGAVTSPQSKRGYTAFVLSKMARLQGASGIHTGTMGFGKAEGEAADRAIAYMITEDAADGPYFHQEWLGMNPTTPIISGGMNALRMPGFFDNLGHSNLIMTAGGGAFGHVDGGAAGAKSLRQAEQCWKQGADPVEFAKDHREFARAFESFPQDADKLYPNWRAKLKPQAA

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 \(c\).

\(|P_{f(4EMV_1)}(2) \setminus P_{f(8XCR_1)}(2)|=51\), \(|P_{f(8XCR_1)}(2) \setminus P_{f(4EMV_1)}(2)|=110\). 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 1:1000010100000011011011011000111011000111100111011001100110111001010100010101000101110110111110101110110111011011000011101110011011001101010001110000100110110010011011000010010111010110000100001000100011110010101000000011010000
Pair \(Z_2\) Length of longest common subsequence
4EMV_1,8XCR_1 161 4
4EMV_1,5HQL_1 169 4
8XCR_1,5HQL_1 160 4

Newick tree

 
[
	4EMV_1:83.34,
	[
		8XCR_1:80,5HQL_1:80
	]:3.34
]

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 \((0.85)(0.8)(\frac{645 }{\log_{20} 645}-\frac{226}{\log_{20}226})=118.\)
Status Protein1 Protein2 d d1/2
Query variables 4EMV_1 8XCR_1 147 111.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]