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Parikh vectors
6GLK_1 5FKN_1 8QXW_1 Letter Amino acid
13 16 33 E Glutamic acid
17 12 35 G Glycine
9 7 15 H Histidine
7 3 8 M Methionine
6 23 49 A Alanine
1 7 27 N Asparagine
14 13 28 D Aspartic acid
6 8 38 I Isoleucine
9 7 25 P Proline
4 2 7 W Tryptophan
10 6 24 K Lycine
10 11 32 T Threonine
6 5 6 Y Tyrosine
13 9 36 V Valine
8 16 13 R Arginine
12 11 25 Q Glutamine
17 33 55 L Leucine
2 0 6 C Cysteine
12 7 15 F Phenylalanine
6 11 33 S Serine 

6GLK_1|Chain A|7,8-dihydro-8-oxoguanine triphosphatase|Homo sapiens (9606)
>5FKN_1|Chains A, B|TETRACYCLINE REPRESSOR, CLASS D, T103A MUTANT|ESCHERICHIA COLI (83333)
>8QXW_1|Chain A|Importin subunit alpha-1|Mus musculus (10090)
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 \(w=f(c)\)
6GLK , Knot 84 182 0.80 40 131 173 
MKHHHHHHPMSDYDIPTTENLYFQGAMGASRLYTLVLVLQPQRVLLGMKKRGFGAGRWAGFGGKVQEGETIEDGARRELQEESGLTVDALHKVGQIVFEFVGEPELMDVHVFCTDSIQGTPVESDEMRPCWFQLDQIPFKDMWPDDSYWFPLLLQKKKFHGYFKFQGQDTILDYTLREVDTV
5FKN , Knot 95 207 0.81 38 139 194 
SRLNRESVIDAALELLNETGIDGLTTRKLAQKLGIEQPTLYWHVKNKRALLDALAVEILARHHDYSLPAAGESWQSFLRNNAMSFRRALLRYRDGAKVHLGARPDEKQYDTVETQLRFMTENGFSLRDGLYAISAVSHFTLGAVLEQQEHTAALTDRPAAPDENLPPLLREALQIMDSDDGEQAFLHGLESLIRGFEVQLTALLQIV
8QXW , Knot 206 510 0.84 40 248 467 
MHHHHHHSSGLVPRGSGMLETAAALFERNHMDSPDLGTDDDDLAMADIGSNQGTVNWSVEDIVKGINSNNLESQLQATQAARKLLSREKQPPIDNIIRAGLIPKFVSFLGKTDCSPIQFESAWALTNIASGTSEQTKAVVDGGAIPAFISLLASPHAHISEQAVWALGNIAGDGSAFRDLVIKHGAIDPLLALLAVPDLSTLACGYLRNLTWTLSNLCRNKNPAPPLDAVEQILPTLVRLLHHNDPEVLADSCWAISYLTDGPNERIEMVVKKGVVPQLVKLLGATELPIVTPALRAIGNIVTGTDEQTQKVIDAGALAVFPSLLTNPKTNIQKEATWTMSNITAGRQDQIQQVVNHGLVPFLVGVLSKADFKTQKEAAWAITNYTSGGTVEQIVYLVHCGIIEPLMNLLSAKDTKIIQVILDAISNIFQAAEKLGETEKLSIMIEECGGLDKIEALQRHENESVYKASLNLIEKYFSVEEEEDQNVVPETTSEGFAFQVQDGAPGTFNF

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 \(c\).

\(|P_{f(6GLK_1)}(2) \setminus P_{f(5FKN_1)}(2)|=65\), \(|P_{f(5FKN_1)}(2) \setminus P_{f(6GLK_1)}(2)|=73\). 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 1:10000000110000110000101011111001001111101001111100011111011111101001001001100010000110101100110111011101011010110000101011000010101101001110011100001111110000101010101000110001001001
Pair \(Z_2\) Length of longest common subsequence
6GLK_1,5FKN_1 138 3
6GLK_1,8QXW_1 181 6
5FKN_1,8QXW_1 181 4

Newick tree

 
[
	8QXW_1:96.60,
	[
		6GLK_1:69,5FKN_1:69
	]:27.60
]

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 \((0.85)(0.8)(\frac{389 }{\log_{20} 389}-\frac{182}{\log_{20}182})=61.6\)
Status Protein1 Protein2 d d1/2
Query variables 6GLK_1 5FKN_1 74 70
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]