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Parikh vectors
8CFB_1 7PET_1 5BVK_1 Letter Amino acid
31 7 18 E Glutamic acid
31 7 12 I Isoleucine
15 3 11 M Methionine
30 4 21 D Aspartic acid
18 8 17 Q Glutamine
41 7 25 G Glycine
12 2 6 H Histidine
34 13 11 K Lycine
19 6 14 S Serine
12 3 8 Y Tyrosine
35 6 22 V Valine
21 1 12 N Asparagine
11 4 16 F Phenylalanine
27 10 11 T Threonine
46 19 27 A Alanine
8 0 8 C Cysteine
41 12 37 L Leucine
16 6 19 P Proline
7 0 5 W Tryptophan
17 18 24 R Arginine 

8CFB_1|Chains A, B, C, D|Adenosylhomocysteinase|Pseudomonas aeruginosa PAO1 (208964)
>7PET_1|Chains A, DA[auth E], E[auth K], I[auth O], M[auth a], R[auth e], V[auth k], Z[auth o]|Histone H3.2|Homo sapiens (9606)
>5BVK_1|Chain A|Epithelial discoidin domain-containing receptor 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 \(w=f(c)\)
8CFB , Knot 195 472 0.84 40 247 448 
SNAMSAVMTPAGFTDYKVADITLAAWGRRELIIAESEMPALMGLRRKYAGQQPLKGAKILGCIHMTIQTGVLIETLVALGAEVRWSSCNIFSTQDQAAAAIAAAGIPVFAWKGETEEEYEWCIEQTILKDGQPWDANMVLDDGGDLTEILHKKYPQMLERIHGITEETTTGVHRLLDMLKNGTLKVPAINVNDSVTKSKNDNKYGCRHSLNDAIKRGTDHLLSGKQALVIGYGDVGKGSSQSLRQEGMIVKVAEVDPICAMQACMDGFEVVSPYKNGINDGTEASIDAALLGKIDLIVTTTGNVNVCDANMLKALKKRAVVCNIGHFDNEIDTAFMRKNWAWEEVKPQVHKIHRTGKDGFDAHNDDYLILLAEGRLVNLGNATGHPSRIMDGSFANQVLAQIHLFEQKYADLPAAEKAKRLSVEVLPKKLDEEVALEMVKGFGGVVTQLTPKQAEYIGVSVEGPFKPDTYRY
7PET , Knot 65 136 0.78 36 102 128 
MARTKQTARKSTGGKAPRKQLATKAARKSAPATGGVKKPHRYRPGTVALREIRRYQKSTELLIRKLPFQRLVREIAQDFKTDLRFQSSAVMALQEASEAYLVGLFEDTNLAAIHAKRVTIMPKDIQLARRIRGERA
5BVK , Knot 142 324 0.84 40 200 312 
GAMGSAVGDGPPRVDFPRSRLRFKEKLGEGQFGEVHLCEVDSPQDLVSLDFPLNVRKGHPLLVAVKILRPDATKNARNDFLKEVKIMSRLKDPNIIRLLGVCVQDDPLCMITDYMENGDLNQFLSAHQLEDKAAEGAPGDGQAAQGPTISYPMLLHVAAQIASGMRYLATLNFVHRDLATRNCLVGENFTIKIADFGMSRNLYAGDYYRVQGRAVLPIRWMAWECILMGKFTTASDVWAFGVTLWEVLMLCRAQPFGQLTDEQVIENAGEFFRDQGRQVYLSRPPACPQGLYELMLRCWSRESEQRPPFSQLHRFLAEDALNTV

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 \(c\).

\(|P_{f(8CFB_1)}(2) \setminus P_{f(7PET_1)}(2)|=166\), \(|P_{f(7PET_1)}(2) \setminus P_{f(8CFB_1)}(2)|=21\). 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 1:0011011101111000011010111110001111000111111100001100110110111010101001111001111110101000011000001111111111111110100000001010001100101101011100110100110000101100101100000011001101100101011110100010000000001000010011001000110100111110101101000010001111011010110110101011011010001100100101011111010111000101010010110110001110011010001001110001110010101001000100110100000111110101101101010100110101100111010110000101111001001010111001000111011011111100101001001110101110100000
Pair \(Z_2\) Length of longest common subsequence
8CFB_1,7PET_1 187 4
8CFB_1,5BVK_1 163 3
7PET_1,5BVK_1 182 3

Newick tree

 
[
	7PET_1:95.57,
	[
		8CFB_1:81.5,5BVK_1:81.5
	]:14.07
]

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 \((0.85)(0.8)(\frac{608 }{\log_{20} 608}-\frac{136}{\log_{20}136})=136.\)
Status Protein1 Protein2 d d1/2
Query variables 8CFB_1 7PET_1 175 110
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]