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Parikh vectors
6WAD_1 8VST_1 6BDX_1 Letter Amino acid
30 12 15 R Arginine
9 3 10 H Histidine
40 24 21 L Leucine
9 12 5 M Methionine
12 14 8 F Phenylalanine
16 2 7 P Proline
29 12 35 A Alanine
6 3 2 C Cysteine
39 18 13 E Glutamic acid
11 33 19 I Isoleucine
14 14 18 T Threonine
8 1 1 W Tryptophan
30 9 25 V Valine
17 10 19 D Aspartic acid
18 7 5 Q Glutamine
37 8 33 G Glycine
4 21 10 N Asparagine
13 22 11 K Lycine
24 28 9 S Serine
10 12 4 Y Tyrosine 

6WAD_1|Chain A|Protein arginine N-methyltransferase 6|Homo sapiens (9606)
>8VST_1|Chains A, B|Regulatory protein Rns|Escherichia coli (562)
>6BDX_1|Chain A|4-hydroxy-tetrahydrodipicolinate reductase|Neisseria gonorrhoeae (485)
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 \(w=f(c)\)
6WAD , Knot 154 376 0.81 40 207 355 
GMSQPKKRKLESGGGGEGGEGTEEEDGAEREAALERPRRTKRERDQLYYECYSDVSVHEEMIADRVRTDAYRLGILRNWAALRGKTVLDVGAGTGILSIFCAQAGARRVYAVEASAIWQQAREVVRFNGLEDRVHVLPGPVETVELPEQVDAIVSEWMGYGLLHESMLSSVLHARTKWLKEGGLLLPASAELFIVPISDQMLEWRLGFWSQVKQHYGVDMSCLEGFATRCLMGHSEIVVQGLSGEDVLARPQRFAQLELSRAGLEQELEAGVGGRFRCSCYGSAPMHGFAIWFQVTFPGGESEKPLVLSTSPFHPATHWKQALLYLNEPVQVEQDTDVSGEITLLPSRDNPRRLRVLLRYKVGDQEEKTKDFAMED
8VST , Knot 117 265 0.82 40 160 252 
MDFKYTEEKETIKINNIMIAKYTVLYTSNCIMDIYSEEEKITCFSNRLVFLERGVNISVRMQKQILSEKPYVAFALNGDMLRHLKDALMIIYGMSKIDTNACRSMSRKIMTTEVNKTLLDELKNINSHDNSAFISSLIYLISKLENNEKIIESIYISSVSFFSDKVRNLIEKDLSRKWTLGIIADAFNASEITIRKRLESENTNFNQILMQLRMSKAALLLLENSYQISQISNMIGISSASYFIRIFNKHYGVTPKQFFTYFKGG
6BDX , Knot 114 270 0.78 40 153 246 
GMIPLKIAIAGANGRMGRVLVEAVNNHPDTVLSGALEHSGSEALGLDAGYAVGLKTGIAISDDVDAVLAQSDVLIDFTRPEPTLKHLQKCVEKQVNIIIGTTGFDDAGKAAIRAAAEKTGIVFAANFSVGVNLTFHILDTVARVLNEGYDIEIIEGHHRHKVDAPSGTALRMGEVIAGALGRDLKQCAVYGREGHTGPRDPSTIGFATVRAGDIVGDHTALFATDGERVEITHKAGSRMTFAAGAVRAAVWVNGKTGLYDMQDVLGLNNR

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 \(c\).

\(|P_{f(6WAD_1)}(2) \setminus P_{f(8VST_1)}(2)|=125\), \(|P_{f(8VST_1)}(2) \setminus P_{f(6WAD_1)}(2)|=78\). 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 1:1100100001001111011010000011000111001000000000100000001010001110010001001111001111010011011110111011010111001011010111001001101011000101111110010110010111001110111000110011010001100111111101011111100011010111100100001101001011100011100011101101001110100110101001110001011111010000010111011111101011110000111100011011001001110100110100000101010111000010010111000110000000011100
Pair \(Z_2\) Length of longest common subsequence
6WAD_1,8VST_1 203 4
6WAD_1,6BDX_1 164 4
8VST_1,6BDX_1 169 3

Newick tree

 
[
	8VST_1:96.88,
	[
		6WAD_1:82,6BDX_1:82
	]:14.88
]

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 \((0.85)(0.8)(\frac{641 }{\log_{20} 641}-\frac{265}{\log_{20}265})=105.\)
Status Protein1 Protein2 d d1/2
Query variables 6WAD_1 8VST_1 133 113.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]