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Parikh vectors
6AMR_1 3VUG_1 6AQY_1 Letter Amino acid
6 16 14 H Histidine
8 34 25 L Leucine
8 28 28 K Lycine
1 17 9 R Arginine
11 15 21 N Asparagine
7 21 18 D Aspartic acid
3 12 24 F Phenylalanine
3 5 8 C Cysteine
8 18 22 G Glycine
4 15 11 M Methionine
6 14 21 T Threonine
2 4 5 W Tryptophan
9 15 15 Q Glutamine
5 19 14 P Proline
8 22 16 S Serine
3 17 13 Y Tyrosine
6 26 22 V Valine
7 20 21 A Alanine
8 26 20 E Glutamic acid
15 26 14 I Isoleucine 

6AMR_1|Chain A|Thioredoxin|Ehrlichia chaffeensis (strain ATCC CRL-10679 / Arkansas) (205920)
>3VUG_1|Chain A|Mitogen-activated protein kinase 8|Homo sapiens (9606)
>6AQY_1|Chains A, B, C, D, E, F|gdp-l-fucose synthetase|Naegleria fowleri (5763)
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 \(w=f(c)\)
6AMR , Knot 66 128 0.83 40 101 123 
MAHHHHHHMGTLEAQTQGPGSMIEQIGDSEFDNKVTSCNDNILILVDFWAPWCGPCRSLEPQLEKLAQQYTENVKIYKINIEDNQDVATQYGVSAIPTILMFKNGKKLSQVIGADISKIISEINNNIN
3VUG , Knot 158 370 0.84 40 228 356 
MSRSKRDNNFYSVEIGDSTFTVLKRYQNLKPIGSGAQGIVCAAYDAILERNVAIKKLSRPFQNQTHAKRAYRELVLMKCVNHKNIIGLLNVFTPQKSLEEFQDVYIVMELMDANLSQVIQMELDHERMSYLLYQMLCGIKHLHSAGIIHRDLKPSNIVVKSDCTLKILDFGLARTAGTSFMMTPYVVTRYYRAPEVILGMGYKENVDIWSVGCIMGEMIKGGVLFPGTDHIDQWNKVIEQLGTPSPEFMKKLQPTVRTYVENRPKYAGYSFEKLFPDVLFPADSEHNKLKASQARDLLSKMLVIDASKRISVDEALQHPYINVWYDPSEAEAPPPKIPDKQLDEREHTIEEWKELIYKEVMDLEHHHHHH
6AQY , Knot 149 341 0.85 40 220 324 
MAHHHHHHMAQREVSDQPITLTQDDVILVTGGTGLFGKAVEHIVKKEQIKGKWVFLGSKDGDLRDADACKQPFEKYRPTYVIHLAAFVGGLFKNMNFKVSFWLDNVNMNNNILTCCYDFGVKKTISCLSTCVFPDKIEYPITEEKLHEGPPHFSNNAYAYAKRMLDMLGRWYNEKAVNEGKSCLFTSVIPTNLFGPHDNFNVEAGHVLPGLMHKCYKAQQNGTDFVVFGSGKPLRQFLYSHDAARMLLWTMFNYQSEEPIMLCVSEEDEKSIGQVAQTIKDAFNFTGNMVFDTSKADGQYKKTSSNAKFLRLNPTFQYTPFEQAIKETVQWFLENYETARK

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 \(c\).

\(|P_{f(6AMR_1)}(2) \setminus P_{f(3VUG_1)}(2)|=30\), \(|P_{f(3VUG_1)}(2) \setminus P_{f(6AMR_1)}(2)|=157\). 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 1:11000000110101000111011001100010001000000111110111110110001010100110000001010010100000110001101110111100100100111101001100100010
Pair \(Z_2\) Length of longest common subsequence
6AMR_1,3VUG_1 187 6
6AMR_1,6AQY_1 185 9
3VUG_1,6AQY_1 180 6

Newick tree

 
[
	6AMR_1:93.98,
	[
		6AQY_1:90,3VUG_1:90
	]:3.98
]

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 \((0.85)(0.8)(\frac{498 }{\log_{20} 498}-\frac{128}{\log_{20}128})=109.\)
Status Protein1 Protein2 d d1/2
Query variables 6AMR_1 3VUG_1 136 88.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]