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Parikh vectors
7EUE_1 8ROO_1 2ONY_1 Letter Amino acid
13 20 24 G Glycine
3 2 4 M Methionine
5 6 9 F Phenylalanine
6 16 17 S Serine
4 26 20 R Arginine
6 17 14 D Aspartic acid
4 17 14 Q Glutamine
9 23 19 E Glutamic acid
3 7 7 K Lycine
11 16 20 P Proline
15 21 32 A Alanine
4 7 8 I Isoleucine
15 19 31 L Leucine
5 24 11 T Threonine
0 9 6 W Tryptophan
3 12 8 Y Tyrosine
7 11 12 H Histidine
2 4 6 C Cysteine
16 13 20 V Valine
0 6 7 N Asparagine 

7EUE_1|Chains A, B|Cupin domain-containing protein|Streptomyces albus (1888)
>8ROO_1|Chain A|MHC class I antigen|Homo sapiens (9606)
>2ONY_1|Chains A, B|Phenylethanolamine N-methyltransferase|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 \(w=f(c)\)
7EUE , Knot 64 131 0.79 36 94 124 
MTVQDKAAGSDAEIVTALPVPLAVAGHHQPAPFYLTADMFGGLPVQLAGGELSKLVGKPVAAPHVHEVDELYFLVSPEPGQARIEVHLDGVRHELVSPAVMRIPAGSEHCFLTLEATVGSYCFGILVGDRL
8ROO , Knot 123 276 0.83 40 184 266 
GSHSMRYFHTSVSRPGRGEPRFISVGYVDGTQFVRFDSDAASPRTEPRAPWIEQEGPEYWDRNTQISKTNTQTYRESLRNLRGYYNQSEAGSHTLQRMYGCDVGPDGRLLRGHDQSAYDGKDYIALNEDLSSWTAADTAAQITQRKWEAARVAEQLRAYLEGTCVEWLRRHLENGKETLQRADPPKTHVTHHPISDHEATLRCWALGFYPAEITLTWQRDGEDQTQDTELVETRPAGDRTFQKWAAVVVPSGEEQRYTCHVQHEGLPKPLTLRWEP
2ONY , Knot 129 289 0.84 40 177 283 
MSGADRSPNAGAAPDSAPGQAAVASAYQRFEPRAYLRNNYAPPRGDLCNPNGVGPWKLRCLAQTFATGEVSGRTLIDIGSGPTVYQLLSACSHFEDITMTDFLEVNRQELGRWLQEEPGAFNWSMYSQHACLIEGKGECWQDKERQLRARVKRVLPIDVHQPQPLGAGSPAPLPADALVSAFCLEAVSPDLASFQRALDHITTLLRPGGHLLLIGALEESWYLAGEARLTVVPVSEEEVREALVRSGYKVRDLRTYIMPAHLQTGVDDVKGVFFAWAQKVGLEHHHHHH

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 \(c\).

\(|P_{f(7EUE_1)}(2) \setminus P_{f(8ROO_1)}(2)|=35\), \(|P_{f(8ROO_1)}(2) \setminus P_{f(7EUE_1)}(2)|=125\). 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 1:10100011100101101111111111000111101010111111101111010011101111101001001011101011010101010110001101111011110000110101011000111111001
Pair \(Z_2\) Length of longest common subsequence
7EUE_1,8ROO_1 160 3
7EUE_1,2ONY_1 151 4
8ROO_1,2ONY_1 161 4

Newick tree

 
[
	8ROO_1:81.77,
	[
		7EUE_1:75.5,2ONY_1:75.5
	]:6.27
]

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 \((0.85)(0.8)(\frac{407 }{\log_{20} 407}-\frac{131}{\log_{20}131})=83.2\)
Status Protein1 Protein2 d d1/2
Query variables 7EUE_1 8ROO_1 104 76
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]