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Parikh vectors
2XRQ_1 7BNC_1 7VWH_1 Letter Amino acid
5 20 23 A Alanine
1 13 2 W Tryptophan
4 13 8 Y Tyrosine
4 18 13 D Aspartic acid
1 17 10 H Histidine
4 4 8 M Methionine
4 25 17 V Valine
12 21 15 T Threonine
6 16 10 N Asparagine
5 15 13 Q Glutamine
3 30 10 G Glycine
5 40 34 L Leucine
10 13 21 K Lycine
2 11 17 F Phenylalanine
3 20 12 P Proline
4 29 11 R Arginine
2 14 3 C Cysteine
8 22 17 E Glutamic acid
12 12 17 I Isoleucine
8 30 15 S Serine 

2XRQ_1|Chains A[auth D], B[auth E], C[auth F], D[auth G], E[auth H]|HEAT-LABILE ENTEROTOXIN B CHAIN|ESCHERICHIA COLI (562)
>7BNC_1|Chain A|Palmitoleoyl-protein carboxylesterase NOTUM|Homo sapiens (9606)
>7VWH_1|Chains A, B|Peroxisome proliferator-activated receptor delta|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 \(w=f(c)\)
2XRQ , Knot 55 103 0.82 40 86 101 
APQTITELCSEYRNTQIYTINDKILSYTESMAGKREMVIITFKSGETFQVEVPGSQHIDSQKKAIERMKDTLRITYLTETKIDKLCVWNNKTPNSIAAISMKN
7BNC , Knot 162 383 0.83 40 219 364 
ETGSAQQLNEDLRLHLLLNTSVTCNDGSPAGYYLKESRGSRRWLLFLEGGWYCFNRENCDSRYDTMRRLMSSRDWPRTRTGTGILSSQPEENPYWWNANMVFIPYCSSDVWSGASSKSEKNEYAFMGALIIQEVVRELLGRGLSGAKVLLLAGSSAGGTGVLLNVDRVAEQLEKLGYPAIQVRGLADSGWFLDNKQYRHTDCVDTITCAPTEAIRRGIRYWNGVVPERCRRQFQEGEEWNCFFGYKVYPTLRSPVFVVQWLFDEAQLTVDNVHLTGQPVQEGLRLYIQNLGRELRHTLKDVPASFAPACLSHEIIIRSHWTDVQVKGTSLPRALHCWDRSLHDSHKASKTPLKGCPVHLVDSCPWPHCNPSCPTGTKHHHHHH
7VWH , Knot 129 276 0.87 40 186 267 
GSHMPQVADLKAFSKHIYNAYLKNFNMTKKKARSILTGKASHTAPFVIHDIETLWQAEKGLVWKQLVNGLPPYKEISVHVFYRCQCTTVETVRELTEFAKSIPSFSSLFLNDQVTLLKYGVHEAIFAMLASIVNKDGLLVANGSGFVTREFLRSLRKPFSDIIEPKFEFAVKFNALELDDSDLALFIAAIILCGDRPGLMNVPRVEAIQDTILRALEFHLQANHPDAQYLFPKLLQKMADLRQLVTEHAQMMQRIKKTETETSLHPLLQEIYKDMY

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 \(c\).

\(|P_{f(2XRQ_1)}(2) \setminus P_{f(7BNC_1)}(2)|=39\), \(|P_{f(7BNC_1)}(2) \setminus P_{f(2XRQ_1)}(2)|=172\). 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 1:1100100100000000100100011000001110001111010010010101110001000001100100010100100001001011000010011110100
Pair \(Z_2\) Length of longest common subsequence
2XRQ_1,7BNC_1 211 3
2XRQ_1,7VWH_1 178 4
7BNC_1,7VWH_1 193 4

Newick tree

 
[
	7BNC_1:10.82,
	[
		2XRQ_1:89,7VWH_1:89
	]:15.82
]

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 \((0.85)(0.8)(\frac{486 }{\log_{20} 486}-\frac{103}{\log_{20}103})=114.\)
Status Protein1 Protein2 d d1/2
Query variables 2XRQ_1 7BNC_1 147 92.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]