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Parikh vectors
5XJR_1 4LYL_1 8RQU_1 Letter Amino acid
7 4 2 C Cysteine
16 11 16 E Glutamic acid
10 17 26 G Glycine
3 13 3 H Histidine
5 7 6 W Tryptophan
6 6 4 Y Tyrosine
14 11 16 D Aspartic acid
20 13 10 S Serine
19 18 16 V Valine
12 15 5 K Lycine
16 11 25 R Arginine
19 9 6 Q Glutamine
8 7 12 I Isoleucine
27 21 31 L Leucine
2 3 5 M Methionine
8 9 25 T Threonine
15 16 37 A Alanine
7 11 3 F Phenylalanine
19 14 13 P Proline
8 7 5 N Asparagine 

5XJR_1|Chain A[auth 2]|Gem-associated protein 2|Homo sapiens (9606)
>4LYL_1|Chains A, C, E, G, I, K, M, O|Uracil-DNA glycosylase|Gadus morhua (8049)
>8RQU_1|Chains A, B|Beta-lactamase TEM-1|Escherichia coli (562)
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 \(w=f(c)\)
5XJR , Knot 110 241 0.83 40 168 233 
FDPSVPPRTPQEYLRRVQIEAAQCPDVVVAQIDPKKLKRKQSVNISLSGCQPAPEGYSPTLQWQQQQVAQFSTVRQNVNKHRSHWKSQQLDSNVTMPKSEDEEGWKKFCLGEKLCADGAVGPATNESPGIDYVQIGFPPLLSIVSRMNQATVTSVLEYLSNWFGERDFTPELGRWLYALLACLEKPLLPEAHSLIRQLARRCSEVRLLVDSKDDERVPALNLLICLVSRYFDQRDLADEPS
4LYL , Knot 106 223 0.85 40 163 219 
MEFFGETWRRELAAEFEKPYFKQLMSFVADERSRHTVYPPADQVYSWTEMCDIQDVKVVILGQDPYHGPNQAHGLCFSVQKPVPPPPSLVNIYKELCTDIDGFKHPGHGDLSGWAKQGVLLLNAVLTVRAHQANSHKDRGWETFTDAVIKWLSVNREGVVFLLWGSYAHKKGATIDRKRHHVLQAVHPSPLSAHRGFLGCKHFSKANGLLKLSGTEPINWRAL
8RQU , Knot 116 266 0.81 40 158 250 
GSHMPETLVTVRRIEAQLGARVGVAVLDTGSGRSWEGYRADERFPMASTFKVLACGALLSRVDAGQEDLDRRIRYTQDELVTYSPVTEKHLDDGMTLRALCEATITTSDNTAANLILEALGGPKALTRFLRAIGDPVTRLDRWETALNEATPGDVRDTTTPRAMAATLRTLLLGDALTPASRQQLIAWLEANQVGGPLLRAGLPAGWRIGDKTGAGGRGTRGIVAIVWPPGRAPLIAAVYLTESEASMDERNAAIAEIGAALVKHW

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 \(c\).

\(|P_{f(5XJR_1)}(2) \setminus P_{f(4LYL_1)}(2)|=91\), \(|P_{f(4LYL_1)}(2) \setminus P_{f(5XJR_1)}(2)|=86\). 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 1:1010111001000100101011001011110101001000001010101001110100101010000110100100010000001000010001011000000110010110010101111110000111001011111110110010010100110010011100010101101101111010011110100110011000001011100000001111011101100010000110010
Pair \(Z_2\) Length of longest common subsequence
5XJR_1,4LYL_1 177 3
5XJR_1,8RQU_1 146 3
4LYL_1,8RQU_1 173 4

Newick tree

 
[
	4LYL_1:91.83,
	[
		5XJR_1:73,8RQU_1:73
	]:18.83
]

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 \((0.85)(0.8)(\frac{464 }{\log_{20} 464}-\frac{223}{\log_{20}223})=69.9\)
Status Protein1 Protein2 d d1/2
Query variables 5XJR_1 4LYL_1 89 86
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]