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Parikh vectors
4XUR_1 7HJN_1 5PGW_1 Letter Amino acid
6 5 12 R Arginine
9 7 10 N Asparagine
0 0 4 C Cysteine
19 13 18 E Glutamic acid
6 6 9 Q Glutamine
12 3 16 I Isoleucine
10 10 17 K Lycine
9 0 5 P Proline
8 5 8 D Aspartic acid
3 1 10 H Histidine
10 11 26 L Leucine
3 3 16 M Methionine
14 6 26 S Serine
10 8 25 V Valine
16 23 27 A Alanine
15 6 22 G Glycine
6 7 9 F Phenylalanine
15 8 15 T Threonine
4 0 1 W Tryptophan
6 4 10 Y Tyrosine 

4XUR_1|Chains A, B, C|Endo-1,4-beta-xylanase C|Paenibacillus barcinonensis (198119)
>7HJN_1|Chain A|De novo designed ABLE protein|synthetic construct (32630)
>5PGW_1|Chains A, B, C[auth D], D[auth E]|Corticosteroid 11-beta-dehydrogenase isozyme 1|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 \(w=f(c)\)
4XUR , Knot 87 181 0.83 38 138 178 
APENPGEPGEAGQALFKADFEDGNIGNWRARGTEKLEVVSGIGHNSNRSLKTSSRSETYHGPLVEVLPYLQKGSTVHISFWAMYDEGPATQVINGSLEKEFNRDTANLEYAMFASTTLNKGQWKKIEADIIVPAESTGISGLRMYAETPWKQSSEVTETDTIPFYVDDVQITATEAIAIEK
7HJN , Knot 59 126 0.75 34 92 117 
SVKSEYAEAAAVGQEAVAVFNTMKAAFQNGDKEAVAQYLARLASLYTRHEELLNRILEKARREGNKEAVTLMNEFTATFQTGKSIFNAMVAAFKNGDDDSFESYLQALEKVTAKGETLADQIAKAL
5PGW , Knot 129 286 0.85 40 183 279 
GSHMASMTGGQQMGRGSNEEFRPEMLQGKKVIVTGASKGIGREMAYHLAKMGAHVVVTARSKETLQKVVSHCLELGAASAHYIAGTMEDMTFAEQFVAQAGKLMGGLDMLILNHITNTSLNLFHDDIHHVRKSMEVNFLSYVVLTVAALPMLKQSNGSIVVVSSLAGKVAYPMVAAYSASKFALDGFFSSIRKEYSVSRVNVSITLCVLGLIDTETAMKAVSGIVHMQAAPKEECALEIIKGGALRQEEVYYDSSRWTTLLIRNPCRKILEELYSTSYNMDRFINK

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 \(c\).

\(|P_{f(4XUR_1)}(2) \setminus P_{f(7HJN_1)}(2)|=96\), \(|P_{f(7HJN_1)}(2) \setminus P_{f(4XUR_1)}(2)|=50\). 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 1:1100110110110111010100101101010100010110111000000100000000001111011101001001010111100011100110101000100001010011110001001010010101111100011011010100110000010000011101001010100111100
Pair \(Z_2\) Length of longest common subsequence
4XUR_1,7HJN_1 146 3
4XUR_1,5PGW_1 171 4
7HJN_1,5PGW_1 171 4

Newick tree

 
[
	5PGW_1:89.27,
	[
		4XUR_1:73,7HJN_1:73
	]:16.27
]

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 \((0.85)(0.8)(\frac{307 }{\log_{20} 307}-\frac{126}{\log_{20}126})=56.1\)
Status Protein1 Protein2 d d1/2
Query variables 4XUR_1 7HJN_1 74 60
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]