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Parikh vectors
3JQW_1 8XEM_1 7YDJ_1 Letter Amino acid
11 4 9 Y Tyrosine
5 10 25 A Alanine
14 6 14 G Glycine
10 9 26 I Isoleucine
3 12 30 L Leucine
2 5 11 M Methionine
1 3 2 W Tryptophan
3 7 28 R Arginine
0 0 5 C Cysteine
7 11 17 E Glutamic acid
1 4 7 H Histidine
4 9 21 F Phenylalanine
9 5 25 V Valine
7 8 14 T Threonine
10 11 16 N Asparagine
8 5 26 D Aspartic acid
3 12 17 Q Glutamine
7 8 26 K Lycine
8 1 6 P Proline
8 9 20 S Serine 

3JQW_1|Chains A, B, C|ColH protein|Clostridium histolyticum (1498)
>8XEM_1|Chains A, B, C, D|HEPN|Legionella pneumophila (446)
>7YDJ_1|Chain A|engineered mini-G12|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 \(w=f(c)\)
3JQW , Knot 63 121 0.83 38 100 119 
QVYPIGTEKEPNNSKETASGPIVPGIPVSGTIENTSDQDYFYFDVITPGEVKIDINKLGYGGATWVVYDENNNAVSYATDDGQNLSGKFKADKPGRYYIHLYMFNGSYMPYRINIEGSVGR
8XEM , Knot 72 139 0.85 38 111 133 
MTNIDVRWQQRLNNYARALQQLSLAVNLAQTRPLSDLEKQGLIQAFEFTHELAWNVMKDYFFFQGNSAITGSRDATRESFNKGLIKEGEIWMEMIKSRNQTSHTYNQSVADEIVKNIINFYHTSFQAFLEKMQGLKEHE
7YDJ , Knot 149 345 0.84 40 208 326 
MMGCTLSAEDKAAVERSKMIDALLARERRAVRRLVKILLLGADNSGKSTFLKQMRIIHVNGYSEEECKQYKAVVYSNTIQSIIAIIRAMGRLKIDFGDSARADDARQLFVLAGAAEEGFMTAELAGVIKRLWKDSGVQACFNRSREYQLNDSAAYYLNDLDRIAQPNYIPTQQDVLRTRVKTKGIVEHDFVIKKIPFKMVDVGAQRSQRQKWFQCFDGITSILFMVDSSDYNRLVESMNDFETIVNNKLFFNVSIILFLNKMDLLVEKVKTVSIKKHFPDFRGDPHRLEDVQRYLVQCFDRKRRNRSKPLFHHFTTSIDTENARFIFHAVKDTILQENLKDIMLQ

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 \(c\).

\(|P_{f(3JQW_1)}(2) \setminus P_{f(8XEM_1)}(2)|=71\), \(|P_{f(8XEM_1)}(2) \setminus P_{f(3JQW_1)}(2)|=82\). 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 1:0101110000100000010111111111010100000000101011011010101001101110111000000110010001001010101001100010101101001100101010110
Pair \(Z_2\) Length of longest common subsequence
3JQW_1,8XEM_1 153 3
3JQW_1,7YDJ_1 194 3
8XEM_1,7YDJ_1 189 3

Newick tree

 
[
	7YDJ_1:10.36,
	[
		3JQW_1:76.5,8XEM_1:76.5
	]:24.86
]

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 \((0.85)(0.8)(\frac{260 }{\log_{20} 260}-\frac{121}{\log_{20}121})=43.8\)
Status Protein1 Protein2 d d1/2
Query variables 3JQW_1 8XEM_1 58 54.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]