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Parikh vectors
7PWA_1 1ZUE_1 8SOU_1 Letter Amino acid
9 3 12 R Arginine
13 1 14 D Aspartic acid
15 3 8 K Lycine
16 1 22 T Threonine
15 1 19 V Valine
17 3 17 N Asparagine
3 6 5 C Cysteine
13 0 14 L Leucine
16 3 36 S Serine
2 2 2 W Tryptophan
12 2 17 Y Tyrosine
12 1 33 G Glycine
13 2 6 F Phenylalanine
11 1 9 P Proline
11 2 33 A Alanine
10 2 7 Q Glutamine
8 4 5 E Glutamic acid
13 1 4 H Histidine
6 1 11 I Isoleucine
6 3 5 M Methionine 

7PWA_1|Chains A, B|Protein mono-ADP-ribosyltransferase PARP15|Homo sapiens (9606)
>1ZUE_1|Chain A|Defensin-like peptide 2/4|null
>8SOU_1|Chain A|Proteinase K|Parengyodontium album (37998)
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 \(w=f(c)\)
7PWA , Knot 103 221 0.83 40 159 215 
MHHHHHHSSGVDLGTENLYFQSMNLPEHWTDMNHQLFCMVQLEPGQSEYNTIKDKFTRTCSSYAIEKIERIQNAFLWQSYQVKKRQMDIKNDHKNNERLLFHGTDADSVPYVNQHGFNRSCAGKNAVSYGKGTYFAVDASYSAKDTYSKPDSNGRKHMYVVRVLTGVFTKGRAGLVTPPPKNPHNPTDLFDSVTNNTRSPKLFVVFFDNQAYPEYLITFTA
1ZUE , Knot 27 42 0.80 38 39 40 
IMFFEMQACWSHSGVCRDKSERNCKPMAWTYCENRNQKCCEY
8SOU , Knot 120 279 0.80 40 170 266 
AAQTNAPWGLARISSTSPGTSTYYYDESAGQGSCVYVIDTGIEASHPEFEGRAQMVKTYYYSSRDGNGHGTHCAGTVGSRTYGVAKKTQLFGVKVLDDNGSGQYSTIIAGMDFVASDKNNRNCPKGVVASLSLGGGYSSSVNSAAARLQSSGVMVAVAAGNNNADARNYSPASEPSVCTVGASDRYDRRSSFSNYGSVLDIFGPGTDILSTWIGGSTRSISGTSMATPHVAGLAAYLMTLGKTTAASACRYIADTANKGDLSNIPFGTVNLLAYNNYQA

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 \(c\).

\(|P_{f(7PWA_1)}(2) \setminus P_{f(1ZUE_1)}(2)|=144\), \(|P_{f(1ZUE_1)}(2) \setminus P_{f(7PWA_1)}(2)|=24\). 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 1:10000000011011000101001011001001000110110101100000010001000000011001001001111000010000101000000000111010010011010001100001100110010100111010001000000100010001011011011100101111011100100100110010000001011111100010100110101
Pair \(Z_2\) Length of longest common subsequence
7PWA_1,1ZUE_1 168 3
7PWA_1,8SOU_1 177 4
1ZUE_1,8SOU_1 177 3

Newick tree

 
[
	8SOU_1:89.94,
	[
		7PWA_1:84,1ZUE_1:84
	]:5.94
]

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 \((0.85)(0.8)(\frac{263 }{\log_{20} 263}-\frac{42}{\log_{20}42})=73.2\)
Status Protein1 Protein2 d d1/2
Query variables 7PWA_1 1ZUE_1 94 56
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]