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Parikh vectors
1WMG_1 4BSB_1 9ENX_1 Letter Amino acid
10 33 26 G Glycine
5 17 14 K Lycine
4 15 11 P Proline
16 29 16 S Serine
10 18 32 A Alanine
5 20 18 R Arginine
4 19 6 N Asparagine
4 13 11 Q Glutamine
0 6 9 H Histidine
2 13 11 F Phenylalanine
3 23 25 T Threonine
8 13 17 D Aspartic acid
1 9 2 C Cysteine
4 15 19 E Glutamic acid
4 18 13 I Isoleucine
12 23 24 L Leucine
4 6 8 M Methionine
2 3 3 W Tryptophan
3 8 5 Y Tyrosine
2 20 19 V Valine 

1WMG_1|Chains A, B, C, D, E, F|netrin receptor Unc5h2|Mus musculus (10090)
>4BSB_1|Chain A|HEMAGGLUTININ|INFLUENZA VIRUS A/ANHUI/1/2013 (H7N9) (11320)
>9ENX_1|Chains A, B|beta-lactamase|Klebsiella pneumoniae (573)
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 \(w=f(c)\)
1WMG , Knot 54 103 0.81 38 80 98 
GSSGSSGYAFKIPLSIRQKICSSLDAPNSRGNDWRLLAQKLSMDRYLNYFATKASPTGVILDLWEARQQDDGDLNSLASALEEMGKSEMLVAMATDGSGPSSG
4BSB , Knot 139 321 0.83 40 209 313 
DKICLGHHALSNGTKVNTLTERGVEVVNATETVERTNIPRICSKGKRTVDLGQCGLLGTITGPPQCDQFLEFSADLIIERREGSDVCYPGKFVNEEALRQILRESGGIDKEAMGFTYSGIRTNGTTSACRRSGSSFYAEMKWLLSNTDNAAFPQMTKSYKNTRKSPALIVWGIHHSVSTAEQTKLYGSGNKLVTVGSSNYQQSFVPSPGARPQVNGLSGRIDFHWLMLNPNDTVTFSFNGAFIAPDRASFLRGKSMGIQSGVQVDANCEGDCYHSGGTIISNLPFQNIDSRAVGKCPRYVKQRSLLLATGMKNVPEIPKGR
9ENX , Knot 124 289 0.81 40 172 275 
MAHHHHHHSSGLEVLFQGPASEKLTFKTDLEKLEREKAAQIGVAIVDPQGEIVAGHRMAQRFAMCSTFKFPLAALVFERIDSGTERGDRKLSYGPDMIVEWSPATERFLASGHMTVLEAAQAAVQLSDNGATNLLLREIGGPAAMTQYFRKIGDSVSRLDRKEPEMGDNTPGDLRDTTTPIAMARTVAKVLYGGALTSTSTHTIERWLIGNQTGDATLRAGFPKDWVVGEKTGTCANGGRNDIGFFKAQERDYAVAVYTTAPKLSAVERDELVASVGQVITQLILSTDK

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 \(c\).

\(|P_{f(1WMG_1)}(2) \setminus P_{f(4BSB_1)}(2)|=26\), \(|P_{f(4BSB_1)}(2) \setminus P_{f(1WMG_1)}(2)|=155\). 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 1:1001001011011101000100010110001001011100101000100110010101111011010000010100110110011000111111001011001
Pair \(Z_2\) Length of longest common subsequence
1WMG_1,4BSB_1 181 4
1WMG_1,9ENX_1 156 3
4BSB_1,9ENX_1 161 4

Newick tree

 
[
	4BSB_1:88.04,
	[
		1WMG_1:78,9ENX_1:78
	]:10.04
]

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 \((0.85)(0.8)(\frac{424 }{\log_{20} 424}-\frac{103}{\log_{20}103})=97.5\)
Status Protein1 Protein2 d d1/2
Query variables 1WMG_1 4BSB_1 125 80
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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