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Parikh vectors
3WTH_1 3DEA_1 5BVR_1 Letter Amino acid
7 8 6 Y Tyrosine
27 17 11 S Serine
14 14 12 T Threonine
5 1 5 W Tryptophan
5 19 12 G Glycine
3 1 6 H Histidine
8 11 16 N Asparagine
8 7 10 Q Glutamine
16 7 12 E Glutamic acid
12 15 30 L Leucine
1 3 5 M Methionine
11 10 8 P Proline
4 4 3 C Cysteine
14 15 18 I Isoleucine
7 6 14 K Lycine
6 8 9 F Phenylalanine
22 12 11 V Valine
12 27 13 A Alanine
15 8 16 R Arginine
17 8 19 D Aspartic acid 

3WTH_1|Chains A, B, C, D, E|Acetylcholine-binding protein|Lymnaea stagnalis (6523)
>3DEA_1|Chains A, B|Cutinase|Glomerella cingulata (474922)
>5BVR_1|Chain A|Alpha-actinin-like protein 1|Schizosaccharomyces pombe (strain 972 / ATCC 24843) (284812)
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 \(w=f(c)\)
3WTH , Knot 98 214 0.82 40 140 207 
EAEAADRADILYNIRQTSRPDVIPTQRDRPVAVSVSLKFINILEVNEITNEVDVVFWQRTTWSDRTLAWDSSHSPDQVSVPISSLWVPDLAAYNAISKPEVLTPQLARVVSDGEVLYMPSIRQRFSCDVSGVDTESGATCRIKIGSWTHHSREISVDPTTENSDDSEYFSQYSRFEILDVTQKKNSVTYSCCPEAYEDVEVSLNFRKKGRSEIL
3DEA , Knot 94 201 0.82 40 138 196 
AMAISDPQSSTRNELETGSSSACPKVIYIFARASTEPGNMGISAGPIVADALERIYGANDVWVQGVGGPYLADLASNFLPDGTSSAAINEARRLFTLANTKCPNAAIVSGGYSQGTAVMAGSISGLSTTIKNQIKGVVLFGYTKNLQNLGRIPNFETSKTEVYCDIADAVCYGTLFILPAHFLYQTDAAVAAPRFLQARIG
5BVR , Knot 106 236 0.81 40 158 228 
GSMQANQWQSVQNRTFTKWFNTKLSSRDLPSVFDLRKDLSDGILLIQLLEIIGDENLGRYNRNPRMRVHRLENVNKALEYIKSKGMPLTNIGPADIVDGNLKLILGLIWTLILRFTIADINEEGLTAKEGLLLWCQRKTANYHPEVDVQDFTRSWTNGLAFCALIHQHRPDLLDYNKLDKKNHRANMQLAFDIAQKSIGIPRLIEVEDVCDVDRPDERSIMTYVAEYFHAFSTLDK

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 \(c\).

\(|P_{f(3WTH_1)}(2) \setminus P_{f(3DEA_1)}(2)|=82\), \(|P_{f(3DEA_1)}(2) \setminus P_{f(3WTH_1)}(2)|=80\). 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 1:0101100101100100000101110000011110101011011010010001011110000100001110000010010111001111011100110010110101101100101101101000100010110000110001011010000001010100000000001000001011010000001000001010001010101000100011
Pair \(Z_2\) Length of longest common subsequence
3WTH_1,3DEA_1 162 3
3WTH_1,5BVR_1 166 4
3DEA_1,5BVR_1 174 4

Newick tree

 
[
	5BVR_1:86.32,
	[
		3WTH_1:81,3DEA_1:81
	]:5.32
]

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 \((0.85)(0.8)(\frac{415 }{\log_{20} 415}-\frac{201}{\log_{20}201})=63.0\)
Status Protein1 Protein2 d d1/2
Query variables 3WTH_1 3DEA_1 80 77.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]