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Parikh vectors
3MXF_1 4HRX_1 3WPA_1 Letter Amino acid
2 4 0 C Cysteine
8 14 9 E Glutamic acid
7 20 28 I Isoleucine
5 8 1 M Methionine
7 10 8 Y Tyrosine
5 31 45 V Valine
12 11 59 N Asparagine
9 10 20 Q Glutamine
2 20 45 G Glycine
2 25 23 S Serine
5 17 45 A Alanine
1 15 7 H Histidine
12 8 29 K Lycine
5 14 2 F Phenylalanine
12 13 4 P Proline
8 8 44 T Threonine
6 21 32 D Aspartic acid
13 27 21 L Leucine
3 2 1 W Tryptophan
3 10 4 R Arginine 

3MXF_1|Chain A|Bromodomain-containing protein 4|Homo sapiens (9606)
>4HRX_1|Chain A|Hydrolase, alpha/beta fold family protein|Arabidopsis thaliana (3702)
>3WPA_1|Chain A|Trimeric autotransporter adhesin|Acinetobacter (710648)
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 \(w=f(c)\)
3MXF , Knot 66 127 0.84 40 110 123 
SMNPPPPETSNPNKPKRQTNQLQYLLRVVLKTLWKHQFAWPFQQPVDAVKLNLPDYYKIIKTPMDMGTIKKRLENNYYWNAQECIQDFNTMFTNCYIYNKPGDDIVLMAEALEKLFLQKINELPTEE
4HRX , Knot 127 288 0.83 40 174 275 
HHHHHHLESTSLYKKAGFMGVVEEAHNVKVIGSGEATIVLGHGFGTDQSVWKHLVPHLVDDYRVVLYDNMGAGTTNPDYFDFDRYSNLEGYSFDLIAILEDLKIESCIFVGHSVSAMIGVLASLNRPDLFSKIVMISASPRYVNDVDYQGGFEQEDLNQLFEAIRSNYKAWCLGFAPLAVGGDMDSIAVQEFSRTLFNMRPDIALSVGQTIFQSDMRQILPFVTVPCHILQSVKDLAVPVVVSEYLHANLGCESVVEVIPSDGHLPQLSSPDSVIPVILRHIRNDIAM
3WPA , Knot 155 427 0.73 38 152 353 
MKQIEDKIEEILSKIYHIENEIARIKKLIKAVGNQVVTTQTTLVNSLGGNAKVNADGTITGPTYNVAQGNQTNVGDALTALDNAINTAATTSKSTVSNGQNIVVSKSKNADGSDNYEVSTAKDLTVDSVKAGDTVLNNAGITIGNNAVVLNNTGLTISGGPSVTLAGIDAGNKTIQNVANAVNATDAVNKGQLDSAINNVNNNVNELANNAVKYDDASKDKITLGGGATGTTITNVKDGTVAQGSKDAVNGGQLWNVQQQVDQNTTDISNIKNDINNGTVGLVQQAGKDAPVTVAKDTGGTTVNVAGTDGNRVVTGVKEGAVNATSKDAVNGSQLNTTNQAVVNYLGGGAGYDNITGSFTAPSYTVGDSKYNNVGGAIDALNQADQALNSKIDNVSNKLDNAFRITNNRIDDVEKKANAGIHHHHHH

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 \(c\).

\(|P_{f(3MXF_1)}(2) \setminus P_{f(4HRX_1)}(2)|=57\), \(|P_{f(4HRX_1)}(2) \setminus P_{f(3MXF_1)}(2)|=121\). 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 1:0101111000010010000001001101110011000111110011011010110000110011011010001000001010001001001100001000110011111011001110010011000
Pair \(Z_2\) Length of longest common subsequence
3MXF_1,4HRX_1 178 4
3MXF_1,3WPA_1 164 3
4HRX_1,3WPA_1 152 6

Newick tree

 
[
	3MXF_1:88.53,
	[
		3WPA_1:76,4HRX_1:76
	]:12.53
]

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 \((0.85)(0.8)(\frac{415 }{\log_{20} 415}-\frac{127}{\log_{20}127})=86.8\)
Status Protein1 Protein2 d d1/2
Query variables 3MXF_1 4HRX_1 108 78
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]