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Parikh vectors
3SSR_1 5DUV_1 4AZU_1 Letter Amino acid
10 6 4 E Glutamic acid
5 11 10 L Leucine
3 5 6 M Methionine
5 6 10 S Serine
0 2 1 W Tryptophan
17 17 10 V Valine
13 7 7 A Alanine
0 0 3 C Cysteine
2 10 6 Q Glutamine
9 17 11 G Glycine
3 7 11 K Lycine
3 9 2 Y Tyrosine
3 6 11 D Aspartic acid
7 5 4 I Isoleucine
5 4 10 T Threonine
8 5 1 R Arginine
3 8 7 N Asparagine
8 5 4 H Histidine
2 12 6 F Phenylalanine
4 13 4 P Proline 

3SSR_1|Chains A, B|Carbon dioxide concentrating mechanism protein|Thermosynechococcus elongatus (197221)
>5DUV_1|Chains A, B, C, D|Galectin-4|Homo sapiens (9606)
>4AZU_1|Chains A, B, C, D|AZURIN|Pseudomonas aeruginosa (287)
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 \(w=f(c)\)
3SSR , Knot 55 110 0.78 36 87 103 
MPIAVGMIETRGFPAVVEAADAMVKAARVTLVGYEKIGSGRVTVIVRGDVSEVQASVAAGVDSAKRVNGGEVLSTHIIARPHENLEYVLPIRYTEAVEQFRNLEHHHHHH
5DUV , Knot 75 155 0.81 38 115 151 
MAYVPAPGYQPTYNPTLPYYQPIPGGLNVGMSVYIQGVASEHMKRFFVNFVVGQDPGSDVAFHFNPRFDGWDKVVFNTLQGGKWGSEERKRSMPFKKGAAFELVFIVLAEHYKVVVNGNPFYEYGHRLPLQMVTHLQVDGDLQLQSINFIGGQPL
4AZU , Knot 68 128 0.86 40 105 126 
AECSVDIQGNDQMQFNTNAITVDKSCKQFTVNLSHPGNLPKNVMGHNWVLSTAADMQGVVTDGMASGLDKDYLKPDDSRVIAHTKLIGSGEKDSVTFDVSKLKEGEQYMFFCTFPGHSALMKGTLTLK

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 \(c\).

\(|P_{f(3SSR_1)}(2) \setminus P_{f(5DUV_1)}(2)|=53\), \(|P_{f(5DUV_1)}(2) \setminus P_{f(3SSR_1)}(2)|=81\). 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 1:11111111000111111011011101101011100011010101110101001010111110010010110110001110100010011110000110010010000000
Pair \(Z_2\) Length of longest common subsequence
3SSR_1,5DUV_1 134 3
3SSR_1,4AZU_1 128 4
5DUV_1,4AZU_1 154 3

Newick tree

 
[
	5DUV_1:74.69,
	[
		3SSR_1:64,4AZU_1:64
	]:10.69
]

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 \((0.85)(0.8)(\frac{265 }{\log_{20} 265}-\frac{110}{\log_{20}110})=49.0\)
Status Protein1 Protein2 d d1/2
Query variables 3SSR_1 5DUV_1 62 51.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]