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Parikh vectors
6MQS_1 7ICM_1 3UIF_1 Letter Amino acid
6 0 15 D Aspartic acid
8 0 16 Q Glutamine
8 0 19 E Glutamic acid
7 0 17 I Isoleucine
22 3 17 T Threonine
9 0 13 Y Tyrosine
10 1 32 A Alanine
7 0 18 N Asparagine
24 0 33 V Valine
17 0 26 L Leucine
10 0 25 K Lycine
5 0 11 W Tryptophan
20 1 23 G Glycine
3 0 11 H Histidine
38 0 24 S Serine
4 0 14 F Phenylalanine
14 0 17 P Proline
0 0 4 M Methionine
8 0 13 R Arginine
5 2 0 C Cysteine

6MQS_1|Chains A, C|antibody A12V163-a.01 heavy chain|Macaca mulatta (9544)
>7ICM_1|Chain A[auth T]|DNA (5'-D(*CP*AP*TP*CP*TP*GP*T)-3')|
>3UIF_1|Chain A|Sulfonate ABC transporter, periplasmic sulfonate-binding protein SsuA|Methylobacillus flagellatus (265072)
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 \(w=f(c)\)
6MQS , Knot 100 225 0.80 38 144 210
QVQLQESGPGLVKPSETLSLTCGVSGGSISDAYYWSWIRQSPGKRLEWIGYIFGSNGGTRYNPSLRSRVSISIDTSKNQLSLKLTSVTAADTAVYYCVREGVPTEATTGDHWGQGVPVTVSSASTKGPSVFPLAPSSRSTSESTAALGCLVKDYFPEPVTVSWNSGSLTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYVCNVNHKPSNTKVDKRVEIKTC
7ICM , Knot 6 7 0.55 8 6 5
CATCTGT
3UIF , Knot 156 348 0.87 38 210 332
MVEDLKVVRIASVATNVGGKTVYAGSASLVVNGAFPEELRKQGIKVEWVPAAMASVGPVINEGFASGKIDFGIYGDLPPIILNASKPTVQLVAPWGTTSNSYLVVPKNSTAKSIKDLKGKKIALHRGRPWELAFSNLLQSEGLTFKDFKIVNVNPQVGAAALASGTVDGFFSLFDSYILEDRGVGKIIWSTKTAPVDWKLMGGVWARNDFVKQNPEITQAIVTAYLKSVHWVAQDENKETYIREYSNKIYPESVNRREYDQDNVSWRQRWSPLYDVALQEHYRKAVAYAQASGLTRTQADVQQMLNPHFVATALKELKLEGFWTPNAENLYFQSHHHHHHWSHPQFEK

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 \(c\).

\(|P_{f(6MQS_1)}(2) \setminus P_{f(7ICM_1)}(2)|=140\), \(|P_{f(7ICM_1)}(2) \setminus P_{f(6MQS_1)}(2)|=2\). 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 1:010100011111010001010011011010010010110001100101110111001100001010001010100000010101001011001100010011100100100110111101001000110111111000000000111101100011011010100101001100111110001100100110110001100001001000100001000101000
Pair \(Z_2\) Length of longest common subsequence
6MQS_1,7ICM_1 142 2
6MQS_1,3UIF_1 164 3
7ICM_1,3UIF_1 212 2

Newick tree

 
[
	3UIF_1:10.45,
	[
		6MQS_1:71,7ICM_1:71
	]:30.45
]

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 \((0.85)(0.8)(\frac{232 }{\log_{20} 232}-\frac{7}{\log_{20}7})=79.4\)
Status Protein1 Protein2 d d1/2
Query variables 6MQS_1 7ICM_1 97 49.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]