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Parikh vectors
1CTQ_1 4HYU_1 1DSF_1 Letter Amino acid
5 12 6 F Phenylalanine
3 19 6 P Proline
0 4 1 W Tryptophan
3 7 3 C Cysteine
11 18 10 G Glycine
11 26 4 I Isoleucine
14 21 6 D Aspartic acid
11 34 13 L Leucine
15 26 9 V Valine
4 15 1 M Methionine
8 20 16 S Serine
11 14 8 T Threonine
11 20 2 A Alanine
4 15 2 N Asparagine
11 15 6 Q Glutamine
8 28 6 K Lycine
9 17 3 Y Tyrosine
11 17 4 R Arginine
13 25 3 E Glutamic acid
3 16 3 H Histidine 

1CTQ_1|Chain A|PROTEIN (TRANSFORMING PROTEIN P21/H-RAS-1)|Homo sapiens (9606)
>4HYU_1|Chain A|Mitogen-activated protein kinase 8|Homo sapiens (9606)
>1DSF_1|Chain A[auth L]|ANTICANCER ANTIBODY B1|Mus musculus (10090)
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 \(w=f(c)\)
1CTQ , Knot 79 166 0.81 38 130 163 
MTEYKLVVVGAGGVGKSALTIQLIQNHFVDEYDPTIEDSYRKQVVIDGETCLLDILDTAGQEEYSAMRDQYMRTGEGFLCVFAINNTKSFEDIHQYREQIKRVKDSDDVPMVLVGNKCDLAARTVESRQAQDLARSYGIPYIETSAKTRQGVEDAFYTLVREIRQH
4HYU , Knot 158 369 0.84 40 230 355 
MSRSKRDNNFYSVEIGDSTFTVLKRYQNLKPIGSGAQGIVCAAYDAILERNVAIKKLSRPFQNQTHAKRAYRELVLMKCVNHKNIIGLLNVFTPQKSLEEFQDVYIVMELMDANLCQVIQMELDHERMSYLLYQMLCGIKHLHSAGIIHRDLKPSNIVVKSDCTLKILDFGLARTAGTSFMMTPYVVTRYYRAPEVILGMGYKENVDIWSVGCIMGEMIKGGVLFPGTDHIDQWNKVIEQLGTPCPEFMKKLQPTVRTYVENRPKYAGYSFEKLFPDVLFPADSEHNKLKASQARDLLSKMLVIDASKRISVDEALQHPYINVWYDPSEAEAPPPKIPDKQLDEREHTIEEWKELIYKEVMDLHHHHHH
1DSF , Knot 57 112 0.80 40 94 106 
DVVMTQTPLSLPVSLGDQASISCRSSQNLVHSDGKTYLHWFLQKPGQSPTLLIYKVSNRFSGVPDRFSGSGSGTDFILKISRVEAEDLGVYFCSQSTHVPLTFGCGTKLELK

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 \(c\).

\(|P_{f(1CTQ_1)}(2) \setminus P_{f(4HYU_1)}(2)|=43\), \(|P_{f(4HYU_1)}(2) \setminus P_{f(1CTQ_1)}(2)|=143\). 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 1:1000011111111110011010110001100001010000000111010001101100110000011000010010111011110000010010000001001000001111111000011100100001001100011101000100001100110011001000
Pair \(Z_2\) Length of longest common subsequence
1CTQ_1,4HYU_1 186 3
1CTQ_1,1DSF_1 164 2
4HYU_1,1DSF_1 200 4

Newick tree

 
[
	4HYU_1:10.95,
	[
		1CTQ_1:82,1DSF_1:82
	]:18.95
]

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 \((0.85)(0.8)(\frac{535 }{\log_{20} 535}-\frac{166}{\log_{20}166})=107.\)
Status Protein1 Protein2 d d1/2
Query variables 1CTQ_1 4HYU_1 136 97.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]