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Parikh vectors
2HKH_1 5ITF_1 3NUF_1 Letter Amino acid
12 11 3 N Asparagine
9 11 10 E Glutamic acid
10 8 3 F Phenylalanine
34 34 7 S Serine
7 12 16 A Alanine
12 9 7 D Aspartic acid
5 4 0 C Cysteine
5 3 5 H Histidine
2 0 4 M Methionine
19 16 6 T Threonine
3 3 0 W Tryptophan
8 8 1 R Arginine
12 11 6 K Lycine
11 10 4 P Proline
8 8 2 Y Tyrosine
15 14 11 V Valine
8 10 7 I Isoleucine
14 13 4 G Glycine
16 16 14 L Leucine
9 12 9 Q Glutamine 

2HKH_1|Chain A[auth L]|Immunoglobulin Light chain Fab fragment|Mus musculus (10090)
>5ITF_1|Chains A, C|Cetuximab Fab, light chain|MUS MUSCULUS (10090)
>3NUF_1|Chains A, B|PRD-containing transcription regulator|Lactobacillus casei (321967)
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 \(w=f(c)\)
2HKH , Knot 100 219 0.82 40 160 209 
DVVMTQTPLSLPVSLGDQASISCRSSQSLVHSNGNTYLHWYLQKPGQSPNLLIYKVSNRFSGVPDRFSGSGSGTDFTLKISRVEAEDLGVYFCSQSTHVPFTFGSGTKLEIKRADAAPTVSIFPPSSEQLTSGGASVVCFLNNFYPKDINVKWKIDGSERQNGVLNSWTDQDSKDSTYSMSSTLTLTKDEYERHNSYTCEATHKTSTSPIVKSFNRNEC
5ITF , Knot 96 213 0.80 38 145 204 
DILLTQSPVILSVSPGERVSFSCRASQSIGTNIHWYQQRTNGSPRLLIKYASESISGIPSRFSGSGSGTDFTLSINSVESEDIADYYCQQNNNWPTTFGAGTKLELKRTVAAPSVFIFPPSDEQLKSGTASVVCLLNNFYPREAKVQWKVDNALQSGNSQESVTEQDSKDSTYSLSSTLTLSKADYEKHKVYACEVTHQGLSSPVTKSFNRGA
3NUF , Knot 62 119 0.83 36 92 115 
GMTPLDANVELPTEVKAMIEQSSDAQAATALVNYVIKLAAAAEIHFTDLQLQVLTNHLIEMLGRSKSGEQLPAVDPTMFAEVSQKSLDLADQVVQHIGHLEVAEKYVLSIHFEAAQDKI

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 \(c\).

\(|P_{f(2HKH_1)}(2) \setminus P_{f(5ITF_1)}(2)|=69\), \(|P_{f(5ITF_1)}(2) \setminus P_{f(2HKH_1)}(2)|=54\). 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 1:011100011011101100101000000011000100010101001100101110010001011100101010100101010010100111010000001110110100101001011101011110000100111011011001010010101010100000111001000000000001000101000000000000001000000011100100000
Pair \(Z_2\) Length of longest common subsequence
2HKH_1,5ITF_1 123 13
2HKH_1,3NUF_1 160 3
5ITF_1,3NUF_1 149 3

Newick tree

 
[
	3NUF_1:81.89,
	[
		2HKH_1:61.5,5ITF_1:61.5
	]:20.39
]

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 \((0.85)(0.8)(\frac{432 }{\log_{20} 432}-\frac{213}{\log_{20}213})=64.0\)
Status Protein1 Protein2 d d1/2
Query variables 2HKH_1 5ITF_1 66 65.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]