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Parikh vectors
2BTC_1 5AFV_1 6MTY_1 Letter Amino acid
4 6 6 W Tryptophan
10 17 11 Y Tyrosine
17 10 20 V Valine
14 20 10 A Alanine
14 26 12 K Lycine
10 11 19 T Threonine
15 20 6 I Isoleucine
2 12 2 M Methionine
16 15 9 N Asparagine
12 6 5 C Cysteine
3 8 4 H Histidine
14 45 18 L Leucine
8 17 14 P Proline
2 24 5 R Arginine
6 13 8 D Aspartic acid
25 20 20 G Glycine
34 27 39 S Serine
10 16 7 Q Glutamine
4 40 6 E Glutamic acid
3 25 4 F Phenylalanine 

2BTC_1|Chain A[auth E]|PROTEIN (TRYPSIN)|Bos taurus (9913)
>5AFV_1|Chains A, B|CHOLINE KINASE ALPHA|HOMO SAPIENS (9606)
>6MTY_1|Chain A[auth H]|MZ4 Heavy Chain|Homo sapiens (9606)
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 \(w=f(c)\)
2BTC , Knot 101 223 0.81 40 140 216 
IVGGYTCGANTVPYQVSLNSGYHFCGGSLINSQWVVSAAHCYKSGIQVRLGEDNINVVEGNEQFISASKSIVHPSYNSNTLNNDIMLIKLKSAASLNSRVASISLPTSCASAGTQCLISGWGNTKSSGTSYPDVLKCLKAPILSDSSCKSAYPGQITSNMFCAGYLEGGKDSCQGDSGGPVVCSGKLQGIVSWGSGCAQKNKPGVYTKVCNYVSWIKQTIASN
5AFV , Knot 162 378 0.84 40 216 356 
DEQPEPRTRRRAYLWCKEFLPGAWRGLREDEFHISVIRGGLSNMLFQCSLPDTTATLGDEPRKVLLRLYGAILQMRSCNKEGSEQAQKENEFQGAEAMVLESVMFAILAERSLGPKLYGIFPQGRLEQFIPSRRLDTEELSLPDISAEIAEKMATFHGMKMPFNKEPKWLFGTMEKYLKEVLRIKFTEESRIKKLHKLLSYNLPLELENLRSLLESTPSPVVFCHNDCQEGNILLLEGRENSEKQKLMLIDFEYSSYNYRGFDIGNHFCEWMYDYSYEKYPFFRANIRKYPTKKQQLHFISSYLPAFQNDFENLSTEEKSIIKEEMLLEVNRFALASHFLWGLWSIVQAKISSIEFGYMDYAQARFDAYFHQKRKLGV
6MTY , Knot 101 225 0.81 40 148 216 
QVHLQESGPGLVRPSETLSLTCTVSDGSISSYYWSWIRQPPGKGLEWIGSIYYTGSTNYNPSLKSRVTMSVDTSKNQFSLRLNSVTAADTAMYYCAGLDRYSWNEGGDHWGQGILVSVSSASTKGPSVFPLAPSSKSTSGGTAALGCLVKDYFPEPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKKVEPKSCDK

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 \(c\).

\(|P_{f(2BTC_1)}(2) \setminus P_{f(5AFV_1)}(2)|=64\), \(|P_{f(5AFV_1)}(2) \setminus P_{f(2BTC_1)}(2)|=140\). 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 1:1111000110011001010010010110110001110110000011010110001011010001101000110100000010001111010011010001101011000101100011011100000100010110010111100000001011010001101101011000001001111100101011101101010000111000100010110001100
Pair \(Z_2\) Length of longest common subsequence
2BTC_1,5AFV_1 204 4
2BTC_1,6MTY_1 138 3
5AFV_1,6MTY_1 206 4

Newick tree

 
[
	5AFV_1:11.45,
	[
		2BTC_1:69,6MTY_1:69
	]:42.45
]

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 \((0.85)(0.8)(\frac{601 }{\log_{20} 601}-\frac{223}{\log_{20}223})=107.\)
Status Protein1 Protein2 d d1/2
Query variables 2BTC_1 5AFV_1 140 109.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]