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Parikh vectors
6KVF_1 2VIB_1 4ROT_1 Letter Amino acid
20 21 10 T Threonine
22 36 12 V Valine
7 20 15 D Aspartic acid
10 20 14 Q Glutamine
5 28 19 E Glutamic acid
7 14 14 N Asparagine
7 2 1 C Cysteine
13 32 28 L Leucine
37 12 14 S Serine
5 0 8 W Tryptophan
11 15 18 Y Tyrosine
2 15 7 H Histidine
14 19 17 K Lycine
4 16 14 F Phenylalanine
4 22 10 I Isoleucine
4 9 7 M Methionine
15 18 9 P Proline
14 50 21 A Alanine
5 21 8 R Arginine
21 35 22 G Glycine 

6KVF_1|Chains A, B[auth H]|heavy chain|Homo sapiens (9606)
>2VIB_1|Chain A|SERINE HYDROXYMETHYLTRANSFERASE|GEOBACILLUS STEAROTHERMOPHILUS (1422)
>4ROT_1|Chain A|Esterase A|Streptococcus pyogenes (1314)
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 \(w=f(c)\)
6KVF , Knot 97 227 0.77 40 143 208 
MQVQLVQSGAEVKKPGSSVKVSCKASGGTLSSYAISWVRQAPGQGPEWMGWINPNSGGTNYAQKFQGRVTMTRDTSISTAYMELSRLRPDDTAVYYCASGYCSSTSCYDYWGQGTLVTVSSASTKGPSVFPLAPSSKSTSGGTAALGCLVKDYFPEPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKKVEPPKSCDK
2VIB , Knot 168 405 0.83 38 215 379 
MKYLPQQDPQVFAAIEQERKRQHAKIELIASENFVSRAVMEAQGSVLTNKYAEGYPGRRYYGGCEYVDIVEELARERAKQLFGAEHANVQPHSGAQANMAVYFTVLEHGDTVLGMNLSHGGHLTHGSPVNFSGVQYNFVAYGVDPETHVIDYDDVREKARLHRPKLIVAAAAAYPRIIDFAKFREIADEVGAYLMVDMAHIAGLVAAGLHPNPVPYAHFVTTTTHKTLRGPRGGMILCQEQFAKQIDKAIFPGIQGGPLMHVIAAKAVAFGEALQDDFKAYAKRVVDNAKRLASALQNEGFTLVSGGTDNHLLLVDLRPQQLTGKTAEKVLDEVGITVNKNTIPYDPESPFVTSGIRIGTAAVTTRGFGLEEMDEIAAIIGLVLKNVGSEQALEEARQRVAALTD
4ROT , Knot 120 268 0.83 40 182 261 
NLYFQGAMASIAIEYHSVVLGMERKVNVIYPDQSEIPKKDQGDKDIPVLYLLHGMGGNENSWQKRTAIERLLRHTNLIVVMPSTDLGWYTDTAYGLNYYRALSQELPQVLAAFFPNMTQKREKTFVAGLSMGGYGAFKWALKSNRFSYAASFSGALDFSPETNLEGNLGELAYWQGVFGQFEDPDLDKHYLKNMVAESDGKTKFYAWCGYEDFLFATNEKAIADFQAQGLDIDYHKGHGKHEWYYWNQQLEVLLEWLPINYQKEERLS

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 \(c\).

\(|P_{f(6KVF_1)}(2) \setminus P_{f(2VIB_1)}(2)|=62\), \(|P_{f(2VIB_1)}(2) \setminus P_{f(6KVF_1)}(2)|=134\). 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 1:10101100110100110010100010110100011011001110110111110100110001001010101000001001010100101000110001010000000000110101101001000110111111000000110111101100011011010100111001100111110001100100110110001100001001000100001000101100000
Pair \(Z_2\) Length of longest common subsequence
6KVF_1,2VIB_1 196 4
6KVF_1,4ROT_1 171 4
2VIB_1,4ROT_1 171 4

Newick tree

 
[
	2VIB_1:94.01,
	[
		6KVF_1:85.5,4ROT_1:85.5
	]:8.51
]

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 \((0.85)(0.8)(\frac{632 }{\log_{20} 632}-\frac{227}{\log_{20}227})=114.\)
Status Protein1 Protein2 d d1/2
Query variables 6KVF_1 2VIB_1 145 113
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]