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Parikh vectors
3VOD_1 6BBP_1 8OXI_1 Letter Amino acid
9 24 6 A Alanine
7 36 9 R Arginine
6 17 3 Q Glutamine
6 22 7 P Proline
1 9 2 W Tryptophan
6 31 0 N Asparagine
5 8 4 C Cysteine
7 35 5 I Isoleucine
25 49 7 L Leucine
3 13 1 M Methionine
8 36 5 E Glutamic acid
2 24 4 F Phenylalanine
9 29 11 T Threonine
12 26 6 V Valine
2 18 5 Y Tyrosine
8 34 5 D Aspartic acid
7 35 13 G Glycine
3 15 2 H Histidine
12 38 3 K Lycine
6 21 6 S Serine 

3VOD_1|Chains A, B|Multiple antibiotic resistance protein marR|Escherichia coli (83333)
>6BBP_1|Chain A|Cytohesin-3,ADP-ribosylation factor 6|Mus musculus (10090)
>8OXI_1|Chain A|BgtE-5845_p|Blumeria graminis f. sp. tritici (62690)
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 \(w=f(c)\)
3VOD , Knot 69 144 0.79 40 106 136 
MKSTSDLFNEIIPLGRLIHMVNQKKDRLLNEYLSPLDITAAQFKVLCSIRCAACITPVELKKVLSVDLGALTRMLDRLVSKGWVERLPNPNDKRGVLVKLTTGGAAICEQCHQLVGQDLHQELTKNLTADEVATLEYLLKKVLP
6BBP , Knot 211 520 0.84 40 260 480 
MGHHHHHHGSTTQRNKQIAMGRKKFNMDPKKGIQFLIENDLLQSSPEDVAQFLYKGEGLNKTVIGDYLGERDDFNIKVLQAFVELHEFADLNLVQALRQFLWSFRLPGEAQKIDRMMEAFASRYCLCNPGVFQSTDTCYVLSFAIIMLNTSLHNHNVRDKPTAERFITMNRGINEGGDLPEELLRNLYESIKNEPFKIPEDDGNDLTHTFFNPDREGWLLKLGGRVKTWKRRWFILTDNCLYYFEYTTDKEPRGIIPLENLSIREVEDPRKPNCFELYNPSHKGQVIKACKTEADGRVVEGNHVVYRISAPSPEEKEEWMKSIKASISRDPFYDMLATRKRRIANKKGKVLSKIFGNKEMRILMLGLDAAGKTTILYKLKLGQSVTTIPTVGFNVETVTYKNVKFNVWDVGGLDKIRPLWRHYYTGTQGLIFVVDCADRDRIDEARQELHRIINDREMRDAIILIFANKQDLPDAMKPHEIQEKLGLTRIRDRNWYVQPSCATSGDGLYEGLTWLTSNYN
8OXI , Knot 57 104 0.84 38 89 102 
GPLGSMESYWDCKGIPILFRTVHAAVELAFTSQPGSISGYPSICRTTPLRTGPDERRQFPLTDTGARWQGGGITYYVEATRDKRHCEVFGTAGGVYKCTLVLRD

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 \(c\).

\(|P_{f(3VOD_1)}(2) \setminus P_{f(6BBP_1)}(2)|=26\), \(|P_{f(6BBP_1)}(2) \setminus P_{f(3VOD_1)}(2)|=180\). 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 1:100000110011111011011000000110001011010110101100100110101101001101011110011001100111001101000011110100111110000001110010001000101001101001100111
Pair \(Z_2\) Length of longest common subsequence
3VOD_1,6BBP_1 206 4
3VOD_1,8OXI_1 131 3
6BBP_1,8OXI_1 221 3

Newick tree

 
[
	6BBP_1:11.39,
	[
		3VOD_1:65.5,8OXI_1:65.5
	]:51.89
]

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 \((0.85)(0.8)(\frac{664 }{\log_{20} 664}-\frac{144}{\log_{20}144})=149.\)
Status Protein1 Protein2 d d1/2
Query variables 3VOD_1 6BBP_1 193 119.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]