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Parikh vectors
3VWB_1 5VDS_1 4DLF_1 Letter Amino acid
5 23 11 F Phenylalanine
16 28 10 S Serine
0 4 11 W Tryptophan
4 18 19 V Valine
5 20 30 R Arginine
0 10 6 C Cysteine
7 14 17 G Glycine
15 41 38 L Leucine
5 18 27 D Aspartic acid
10 32 18 E Glutamic acid
13 18 5 I Isoleucine
10 18 2 N Asparagine
3 6 4 M Methionine
6 11 11 P Proline
3 14 5 T Threonine
5 11 8 Y Tyrosine
8 14 56 A Alanine
3 12 12 Q Glutamine
8 8 10 H Histidine
17 43 3 K Lycine 

3VWB_1|Chain A|Virulence regulon transcriptional activator virB|Shigella flexneri 2a (42897)
>5VDS_1|Chains A, B|Cyclic GMP-AMP synthase|Homo sapiens (9606)
>4DLF_1|Chain A|Amidohydrolase 2|Burkholderia multivorans (395019)
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 \(w=f(c)\)
3VWB , Knot 68 143 0.78 36 95 136 
MGSSHHHHHHSSGLVPRGSHMAKEHSIRELGIGLNFLKVSGMSYKDIAKKENLSRAKVTRAFQAASVPQEIISLFPIASELNFNDYKILFNYYKGLEKANESLSSTLPILKEEIKDLDTNLPPDIYKKEILNIIKKSKNRKQN
5VDS , Knot 150 363 0.81 40 205 346 
MGASKLRAVLEKLKLSRDDISTAAGMVKGVVDHLLLRLKCDSAFRGVGLLNTGSYYEHVKISAPNEFDVMFKLEVPRIQLEEYSNTRAYYFVKFKRNPKENPLSQFLEGEILSASKMLSKFRKIIKEEINDIKDTDVIMKRKRGGSPAVTLLISEKISVDITLALESKSSWPASTQEGLRIQNWLSAKVRKQLRLKPFYLVPKHAKEGNGFQEETWRLSFSHIEKEILNNHGKSKTCCENKEEKCCRKDCLKLMKYLLEQLKERFKDKKHLDKFSSYHVKTAFFHVCTQNPQDSQWDRKDLGLCFDNCVTYFLQCLRTEKLENYFIPEFNLFSSNLIDKRSKEFLTKQIEYERNNEFPVFDEF
4DLF , Knot 126 303 0.79 40 164 282 
MGALRIDSHQHFWRYRAADYPWIGAGMGVLARDYLPDALHPLMHAQALGASIAVQARAGRDETAFLLELACDEARIAAVVGWEDLRAPQLAERVAEWRGTKLRGFRHQLQDEADVRAFVDDADFARGVAWLQANDYVYDVLVFERQLPDVQAFCARHDAHWLVLDHAGKPALAEFDRDDTALARWRAALRELAALPHVVCKLSGLVTEADWRRGLRASDLRHIEQCLDAALDAFGPQRLMFGSDWPVCLLAASYDEVASLVERWAESRLSAAERSALWGGTAARCYALPEPADARLAENLYFQ

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 \(c\).

\(|P_{f(3VWB_1)}(2) \setminus P_{f(5VDS_1)}(2)|=30\), \(|P_{f(5VDS_1)}(2) \setminus P_{f(3VWB_1)}(2)|=140\). 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 1:11000000000011110100110000100111110110101100001100001001010011011011001101111100101000011100001100100010001111000100100011101000011011000000000
Pair \(Z_2\) Length of longest common subsequence
3VWB_1,5VDS_1 170 5
3VWB_1,4DLF_1 181 4
5VDS_1,4DLF_1 185 3

Newick tree

 
[
	4DLF_1:93.57,
	[
		3VWB_1:85,5VDS_1:85
	]:8.57
]

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 \((0.85)(0.8)(\frac{506 }{\log_{20} 506}-\frac{143}{\log_{20}143})=106.\)
Status Protein1 Protein2 d d1/2
Query variables 3VWB_1 5VDS_1 132 90
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]