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Parikh vectors
2QII_1 7XXP_1 5BVF_1 Letter Amino acid
18 29 14 T Threonine
17 28 21 P Proline
30 32 16 S Serine
8 8 19 Y Tyrosine
18 43 15 V Valine
38 61 25 A Alanine
32 36 19 R Arginine
17 11 12 Q Glutamine
24 32 21 E Glutamic acid
14 7 23 K Lycine
4 9 3 W Tryptophan
8 9 19 N Asparagine
24 31 24 D Aspartic acid
10 22 13 H Histidine
22 19 28 I Isoleucine
27 52 43 L Leucine
17 12 8 M Methionine
20 12 13 F Phenylalanine
5 3 7 C Cysteine
33 52 18 G Glycine 

2QII_1|Chain A|Queuine tRNA-ribosyltransferase|Zymomonas mobilis (542)
>7XXP_1|Chains A, B, C, D, E, F, G, H|N-formimidoyl fortimicin A synthase|Streptomyces luteocolor (285500)
>5BVF_1|Chain A|Mitogen-activated protein kinase 1|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 \(w=f(c)\)
2QII , Knot 161 386 0.82 40 215 372 
MVEATAQETDRPRFSFSIAAREGKARTGTIEMKRGVIRTPAFMPVGTAATVKALKPETVRATGADIILGNTYHLMLRPGAERIAKLGGLHSFMGWDRPILTDSGGYQVMSLSSLTKQSEEGVTFKSHLDGSRHMLSPERSIEIQHLLGSDIVMAFDECTPYPATPSRAASSMERSMRWAKRSRDAFDSRKEQAENAALFGIQQGSVFENLRQQSADALAEIGFDGYAVGGLAVGEGQDEMFRVLDFSVPMLPDDKPHYLMGVGKPDDIVGAVERGIDMFDCVLPTRSGRNGQAFTWDGPINIRNARFSEDLKPLDSECHCAVCQKWSRAYIHHLIRAGEILGAMLMTEHNIAFYQQLMQKIRDSISEGRFSQFAQDFRARYFARNS
7XXP , Knot 205 508 0.83 40 232 474 
MGSSHHSSGLVPRGSHMMSKTADAPGQTDVIVVGNGVLGLSVGVEIARTRPDVRVTLLGKPARQYGATPAAGAMLGAFGEVTAHALASEHGRKKHALAVQAQRLWPEWIESLEATGTAADGRIKTADDTVVLLNTVGHSALDDANFAAVLTALKEANAPHEEIAVESVDWIDPDPNSRPLRALHIEGEGSVDSGILLAALERSFLQAGGRLHPVDATEIRASHGRVEGVVTDDGDFLPAGHVVVAAGARSQRLVAALPGLAHRIPRIYDGVGVSALVDTWDGSGPATVLRTSNRAFACGLHLVPRAGGSVYIGATNAVCLEPRGAASIEETVALFNCATHQLHRGLNGSELRKVQVGSRPAPIDGFPLIGGTSVEGLWMLSGTYRDGLHMSPLLARHVVSLMDGGTGVDGLREFRPERDLISAWSREEILDDVVRHTMATGYEFPWRLPLEWPHMMETFLQGPFAELADRLSDTYTPPADLMTAIMFSEREQQDELIAYYADVHREWH
5BVF , Knot 159 361 0.86 40 234 349 
GPAAAAAAGAGPEMVRGQVFDVGPRYTNLSYIGEGAYGMVCSAYDNVNKVRVAIKKISPFEHQTYCQRTLREIKILLRFRHENIIGINDIIRAPTIEQMKDVYIVQDLMETDLYKLLKTQHLSNDHICYFLYQILRGLKYIHSANVLHRDLKPSNLLLNTTCDLKICDFGLARVADPDHDHTGFLTEYVATRWYRAPEIMLNSKGYTKSIDIWSVGCILAEMLSNRPIFPGKHYLDQLNHILGILGSPSQEDLNCIINLKARNYLLSLPHKNKVPWNRLFPNADSKALDLLDKMLTFNPHKRIEVEQALAHPYLEQYYDPSDEPIAEAPFKFDMELDDLPKEKLKELIFEETARFQPGYRS

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 \(c\).

\(|P_{f(2QII_1)}(2) \setminus P_{f(7XXP_1)}(2)|=66\), \(|P_{f(7XXP_1)}(2) \setminus P_{f(2QII_1)}(2)|=83\). 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 1:11010100000101010111001010010101001110011111110110101101001010110111100001110111001101111001111001110001100110100100000011010001010001101000101001110011111000010110100110010001011000001100000010011111100101100100001011101110101111111101000110110101111100010011111010011111001101100111000100101101011101001010001011000000110001001010011011011111110000111000110010001001010011001010011000
Pair \(Z_2\) Length of longest common subsequence
2QII_1,7XXP_1 149 4
2QII_1,5BVF_1 163 4
7XXP_1,5BVF_1 162 4

Newick tree

 
[
	5BVF_1:83.37,
	[
		2QII_1:74.5,7XXP_1:74.5
	]:8.87
]

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 \((0.85)(0.8)(\frac{894 }{\log_{20} 894}-\frac{386}{\log_{20}386})=135.\)
Status Protein1 Protein2 d d1/2
Query variables 2QII_1 7XXP_1 168 149.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]