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Parikh vectors
4HSH_1 5VHF_1 6CSK_1 Letter Amino acid
27 54 4 L Leucine
21 9 1 F Phenylalanine
30 27 0 S Serine
18 27 0 T Threonine
17 30 0 V Valine
8 18 0 N Asparagine
5 7 0 C Cysteine
10 6 0 H Histidine
16 18 1 M Methionine
17 8 1 P Proline
7 16 0 Y Tyrosine
32 26 2 R Arginine
17 23 0 Q Glutamine
34 14 1 G Glycine
4 5 0 W Tryptophan
24 24 0 D Aspartic acid
22 31 1 I Isoleucine
14 43 5 K Lycine
38 26 3 A Alanine
24 44 0 E Glutamic acid 

4HSH_1|Chain A|Queuine tRNA-ribosyltransferase|Zymomonas mobilis subsp. mobilis (264203)
>5VHF_10|Chain J[auth W]|26S proteasome non-ATPase regulatory subunit 12|Homo sapiens (9606)
>6CSK_1|Chain A|PaDBS1R2|Pyrobaculum aerophilum (178306)
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 \(w=f(c)\)
4HSH , Knot 161 385 0.83 40 215 372 
VEATAQETDRPRFSFSIAAREGKARTGTIEMKRGVIRTPAFMPVGTAATVKALKPETVRATGADIILGNTYHLMLRPGAERIAKLGGLHSFMGWDRPILTDSGGFQVMSLSSLTKQSEEGVTFKSHLDGSRHMLSPERSIEIQHLLGSDIVMAFDECTPYPATPSRAASSMERSMRWAKRSRDAFDSRKEQAENAALFGIQQGSVFENLRQQSADALAEIGFDGYAVGGLAGGEGQDEMFRVLDFSVPMLPDDKPHYLMGVGKPDDIVGAVERGIDMFDCVLPTRSGRNGQAFTWDGPINIRNARFSEDLKPLDSECHCAVCQKWSRAYIHHLIRAGEILGAMLMTEHNIAFYQQLMQKIRDSISEGRFSQFAQDFRARYFARNS
5VHF , Knot 187 456 0.83 40 236 434 
MADGGSERADGRIVKMEVDYSATVDQRLPECAKLAKEGRLQEVIETLLSLEKQTRTASDMVSTSRILVAVVKMCYEAKEWDLLNENIMLLSKRRSQLKQAVAKMVQQCCTYVEEITDLPIKLRLIDTLRMVTEGKIYVEIERARLTKTLATIKEQNGDVKEAASILQELQVETYGSMEKKERVEFILEQMRLCLAVKDYIRTQIISKKINTKFFQEENTEKLKLKYYNLMIQLDQHEGSYLSICKHYRAIYDTPCIQAESEKWQQALKSVVLYVILAPFDNEQSDLVHRISGDKKLEEIPKYKDLLKLFTTMELMRWSTLVEDYGMELRKGSLESPATDVFGSTEEGEKRWKDLKNRVVEHNIRIMAKYYTRITMKRMAQLLDLSVDESEAFLSNLVVNKTIFAKVDRLAGIINFQRPKDPNNLLNDWSQKLNSLMSLVNKTTHLIAKEEMIHNLQ
6CSK , Knot 12 19 0.62 18 16 17 
PMKLLKRLGKKIRLAAAFK

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 \(c\).

\(|P_{f(4HSH_1)}(2) \setminus P_{f(5VHF_1)}(2)|=80\), \(|P_{f(5VHF_1)}(2) \setminus P_{f(4HSH_1)}(2)|=101\). 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 1:1010100000101010111001010010101001110011111110110101101001010110111100001110111001101111001111001110001110110100100000011010001010001101000101001110011111000010110100110010001011000001100000010011111100101100100001011101110101111111101000110110101111100010011111010011111001101100111000100101101011101001010001011000000110001001010011011011111110000111000110010001001010011001010011000
Pair \(Z_2\) Length of longest common subsequence
4HSH_1,5VHF_1 181 4
4HSH_1,6CSK_1 203 3
5VHF_1,6CSK_1 226 3

Newick tree

 
[
	6CSK_1:11.47,
	[
		4HSH_1:90.5,5VHF_1:90.5
	]:21.97
]

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 \((0.85)(0.8)(\frac{841 }{\log_{20} 841}-\frac{385}{\log_{20}385})=122.\)
Status Protein1 Protein2 d d1/2
Query variables 4HSH_1 5VHF_1 156 144
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]