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Parikh vectors
2QWU_1 1PDR_1 5QQE_1 Letter Amino acid
11 8 9 E Glutamic acid
18 3 13 T Threonine
7 3 6 N Asparagine
12 6 12 D Aspartic acid
5 4 5 Q Glutamine
12 7 18 L Leucine
23 7 8 S Serine
4 2 8 Y Tyrosine
19 12 13 A Alanine
15 12 11 G Glycine
3 3 2 H Histidine
2 3 6 F Phenylalanine
7 3 11 P Proline
11 7 16 V Valine
6 8 7 R Arginine
27 8 10 I Isoleucine
15 3 13 K Lycine
9 0 3 M Methionine
1 0 2 W Tryptophan
4 0 5 C Cysteine 

2QWU_1|Chains A, B|Intracellular growth locus, subunit C|Francisella tularensis subsp. tularensis (119856)
>1PDR_1|Chain A|HUMAN DISCS LARGE PROTEIN|Homo sapiens (9606)
>5QQE_1|Chain A|Ras-related C3 botulinum toxin substrate 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 \(w=f(c)\)
2QWU , Knot 94 211 0.79 40 141 201 
MIMSEMITRQQVTSGETIHVRTDPTACIGSHPNCRMFIDSLTIAGEKLDKNIVAIDGGEDVTKADSATAAASVIRMSITPGSINPTISITLGVLIKSNVRTKIEEKVSSILQASATDMKIKLGNSNKKQEYKTDEAWGIMIDLSNLELYPISAKAFSISIEPTELMGVSKDGMRYHIISIDGLTTSQGSLPVCCAASTDKGVAKIGYIAAA
1PDR , Knot 51 99 0.79 34 83 96 
DDEITREPRKVVLHRGSTGLGFNIVGGEDGEGIFISFILAGGPADLSGELRKGDRIISVNSVDLRAASHEQAAAALKNAGQAVTIVAQYRPEEYSRQHA
5QQE , Knot 84 178 0.81 40 140 171 
SMQAIKCVVVGDGAVGKTCLLISYTTNAFPGEYIPTVFDNYSANVMVDGKPVNLGLWDTAGQEDYDRLRPLSYPQTDVFLICFSLVSPASFENVRAKWYPEVRHHCPNTPIILVGTKLDLRDDKDTIEKLKEKKLTPITYPQGLAMAKEIGAVKYLECSALTQRGLKTVFDEAIRAVL

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 \(c\).

\(|P_{f(2QWU_1)}(2) \setminus P_{f(1PDR_1)}(2)|=100\), \(|P_{f(1PDR_1)}(2) \setminus P_{f(2QWU_1)}(2)|=42\). 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 1:1110011000010010010100010101100100011100101110010001111011001001001011101101010110101010101111100010001000100110101001010110000000000001111110100101011010110101010011110001100011010110000101110011000011101101111
Pair \(Z_2\) Length of longest common subsequence
2QWU_1,1PDR_1 142 4
2QWU_1,5QQE_1 165 3
1PDR_1,5QQE_1 143 3

Newick tree

 
[
	5QQE_1:79.15,
	[
		2QWU_1:71,1PDR_1:71
	]:8.15
]

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 \((0.85)(0.8)(\frac{310 }{\log_{20} 310}-\frac{99}{\log_{20}99})=66.1\)
Status Protein1 Protein2 d d1/2
Query variables 2QWU_1 1PDR_1 82 59
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]