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Parikh vectors
3NKC_1 2CTB_1 5KQY_1 Letter Amino acid
3 16 4 D Aspartic acid
2 2 0 C Cysteine
17 21 15 I Isoleucine
26 24 10 L Leucine
3 3 2 M Methionine
18 16 2 F Phenylalanine
4 13 4 N Asparagine
7 14 3 E Glutamic acid
5 8 1 H Histidine
9 10 6 P Proline
13 32 1 S Serine
22 15 6 V Valine
3 12 5 Q Glutamine
10 26 8 T Threonine
6 7 2 W Tryptophan
5 15 7 K Lycine
5 11 4 R Arginine
36 23 13 G Glycine
5 19 1 Y Tyrosine
35 20 5 A Alanine 

3NKC_1|Chains A, B|Aquaporin Z|Escherichia coli (83333)
>2CTB_1|Chain A|CARBOXYPEPTIDASE A|Bos taurus (9913)
>5KQY_1|Chains A, B|Protease E35D-DRV|Human immunodeficiency virus 1 (11676)
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 \(w=f(c)\)
3NKC , Knot 105 234 0.81 40 136 216 
ASHMFRKLAAECFGTFWLVFGGCGSAVLAAGFPELGIGFAGVALAWGLTVLTMAFAVGHISGGHFNPAVTIGLWAGGRFPAKEVVGYVIAQVVGGIVAAALLYLIASGKTGFDAAASGFASNGYGEHSPGGYSMLSALVVELVLSAGFLLVIHGATDKFAPAGFAPIAIGLALTLIGLISIPVTNFSVNPARSTAVAIFQGGWALEQLWFFWVVPIVGGIIGGLIYRTLLEKRD
2CTB , Knot 135 307 0.84 40 191 298 
ARSTNTFNYATYHTLDEIYDFMDLLVAEHPQLVSKLQIGRSYEGRPIYVLKFSTGGSNRPAIWIDLGIHSREWITQATGVWFAKKFTEDYGQDPSFTAILDSMDIFLEIVTNPDGFAFTHSQNRLWRKTRSVTSSSLCVGVDANRNWDAGFGKAGASSSPCSETYHGKYANSEVEVKSIVDFVKDHGNFKAFLSIHSYSQLLLYPYGYTTQSIPDKTELNQVAKSAVEALKSLYGTSYKYGSIITTIYQASGGSIDWSYNQGIKYSFTFELRDTGRYGFLLPASQIIPTAQETWLGVLTIMEHTLNN
5KQY , Knot 53 99 0.82 38 77 95 
PQITLWKRPLVTIKIGGQLKEALLNTGADDTVIEDMSLPGRWKPKMIGGIGGFIKVRQYDQIIIEIAGHKAIGTVLVGPTPVNIIGRNLLTQIGATLNF

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 \(c\).

\(|P_{f(3NKC_1)}(2) \setminus P_{f(2CTB_1)}(2)|=51\), \(|P_{f(2CTB_1)}(2) \setminus P_{f(3NKC_1)}(2)|=106\). 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 1:100110011100110111111101011111111011111111111111011011111101011010111011111110111001110111011111111111011101001101110111001010001110011011110111011111110110001111111111111110111110111001010110001111101111100111111111111111111000110000
Pair \(Z_2\) Length of longest common subsequence
3NKC_1,2CTB_1 157 4
3NKC_1,5KQY_1 147 3
2CTB_1,5KQY_1 180 3

Newick tree

 
[
	2CTB_1:87.79,
	[
		3NKC_1:73.5,5KQY_1:73.5
	]:14.29
]

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 \((0.85)(0.8)(\frac{541 }{\log_{20} 541}-\frac{234}{\log_{20}234})=87.7\)
Status Protein1 Protein2 d d1/2
Query variables 3NKC_1 2CTB_1 116 99
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]