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Parikh vectors
4NBV_1 3QPS_1 2NCS_1 Letter Amino acid
14 3 0 R Arginine
4 2 0 C Cysteine
11 15 0 E Glutamic acid
11 15 1 S Serine
5 9 1 Y Tyrosine
14 7 1 T Threonine
2 1 5 W Tryptophan
10 19 2 N Asparagine
13 17 4 I Isoleucine
9 24 11 K Lycine
9 15 2 F Phenylalanine
4 4 0 P Proline
42 11 1 A Alanine
23 10 1 G Glycine
1 12 0 H Histidine
17 21 3 L Leucine
29 10 1 V Valine
13 6 2 D Aspartic acid
9 11 0 Q Glutamine
6 4 1 M Methionine 

4NBV_1|Chains A, B|3-oxoacyl-[acyl-carrier-protein] reductase putative short-chain dehydrogenases/reductases (SDR) family protein|Cupriavidus taiwanensis (164546)
>3QPS_1|Chain A|CmeR|Campylobacter jejuni (197)
>2NCS_1|Chain A|Envelope glycoprotein gp41|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 \(w=f(c)\)
4NBV , Knot 109 246 0.81 40 146 230 
MKLQGRVAIITGAAAGIGFATAQRFAEDGAIVVLCDVQEARVREAAARLAATGATVSAYRVDVTRRDEVDAMVAAVLAAHQRVDILVNNAGITKDARLAKMTEAQFDAVIDVNLKGVFNCAQAVAGLMTEQGKGVILNASSVVGLYGNFGQTNYAASKFGVIGFTKTWARELGPKGVRVNAVCPGFVNTEILQTVPDKVLDGMTSSCWLRRLAEPAEIASIYAFLASDDASYVNGVAIEASGGMSL
3QPS , Knot 94 216 0.78 40 137 206 
HHHHHHMNSNRTPSQKVLARQEKIKAVALELFLTKGYQETSLSDIIKLSGGSYSNIYDGFKSKEGLFFEILDDICKKHFHLIYSKTQEIKNGTLKEILTSFGLAFIEIFNQPEAVAFGKIIYSQVYDKDRHLANWIENNQQNFSYNILMGFFKQQNNSYMKKNAEKLAVLFCTMLKEPYHHLNVLINAPLKNKKEQKEHVEFVVNVFLNGINSSKA
2NCS , Knot 18 36 0.59 28 27 30 
KKKKDKWASLWNWFDITNWLWYIKLFIMIVGKKKKK

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 \(c\).

\(|P_{f(4NBV_1)}(2) \setminus P_{f(3QPS_1)}(2)|=91\), \(|P_{f(3QPS_1)}(2) \setminus P_{f(4NBV_1)}(2)|=82\). 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 1:101010111101111111110100110011111100100101001110111011010100101000001011111111100010111001110001011010010101110101011100101111110001011110100111101011000011001111110001100111011010110111100011001100110110000110011011011010111100010010111101011101
Pair \(Z_2\) Length of longest common subsequence
4NBV_1,3QPS_1 173 3
4NBV_1,2NCS_1 145 2
3QPS_1,2NCS_1 142 3

Newick tree

 
[
	4NBV_1:82.53,
	[
		2NCS_1:71,3QPS_1:71
	]:11.53
]

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 \((0.85)(0.8)(\frac{462 }{\log_{20} 462}-\frac{216}{\log_{20}216})=71.5\)
Status Protein1 Protein2 d d1/2
Query variables 4NBV_1 3QPS_1 92 85.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]