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Parikh vectors
4HLB_1 2LXR_1 2CEJ_1 Letter Amino acid
11 5 4 R Arginine
5 3 13 G Glycine
6 7 12 L Leucine
8 7 2 F Phenylalanine
7 4 1 S Serine
10 4 6 V Valine
1 4 3 N Asparagine
14 3 4 D Aspartic acid
0 0 2 C Cysteine
9 9 4 E Glutamic acid
8 7 6 K Lycine
0 1 2 W Tryptophan
2 3 6 Q Glutamine
6 6 13 I Isoleucine
1 3 2 M Methionine
1 3 6 P Proline
7 1 8 T Threonine
3 0 1 Y Tyrosine
16 5 3 A Alanine
0 1 1 H Histidine 

4HLB_1|Chain A|Uncharacterized protein|Desulfovibrio piger (411464)
>2LXR_1|Chain A|NADH dehydrogenase I subunit E|Helicobacter pylori (85962)
>2CEJ_1|Chains A, B|POL PROTEIN|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 \(w=f(c)\)
4HLB , Knot 57 115 0.78 34 83 110 
GAEQQADTVTENSDSEVFVDDSDRFTAFEEELLARYADKGIRSVDVAAYAKGIDIVFVAADRKMTRAEFSAIASRSIRELKERFGFDKDVPIGAVLDYKKDAATDTRTRFVLKLR
2LXR , Knot 43 76 0.81 36 68 74 
MKRFDLRPLKAGIFERLEELIEKEMQPNEVAIFMFEVGDFSNIPKSAEFIQSKGHELLNSLRFNQADWTIVVRKKA
2CEJ , Knot 52 99 0.80 40 81 94 
PQITLWQRPLVTIKIGGQLKEALLDTGADDTVLEEMSLPGRWKPKMIGGIGGFIKVRQYDQILIEICGHKAIGTVLVGPTPVNIIGRNLLTQIGCTLNF

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 \(c\).

\(|P_{f(4HLB_1)}(2) \setminus P_{f(2LXR_1)}(2)|=58\), \(|P_{f(2LXR_1)}(2) \setminus P_{f(4HLB_1)}(2)|=43\). 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 1:1100010010000000111000001011000111001001100101110101101111110001001010111000100100011100011111110000011000000111010
Pair \(Z_2\) Length of longest common subsequence
4HLB_1,2LXR_1 101 3
4HLB_1,2CEJ_1 130 3
2LXR_1,2CEJ_1 103 3

Newick tree

 
[
	2CEJ_1:61.11,
	[
		4HLB_1:50.5,2LXR_1:50.5
	]:10.61
]

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 \((0.85)(0.8)(\frac{191 }{\log_{20} 191}-\frac{76}{\log_{20}76})=38.3\)
Status Protein1 Protein2 d d1/2
Query variables 4HLB_1 2LXR_1 44 36.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]