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Parikh vectors
5PBV_1 6UQV_1 2BHD_1 Letter Amino acid
6 9 16 N Asparagine
6 24 34 G Glycine
11 17 21 S Serine
8 10 20 T Threonine
4 11 18 Y Tyrosine
8 21 34 V Valine
2 18 18 Q Glutamine
9 14 42 E Glutamic acid
4 5 28 I Isoleucine
12 14 23 D Aspartic acid
13 28 39 L Leucine
6 3 11 M Methionine
6 15 18 P Proline
4 42 35 A Alanine
5 24 33 R Arginine
2 2 4 C Cysteine
8 10 14 H Histidine
14 4 13 K Lycine
8 10 15 F Phenylalanine
2 6 4 W Tryptophan 

5PBV_1|Chain A|Bromodomain adjacent to zinc finger domain protein 2B|Homo sapiens (9606)
>6UQV_1|Chain A|ChoE|Pseudomonas aeruginosa (strain ATCC 15692 / DSM 22644 / CIP 104116 / JCM 14847 / LMG 12228 / 1C / PRS 101 / PAO1) (208964)
>2BHD_1|Chain A|XAA-PRO AMINOPEPTIDASE|ESCHERICHIA COLI (562)
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 \(w=f(c)\)
5PBV , Knot 66 138 0.78 40 109 129 
MHHHHHHSSGVDLGTENLYFQSMSVKKPKRDDSKDLALCSMILTEMETHEDAWPFLLPVNLKLVPGYKKVIKKPMDFSTIREKLSSGQYPNLETFALDVRLVFDNCETFNEDDSDIGRAGHNMRKYFEKKWTDTFKVS
6UQV , Knot 128 287 0.84 40 177 273 
HTSPLLAPVRQIHAFGDSYSDNGESQRLTREMLAKGIAGAQALPGEVYWQGRWSNGPTAVEVLARQLGAQLADHAVGGAKSGADNYYGWMSAYRHTGLAGQVDAYLATLDGKPVDGQALHFIFVSANDFFEHEDFAGEQPLEQLAGSSVANIRAAVQRLGEAGARRFLVVSSTDLSVVPAVVAGNRVERAQRYLQAVNASLPIQLAALRKTRGLELSWFDHLTFSRHLRRNPARYGLVELDAPCQPTQPSVRPACANPDQYYFWDEWHPTRRVHQLAGEAMAARYAR
2BHD , Knot 180 440 0.83 40 231 411 
SEISRQEFQRRRQALVEQMQPGSAALIFAAPEVTRSADSEYPYRQNSDFWYFTGFNEPEAVLVLIKSDDTHNHSVLFNRVRDLTAEIWFGRRLGQDAAPEKLGVDRALAFSEINQQLYQLLNGLDVVYHAQGEYAYADVIVNSALEKLRKGSRQNLTAPATMIDWRPVVHEMRLFKSPEEIAVLRRAGEITAMAHTRAMEKCRPGMFEYHLEGEIHHEFNRHGARYPSYNTIVGSGENGCILHYTENECEMRDGDLVLIDAGCEYKGYAGDITRTFPVNGKFTQAQREIYDIVLESLETSLRLYRPGTSILEVTGEVVRIMVSGLVKLGILKGDVDELIAQNAHRPFFMHGLSHWLGLDVHDVGVYGQDRSRILEPGMVLTVEPGLYIAPDAEVPEQYRGIGIRIEDDIVITETGNENLTASVVKKPEEIEALMVAARKQ

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 \(c\).

\(|P_{f(5PBV_1)}(2) \setminus P_{f(6UQV_1)}(2)|=60\), \(|P_{f(6UQV_1)}(2) \setminus P_{f(5PBV_1)}(2)|=128\). 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 1:100000000110110001010010100100000001110011100100000111111110101111000110011010010001001001010011101011100000100000011011001000100010001010
Pair \(Z_2\) Length of longest common subsequence
5PBV_1,6UQV_1 188 3
5PBV_1,2BHD_1 180 4
6UQV_1,2BHD_1 168 3

Newick tree

 
[
	5PBV_1:94.54,
	[
		2BHD_1:84,6UQV_1:84
	]:10.54
]

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 \((0.85)(0.8)(\frac{425 }{\log_{20} 425}-\frac{138}{\log_{20}138})=85.9\)
Status Protein1 Protein2 d d1/2
Query variables 5PBV_1 6UQV_1 113 82.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]