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Parikh vectors
3DHB_1 1RPN_1 8SVX_1 Letter Amino acid
27 20 26 E Glutamic acid
19 28 24 G Glycine
11 12 7 Y Tyrosine
17 28 13 V Valine
8 28 19 R Arginine
10 7 6 N Asparagine
4 2 6 C Cysteine
12 13 19 F Phenylalanine
5 9 13 M Methionine
16 11 17 P Proline
15 18 31 S Serine
13 18 19 T Threonine
1 6 3 W Tryptophan
6 17 23 Q Glutamine
26 30 44 L Leucine
14 12 16 K Lycine
20 13 17 I Isoleucine
10 29 17 A Alanine
11 22 14 D Aspartic acid
9 12 17 H Histidine 

3DHB_1|Chain A|N-Acyl Homoserine Lactone Hydrolase|Bacillus thuringiensis serovar kurstaki (29339)
>1RPN_1|Chains A, B, C, D|GDP-mannose 4,6-dehydratase|Pseudomonas aeruginosa (287)
>8SVX_1|Chains A, B|Nuclear receptor subfamily 1 group I member 2, Nuclear receptor coactivator 1 fusion protein,Nuclear receptor coactivator 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 \(w=f(c)\)
3DHB , Knot 116 254 0.84 40 168 243 
GRISMTVKKLYFIPAGRCMLDHSSVNSALTPGKLLNLPVWCYLLETEEGPILVDTGMPESAVNNEGLFNGTFVEGQILPKMTEEDRIVNILKRVGYEPDDLLYIISSHLHFDHAGGNGAFTNTPIIVQRTEYEAALHREEYMKECILPHLNYKIIEGDYEVVPGVQLLYTPGHSPGHQSLFIETEQSGSVLLTIDASYTKENFEDEVPFAGFDPELALSSIKRLKEVVKKEKPIIFFGHDIEQEKSCRVFPEYI
1RPN , Knot 150 335 0.86 40 205 318 
MRGSHHHHHHGSMTRSALVTGITGQDGAYLAKLLLEKGYRVHGLVARRSSDTRWRLRELGIEGDIQYEDGDMADACSVQRAVIKAQPQEVYNLAAQSFVGASWNQPVTTGVVDGLGVTHLLEAIRQFSPETRFYQASTSEMFGLIQAERQDENTPFYPRSPYGVAKLYGHWITVNYRESFGLHASSGILFNHESPLRGIEFVTRKVTDAVARIKLGKQQELRLGNVDAKRDWGFAGDYVEAMWLMLQQDKADDYVVATGVTTTVRDMCQIAFEHVGLDYRDFLKIDPAFFRPAEVDVLLGNPAKAQRVLGWKPRTSLDELIRMMVEADLRRVSRE
8SVX , Knot 148 351 0.82 40 208 327 
HHHHHHGSERTGTQPLGVQGLTEEQRMMIRELMDAQMKTFDTTFSHFKNFRLPGVLSSGCELPESLQAPSREEAAKWSQVRKDLCSLKVSLQLRGEDGSVWNYKPPADSGGKEIFSLLPHMADMSTYMFKGIISFAKVISYFRDLPIEDQISLLKGAAFELCQLRFNTVFNAETGTWECGRLSYCLEDTAGGFQQLLLEPMLKFHYMLKKLQLHEEEYVLMQAISLFSPDRPGVLQHRVVDQLQEQFAITLKSYIECNRPQPAHRFLFLKIMAMLTELRSINAQHTQRLLRIQDIHPFATPLMQEVFGITGSSGGSGGSSHSSLTERHKILHRLLQEGSPSDITTLSVEPD

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 \(c\).

\(|P_{f(3DHB_1)}(2) \setminus P_{f(1RPN_1)}(2)|=71\), \(|P_{f(1RPN_1)}(2) \setminus P_{f(3DHB_1)}(2)|=108\). 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 1:10101010010111110011000010011011011011110011000011111001110011000111010110101110100000110110011001001101100010100111011100011110000001110000010001110100011010001111101100110011000111000001011101010000001000111111010111001001001100001111110010000000111001
Pair \(Z_2\) Length of longest common subsequence
3DHB_1,1RPN_1 179 4
3DHB_1,8SVX_1 174 3
1RPN_1,8SVX_1 177 8

Newick tree

 
[
	1RPN_1:89.65,
	[
		3DHB_1:87,8SVX_1:87
	]:2.65
]

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 \((0.85)(0.8)(\frac{589 }{\log_{20} 589}-\frac{254}{\log_{20}254})=94.6\)
Status Protein1 Protein2 d d1/2
Query variables 3DHB_1 1RPN_1 121 106.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]