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Parikh vectors
9BHC_1 4JUS_1 5LRU_1 Letter Amino acid
20 14 23 V Valine
16 4 16 D Aspartic acid
12 3 13 Q Glutamine
37 6 41 L Leucine
10 0 10 M Methionine
16 4 10 F Phenylalanine
14 14 16 P Proline
2 0 10 W Tryptophan
27 17 22 A Alanine
4 0 4 C Cysteine
21 5 18 G Glycine
9 3 15 K Lycine
9 0 10 N Asparagine
19 9 28 E Glutamic acid
7 7 16 H Histidine
10 1 8 Y Tyrosine
23 6 15 R Arginine
30 2 10 I Isoleucine
24 7 23 S Serine
17 2 10 T Threonine 

9BHC_1|Chains A, B, C, D|Undecaprenyl-phosphate 4-deoxy-4-formamido-L-arabinose transferase|Salmonella enterica subsp. enterica serovar Typhimurium (90371)
>4JUS_1|Chains A, B, C, D, E, F, G, H|Heat shock protein beta-6|Homo sapiens (9606)
>5LRU_1|Chain A|OTU domain-containing protein 7B|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 \(w=f(c)\)
9BHC , Knot 139 327 0.82 40 196 309 
MFDAAPIKKVSVVIPVYNEQESLPELIRRTTTACESLGKAWEILLIDDGSSDSSAELMVKASQEADSHIISILLNRNYGQHAAIMAGFSHVSGDLIITLDADLQNPPEEIPRLVAKADEGFDVVGTVRQNRQDSLFRKSASKIINLLIQRTTGKAMGDYGCMLRAYRRPIIDTMLRCHERSTFIPILANIFARRATEIPVHHAEREFGDSKYSFMRLINLMYDLVTCLTTTPLRLLSLLGSVIAIGGFSLSVLLIVLRLALGPQWAAEGVFMLFAVLFTFIGAQFIGMGLLGEYIGRIYNDVRARPRYFVQQVIYPESTPFTEESHQ
4JUS , Knot 51 104 0.76 32 72 99 
APSVALPVAQVPTDPGHFSVLLDVKHFSPEEIAVKVVGEHVEVHARHEERPDEHGFVAREFHRRYRLPPGVDPAAVTSALSPEGVLSIQAAPASAQAPPPAAAK
5LRU , Knot 138 318 0.83 40 192 297 
MEMPICAFQLPDLTVYNEDFRSFIERDLIEQSMLVALEQAGRLNWWVSVDPTSQRLLPLATTGDGNCLLHAASLGMWGFHDRDLMLRKALYALMEKGVEKEALKRRWRWQQTQQNKESGLVYTEDEWQKEWNELIKLASSEPRMHLGTNGANCGGVESSEEPVYESLEEFHVFVLAHVLRRPIVVVADTMLRDSGGEAFAPIPFGGIYLPLEVPASQCHRSPLVLAYDQAHFSALVSMEQKENTKEQAVIPLTDSEYKLLPLHFAVDPGKGWEWGKDDSDNVRLASVILSLEVKLHLLHSYMNVKWIPLSSKHHHHHH

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 \(c\).

\(|P_{f(9BHC_1)}(2) \setminus P_{f(4JUS_1)}(2)|=146\), \(|P_{f(4JUS_1)}(2) \setminus P_{f(9BHC_1)}(2)|=22\). 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 1:110111100101111100000011011000001000110110111100100000101110100010001101110000100111111100101011101010100110011011101001101110100000001100010011011100001011100101101000111001100000001111110111001001110010001100000110110110011001000110110111011111110101111110111110111011111111110111101111111100110100010101001100110100011000000
Pair \(Z_2\) Length of longest common subsequence
9BHC_1,4JUS_1 168 4
9BHC_1,5LRU_1 180 5
4JUS_1,5LRU_1 172 3

Newick tree

 
[
	5LRU_1:89.32,
	[
		9BHC_1:84,4JUS_1:84
	]:5.32
]

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 \((0.85)(0.8)(\frac{431 }{\log_{20} 431}-\frac{104}{\log_{20}104})=99.1\)
Status Protein1 Protein2 d d1/2
Query variables 9BHC_1 4JUS_1 125 80
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]