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Parikh vectors
5BQA_1 7MKC_1 4RVS_1 Letter Amino acid
0 6 8 H Histidine
3 11 1 M Methionine
6 4 11 F Phenylalanine
5 9 19 S Serine
0 5 3 W Tryptophan
2 9 4 N Asparagine
1 11 16 R Arginine
1 8 17 D Aspartic acid
1 4 9 Y Tyrosine
6 15 38 V Valine
10 19 46 A Alanine
1 17 17 E Glutamic acid
9 15 10 I Isoleucine
2 11 9 K Lycine
1 2 2 C Cysteine
10 18 36 G Glycine
12 18 30 L Leucine
2 18 18 P Proline
3 16 26 T Threonine
1 15 9 Q Glutamine 

5BQA_1|Chains A, B, C, D, E, F[auth K], G[auth L], H[auth M], I[auth N], J[auth O]|ATP synthase subunit 9, mitochondrial|Saccharomyces cerevisiae (4932)
>7MKC_1|Chain A|capsid protein|Human immunodeficiency virus 1 (11676)
>4RVS_1|Chains A, B|Probable quinone reductase Qor (NADPH:quinone reductase) (Zeta-crystallin homolog protein)|Mycobacterium tuberculosis (83332)
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 \(w=f(c)\)
5BQA , Knot 40 76 0.76 36 57 72 
MQLVLAAKYIGAGISTIGLLGAGIGIAIVFAALINGVSRNPSIKDTVFPMAILGFALSEATGLFCLMVSFLLLFGV
7MKC , Knot 103 231 0.81 40 158 221 
PIVQNLQGQMVHQAISPRTLNAWVKVVEEKAFSPEVIPMFSALSEGATPQDLNTMLNTVGGHQAAMQMLKETIDEEAAEWDRLHPVHAGPIAPGQMREPRGSDIAGTTSTLQEQIGWMTHNPPIPVGEIYKRWIILGLNKIVRMYSPTSILDIRQGPKEPFRDYVDRFYKTLRAEQASQEVKNWMTETLLVQNANPDCKTILKALGPGATLEEMMTACQGVGGPGHKARVL
4RVS , Knot 138 329 0.81 40 175 304 
SMHAIEVTETGGPGVLRHVDQPQPQPGHGELLIKAEAIGVNFIDTYFRSGQYPRELPFVIGSEVCGTVEAVGPGVTAADTAISVGDRVVSASANGAYAEFCTAPASLTAKVPDDVTSEVAASALLKGLTAHYLLKSVYPVKRGDTVLVHAGAGGVGLILTQWATHLGVRVITTVSTAEKAKLSKDAGADVVLDYPEDAWQFAGRVRELTGGTGVQAVYDGVGATTFDASLASLAVRGTLALFGAASGPVPPVDPQRLNAAGSVYLTRPSLFHFTRTGEEFSWRAAELFDAIGNEAITVAVGGRYPLADALRAHQDLEARKTVGSVVLLP

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 \(c\).

\(|P_{f(5BQA_1)}(2) \setminus P_{f(7MKC_1)}(2)|=29\), \(|P_{f(7MKC_1)}(2) \setminus P_{f(5BQA_1)}(2)|=130\). 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 1:1011111001111100111111111111111111011000101000111111111110010111011101111111
Pair \(Z_2\) Length of longest common subsequence
5BQA_1,7MKC_1 159 4
5BQA_1,4RVS_1 164 3
7MKC_1,4RVS_1 159 4

Newick tree

 
[
	4RVS_1:81.17,
	[
		5BQA_1:79.5,7MKC_1:79.5
	]:1.67
]

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 \((0.85)(0.8)(\frac{307 }{\log_{20} 307}-\frac{76}{\log_{20}76})=73.4\)
Status Protein1 Protein2 d d1/2
Query variables 5BQA_1 7MKC_1 92 60.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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