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Parikh vectors
5PGD_1 7FNG_1 5QQV_1 Letter Amino acid
2 9 19 Q Glutamine
8 21 19 T Threonine
2 5 4 W Tryptophan
6 8 15 M Methionine
8 10 22 F Phenylalanine
6 10 21 P Proline
8 14 33 V Valine
4 18 36 A Alanine
2 1 10 C Cysteine
4 23 22 I Isoleucine
14 19 21 K Lycine
5 11 24 R Arginine
6 7 37 G Glycine
8 5 26 H Histidine
4 7 15 Y Tyrosine
11 18 37 S Serine
6 14 15 N Asparagine
12 14 23 D Aspartic acid
9 15 25 E Glutamic acid
13 29 45 L Leucine 

5PGD_1|Chain A|Bromodomain adjacent to zinc finger domain protein 2B|Homo sapiens (9606)
>7FNG_1|Chain A|Pre-mRNA-splicing factor 8|Saccharomyces cerevisiae S288C (559292)
>5QQV_1|Chains A, B|5-aminolevulinate synthase, erythroid-specific, mitochondrial|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 \(w=f(c)\)
5PGD , Knot 66 138 0.78 40 109 129 
MHHHHHHSSGVDLGTENLYFQSMSVKKPKRDDSKDLALCSMILTEMETHEDAWPFLLPVNLKLVPGYKKVIKKPMDFSTIREKLSSGQYPNLETFALDVRLVFDNCETFNEDDSDIGRAGHNMRKYFEKKWTDTFKVS
7FNG , Knot 117 258 0.84 40 175 251 
GAMNSSNYAELFNNDIKLFVDDTNVYRVTVHKTFEGNVATKAINGCIFTLNPKTGHLFLKIIHTSVWAGQKRLSQLAKWKTAEEVSALVRSLPKEEQPKQIIVTRKAMLDPLEVHMLDFPNIAIRPTELRLPFSAAMSIDKLSDVVMKATEPQMVLFNIYDDWLDRISSYTAFSRLTLLLRALKTNEESAKMILLSDPTITIKSYHLWPSFTDEQWITIESQMRDLILTEYGRKYNVNISALTQTEIKDIILGQNIKA
5QQV , Knot 196 469 0.85 40 260 444 
MGHHHHHHSSGVDLGTENLYFQSMFSYDQFFRDKIMEKKQDHTYRVFKTVNRWADAYPFAQHFSEASVASKDVSVWCSNDYLGMSRHPQVLQATQETLQRHGVGAGGTRNISGTSKFHVELEQELAELHQKDSALLFSSCFVANDSTLFTLAKILPGCEIYSDAGNHASMIQGIRNSGAAKFVFRHNDPDHLKKLLEKSNPKIPKIVAFETVHSMDGAICPLEELCDVSHQYGALTFVDEVHAVGLYGSRGAGIGERDGIMHKIDIISGTLGKAFGCVGGYIASTRDLVDMVRSYAAGFIFTTSLPPMVLSGALESVRLLKGEEGQALRRAHQRNVKHMRQLLMDRGLPVIPCPSHIIPIRVGNAALNSKLCDLLLSKHGIYVQAINYPTVPRGEELLRLAPSPHHSPQMMEDFVEKLLLAWTAVGLPLQDVSVAACNFCRRPVHFELMSEWERSYFGNMGPQYVTTYA

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 \(c\).

\(|P_{f(5PGD_1)}(2) \setminus P_{f(7FNG_1)}(2)|=53\), \(|P_{f(7FNG_1)}(2) \setminus P_{f(5PGD_1)}(2)|=119\). 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
5PGD_1,7FNG_1 172 4
5PGD_1,5QQV_1 203 22
7FNG_1,5QQV_1 195 3

Newick tree

 
[
	5QQV_1:10.63,
	[
		5PGD_1:86,7FNG_1:86
	]:17.63
]

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 \((0.85)(0.8)(\frac{396 }{\log_{20} 396}-\frac{138}{\log_{20}138})=77.8\)
Status Protein1 Protein2 d d1/2
Query variables 5PGD_1 7FNG_1 99 75.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]