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Parikh vectors
7UMQ_1 3SCK_1 3FDR_1 Letter Amino acid
3 32 7 V Valine
7 38 5 A Alanine
0 48 5 E Glutamic acid
4 27 5 P Proline
1 29 3 Y Tyrosine
3 20 1 M Methionine
1 26 4 F Phenylalanine
4 35 7 S Serine
1 18 5 R Arginine
0 8 2 C Cysteine
16 32 7 G Glycine
2 21 2 I Isoleucine
2 35 4 T Threonine
3 20 1 W Tryptophan
2 60 14 L Leucine
4 33 2 K Lycine
2 39 6 N Asparagine
0 31 9 D Aspartic acid
3 30 3 Q Glutamine
4 21 2 H Histidine 

7UMQ_1|Chains A, B, C, D, E, F, G, H, I, J|Major prion protein|Homo sapiens (9606)
>3SCK_1|Chains A, B|Angiotensin-converting enzyme 2 chimera|Paguma larvata (9675)
>3FDR_1|Chain A|Tudor and KH domain-containing protein|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 \(w=f(c)\)
7UMQ , Knot 31 62 0.68 34 47 55 
GWGQPHGGGWGQGGGTHSQWNKPSKPKTNMKHMAGAAAAGAVVGGLGGYVLGSAMSRPIIHF
3SCK , Knot 243 603 0.86 40 291 574 
STTEELAKTFLETFNYEAQELSYQSSVASWNYNTNITEENVQNMNNAGDKWSAFLKEQSTLAQTYPLQEIQNLTVKLQLQALQQNGSSVLSEDKSKRLNTILNTMSTIYSTGKVCNPDNPQECLLLEPGLNEIMANSLDYNERLWAWESWRSEVGKQLRPLYEEYVVLKNEMARANHYEDYGDYWRGDYEVNGVDGYDYSRGQLIEDVEHTFEEIKPLYEHLHAYVRAKLMNAYPSYISPIGCLPAHLLGDMWGRFWTNLYSLTVPFGQKPNIDVTDAMVDQAWDAQRIFKEAEKFFVSVGLPNMTQGFWENSMLTDPGNVQKAVCHPTAWDLGKGDFRILMCTKVTMDDFLTAHHEMGHIQYDMAYAAQPFLLRNGANEGFHEAVGEIMSLSAATPKHLKSIGLLSPDFQEDNETEINFLLKQALTIVGTLPFTYMLEKWRWMVFKGEIPKDQWMKKWWEMKREIVGVVEPVPHDETYCDPASLFHVSNDYSFIRYYTRTLYQFQFQEALCQAAKHEGPLHKCDISNSTEAGQKLFNMLRLGKSEPWTLALENVVGAKNMNVRPLLNYFEPLFTWLKDQNKNSFVGWSTDWSPYADHHHHHH
3FDR , Knot 49 94 0.79 40 77 92 
GSRSLQLDKLVNEMTQHYENSVPEDLTVHVGDIVAAPLPTNGSWYRARVLGTLENGNLDLYFVDFGDNGDCPLKDLRALRSDFLSLPFQAIECS

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 \(c\).

\(|P_{f(7UMQ_1)}(2) \setminus P_{f(3SCK_1)}(2)|=9\), \(|P_{f(3SCK_1)}(2) \setminus P_{f(7UMQ_1)}(2)|=253\). 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 1:11101011111011100001001001000100111111111111111101110110011101
Pair \(Z_2\) Length of longest common subsequence
7UMQ_1,3SCK_1 262 3
7UMQ_1,3FDR_1 108 3
3SCK_1,3FDR_1 240 3

Newick tree

 
[
	3SCK_1:14.66,
	[
		7UMQ_1:54,3FDR_1:54
	]:87.66
]

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 \((0.85)(0.8)(\frac{665 }{\log_{20} 665}-\frac{62}{\log_{20}62})=177.\)
Status Protein1 Protein2 d d1/2
Query variables 7UMQ_1 3SCK_1 234 127.5
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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