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Parikh vectors
8FZC_1 4ZBJ_1 3PZR_1 Letter Amino acid
5 2 4 C Cysteine
14 9 26 I Isoleucine
18 18 35 L Leucine
11 5 20 T Threonine
2 3 6 H Histidine
18 15 16 P Proline
5 2 8 W Tryptophan
5 7 5 Y Tyrosine
16 20 27 V Valine
8 11 23 D Aspartic acid
18 3 14 Q Glutamine
17 18 21 E Glutamic acid
12 7 19 K Lycine
5 7 10 F Phenylalanine
6 15 21 S Serine
16 6 32 A Alanine
11 7 19 R Arginine
12 11 13 N Asparagine
17 9 34 G Glycine
8 0 17 M Methionine 

8FZC_1|Chains A, B, C|Spacer peptide 2|Human immunodeficiency virus type 2 (ISOLATE ROD) (11720)
>4ZBJ_1|Chain A|Histone chaperone ASF1|Saccharomyces cerevisiae (strain ATCC 204508 / S288c) (559292)
>3PZR_1|Chains A, B|Aspartate-semialdehyde dehydrogenase|Vibrio cholerae (666)
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 \(w=f(c)\)
8FZC , Knot 107 224 0.86 40 163 219 
SPRTLNAWVKLVEEKKFGAEVVPGFQALSEGCTPYDINQMLNCVGDHQAAMQIIREIINEEAAEWDVQHPIPGPLPAGQLREPRGSDIAGTTSTVEEQIQWMFRPQNPVPVGNIYRRWIQIGLQKCVRMYNPTNILDIKQGPKEPFQSYVDRFYKSLRAEQTDPAVKNWMTQTLLVQNANPDCKLVLKGLGMNPTLEEMLTACQGVGGPGQKARLMAEALKEVI
4ZBJ , Knot 80 175 0.78 38 121 172 
PLGSPNSSIVSLLGIKVLNNPAKFTDPYEFEITFECLESLKHDLEWKLTYVGSSRSLDHDQELDSILVGPVPVGVNKFVFSADPPSAELIPASELVSVTVILLSCSYDGREFVRVGYYVNNEYDEEELRENPPAKVQVDHIVRNILAEKPRVTRFNIVWDNENEGDLYPPEQPGV
3PZR , Knot 157 370 0.83 40 214 357 
MRVGLVGWRGMVGSVLMQRMVEERDFDLIEPVFFSTSQIGVPAPNFGKDAGMLHDAFDIESLKQLDAVITCQGGSYTEKVYPALRQAGWKGYWIDAASTLRMDKEAIITLDPVNLKQILHGIHHGTKTFVGGNCTVSLMLMALGGLYERGLVEWMSAMTYQAASGAGAQNMRELISQMGVINDAVSSELANPASSILDIDKKVAETMRSGSFPTDNFGVPLAGSLIPWIDVKRDNGQSKEEWKAGVEANKILGLQDSPVPIDGTCVRIGAMRCHSQALTIKLKQNIPLDEIEEMIATHNDWVKVIPNERDITARELTPAKVTGTLSVPVGRLRKMAMGDDFLNAFTVGDQLLWGAAEPLRRTLRIILAEK

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 \(c\).

\(|P_{f(8FZC_1)}(2) \setminus P_{f(4ZBJ_1)}(2)|=108\), \(|P_{f(4ZBJ_1)}(2) \setminus P_{f(8FZC_1)}(2)|=66\). 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 1:01001011101100001110111110110010010010011001100011101100110001101010011111111101001010011100001000101110100111110100011011100010100100110100110011000100100010100001110011000111001010001110111101010011010011111100101110110011
Pair \(Z_2\) Length of longest common subsequence
8FZC_1,4ZBJ_1 174 5
8FZC_1,3PZR_1 169 4
4ZBJ_1,3PZR_1 177 5

Newick tree

 
[
	4ZBJ_1:88.81,
	[
		8FZC_1:84.5,3PZR_1:84.5
	]:4.31
]

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 \((0.85)(0.8)(\frac{399 }{\log_{20} 399}-\frac{175}{\log_{20}175})=66.6\)
Status Protein1 Protein2 d d1/2
Query variables 8FZC_1 4ZBJ_1 87 76
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]