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Parikh vectors
8AOZ_1 5CYB_1 5LUD_1 Letter Amino acid
34 10 7 L Leucine
3 3 5 M Methionine
17 2 15 F Phenylalanine
7 1 1 W Tryptophan
27 15 9 V Valine
52 12 9 A Alanine
3 1 4 C Cysteine
21 6 6 P Proline
3 2 2 Y Tyrosine
6 9 9 N Asparagine
19 15 7 D Aspartic acid
10 23 3 Q Glutamine
8 7 10 I Isoleucine
19 8 8 S Serine
19 15 11 T Threonine
27 5 6 R Arginine
21 2 12 E Glutamic acid
30 10 23 G Glycine
8 0 4 H Histidine
7 6 14 K Lycine 

8AOZ_1|Chain A[auth AAA]|Beta-ribofuranosylaminobenzene 5'-phosphate synthase|Streptomyces kaniharaensis (212423)
>5CYB_1|Chain A|Lipoprotein|Streptococcus pneumoniae (1313)
>5LUD_1|Chain A|Peptidyl-prolyl cis-trans isomerase|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 \(w=f(c)\)
8AOZ , Knot 140 341 0.79 40 180 320 
MALDRPDRSGAVRVSAPARLSFTAISLDGSSLRRNGIAAMAVDRPGLTAEVREAADGIVAVTGTAEETARELAAALEALRKLWDGPAARVDVLEALPQHSGFGSKTSTLLAVGHAYGRLCGVEPDLRELARTLGRGRTSGASTGLSAYGGFLVDGGHVNPPDFAEAPQKYLRPSRFAQQVAPPKPVVRLDFPDWPVLVLLTHGRHLGGQEELEWFHSVAPIPAEESWRTSHLVFMGLAPAVLEQDFDAFCAAVNEITFTGHFKQAQIAFQGDAVADVLEAGRAAPSVDAIALSVTGPACFAFTKRPEDAERWAWELKNRGLIRDFWFTRANNQGLATTVVS
5CYB , Knot 65 152 0.71 38 96 134 
MKLKRFTLSLASLASFSLLVACSQRAQQVQQPVAQQQVQQPAQQNTNTANAGGNQNQAAPVQNQPVAQPTDIDGTYTGQDDGDRITLVVTGTTGTWTELESDGDQKVKQVTLDSANQRMIIGDDVKIYTVNGNQIVVDDMDRDPSDQIVLTK
5LUD , Knot 82 165 0.84 40 129 161 
MVNPTVFFDIAVDGEPLGRVSFELFADKVPKTAENFRALSTGEKGFGYKGSCFHRIIPGFMCQGGDFTRHNGTGGKSIYGEKFEDENFILKHTGPGILSMANAGPNTNGSQFFICTAKTEWLDGKHVVFGKVKEGMNIVEAMERFGSRNGKTSKKITIADCGQLE

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 \(c\).

\(|P_{f(8AOZ_1)}(2) \setminus P_{f(5CYB_1)}(2)|=121\), \(|P_{f(5CYB_1)}(2) \setminus P_{f(8AOZ_1)}(2)|=37\). 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 1:11100100011101011101010110101001000111111100111010100110111110101000100111110110011011110101101110001110000011111010101011010100110011010001100110101111101101011011011000101001100111101110101101111111001001110001011001111110001000011111111111000101101110010101010010111010111011011011101011110101110111000100100111010001110011100100011100110
Pair \(Z_2\) Length of longest common subsequence
8AOZ_1,5CYB_1 158 4
8AOZ_1,5LUD_1 169 3
5CYB_1,5LUD_1 149 4

Newick tree

 
[
	8AOZ_1:84.08,
	[
		5CYB_1:74.5,5LUD_1:74.5
	]:9.58
]

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 \((0.85)(0.8)(\frac{493 }{\log_{20} 493}-\frac{152}{\log_{20}152})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 8AOZ_1 5CYB_1 126 89
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]