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Parikh vectors
4RPC_1 1YZS_1 1AZN_1 Letter Amino acid
10 3 4 H Histidine
6 4 10 T Threonine
1 1 1 W Tryptophan
16 10 10 V Valine
12 1 7 N Asparagine
11 12 4 P Proline
12 5 2 Y Tyrosine
17 7 4 I Isoleucine
32 12 10 L Leucine
10 11 10 S Serine
26 10 8 A Alanine
15 6 1 R Arginine
6 8 6 Q Glutamine
21 14 11 G Glycine
10 0 6 M Methionine
6 2 5 F Phenylalanine
9 8 11 D Aspartic acid
4 1 3 C Cysteine
26 3 4 E Glutamic acid
13 3 11 K Lycine 

4RPC_1|Chains A, B|putative alpha/beta hydrolase|Desulfitobacterium hafniense (272564)
>1YZS_1|Chain A|sulfiredoxin|Homo sapiens (9606)
>1AZN_1|Chains A, B, C, D|AZURIN|Pseudomonas aeruginosa (287)
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 \(w=f(c)\)
4RPC , Knot 117 263 0.82 40 165 246 
SNAMSMKKGNCKLHRMGEPHGPKVLLIHGAGFYWQTCFARIIRDLKDRYCLLIPELEGHTAHPREYMVSVEETAGKLGEALEELRVDKVQAIYGVSLGASVAVEMAIRGEIKVMNLLLDGGQYEGMGEMTEQYANIMADAFLNLLAGEHLPSPVKENMGFAANNDVEVLQPLIYEHITREALLHALLAAYRYDLKAKNARVDARVSVLIGGNEIYGAQFTPLLAEISRHPLDIYEFPNRGHAEVLSKEPEKISRLIREILNYC
1YZS , Knot 57 121 0.75 38 88 113 
GAPEGPGPSGGAQGGSIHSGRIAAVHNVPLSVLIRPLPSVLDPAKVQSLVDTIREDPDSVPPIDVLWIKGAQGGDYFYSFGGCHRYAAYQQLQRETIPAKLVQSTLSDLRVYLGASTPDLQ
1AZN , Knot 68 128 0.86 40 105 126 
AECSVDIQGNDQMQFNTNAITVDKSCKQFTVNLSHPGNLPKNVMGHNWVLSTAADMQGVVTDGMASGLDKDYLKPDDSRVIAHTKLIGSGEKDSVTFDVSKLKEGEQYMFFCTAPGHSALMKGTLTLK

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 \(c\).

\(|P_{f(4RPC_1)}(2) \setminus P_{f(1YZS_1)}(2)|=122\), \(|P_{f(1YZS_1)}(2) \setminus P_{f(4RPC_1)}(2)|=45\). 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 1:00110100100010011010110111101111010001101100100000111101010010100011010001101101100101001011011011101110111010101101110110001110100001011101110111100110110001111100010110111000100011101111100001010010101010111110010110101111010001101001100101011000100100110011000
Pair \(Z_2\) Length of longest common subsequence
4RPC_1,1YZS_1 167 4
4RPC_1,1AZN_1 162 3
1YZS_1,1AZN_1 135 3

Newick tree

 
[
	4RPC_1:86.62,
	[
		1AZN_1:67.5,1YZS_1:67.5
	]:19.12
]

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 \((0.85)(0.8)(\frac{384 }{\log_{20} 384}-\frac{121}{\log_{20}121})=80.0\)
Status Protein1 Protein2 d d1/2
Query variables 4RPC_1 1YZS_1 102 72.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]