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Parikh vectors
1PLC_1 3NNQ_1 5CCQ_1 Letter Amino acid
0 3 5 R Arginine
7 6 9 D Aspartic acid
8 6 6 E Glutamic acid
2 11 5 H Histidine
2 7 3 Y Tyrosine
7 5 7 A Alanine
1 4 3 Q Glutamine
5 4 7 P Proline
9 4 15 V Valine
6 4 9 N Asparagine
1 2 4 C Cysteine
10 2 22 G Glycine
6 12 10 L Leucine
6 11 14 K Lycine
2 4 3 M Methionine
7 5 11 F Phenylalanine
12 9 11 S Serine
6 4 8 I Isoleucine
2 10 11 T Threonine
0 1 1 W Tryptophan 

1PLC_1|Chain A|PLASTOCYANIN|Populus nigra (3691)
>3NNQ_1|Chains A, B|N-terminal domain of Moloney murine leukemia virus integrase|Moloney murine leukemia virus (11801)
>5CCQ_1|Chain A|Peptidyl-prolyl cis-trans isomerase F, 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 \(w=f(c)\)
1PLC , Knot 50 99 0.77 36 81 95 
IDVLLGADDGSLAFVPSEFSISPGEKIVFKNNAGFPHNIVFDEDSIPSGVDASKISMSEEDLLNAKGETFEVALSNKGEYSFYCSPHQGAGMVGKVTVN
3NNQ , Knot 59 114 0.81 40 99 108 
MIENSSPYTSEHFHYTVTDIKDLTKLGAIYDKTKKYWVYQGKPVMPDQFTFELLDFLHQLTHLSFSKMKALLERSHSPYYMLNRDRTLKNITETCKACAQVNASKSLEHHHHHH
5CCQ , Knot 77 164 0.79 40 119 158 
GNPLVYLDVDANGKPLGRVVLELKADVVPKTAENFRALCTGEKGFGYKGSTFHRVIPSFMCQAGDFTNHNGTGGKSIYGSRFPDENFTLKHVGPGVLSMANAGPNTNGSQFFICTIKTDWLDGKHVVFGHVIEGMDVVKKIESFGSKSGRTSKKIVITDCGQLS

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 \(c\).

\(|P_{f(1PLC_1)}(2) \setminus P_{f(3NNQ_1)}(2)|=55\), \(|P_{f(3NNQ_1)}(2) \setminus P_{f(1PLC_1)}(2)|=73\). 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 1:101111100101111100101011001110001111001110000110110100101000011010100101110001000100010011111101010
Pair \(Z_2\) Length of longest common subsequence
1PLC_1,3NNQ_1 128 3
1PLC_1,5CCQ_1 128 3
3NNQ_1,5CCQ_1 152 3

Newick tree

 
[
	5CCQ_1:72.22,
	[
		1PLC_1:64,3NNQ_1:64
	]:8.22
]

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 \((0.85)(0.8)(\frac{213 }{\log_{20} 213}-\frac{99}{\log_{20}99})=37.0\)
Status Protein1 Protein2 d d1/2
Query variables 1PLC_1 3NNQ_1 48 43.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]