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Parikh vectors
2PBP_1 5CPI_1 1KDD_1 Letter Amino acid
3 3 0 Y Tyrosine
19 18 1 R Arginine
1 2 1 C Cysteine
29 12 5 L Leucine
6 10 0 T Threonine
4 0 1 W Tryptophan
10 4 0 D Aspartic acid
24 7 3 E Glutamic acid
12 4 0 F Phenylalanine
10 6 1 S Serine
35 18 4 A Alanine
7 1 2 N Asparagine
21 8 0 G Glycine
13 7 0 I Isoleucine
19 6 3 V Valine
11 8 1 Q Glutamine
0 3 1 H Histidine
13 13 12 K Lycine
10 3 0 M Methionine
11 6 0 P Proline 

2PBP_1|Chain A|Enoyl-CoA hydratase subunit I|Geobacillus kaustophilus (235909)
>5CPI_1|Chains A, E|Histone H3.1|Homo sapiens (9606)
>1KDD_1|Chains A[auth B], C[auth D], E|GCN4 ACID BASE HETERODIMER BASE-d12La16L|null
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 \(w=f(c)\)
2PBP , Knot 114 258 0.81 38 157 243 
MSEFVSIAARQEGAVGIIELARPDVLNALSRQMVAEIVAAVEAFDRNEKVRVIVLTGRGRAFAAGADIQEMAKDDPIRLEWLNQFADWDRLSIVKTPMIAAVNGLALGGGFELALSCDLIVASSAAEFGFPEVNLGVMPGAGGTQRLTKLIGPKRALEWLWTGARMSAKEAEQLGIVNRVVSPELLMEETMRLAGRLAEQPPLALRLIKEAVQKAVDYPLYEGMQFERKNFYLLFASEDQKEGMAAFLEKRKPRFQGK
5CPI , Knot 68 139 0.80 38 107 131 
GSHMARTKQTARKSTGGKAPRKQLATKAARKSAPATGGVKKPHRYRPGTVALREIRRYQKSTELLIRKLPFQRLVREIAQDFKTDLRFQSSAVMALQEACEAYLVGLFEDTNLCAIHAKRVTIMPKDIQLARRIRGERA
1KDD , Knot 19 36 0.63 26 27 34 
XKVKQLKAKVEELKSKLWHLKNKVARLKKKNAECKA

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 \(c\).

\(|P_{f(2PBP_1)}(2) \setminus P_{f(5CPI_1)}(2)|=97\), \(|P_{f(5CPI_1)}(2) \setminus P_{f(2PBP_1)}(2)|=47\). 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 1:100110111000111111011010110110001110111110110000010111101010111111010011000110101100110100101100111111011111111011100011110011011110101111111110001001111001101110110101001001111001101011100010111011001111101100110011001100110100001011110000001111110000101010
Pair \(Z_2\) Length of longest common subsequence
2PBP_1,5CPI_1 144 3
2PBP_1,1KDD_1 152 2
5CPI_1,1KDD_1 110 3

Newick tree

 
[
	2PBP_1:79.36,
	[
		5CPI_1:55,1KDD_1:55
	]:24.36
]

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 \((0.85)(0.8)(\frac{397 }{\log_{20} 397}-\frac{139}{\log_{20}139})=77.7\)
Status Protein1 Protein2 d d1/2
Query variables 2PBP_1 5CPI_1 97 73.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]