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Parikh vectors
2DWD_1 8SUB_1 1BOZ_1 Letter Amino acid
5 28 10 N Asparagine
8 28 16 E Glutamic acid
14 41 19 L Leucine
2 4 6 M Methionine
5 27 8 F Phenylalanine
15 12 12 P Proline
31 26 12 S Serine
5 14 8 R Arginine
7 21 9 D Aspartic acid
16 35 14 G Glycine
21 15 7 T Threonine
9 23 6 Y Tyrosine
21 26 14 V Valine
18 29 5 A Alanine
4 3 1 C Cysteine
4 11 3 H Histidine
6 3 3 W Tryptophan
9 16 7 Q Glutamine
6 24 9 I Isoleucine
13 29 17 K Lycine 

2DWD_1|Chain A|ANTIBODY FAB HEAVY CHAIN|Mus musculus (10090)
>8SUB_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|SIR2-like domain-containing protein|Escherichia coli (562)
>1BOZ_1|Chain A|PROTEIN (DIHYDROFOLATE REDUCTASE)|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 \(w=f(c)\)
2DWD , Knot 100 219 0.82 40 147 205 
QVQLQQPGAELVKPGASVKLSCKASGYTFTSDWIHWVKQRPGHGLEWIGEIIPSYGRANYNEKIQKKATLTADKSSSTAFMQLSSLTSEDSAVYYCARERGDGYFAVWGAGTTVTVSSAKTTPPSVYPLAPGSAAQTNSMVTLGCLVKGYFPEPVTVTWNSGSLSSGVHTFPAVLQSDLYTLSSSVTVPSSSWPSETVTCNVAHPASSTKVDKKIVPRD
8SUB , Knot 173 415 0.83 40 221 389 
MSIYQGGNKLNEDDFRSHVYSLCQLDNVGVLLGAGASVGCGGKTMKDVWKSFKQNYPELLGALIDKYLLVSQIDSDNNLVNVELLIDEATKFLSVAKTRRCEDEEEEFRKILSSLYKEVTKAALLTGEQFREKNQGKKDAFKYHKELISKLISNRQPGQSAPAIFTTNYDLALEWAAEDLGIQLFNGFSGLHTRQFYPQNFDLAFRNVNAKGEARFGHYHAYLYKLHGSLTWYQNDSLTVNEVSASQAYDEYINDIINKDDFYRGQHLIYPGANKYSHTIGFVYGEMFRRFGEFISKPQTALFINGFGFGDYHINRIILGALLNPSFHVVIYYPELKEAITKVSKGGGSEAEKAIVTLKNMAFNQVTVVGGGSKAYFNSFVEHLPYPVLFPRDNIVDELVEAIANLSKGEGNVPF
1BOZ , Knot 85 186 0.79 40 136 177 
VGSLNCIVAVSQNMGIGKNGDLPWPPLRNEGRYFQRMTTTSSVEGKQNLVIMGKKTWFSIPEKNRPLKGRINLVLSRELKEPPQGAHFLSRSLDDALKLTEQPELANKVDMVWIVGGSSVYKEAMNHPGHLKLFVTRIMQDFESDTFFPEIDLEKYKLLPEYPGVLSDVQEEKGIKYKFEVYEKND

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 \(c\).

\(|P_{f(2DWD_1)}(2) \setminus P_{f(8SUB_1)}(2)|=53\), \(|P_{f(8SUB_1)}(2) \setminus P_{f(2DWD_1)}(2)|=127\). 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 1:010100111011011101010001010010001101100011011011101110010100000100010101000000111010010000011000100010101111111001010010001101011111011000011011011010110110101001010011001111100010010001011000110001000110110000100011100
Pair \(Z_2\) Length of longest common subsequence
2DWD_1,8SUB_1 180 4
2DWD_1,1BOZ_1 161 3
8SUB_1,1BOZ_1 173 4

Newick tree

 
[
	8SUB_1:90.70,
	[
		2DWD_1:80.5,1BOZ_1:80.5
	]:10.20
]

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 \((0.85)(0.8)(\frac{634 }{\log_{20} 634}-\frac{219}{\log_{20}219})=117.\)
Status Protein1 Protein2 d d1/2
Query variables 2DWD_1 8SUB_1 147 111.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]