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Parikh vectors
2FWW_1 8GTR_1 8KBF_1 Letter Amino acid
12 20 10 R Arginine
13 11 6 Q Glutamine
12 12 9 H Histidine
4 9 5 M Methionine
11 16 6 T Threonine
9 6 1 W Tryptophan
10 43 13 S Serine
10 15 4 Y Tyrosine
8 9 1 C Cysteine
9 16 14 E Glutamic acid
21 10 10 G Glycine
11 21 7 I Isoleucine
20 71 8 L Leucine
4 19 3 F Phenylalanine
14 24 6 A Alanine
15 19 7 D Aspartic acid
9 19 10 K Lycine
23 20 3 P Proline
5 11 7 N Asparagine
25 21 8 V Valine 

2FWW_1|Chains A, B, C, D|Tryptase beta-2|Homo sapiens (9606)
>8GTR_1|Chains A, B, C, D, E, F, G|Pannexin-3|Homo sapiens (9606)
>8KBF_1|Chains A, B|Thoeris anti-defense 1|Clostridium botulinum (1491)
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 \(w=f(c)\)
2FWW , Knot 116 245 0.86 40 180 239 
IVGGQEAPRSKWPWQVSLRVHGPYWMHFCGGSLIHPQWVLTAAHCVGPDVKDLAALRVQLREQHLYYQDQLLPVSRIIVHPQFYTAQIGADIALLELEEPVKVSSHVHTVTLPPASETFPPGMPCWVTGWGDVDNDERLPPPFPLKQVKVPIMENHICDAKYHLGAYTGDDVRIVRDDMLCAGNTRRDSCQGDSGGPLVCKVNGTWLQAGVVSWGEGCAQPNRPGIYTRVTYYLDWIHHYVPKKP
8GTR , Knot 159 392 0.80 40 216 362 
MSLAHTAAEYMLSDALLPDRRGPRLKGLRLELPLDRIVKFVAVGSPLLLMSLAFAQEFSSGSPISCFSPSNFSIRQAAYVDSSCWDSLLHHKQDGPGQDKMKSLWPHKALPYSLLALALLMYLPVLLWQYAAVPALSSDLLFIISELDKSYNRSIRLVQHMLKIRQKSSDPYVFWNELEKARKERYFEFPLLERYLACKQRSHSLVATYLLRNSLLLIFTSATYLYLGHFHLDVFFQEEFSCSIKTGLLSDETHVPNLITCRLTSLSIFQIVSLSSVAIYTILVPVIIYNLTRLCRWDKRLLSVYEMLPAFDLLSRKMLGCPINDLNVILLFLRANISELISFSWLSVLCVLKDTTTQKHNIDTVVDFMTLLAGLEPSKPKHLTNSACDEHP
8KBF , Knot 68 138 0.81 40 108 133 
MGSSHHHHHHSQDPMKELSTIQKREKLNTVERIGSEGPGGAYHEYVIKSNSMDSQGNYDVYETIKFQKGARKEEKSQHGVIDSDLLEIVRDRLKSFQAGPFSSRENACALTHVEEALMWMNRRVEDRIERNVLGTNTK

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 \(c\).

\(|P_{f(2FWW_1)}(2) \setminus P_{f(8GTR_1)}(2)|=79\), \(|P_{f(8GTR_1)}(2) \setminus P_{f(2FWW_1)}(2)|=115\). 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 1:11110011000111010101011011010110110101110110011101001111010100001000001111001110101001011101111010011010001001011110001111110110111010000011111110010111100010010001110010010110001101100000000100111110010101101111011010101001110001000101100011001
Pair \(Z_2\) Length of longest common subsequence
2FWW_1,8GTR_1 194 3
2FWW_1,8KBF_1 178 4
8GTR_1,8KBF_1 186 4

Newick tree

 
[
	8GTR_1:96.94,
	[
		2FWW_1:89,8KBF_1:89
	]:7.94
]

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 \((0.85)(0.8)(\frac{637 }{\log_{20} 637}-\frac{245}{\log_{20}245})=110.\)
Status Protein1 Protein2 d d1/2
Query variables 2FWW_1 8GTR_1 140 115
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]