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Parikh vectors
8SRO_1 9EOF_1 4NAY_1 Letter Amino acid
5 25 12 I Isoleucine
6 10 2 M Methionine
14 17 8 F Phenylalanine
6 22 10 D Aspartic acid
11 15 14 H Histidine
21 71 16 L Leucine
17 18 11 K Lycine
13 37 3 P Proline
8 18 8 T Threonine
6 11 8 Y Tyrosine
12 31 3 V Valine
10 22 2 Q Glutamine
12 18 6 G Glycine
7 3 2 W Tryptophan
7 11 1 C Cysteine
15 34 6 S Serine
9 13 10 N Asparagine
20 18 9 E Glutamic acid
22 32 5 A Alanine
15 23 9 R Arginine 

8SRO_1|Chains A, B, C, D|Forkhead box protein P3|Mus musculus (10090)
>9EOF_1|Chain A[auth C]|Integrator complex subunit 15|Homo sapiens (9606)
>4NAY_1|Chain A|Metallothiol transferase FosB|Staphylococcus aureus subsp. aureus (158879)
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 \(w=f(c)\)
8SRO , Knot 112 236 0.86 40 177 231 
GSYPLLANGVCKWPGCEKVFEEPEEFLKHCQADHLLDEKGKAQCLLQREVVQSLEQQLELEKEKLGAMQAHLAGKMALAKAPSVASMDKSSCCIVATSTQGSVLPAWSAPREAPDGGLFAVRRHLWGSHGNSSFPEFFHNMDYFKYHNMRPPFTYATLIRWAILEAPERQRTLNEIYHWFTRMFAYFRNHPATWKNAIRHNLSLHKCFVRVESEKGAVWTVDEFEFRKKRSQRPNK
9EOF , Knot 184 449 0.83 40 228 423 
MSDIRHSLLRRDALSAAKEVLYHLDIYFSSQLQSAPLPIVDKGPVELLEEFVFQVPKERSAQPKRLNSLQELQLLEIMCNYFQEQTKDSVRQIIFSSLFSPQGNKADDSRMSLLGKLVSMAVAVCRIPVLECAASWLQRTPVVYCVRLAKALVDDYCCLVPGSIQTLKQIFSASPRFCCQFITSVTALYDLSSDDLIPPMDLLEMIVTWIFEDPRLILITFLNTPIAANLPIGFLELTPLVGLIRWCVKAPLAYKRKKKPPLSNGHVSNKVTKDPGVGMDRDSHLLYSKLHLSVLQVLMTLQLHLTEKNLYGRLGLILFDHMVPLVEEINRLADELNPLNASQEIELSLDRLAQALQVAMASGALLCTRDDLRTLCSRLPHNNLLQLVISGPVQQSPHAALPPGFYPHIHTPPLGYGAVPAHPAAHPALPTHPGHTFISGVTFPFRPIR
4NAY , Knot 67 145 0.76 40 107 138 
MHHHHHHLKSINHICFSVRNLNDSIHFYRDILLGKLLLTGKKTAYFELAGLWIALNEEKDIPRNEIHFSYTHIAFTIDDSEFKYWHQRLKDNNVNILEGRVRDIRDRQSIYFTDPDGHKLELHTGTLENRLNYYKEAKPHMTFYK

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 \(c\).

\(|P_{f(8SRO_1)}(2) \setminus P_{f(9EOF_1)}(2)|=61\), \(|P_{f(9EOF_1)}(2) \setminus P_{f(8SRO_1)}(2)|=112\). 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 1:10011110110011100011001001100001001100010100110001100100010100001111010111011110110110100000011100001011111011001101111110001110010001101100100100001011100101101111011000001001001100111010001101001100010100011010000111101001010000000100
Pair \(Z_2\) Length of longest common subsequence
8SRO_1,9EOF_1 173 4
8SRO_1,4NAY_1 182 3
9EOF_1,4NAY_1 203 5

Newick tree

 
[
	4NAY_1:99.47,
	[
		8SRO_1:86.5,9EOF_1:86.5
	]:12.97
]

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 \((0.85)(0.8)(\frac{685 }{\log_{20} 685}-\frac{236}{\log_{20}236})=125.\)
Status Protein1 Protein2 d d1/2
Query variables 8SRO_1 9EOF_1 158 121.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]