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Parikh vectors
6ALB_1 3ULX_1 2CVP_1 Letter Amino acid
5 3 15 M Methionine
11 19 20 R Arginine
8 6 28 N Asparagine
17 1 24 Q Glutamine
15 11 34 E Glutamic acid
13 2 4 C Cysteine
17 13 24 P Proline
11 7 29 V Valine
19 15 56 L Leucine
18 14 41 K Lycine
11 2 35 S Serine
11 5 32 T Threonine
6 21 42 A Alanine
18 12 25 D Aspartic acid
9 15 40 G Glycine
11 4 34 I Isoleucine
4 6 12 W Tryptophan
14 8 13 Y Tyrosine
3 3 21 H Histidine
12 7 28 F Phenylalanine 

6ALB_1|Chain A|CREB-binding protein|Homo sapiens (9606)
>3ULX_1|Chain A|Stress-induced transcription factor NAC1|Oryza sativa subsp. japonica (39947)
>2CVP_1|Chains A, B|Glucose-6-phosphate isomerase|Mus musculus (10090)
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 \(w=f(c)\)
6ALB , Knot 106 233 0.82 40 173 227 
SRKKIFKPEELRQALMPTLEALYRQDPESLPFRQPVDPQLLGIPDYFDIVKNPMDLSTIKRKLDTGQYQEPWQYVDDVWLMFNNAWLYNRKTSRVYKFCSKLAEVFEQEIDPVMQSLGYCCGRKYEFSPQTLCCYGKQLCTIPRDAAYYSYQNRYHFCEKCFTEIQGENVTLGDDPSQPQTTISKDQFEKKKNDTLDPEPFVDCKECGRKMHQICVLHYDIIWPSGFVCDNCL
3ULX , Knot 80 174 0.79 40 124 166 
MGMRRERDAEAELNLPPGFRFHPTDDELVEHYLCRKAAGQRLPVPIIAEVDLYKFDPWDLPERALFGAREWYFFTPRDRKYPNGSRPNRAAGNGYWKATGADKPVAPRGRTLGIKKALVFYAGKAPRGVKTDWIMHEYRLADAGRAAAGAKKGSLRLDDWVLCRLYNKKNEWEK
2CVP , Knot 225 557 0.85 40 271 531 
MAALTRNPQFQKLLEWHRANSANLKLRELFEADPERFNNFSLNLNTNHGHILVDYSKNLVSKEVMQMLVELAKSRGVEAARDNMFSGSKINYTEDRAVLHVALRNRSNTPIKVDGKDVMPEVNRVLDKMKSFCQRVRSGDWKGYTGKSITDIINIGIGGSDLGPLMVTEALKPYSKGGPRVWFVSNIDGTHIAKTLASLSPETSLFIIASKTFTTQETITNAETAKEWFLEAAKDPSAVAKHFVALSTNTAKVKEFGIDPQNMFEFWDWVGGRYSLWSAIGLSIALHVGFDHFEQLLSGAHWMDQHFLKTPLEKNAPVLLALLGIWYINCYGCETHALLPYDQYMHRFAAYFQQGDMESNGKYITKSGARVDHQTGPIVWGEPGTNGQHAFYQLIHQGTKMIPCDFLIPVQTQHPIRKGLHHKILLANFLAQTEALMKGKLPEEARKELQAAGKSPEDLEKLLPHKVFEGNRPTNSIVFTKLTPFILGALIAMYEHKIFVQGIMWDINSFDQWGVELGKQLAKKIEPELEGSSAVTSHDSSTNGLISFIKQQRDTKL

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 \(c\).

\(|P_{f(6ALB_1)}(2) \setminus P_{f(3ULX_1)}(2)|=116\), \(|P_{f(3ULX_1)}(2) \setminus P_{f(6ALB_1)}(2)|=67\). 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 1:00001101001001111010110000100111001101011111001011001101001000100100001100100111110011100000001001000110110001011100110001000010100100010010011001100000000010000100101001011001001000100001000000010101110000010010010110001111011100001
Pair \(Z_2\) Length of longest common subsequence
6ALB_1,3ULX_1 183 3
6ALB_1,2CVP_1 196 4
3ULX_1,2CVP_1 219 3

Newick tree

 
[
	2CVP_1:10.72,
	[
		6ALB_1:91.5,3ULX_1:91.5
	]:16.22
]

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 \((0.85)(0.8)(\frac{407 }{\log_{20} 407}-\frac{174}{\log_{20}174})=69.2\)
Status Protein1 Protein2 d d1/2
Query variables 6ALB_1 3ULX_1 91 78.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]