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Parikh vectors
1FCX_1 2RAS_1 3SDA_1 Letter Amino acid
15 10 8 I Isoleucine
36 19 27 L Leucine
5 14 21 V Valine
15 26 18 A Alanine
17 19 15 E Glutamic acid
3 3 17 H Histidine
9 9 11 F Phenylalanine
15 5 13 P Proline
19 9 14 T Threonine
1 2 14 W Tryptophan
4 8 8 Y Tyrosine
15 18 13 R Arginine
6 2 5 C Cysteine
11 13 28 S Serine
3 4 9 N Asparagine
11 11 9 M Methionine
13 14 20 G Glycine
13 4 14 K Lycine
14 16 15 D Aspartic acid
10 6 23 Q Glutamine 

1FCX_1|Chain A|RETINOIC ACID RECEPTOR GAMMA-1|Homo sapiens (9606)
>2RAS_1|Chains A, B|Transcriptional regulator, TetR family|Novosphingobium aromaticivorans (279238)
>3SDA_1|Chain A|Antigen-presenting glycoprotein CD1d1|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 \(w=f(c)\)
1FCX , Knot 109 235 0.84 40 159 228 
SPQLEELITKVSKAHQETFPSLCQLGKYTTNSSADHRVQLDLGLWDKFSELATKCIIKIVEFAKRLPGFTGLSIADQITLLKAACLDILMLRICTRYTPEQDTMTFSDGLTLNRTQMHNAGFGPLTDLVFAFAGQLLPLEMDDTETGLLSAICLICGDRMDLEEPEKVDKLQEPLLEALRLYARRRRPSQPYMFPRMLMKITDLRGISTKGAERAITLKMEIPGPMPPLIREMLE
2RAS , Knot 102 212 0.86 40 145 207 
GMASSGTEHDAMRARLVDVAQAIVEERGGAGLTLSELAARAGISQANLSRYFETREDLMEAIADYWFHPMVEIMEDVLASDLPPRRKMYEFFARRFVVMRRKWEADPVKLQTYIEVGNDYFEQVRSYIDLADHYLGEIIGEAMSDGAFSGLEVDETISLVNQMCAPYCALNTMTTFMERLSEDKLARIVDAVFDGLSAQDRGARSLTGLRAA
3SDA , Knot 131 302 0.82 40 189 286 
SEAQQKNYTFRCLQMSSFANRSWSRTDSVVWLGDLQTHRWSNDSATISFTKPWSQGKLSNQQWEKLQHMFQVYRVSFTRDIQELVKMMSPKEDYPIEIQLSAGCEMYPGNASESFLHVAFQGKYVVRFWGTSWQTVPGAPSWLDLPIKVLNADQGTSATVQMLLNDTCPLFVRGLLEAGKSDLEKQEKPVAWLSSVPSSAHGHRQLVCHVSGFYPKPVWVMWMRGDQEQQGTHRGDFLPNADETWYLQATLDVEAGEEAGLACRVKHSSLGGQDIILYWGSLHHILDAQKMVWNHRHHHHHH

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 \(c\).

\(|P_{f(1FCX_1)}(2) \setminus P_{f(2RAS_1)}(2)|=86\), \(|P_{f(2RAS_1)}(2) \setminus P_{f(1FCX_1)}(2)|=72\). 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 1:0101001100100100001101001100000001000101011110010011000110110110011110110110010110110101111010000010000101001101000010011111100111111101111010000011101101101001010010010010011101101010000100101110111010010110001100110101011111111100110
Pair \(Z_2\) Length of longest common subsequence
1FCX_1,2RAS_1 158 4
1FCX_1,3SDA_1 194 4
2RAS_1,3SDA_1 180 3

Newick tree

 
[
	3SDA_1:97.94,
	[
		1FCX_1:79,2RAS_1:79
	]:18.94
]

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 \((0.85)(0.8)(\frac{447 }{\log_{20} 447}-\frac{212}{\log_{20}212})=68.5\)
Status Protein1 Protein2 d d1/2
Query variables 1FCX_1 2RAS_1 86 82
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]