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Parikh vectors
2ROX_1 1EXE_1 7SDR_1 Letter Amino acid
5 1 23 Y Tyrosine
5 3 13 D Aspartic acid
12 8 16 E Glutamic acid
10 7 19 G Glycine
8 13 24 K Lycine
8 4 12 P Proline
12 11 22 A Alanine
4 2 7 R Arginine
4 0 9 H Histidine
7 8 26 L Leucine
11 6 16 S Serine
12 8 20 V Valine
3 3 16 N Asparagine
0 6 18 Q Glutamine
5 4 12 F Phenylalanine
12 7 33 T Threonine
2 0 2 W Tryptophan
1 0 11 C Cysteine
5 6 11 I Isoleucine
1 2 8 M Methionine 

2ROX_1|Chains A, B|TRANSTHYRETIN|Homo sapiens (9606)
>1EXE_1|Chains A, B|TRANSCRIPTION FACTOR 1|Bacillus phage SPO1 (10685)
>7SDR_1|Chains A, B, C|Non-structural protein 3|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
2ROX , Knot 65 127 0.82 38 101 125 
GPTGTGESKCPLMVKVLDAVRGSPAINVAVHVFRKAADDTWEPFASGKTSESGELHGLTTEEEFVEGIYKVEIDTKSYWKALGISPFHEHAEVVFTANDSGPRRYTIAALLSPYSYSTTAVVTNPKE
1EXE , Knot 50 99 0.77 34 79 96 
MNKTELIKAIAQDTGLTQVSVSKMLASFEKIITETVAKGDKVQLTGFLNIKPVARQARKGFNPQTQEALEIAPSVGVSVKPGESLKKAAEGLKYEDFAK
7SDR , Knot 142 318 0.85 40 203 304 
SNAEVRTIKVFTTVDNINLHTQVVDMSMTYGQQFGPTYLDGADVTKIKPHNSHEGKTFYVLPNDDTLRVEAFEYYHTTDPSFLGRYMSALNHTKKWKYPQVNGLTSIKWADNNCYLATALLTLQQIELKFNPPALQDAYYRARAGEAANFCALILAYCNKTVGELGDVRETMSYLFQHANLDSCKRVLNVVCKTCGQQQTTLKGVEAVMYMGTLSYEQFKKGVQIPCTCGKQATKYLVQQESPFVMMSAPPAQYELKHGTFTCASEYTGNYQCGHYKHITSKETLYCIDGALLTKSSEYKGPITDVFYKENSYTTTIK

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 \(c\).

\(|P_{f(2ROX_1)}(2) \setminus P_{f(1EXE_1)}(2)|=76\), \(|P_{f(1EXE_1)}(2) \setminus P_{f(2ROX_1)}(2)|=54\). 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 1:1101010000111101101101011101110110011000101110100000101011000001101100101000001011110110001011101000110000111110100000011100100
Pair \(Z_2\) Length of longest common subsequence
2ROX_1,1EXE_1 130 3
2ROX_1,7SDR_1 178 3
1EXE_1,7SDR_1 178 3

Newick tree

 
[
	7SDR_1:95.67,
	[
		2ROX_1:65,1EXE_1:65
	]:30.67
]

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 \((0.85)(0.8)(\frac{226 }{\log_{20} 226}-\frac{99}{\log_{20}99})=41.0\)
Status Protein1 Protein2 d d1/2
Query variables 2ROX_1 1EXE_1 52 45.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]