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Parikh vectors
2OMA_1 7LXA_1 4WTM_1 Letter Amino acid
4 2 0 C Cysteine
18 9 0 E Glutamic acid
17 11 2 G Glycine
5 7 0 H Histidine
21 16 0 L Leucine
24 9 0 V Valine
9 12 0 N Asparagine
17 11 0 T Threonine
34 16 1 A Alanine
13 13 0 R Arginine
13 5 0 Q Glutamine
16 10 0 I Isoleucine
13 13 0 K Lycine
7 5 0 F Phenylalanine
12 6 0 S Serine
6 6 0 Y Tyrosine
5 10 0 D Aspartic acid
2 5 0 M Methionine
9 3 0 P Proline
5 3 0 W Tryptophan 

2OMA_1|Chains A, B|Triosephosphate isomerase|Trypanosoma cruzi (5693)
>7LXA_1|Chain A|Lysozyme|Enterobacteria phage T4 (10665)
>4WTM_1|Chain A[auth T]|RNA TEMPLATE UAGG|synthetic construct (32630)
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 \(w=f(c)\)
2OMA , Knot 111 250 0.81 40 160 238 
ASKPQPIAAANWKCNGSESLLVPLIETLNAATFDHDVQCVVAPTFLHIPMTKARLTNPKFQIAAQNAITRSGAFTGEVSLQILKDYGISWVVLGHSERRLYYGETNEIVAEKVAQACAAGFHVIVCVGETNEEREAGRTAAVVLTQLAAVAQKLSKEAWSRVVIAYEPVWAIGTGKVATPQQAQEVHELLRRWVRSKLGTDIAAQLRILYGGSVTAKNARTLYQMRDINGFLVGGASLKPEFVEIIEATK
7LXA , Knot 79 172 0.78 40 122 162 
MNIFEMLRIDEGLRLKIYKDTEGYYTIGIGHLLTKSPSLNAAKSELDKAIGRNCNGVITKDEAEKLFNQDVDAAVRGILRNAKLKPVYDSLDAVRRCAAINMVFQMGETGVAGFTNSLRMLQQKRWDEAAVNLAKSRWYNQTPNRAKRVITTFRTGTWDAYKNLLEHHHHHH
4WTM , Knot 4 4 0.46 6 3 2 
UAGG

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 \(c\).

\(|P_{f(2OMA_1)}(2) \setminus P_{f(7LXA_1)}(2)|=98\), \(|P_{f(7LXA_1)}(2) \setminus P_{f(2OMA_1)}(2)|=60\). 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 1:1001011111010001000111111001011010001001111011011100101001010111001100011101010101100011011111000001001000011100110101111011101100000001100111110011111001000110011110011111101011010010010011001100011001110101101101010010010010010111111101010110110100
Pair \(Z_2\) Length of longest common subsequence
2OMA_1,7LXA_1 158 4
2OMA_1,4WTM_1 159 2
7LXA_1,4WTM_1 123 2

Newick tree

 
[
	2OMA_1:84.34,
	[
		7LXA_1:61.5,4WTM_1:61.5
	]:22.84
]

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 \((0.85)(0.8)(\frac{422 }{\log_{20} 422}-\frac{172}{\log_{20}172})=74.1\)
Status Protein1 Protein2 d d1/2
Query variables 2OMA_1 7LXA_1 94 79
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]