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Parikh vectors
4HXB_1 5EHA_1 5EEW_1 Letter Amino acid
11 14 2 N Asparagine
9 10 6 I Isoleucine
3 6 0 W Tryptophan
13 7 7 V Valine
9 7 4 R Arginine
13 8 4 D Aspartic acid
13 6 7 K Lycine
9 7 3 F Phenylalanine
29 11 4 S Serine
5 1 0 C Cysteine
10 8 6 E Glutamic acid
15 16 7 G Glycine
16 12 4 L Leucine
13 7 0 P Proline
19 11 6 T Threonine
8 10 5 A Alanine
9 6 2 Q Glutamine
4 1 4 H Histidine
2 1 1 M Methionine
9 4 2 Y Tyrosine 

4HXB_1|Chain A[auth L]|6B9 FAB LIGHT CHAIN|MUS MUSCULUS (10090)
>5EHA_1|Chain A|Lectin-like fold protein|Agaricus bisporus (5341)
>5EEW_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V|Transcription attenuation protein MtrB|Geobacillus stearothermophilus (1422)
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 \(w=f(c)\)
4HXB , Knot 101 219 0.82 40 159 213 
DVLMTQTPLSLPVSLGDQASISCRSSQTIVHSNGNTYLEWYLQKPGQSPKLLIYKVSNRFPGVPDRFSGSGSGTDFTLKIRRVEAEDLGVYYCFQASHDPPTFGGGTKLEIKRADAAPTVSIFPPSSEQLTSGGASVVCFLNNFYPKDINVKWKIDGSERQNGVLNSWTDQDSKDSTYSMSSTLTLTKDEYERHNSYTCEATHKTSTSPIVKSFNRNEC
5EHA , Knot 75 153 0.82 40 120 147 
ARKIPLDLPGTRILNGANWANNSATENLATNSGTLIIFDQSTPGQDADRWLIHNYLDGYKIFNMGSNNWASVSRGNTVLGVSEFDGQTCKWSIEYSGNGEEFWIRVPREGGGGAVWTIKPASSQGPTTVFLDLLKETDPNQRIKFAVENLYFQ
5EEW , Knot 42 74 0.81 34 63 72 
MYTNSDFVVIKALEDGVNVIGLTRGADTRFHHSEKLDKGEVLIAQFTEHTSAIKVRGKAYIQTRHGVIESEGKK

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 \(c\).

\(|P_{f(4HXB_1)}(2) \setminus P_{f(5EHA_1)}(2)|=93\), \(|P_{f(5EHA_1)}(2) \setminus P_{f(4HXB_1)}(2)|=54\). 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 1:011100011011101100101000000011000100010101001100101110010001111100101010100101010010100111000101000110111100101001011101011110000100111011011001010010101010100000111001000000000001000101000000000000001000000011100100000
Pair \(Z_2\) Length of longest common subsequence
4HXB_1,5EHA_1 147 3
4HXB_1,5EEW_1 174 2
5EHA_1,5EEW_1 145 3

Newick tree

 
[
	4HXB_1:83.03,
	[
		5EHA_1:72.5,5EEW_1:72.5
	]:10.53
]

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 \((0.85)(0.8)(\frac{372 }{\log_{20} 372}-\frac{153}{\log_{20}153})=66.0\)
Status Protein1 Protein2 d d1/2
Query variables 4HXB_1 5EHA_1 84 70.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]