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Parikh vectors
2BGA_1 6OBJ_1 2XKS_1 Letter Amino acid
11 34 2 A Alanine
16 16 5 N Asparagine
13 13 5 I Isoleucine
23 6 8 K Lycine
4 17 5 F Phenylalanine
5 20 5 P Proline
5 10 6 Y Tyrosine
23 27 7 V Valine
4 31 5 R Arginine
5 15 3 Q Glutamine
16 27 8 E Glutamic acid
7 3 4 H Histidine
25 33 7 L Leucine
2 8 1 M Methionine
13 21 9 S Serine
18 16 5 T Threonine
11 13 7 D Aspartic acid
2 2 2 C Cysteine
4 5 2 W Tryptophan
20 22 3 G Glycine 

2BGA_1|Chains A, B|BETA-LACTAMASE II|BACILLUS CEREUS (1396)
>6OBJ_1|Chains A, C[auth B]|SgraIR restriction enzyme|Streptomyces griseus (1911)
>2XKS_1|Chain A|BETA-2-MICROGLOBULIN|HOMO SAPIENS (9606)
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 \(w=f(c)\)
2BGA , Knot 101 227 0.80 40 148 217 
SQKVEKTVIKNETGTISISQLNKNVWVHTELGSFNGEAVPSNGLVLNTSKGLVLVDSSWDDKLTKELIEMVEKKFQKRVTDVIITHAHADCIGGIKTLKERGIKAHSTALTAELAKKNGYEEPLGDLQTVTNLKFGNMKVETFYPGKGHTEDNIVVWLPQYNILVGGCLVKSTSAKDLGNVADAYVNEWSTSIENVLKRYRNINAVVPGHGEVGDKGLLLHTLDLLK
6OBJ , Knot 146 339 0.83 40 199 329 
MPFTYSIEATRNLATTERCIQDIRNAPVRNRSTQFQLAQQNMLAYTFGEVIPGFASAGINGMDYRDVIGRPVENAVTEGTHFFRDDFRVDSNAKAKVAGDIFEIVSSAVMWNCAARWNSLMVGEGWRSQPRYSRPTLSPSPRRQVAVLNLPRSFDWVSLLVPESQEVIEEFRAGLRKDGLGLPTSTPDLAVVVLPEEFQNDEMWREEIAGLTRPNQILLSGAYQRLQGRVQPGEISLAVAFKRSLRSDRLYQPLYEANVMQLLLEGKLGAPKVEFEVHTLAPEGTNAFVTYEAASLYGLAEGRSAVHRAIRELYVPPTAADLARRFFAFLNERMELVNG
2XKS , Knot 52 99 0.80 40 83 96 
IQRTPKIQVYSRHPAENGKSNFLNCYVSGFHPSDIEVDLLKNGERIEKVEHSDLSFSKDWSFYLLYYTEFTPTEKDEYACRVNHVTLSQPKIVKWDRDM

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 \(c\).

\(|P_{f(2BGA_1)}(2) \setminus P_{f(6OBJ_1)}(2)|=60\), \(|P_{f(6OBJ_1)}(2) \setminus P_{f(2BGA_1)}(2)|=111\). 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 1:00010001100001010100100011100011010101110011110000111110001000100011011000100010011100101001111001000110100011010110001000111010010010110101001011010000011111100011111011000010011011010100100010011000001011111010110011110010110
Pair \(Z_2\) Length of longest common subsequence
2BGA_1,6OBJ_1 171 3
2BGA_1,2XKS_1 159 4
6OBJ_1,2XKS_1 180 3

Newick tree

 
[
	6OBJ_1:90.37,
	[
		2BGA_1:79.5,2XKS_1:79.5
	]:10.87
]

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 \((0.85)(0.8)(\frac{566 }{\log_{20} 566}-\frac{227}{\log_{20}227})=96.6\)
Status Protein1 Protein2 d d1/2
Query variables 2BGA_1 6OBJ_1 130 104
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]