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Parikh vectors
2BJX_1 2RHR_1 3WUZ_1 Letter Amino acid
1 10 4 H Histidine
8 7 5 K Lycine
0 4 2 M Methionine
3 9 6 P Proline
0 2 4 W Tryptophan
11 33 3 A Alanine
5 7 3 N Asparagine
10 8 4 D Aspartic acid
1 6 6 Y Tyrosine
0 4 1 C Cysteine
5 5 9 Q Glutamine
6 35 9 G Glycine
9 23 9 L Leucine
8 13 15 S Serine
1 19 9 R Arginine
11 23 5 E Glutamic acid
6 10 6 I Isoleucine
10 5 7 F Phenylalanine
6 19 7 T Threonine
9 35 6 V Valine 

2BJX_1|Chain A|PROTEIN (PROTEIN DISULFIDE ISOMERASE)|Homo sapiens (9606)
>2RHR_1|Chains A, B|Actinorhodin Polyketide Ketoreductase|Streptomyces coelicolor (1902)
>3WUZ_1|Chain A|Paired immunoglobulin-like type 2 receptor alpha|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 \(w=f(c)\)
2BJX , Knot 55 110 0.78 34 90 106 
AATTLPDGAAAESLVESSEVAVIGFFKDVESDSAKQFLQAAEAIDDIPFGITSNSDVFSKYQLDKDGVVLFKKFDEGRNNFEGEVTKENLLDFIKHNQLPLVIEFTEQTA
2RHR , Knot 115 277 0.77 40 151 259 
HHHHHHSSGLVPRGSHMATQDSEVALVTGATSGIGLEIARRLGKEGLRVFVCARGEEGLRTTLKELREAGVEADGRTCDVRSVPEIEALVAAVVERYGPVDVLVNNAGRLGGGATAELADELWLDVVETNLTGVFRVTKQVLKAGGMLERGTGRIVNIASTGGKQGVVHAAPYSASKHGVVGFTKALGLELARTGITVNAVCPGFVETPMAASVREHYSDIWEVSTEEAFDRITARVPIGRYVQPSEVAEMVAYLIGPGAAAVTAQALNVCGGLGNY
3WUZ , Knot 62 120 0.82 40 101 116 
MLYGVTQPKHLSASMGGSVEIPFSFYYPWELATAPDVRISWRRGHFHGQSFYSTRPPSIHKDYVNRLFLNWTEGQKSGFLRISNLQKQDQSVYFCRVELDTRSSGRQQWQSIEGTKLSIT

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 \(c\).

\(|P_{f(2BJX_1)}(2) \setminus P_{f(2RHR_1)}(2)|=42\), \(|P_{f(2RHR_1)}(2) \setminus P_{f(2BJX_1)}(2)|=103\). 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 1:11001101111001100001111111001000010011011011001111100000110000100011111001001000101010000110110000111110100001
Pair \(Z_2\) Length of longest common subsequence
2BJX_1,2RHR_1 145 4
2BJX_1,3WUZ_1 135 3
2RHR_1,3WUZ_1 152 3

Newick tree

 
[
	2RHR_1:76.39,
	[
		2BJX_1:67.5,3WUZ_1:67.5
	]:8.89
]

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 \((0.85)(0.8)(\frac{387 }{\log_{20} 387}-\frac{110}{\log_{20}110})=84.6\)
Status Protein1 Protein2 d d1/2
Query variables 2BJX_1 2RHR_1 104 72.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]