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Parikh vectors
5BVZ_1 4RLB_1 4DGM_1 Letter Amino acid
4 10 13 P Proline
1 7 5 W Tryptophan
8 17 22 Y Tyrosine
15 6 12 Q Glutamine
14 31 16 G Glycine
3 7 12 H Histidine
15 11 19 I Isoleucine
11 22 17 N Asparagine
2 8 12 F Phenylalanine
23 16 25 V Valine
7 14 28 K Lycine
26 25 15 A Alanine
10 5 19 R Arginine
20 22 25 D Aspartic acid
24 17 36 L Leucine
20 10 8 T Threonine
0 0 3 C Cysteine
12 7 21 E Glutamic acid
1 5 8 M Methionine
17 13 10 S Serine 

5BVZ_1|Chains A, B, C, D, E, F|DNA stabilization protein|Enterobacteria phage HK620 (155148)
>4RLB_1|Chains A, B|Carbapenem-associated resistance protein|Acinetobacter baumannii (470)
>4DGM_1|Chain A|Casein kinase II subunit alpha|Zea mays (4577)
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 \(w=f(c)\)
5BVZ , Knot 103 233 0.80 38 138 223 
MADSNLNEPVIIQATRLDTSILPRNIFSQSYLLYVIAQGTDVGNVANKANEAGQGAYDAQVKNDEQDVELADHDARIAANTKAINILEVRLTTAEGKIVVLRSDVDYLLDEVIDIQAHLVTVDQRLDGVESDVSDIKSDYVSKTVTESQSLASPLDVKTSYSVDGIQVVGARQTGWTAATGTPLLGSFNANQSYTVGTTYTQSEVAALATGLEQARQRILALETALRLHGLID
4RLB , Knot 107 253 0.78 38 154 240 
MKHHHHHHPMSDYDIPTTENLYFQGAMDEAVVHDSYAFDKNQLIPVGARAEVGTTGYGGALLWQANPYVGLALGYNGGDISWSDDLSINGTKYDMDMDNKLAYLNAEIRPWGASTNPWAQGLYVAAGAAYVDNQYDLTKNVGTNASVEIDGNRFNGGANGVSIAGNLKYDNDIAPYIGFGFAPKFSKNWGVFGEVGAYYSGNPKVSLASNNDALIGSDGRTLGKTLDDQERKIANDDKYKWLPVGKVGVNFYW
4DGM , Knot 141 326 0.83 40 200 311 
SKARVYADVNVLRPKEYWDYEALTVQWGEQDDYEVVRKVGRGKYSEVFEGINVNNNEKCIIKILKPVKKKKIKREIKILQNLMGGPNIVKLLDIVRDQHSKTPSLIFEYVNNTDFKVLYPTLTDYDIRYYIYELLKALDYCHSQGIMHRDVKPHNVMIDHELRKLRLIDWGLAEFYHPGKEYNVRVASRYFKGPELLVDLQDYDYSLDMWSLGCMFAGMIFRKEPFFYGHDNHDQLVKIAKVLGTDGLNVYLNKYRIELDPQLEALVGRHSRKPWLKFMNADNQHLVSPEAIDFLDKLLRYDHQERLTALEAMTHPYFQQVRAAEN

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 \(c\).

\(|P_{f(5BVZ_1)}(2) \setminus P_{f(4RLB_1)}(2)|=65\), \(|P_{f(4RLB_1)}(2) \setminus P_{f(5BVZ_1)}(2)|=81\). 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 1:11000100111101001000111001100001101110100110110010011011001010000001011000101110001101101010010101111000100110011010101101000101100010010000100010000011011010000010110111100011011010111101010000011000000011111011001000111100110101110
Pair \(Z_2\) Length of longest common subsequence
5BVZ_1,4RLB_1 146 4
5BVZ_1,4DGM_1 172 3
4RLB_1,4DGM_1 170 4

Newick tree

 
[
	4DGM_1:89.28,
	[
		5BVZ_1:73,4RLB_1:73
	]:16.28
]

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 \((0.85)(0.8)(\frac{486 }{\log_{20} 486}-\frac{233}{\log_{20}233})=72.9\)
Status Protein1 Protein2 d d1/2
Query variables 5BVZ_1 4RLB_1 91 86
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]