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Parikh vectors
7RMJ_1 4ZQR_1 1CBN_1 Letter Amino acid
11 22 2 R Arginine
16 15 1 E Glutamic acid
18 20 4 P Proline
9 28 3 S Serine
10 6 3 N Asparagine
18 51 4 G Glycine
6 7 0 H Histidine
4 5 1 F Phenylalanine
16 22 6 T Threonine
4 9 2 Y Tyrosine
15 49 2 V Valine
20 56 5 A Alanine
4 2 6 C Cysteine
18 36 1 L Leucine
10 17 0 M Methionine
4 1 0 W Tryptophan
7 25 1 D Aspartic acid
15 10 0 Q Glutamine
15 12 5 I Isoleucine
11 14 0 K Lycine 

7RMJ_1|Chains A, B, C|CAPSID PROTEIN P24|Human immunodeficiency virus 1 (11676)
>4ZQR_1|Chains A, B, C, D|Inosine-5'-monophosphate dehydrogenase,Inosine-5'-monophosphate dehydrogenase|Mycobacterium tuberculosis (strain ATCC 25618 / H37Rv) (83332)
>1CBN_1|Chain A|CRAMBIN|Crambe hispanica subsp. abyssinica (3721)
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 \(w=f(c)\)
7RMJ , Knot 106 231 0.83 40 158 223 
PIVQNLQGQMVHQCISPRTLNAWVKVVEEKAFSPEVIPMFSALSCGATPQDLNTMLNTVGGHQAAMQMLKETINEEAAEWDRLHPVHAGPIAPGQMREPRGSDIAGTTSTLQEQIGWMTHNPPIPVGEIYKRWIILGLNKIVRMYSPTSILDIRQGPKEPFRDYVDRFYKTLRAEQASQEVKNAATETLLVQNANPDCKTILKALGPGATLEEMMTACQGVGGPGHKARVL
4ZQR , Knot 164 407 0.80 40 191 376 
SNAMSRGMSGLEDSSDLVVSPYVRMGGLTTDPVPTGGDDPHKVAMLGLTFDDVLLLPAASDVVPATADTSSQLTKKIRLKVPLVSSAMDTVTESRMAIAMARAGGMGVLHRNLPVAEQAGQVEMVKRSGGLLVGAAVGVGGDAWVRAMMLVDAGVDVLVVDTAHAHNRLVLDMVGKLKSEVGDRVEVVGGNVATRSAAAALVDAGADAVKVGVGPGSICTTRVVAGVGAPQITAILEAVAACRPAGVPVIADGGLQYSGDIAKALAAGASTAMLGSLLAGTAEAPGELIFVNGKQYKSYRGMGSLGAMRGRGGATSYSKDRYFADDALSEDKLVPEGIEGRVPFRGPLSSVIHQLTGGLRAAMGYTGSPTIEVLQQAQFVRITPAGLKESHPHDVAMTVEAPNYYAR
1CBN , Knot 27 46 0.75 30 39 44 
TTCCPSIVARSNFNVCRLPGTSEAICATYTGCIIIPGATCPGDYAN

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 \(c\).

\(|P_{f(7RMJ_1)}(2) \setminus P_{f(4ZQR_1)}(2)|=62\), \(|P_{f(4ZQR_1)}(2) \setminus P_{f(7RMJ_1)}(2)|=95\). 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 1:111001010110001010010111011000110101111101100110100100110011100111011000100011010010110111111101001010011100001000111100011111101000111111001101001001101001100110001001000101001000100110001110010100001101111110100110100111111001011
Pair \(Z_2\) Length of longest common subsequence
7RMJ_1,4ZQR_1 157 4
7RMJ_1,1CBN_1 165 4
4ZQR_1,1CBN_1 184 3

Newick tree

 
[
	1CBN_1:90.14,
	[
		7RMJ_1:78.5,4ZQR_1:78.5
	]:11.64
]

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 \((0.85)(0.8)(\frac{638 }{\log_{20} 638}-\frac{231}{\log_{20}231})=114.\)
Status Protein1 Protein2 d d1/2
Query variables 7RMJ_1 4ZQR_1 140 112
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]