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Parikh vectors
7RBJ_1 3DLI_1 9EYB_1 Letter Amino acid
1 0 3 W Tryptophan
12 13 3 Y Tyrosine
18 13 2 V Valine
26 24 11 L Leucine
36 20 8 K Lycine
27 27 5 E Glutamic acid
14 9 3 P Proline
21 13 6 D Aspartic acid
7 4 1 M Methionine
21 17 2 S Serine
18 5 2 A Alanine
19 12 2 R Arginine
13 1 2 Q Glutamine
22 13 4 G Glycine
15 11 0 H Histidine
23 22 1 I Isoleucine
14 15 1 F Phenylalanine
19 7 4 T Threonine
12 8 3 N Asparagine
3 6 1 C Cysteine

7RBJ_1|Chain A|DNA polymerase beta|Homo sapiens (9606)
>3DLI_1|Chains A, B, C|methyltransferase|Archaeoglobus fulgidus (2234)
>9EYB_1|Chain A[auth AAA]|Spectrin alpha chain, non-erythrocytic 1|Gallus gallus (9031)
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 \(w=f(c)\)
7RBJ , Knot 147 341 0.83 40 200 324
MSKRKAPQETLNGGITDMLTELANFEKNVSQAIHKYNAYRKAASVIAKYPHKIKSGAEAKKLPGVGTKIAEKIDEFLATGKLRKLEKIRQDDTSSSINFLTRVSGIGPSAARKFVDEGIKTLEDLRKNEDKLNHHQRIGLKYFGDFEKRIPREEMLQMQDIVLNEVKKVDSEYIATVCGSFRRGAESSGDMDVLLTHPSFTSESTKQPKLLHQVVEQLQKVHFITDTLSKGETKFMGVCQLPSKNDEKEYPHRRIDIRLIPKDQYYCGVLYFTGSDIFNKNMRAHALEKGFTINEYTIRPLGVTGVAGEPLPVDSEKDIFDYIQWKYREPKDRSEHHHHHH
3DLI , Knot 104 240 0.79 38 158 230
MSLSGTDIHTSDYYFLFEEKFRGSRELVKARLRRYIPYFKGCRRVLDIGCGRGEFLELCKEEGIESIGVDINEDMIKFCEGKFNVVKSDAIEYLKSLPDKYLDGVMISHFVEHLDPERLFELLSLCYSKMKYSSYIVIESPNPTSLYSLINFYIDPTHKKPVHPETLKFILEYLGFRDVKIEFFEECEELTKLAKIDSNTVSEEVIRVINENIEKLNRILFGPQDYAIIAKKEGHHHHHH
9EYB , Knot 37 64 0.80 38 58 62
GPMDETGKELLLVLYDYQEKSPRELTCKKGDILTLLNSTNKDWWKVEWNDRQGFLPAAYLKKLD

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 \(c\).

\(|P_{f(7RBJ_1)}(2) \setminus P_{f(3DLI_1)}(2)|=94\), \(|P_{f(3DLI_1)}(2) \setminus P_{f(7RBJ_1)}(2)|=52\). 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 1:10000110001011100110011010001001100001000110111001001001101001111100110010011101010010010000000010110010111101100110011001001000000100000111001101000110001101001110010010000110101010011000101011100101000000010110011001001011000100100011110011000000001000101011100000011101010011000101011001101000010111101111011110000011001010000100000000000
Pair \(Z_2\) Length of longest common subsequence
7RBJ_1,3DLI_1 146 6
7RBJ_1,9EYB_1 178 3
3DLI_1,9EYB_1 156 3

Newick tree

 
[
	9EYB_1:86.95,
	[
		7RBJ_1:73,3DLI_1:73
	]:13.95
]

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 \((0.85)(0.8)(\frac{581 }{\log_{20} 581}-\frac{240}{\log_{20}240})=96.7\)
Status Protein1 Protein2 d d1/2
Query variables 7RBJ_1 3DLI_1 120 102
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]