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Parikh vectors
1IZE_1 8FJD_1 2MLB_1 Letter Amino acid
10 8 2 N Asparagine
14 7 0 Q Glutamine
37 25 7 G Glycine
30 14 13 T Threonine
18 8 1 Y Tyrosine
24 23 2 V Valine
15 31 12 L Leucine
11 28 5 P Proline
49 15 3 S Serine
3 1 0 W Tryptophan
2 8 2 R Arginine
0 7 0 C Cysteine
0 8 1 M Methionine
19 16 1 F Phenylalanine
24 44 3 A Alanine
32 18 4 D Aspartic acid
9 18 8 E Glutamic acid
2 12 1 H Histidine
15 16 7 I Isoleucine
9 18 7 K Lycine 

1IZE_1|Chain A|aspartic proteinase|Aspergillus oryzae (5062)
>8FJD_1|Chains A, B|Chlorophyllase|Triticum aestivum (4565)
>2MLB_1|Chain A|redesigned ubiquitin|synthetic (32630)
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 \(w=f(c)\)
1IZE , Knot 132 323 0.78 36 159 293 
AATGSVTTNPTSNDEEYITQVTVGDDTLGLDFDTGSADLWVFSSQTPSSERSGHDYYTPGSSAQKIDGATWSISYGDGSSASGDVYKDKVTVGGVSYDSQAVESAEKVSSEFTQDTANDGLLGLAFSSINTVQPTPQKTFFDNVKSSLSEPIFAVALKHNAPGVYDFGYTDSSKYTGSITYTDVDNSQGFWGFTADGYSIGSDSSSDSITGIADTGTTLLLLDDSIVDAYYEQVNGASYDSSQGGYVFPSSASLPDFSVTIGDYTATVPGEYISFADVGNGQTFGGIQSNSGIGFSIFGDVFLKSQYVVFDASGPRLGFAAQA
8FJD , Knot 134 325 0.79 40 186 296 
MAAAAPAETMNKSAAGAEVPEAFTSVFQPGKLAVEAIQVDENAAPTPPIPVLIVAPKDAGTYPVAMLLHGFFLHNHFYEHLLRHVASHGFIIVAPQFSISIIPSGDAEDIAAAAKVADWLPDGLPSVLPKGVEPELSKLALAGHSRGGHTAFSLALGHAKTQLTFSALIGLDPVAGTGKSSQLQPKILTYEPSSFGMAMPVLVIGTGLGEEKKNIFFPPCAPKDVNHAEFYRECRPPCYYLVTKDYGHLDMLDDDAPKFITCVCKDGNGCKGKMRRCVAGIMVAFLNAALGEKDADLEAILRDPAVAPTTLDPVEHRVAENLYFQ
2MLB , Knot 39 79 0.72 34 55 71 
MDTINITLPDGKTLTLTVTPEFTVKELAEEIARRLGLSPEDIKLTHNGKTLDPSLTLAEYGITPGSTITLEIKKKGGLE

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 \(c\).

\(|P_{f(1IZE_1)}(2) \setminus P_{f(8FJD_1)}(2)|=66\), \(|P_{f(8FJD_1)}(2) \setminus P_{f(1IZE_1)}(2)|=93\). 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 1:11010100010000000100101100011101001010111100001000001000001100100101101010010100101010000101111000001100100100010000100111111100100101010001100100010011111110001111001100000000101000010000111110101001100000001011100100111100011010000101100000011011100101101010110001011100101101101001111000011110111011100001110101101111101
Pair \(Z_2\) Length of longest common subsequence
1IZE_1,8FJD_1 159 3
1IZE_1,2MLB_1 146 4
8FJD_1,2MLB_1 163 4

Newick tree

 
[
	8FJD_1:82.85,
	[
		1IZE_1:73,2MLB_1:73
	]:9.85
]

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 \((0.85)(0.8)(\frac{648 }{\log_{20} 648}-\frac{323}{\log_{20}323})=90.0\)
Status Protein1 Protein2 d d1/2
Query variables 1IZE_1 8FJD_1 115 112.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]