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Parikh vectors
4DGJ_1 3VGL_1 6ZOS_1 Letter Amino acid
10 9 11 T Threonine
7 6 3 W Tryptophan
14 44 15 A Alanine
20 50 11 G Glycine
14 11 9 P Proline
11 9 9 Q Glutamine
6 1 16 M Methionine
12 20 11 R Arginine
12 21 13 D Aspartic acid
8 5 1 C Cysteine
6 9 7 F Phenylalanine
11 4 5 Y Tyrosine
14 6 8 N Asparagine
15 19 15 E Glutamic acid
5 10 11 K Lycine
16 11 22 S Serine
16 29 13 V Valine
5 12 14 H Histidine
14 19 12 I Isoleucine
19 26 46 L Leucine 

4DGJ_1|Chains A, B, C, D|Enteropeptidase catalytic light chain|Homo sapiens (9606)
>3VGL_1|Chain A|Glucokinase|Streptomyces griseus (455632)
>6ZOS_1|Chains A, B|Estrogen receptor|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 \(w=f(c)\)
4DGJ , Knot 111 235 0.86 40 174 230 
IVGGSDAKEGAWPWVVGLYYDDRLLCGASLVSSDWLVSAAHCVYGRNLEPSKWTAILGLHMKSNLTSPQTVPRLIDEIVINPHYNRRRKDNDIAMMHLEFKVNYTDYIQPISLPEENQVFPPGRNCSIAGWGTVVYQGTTADILQEADVPLLSNERCQQQMPEYNITENMICAGYEEGGIDSCQGDSGGPLMCQENNRWFLAGVTSFGYECALPNRPGVYARVSRFTEWIQSFLH
3VGL , Knot 131 321 0.78 40 171 295 
MGLTIGVDIGGTKIAAGVVDEEGRILSTFKVATPPTAEGIVDAICAAVAGASEGHDVEAVGIGAAGYVDDKRATVLFAPNIDWRHEPLKDKVEQRVGLPVVVENDANAAAWGEYRFGAGQGHDDVICITLGTGLGGGIIIGNKLRRGRFGVAAEFGHIRVVPDGLLCGCGSQGCWEQYASGRALVRYAKQRANATPENAAVLLGLGDGSVDGIEGKHISEAARQGDPVAVDSFRELARWAGAGLADLASLFDPSAFIVGGGVSDEGELVLDPIRKSFRRWLIGGEWRPHAQVLAAQLGGKAGLVGAADLARQGLEHHHHHH
6ZOS , Knot 114 252 0.83 40 161 241 
GSHMALSLTADQMVSALLDAEPPILYSEYDPTRPFSEASMMGLLTNLADRELVHMINWAKRVPGFVDLTLHDQVHLLESAWLEILMIGLVWRSMEHPGKLLFAPNLLLDRNQGKSVEGMVEIFDMLLATSSRFRMMNLQGEEFVCLKSIILLNSGVYTFLSSTLKSLEEKDHIHRVLDKITDTLIHLMAKAGLTLQQQHQRLAQLLLILSHIRHMSNKGMEHLYSMKSKNVVPSYDLLLEMLDAHRLHAPTS

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 \(c\).

\(|P_{f(4DGJ_1)}(2) \setminus P_{f(3VGL_1)}(2)|=84\), \(|P_{f(3VGL_1)}(2) \setminus P_{f(4DGJ_1)}(2)|=81\). 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 1:1111001001111111110000011011011000111011001010010100101111101000100100110110011101000000000011110101010000010110110000111110000111110110010010110010111100000000110001000110110001110000100111110000001111110011000111001110101001001100110
Pair \(Z_2\) Length of longest common subsequence
4DGJ_1,3VGL_1 165 4
4DGJ_1,6ZOS_1 185 3
3VGL_1,6ZOS_1 156 5

Newick tree

 
[
	4DGJ_1:90.62,
	[
		3VGL_1:78,6ZOS_1:78
	]:12.62
]

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 \((0.85)(0.8)(\frac{556 }{\log_{20} 556}-\frac{235}{\log_{20}235})=91.5\)
Status Protein1 Protein2 d d1/2
Query variables 4DGJ_1 3VGL_1 110 99
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]