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Parikh vectors
3JSO_1 4DKO_1 1TWL_1 Letter Amino acid
13 30 17 I Isoleucine
9 17 15 P Proline
17 14 10 A Alanine
17 28 11 G Glycine
5 9 8 H Histidine
22 20 12 L Leucine
9 19 4 S Serine
1 7 11 Y Tyrosine
19 21 12 V Valine
12 13 18 D Aspartic acid
16 21 18 E Glutamic acid
6 36 2 N Asparagine
6 29 3 T Threonine
13 15 1 Q Glutamine
10 26 19 K Lycine
5 7 4 M Methionine
6 11 10 F Phenylalanine
1 7 3 W Tryptophan
15 7 8 R Arginine
0 16 0 C Cysteine 

3JSO_1|Chains A, B|LexA repressor|Escherichia coli K-12 (83333)
>4DKO_1|Chains A, B[auth C]|HIV-1 gp120 core|HUMAN IMMUNODEFICIENCY VIRUS TYPE 1 (11686)
>1TWL_1|Chain A|Inorganic pyrophosphatase|Pyrococcus furiosus (2261)
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 \(w=f(c)\)
3JSO , Knot 95 202 0.83 38 137 187 
MKALTARQQEVFDLIRDHISQTGMPPTRAEIAQRLGFRSPNAAEEHLKALARKGVIEIVSGASRGIRLLQEEEEGLPLVGRVAAGEPLLAQQHIEGHYQVDPSLFKPNADFLLRVSGMSMKDIGIMDGDLLAVHKTQDVRNGQVVVARIDDEVTVARLKKQGNKVELLPENSEFKPIVVDLRQQSFTIEGLAVGVIRNGDWL
4DKO , Knot 149 353 0.82 40 203 330 
VWKDADTTLFCASDAKAHETEVHNVWATHACVPTDPNPQEIHLENVTENFNMWKNNMVEQMQEDVISLWDQSLQPCVKLTGGSVIKQACPKISFDPIPIHYCTPAGYVILKCNDKNFNGTGPCKNVSSVQCTHGIKPVVSTQLLLNGSLAEEEIIIRSENLTNNAKTIIVHLNKSVEINCTRPSNGGSGSGGDIRKAYCEINGTKWNKVLKQVTEKLKEHFNNKTIIFQPPSGGDLEITMHSFNCRGEFFYCNTTQLFNNTCIGNETMKGCNGTITLPCKIKQIINMWQGTGQAMYAPPIDGKINCVSNITGILLTRDGGANNTSNETFRPGGGNIKDNWRSELYKYKVVQIE
1TWL , Knot 84 186 0.78 38 126 178 
AHHHHHHGSNPFHDLEPGPDVPEVVYAIIEIPKGSRNKYELDKKTGLLKLDRVLYSPFFYPVDYGIIPRTWYEDDDPFDIMVIMREPVYPLTIIEARPIGLFKMIDSGDKDYKVLAVPVEDPYFKDWKDIDDVPKAFLDEIAHFFKRYKELQGKEIIVEGWEGAEAAKREILRAIEMYKEKFGKKE

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 \(c\).

\(|P_{f(3JSO_1)}(2) \setminus P_{f(4DKO_1)}(2)|=60\), \(|P_{f(4DKO_1)}(2) \setminus P_{f(3JSO_1)}(2)|=126\). 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 1:1011010000110110001000111100101100111001011000101110011101101100110110000011111101111011110001010001010110101011101011010011110101111000001001011110100010110100010010111000010111101000010101111111001011
Pair \(Z_2\) Length of longest common subsequence
3JSO_1,4DKO_1 186 3
3JSO_1,1TWL_1 145 3
4DKO_1,1TWL_1 191 3

Newick tree

 
[
	4DKO_1:10.46,
	[
		3JSO_1:72.5,1TWL_1:72.5
	]:27.96
]

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 \((0.85)(0.8)(\frac{555 }{\log_{20} 555}-\frac{202}{\log_{20}202})=101.\)
Status Protein1 Protein2 d d1/2
Query variables 3JSO_1 4DKO_1 131 101.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]