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Parikh vectors
3JVM_1 6OPA_1 4WTK_1 Letter Amino acid
4 9 0 H Histidine
7 28 0 L Leucine
6 12 0 Y Tyrosine
6 34 0 V Valine
5 31 1 G Glycine
6 16 0 F Phenylalanine
9 17 0 D Aspartic acid
2 46 0 N Asparagine
4 19 2 C Cysteine
11 24 0 E Glutamic acid
5 33 0 I Isoleucine
5 22 0 R Arginine
3 20 0 Q Glutamine
12 28 0 K Lycine
8 9 0 M Methionine
7 22 0 P Proline
7 31 0 S Serine
1 43 0 T Threonine
1 9 0 W Tryptophan
11 22 1 A Alanine 

3JVM_1|Chain A|Bromodomain-containing protein 4|Mus musculus (10090)
>6OPA_1|Chain A[auth G]|Envelope glycoprotein gp160|Human immunodeficiency virus 1 (11676)
>4WTK_1|Chain A[auth T]|RNA TEMPLATE AGCC|synthetic construct (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 \(w=f(c)\)
3JVM , Knot 63 120 0.83 40 104 117 
GAMGSKISEQLKCCSGILKEMFAKKHAAYAWPFYKPVDVEALGLHDYCDIIKHPMDMSTIKSKLESREYRDAQEFGADVRLMFSNCYKYNPPDHEVVAMARKLQDVFEMRFAKMPDEPEE
6OPA , Knot 194 475 0.84 40 235 442 
AENLWVTVYYGVPVWKDAETTLFCASDAKAYETEKHNVWATHACVPTDPNPQEIHLENVTEEFNMWKNNMVEQMHTDIISLWDQSLKPCVKLTPLCVTLQCTNVTNNITDDMRGELKNCSFNMTTELRDKKQKVYSLFYRLDVVQINENQGNRSNNSNKEYRLINCNTSAITQACPKVSFEPIPIHYCAPAGFAILKCKDKKFNGTGPCPSVSTVQCTHGIKPVVSTQLLLNGSLAEEEVMIRSENITNNAKNILVQFNTPVQINCTRPNNNTRKSIRIGPGQAFYATGDIIGDIRQAHCNVSKATWNETLGKVVKQLRKHFGNNTIIRFANSSGGDLEVTTHSFNCGGEFFYCNTSGLFNSTWISNTSVQGSNSTGSNDSITLPCRIKQIINMWQRIGQAMYAPPIQGVIRCVSNITGLILTRDGGSTNSTTETFRPGGGDMRDNWRSELYKYKVVKIEPLGVAPTRCKRRVVG
4WTK , Knot 4 4 0.46 6 3 2 
AGCC

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 \(c\).

\(|P_{f(3JVM_1)}(2) \setminus P_{f(6OPA_1)}(2)|=41\), \(|P_{f(6OPA_1)}(2) \setminus P_{f(3JVM_1)}(2)|=172\). 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 1:111100100010000111001110001101111001101011110000011001101001000100000001001110101110000000110001111100100110101101100100
Pair \(Z_2\) Length of longest common subsequence
3JVM_1,6OPA_1 213 3
3JVM_1,4WTK_1 105 2
6OPA_1,4WTK_1 236 2

Newick tree

 
[
	6OPA_1:12.19,
	[
		3JVM_1:52.5,4WTK_1:52.5
	]:73.69
]

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 \((0.85)(0.8)(\frac{595 }{\log_{20} 595}-\frac{120}{\log_{20}120})=138.\)
Status Protein1 Protein2 d d1/2
Query variables 3JVM_1 6OPA_1 174 110.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]