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Parikh vectors
3TKB_1 6OVI_1 4RTO_1 Letter Amino acid
9 4 20 R Arginine
8 8 15 N Asparagine
3 5 5 C Cysteine
17 10 18 K Lycine
8 9 10 T Threonine
8 3 13 D Aspartic acid
7 3 2 W Tryptophan
7 3 20 Y Tyrosine
17 18 20 V Valine
10 25 25 A Alanine
12 7 7 Q Glutamine
12 17 19 E Glutamic acid
7 18 9 I Isoleucine
13 11 17 F Phenylalanine
15 16 15 P Proline
19 20 16 G Glycine
13 10 14 H Histidine
20 21 27 L Leucine
3 6 5 M Methionine
15 14 21 S Serine 

3TKB_1|Chain A|Uracil-DNA glycosylase|Homo sapiens (9606)
>6OVI_1|Chains A, B, C|Keto-deoxy-phosphogluconate aldolase|Legionella pneumophila (446)
>4RTO_1|Chain A|DNA adenine methylase|Escherichia coli (1110693)
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 \(w=f(c)\)
3TKB , Knot 103 223 0.83 40 158 216 
MEFFGESWKKHLSGEFGKPYFIKLMGFVAEERKHYTVYPPPHQVFTWTQMCDIKDVKVVILGQDPYHGPNQAHGLCFSVQRPVPPPPSLENIYKELSTDIEGFVHPGHGDLSGWAKQGVLLLNAVLTVRAHQANSHKERGWEQFTDAVVSWLNQNSNGLVFLLWGSYAQKKGSAIDRKRHHVLQTAHPSPLSVYRGFFGCRHFSKTNELLQKSGKKPIDWREL
6OVI , Knot 100 228 0.79 40 141 210 
MAHHHHHHMSFSEWKTQPGAVFAASPVIPVIVIKELEDALPLAEALFAGGIHVLEVTLRTPVAIKALELLINTFPDELIGAGTVITPGQFHDVVAAGARFAISPGQTRELLIAGQKSEIPLIPGVASVSELMEGLGMGYNHFKFFPAAAAGGIPMLKAISGVFPQVKFCPTGGINSKNYEEYLCLPNVACVGGSWIVPEEAIKNHNWSLITELCMAVSSQKRELCSQN
4RTO , Knot 135 298 0.86 40 199 289 
MGSSHHHHHHSSGLVPRGSHMKKNRAFLKWAGGKYPLLDDIKRHLPKGECLVEPFVGAGSVFLNTDFSRYILADINSDLISLYNIVKMRTDEYVQAARELFVPETNCAEVYYQFREEFNKSQDPFRRAVLFLYLNRYGYNGLCRYNLRGEFNVPFGRYKKPYFPEAELYHFAEKAQNAFFYCESYADSMARADDSSVVYCDPPYAPLSATANFTAYHTNSFTLEQQAHLAEIAEGLVERHIPVLISNHDTMLTREWYQRAKLHVVKVRRSISSNGGTRKKVDELLALYKPGVVSPAKK

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 \(c\).

\(|P_{f(3TKB_1)}(2) \setminus P_{f(6OVI_1)}(2)|=100\), \(|P_{f(6OVI_1)}(2) \setminus P_{f(3TKB_1)}(2)|=83\). 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 1:1011100100010101101011011111100000001011100110100100100101111100100110010110101001111110100100010001011101101010111001111101110101001000000110010011101100000111111110010001011000000110010101101001111000100000110001001101001
Pair \(Z_2\) Length of longest common subsequence
3TKB_1,6OVI_1 183 3
3TKB_1,4RTO_1 167 4
6OVI_1,4RTO_1 168 6

Newick tree

 
[
	6OVI_1:89.22,
	[
		3TKB_1:83.5,4RTO_1:83.5
	]:5.72
]

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 \((0.85)(0.8)(\frac{451 }{\log_{20} 451}-\frac{223}{\log_{20}223})=66.3\)
Status Protein1 Protein2 d d1/2
Query variables 3TKB_1 6OVI_1 84 84
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]