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Parikh vectors
5OVQ_1 3MDX_1 4CEX_1 Letter Amino acid
7 7 2 Q Glutamine
4 7 3 H Histidine
15 4 3 F Phenylalanine
21 8 7 V Valine
19 20 9 A Alanine
3 1 0 C Cysteine
11 16 11 L Leucine
15 5 7 K Lycine
5 5 6 M Methionine
5 0 0 W Tryptophan
9 13 5 R Arginine
18 11 10 E Glutamic acid
17 13 6 T Threonine
10 21 6 G Glycine
15 14 7 I Isoleucine
17 10 5 P Proline
6 9 3 S Serine
7 3 1 Y Tyrosine
7 3 3 N Asparagine
12 8 6 D Aspartic acid 

5OVQ_1|Chains A, B, C, E, K|Peroxiredoxin|Aquifex aeolicus (strain VF5) (224324)
>3MDX_1|Chain A|Deoxyuridine 5'-triphosphate nucleotidohydrolase|Brucella melitensis (359391)
>4CEX_1|Chain A|UREASE SUBUNIT GAMMA|SPOROSARCINA PASTEURII (1474)
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 \(w=f(c)\)
5OVQ , Knot 101 223 0.81 40 146 215 
MEVVSLPRLGEPAPAFEAQTTFGPVKFPDDFKGQWVVLFSHPADFTPVCTTEFVAFAKNYEEFKKRNVQLIGLSVDSNFSHIAWVMNIKEKFGIEIPFPIIADHNMEVAKKYGMIHPAQSTTFTVRALFVIDDKGILRAMIYYPLTTGRNIREVIRLVDALQTADREGVATPADWVPEPQTWEFTEENTKVIVPPPTTYEDAVKRLQEGYECADWYICKKKVA
3MDX , Knot 82 178 0.79 38 125 171 
MAHHHHHHMGTLEAQTQGPGSMTAASSSAPTLGIIRLEHAKGLDLPAYETAGSAGMDLRAAVAEDRQIVLLPGRRTLVPTGLILEIPQGYEVQIRPRSGLAFKNGITCLNTPGTIDSDYRGEVKVLLINLGDDDFRIERGMRIAQAVFAPVIQPKIEERAKISETARGAGGFGSTGTA
4CEX , Knot 51 100 0.78 36 79 97 
MHLNPAEKEKLQIFLASELLLRRKARGLKLNYPEAVAIITSFIMEGARDGKTVAMLMEEGKHVLTRDDVMEGVPEMIDDIQAEATFPDGTKLVTVHNPIS

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 \(c\).

\(|P_{f(5OVQ_1)}(2) \setminus P_{f(3MDX_1)}(2)|=86\), \(|P_{f(3MDX_1)}(2) \setminus P_{f(5OVQ_1)}(2)|=65\). 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 1:1011011011011111010001111011001010111110011010110000111110000010000101111010001001111101000111011111110001011000111011000010101111100011101110011001001001101101100100011101101110100101000000111111000001100100100010101000011
Pair \(Z_2\) Length of longest common subsequence
5OVQ_1,3MDX_1 151 4
5OVQ_1,4CEX_1 157 3
3MDX_1,4CEX_1 146 3

Newick tree

 
[
	5OVQ_1:78.30,
	[
		3MDX_1:73,4CEX_1:73
	]:5.30
]

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 \((0.85)(0.8)(\frac{401 }{\log_{20} 401}-\frac{178}{\log_{20}178})=66.3\)
Status Protein1 Protein2 d d1/2
Query variables 5OVQ_1 3MDX_1 85 74
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]