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Parikh vectors
4CYO_1 5QOR_1 1THQ_1 Letter Amino acid
20 8 11 N Asparagine
15 2 5 Q Glutamine
34 13 11 L Leucine
21 10 10 F Phenylalanine
22 11 8 R Arginine
22 14 6 I Isoleucine
20 11 4 S Serine
16 8 11 T Threonine
7 3 12 W Tryptophan
27 11 10 E Glutamic acid
16 12 15 G Glycine
23 13 5 K Lycine
11 3 5 M Methionine
30 12 10 P Proline
32 5 5 V Valine
23 13 8 D Aspartic acid
4 3 0 C Cysteine
11 2 10 H Histidine
24 5 11 Y Tyrosine
33 8 13 A Alanine

4CYO_1|Chain A|GLYCYLPEPTIDE N-TETRADECANOYLTRANSFERASE|LEISHMANIA MAJOR (5664)
>5QOR_1|Chain A|DCP2 (NUDT20)|Homo sapiens (9606)
>1THQ_1|Chain A|CrcA protein|Escherichia coli (562)
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 \(w=f(c)\)
4CYO , Knot 177 411 0.86 40 245 397
AHAFWSTQPVPQTEDETEKIVFAGPMDEPKTVADIPEEPYPIASTFEWWTPNMEAADDIHAIYELLRDNYVEDDDSMFRFNYSEEFLQWALCPPNYIPDWHVAVRRKADKKLLAFIAGVPVTLRMGTPKYMKVKAQEKGEGEEAAKYDEPRHICEINFLCVHKQLREKRLAPILIKEATRRVNRTNVWQAVYTAGVLLPTPYASGQYFHRSLNPEKLVEIRFSGIPAQYQKFQNPMAMLKRNYQLPSAPKNSGLREMKPSDVPQVRRILMNYLDSFDVGPVFSDAEISHYLLPRDGVVFTYVVENDKKVTDFFSFYRIPSTVIGNSNYNLLNAAYVHYYAATSIPLHQLILDLLIVAHSRGFDVCNMVEILDNRSFVEQLKFGAGDGHLRYYFYNWAYPKIKPSQVALVML
5QOR , Knot 81 167 0.82 40 127 164
SMGVPTYGAIILDETLENVLLVQGYLAKSGWGFPKGKVNKEEAPHDCAAREVFEETGFDIKDYICKDDYIELRINDQLARLYIIPGIPKDTKFNPKTRREIRNIEWFSIEKLPCHRNDMTPKSKLGLAPNKFFMAIPFIRPLRDWLSRRFGDSSDSDNGFSSTGSTP
1THQ , Knot 79 170 0.79 38 120 160
MNADEWMTTFRENIAQTWQQPEHYDLYIPAITWHARFAYDKEKTDRYNERPWGGGFGLSRWDEKGNWHGLYAMAFKDSWNKWEPIAGYGWESTWRPLADENFHLGLGFTAGVTARDNWNYIPLPVLLPLASVGYGPVTFQMTYIPGTYNNGNVYFAWMRFQFLEHHHHHH

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 \(c\).

\(|P_{f(4CYO_1)}(2) \setminus P_{f(5QOR_1)}(2)|=162\), \(|P_{f(5QOR_1)}(2) \setminus P_{f(4CYO_1)}(2)|=44\). 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 1:101110001110000000011111110010011011001011100101101010110010110011000010000011010000011011101100110101110001000111111111101011010010101000101001100001001001011010001000011111100100010000110110011111101010100100010100110101011110000100111110000011011000110010100110100111001001011111001010001110011110011000001001101001100111000001101101000110011100111011111000110100110110000110010111101010001001101010100111111
Pair \(Z_2\) Length of longest common subsequence
4CYO_1,5QOR_1 206 3
4CYO_1,1THQ_1 213 4
5QOR_1,1THQ_1 157 3

Newick tree

 
[
	4CYO_1:11.16,
	[
		5QOR_1:78.5,1THQ_1:78.5
	]:33.66
]

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 \((0.85)(0.8)(\frac{578 }{\log_{20} 578}-\frac{167}{\log_{20}167})=118.\)
Status Protein1 Protein2 d d1/2
Query variables 4CYO_1 5QOR_1 156 107
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]