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Parikh vectors
3LGN_1 1JPR_1 5RTZ_1 Letter Amino acid
9 16 13 K Lycine
3 8 2 M Methionine
4 17 2 R Arginine
8 25 9 D Aspartic acid
8 23 3 Q Glutamine
11 32 6 E Glutamic acid
7 37 19 L Leucine
7 17 5 F Phenylalanine
7 20 6 T Threonine
5 25 22 V Valine
4 22 15 A Alanine
0 5 3 C Cysteine
7 26 11 S Serine
4 16 7 Y Tyrosine
1 13 5 P Proline
2 7 0 W Tryptophan
8 17 16 N Asparagine
5 13 13 G Glycine
3 8 6 H Histidine
7 28 6 I Isoleucine 

3LGN_1|Chains A, B|Heme-degrading monooxygenase isdI|Staphylococcus aureus (158879)
>1JPR_1|Chains A, B|Protein R2 of Ribonucleotide reductase|Escherichia coli (562)
>5RTZ_1|Chains A, B|Non-structural protein 3|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
3LGN , Knot 55 110 0.78 38 89 105 
AHMFMAENRLQLQKGSAEETIERFYNRQGIETIEGFQQMFVTKTLNTEDTDEVKILTIWESEDSFNNWLNSDVFKEAHKNVRLKSDDDGQQSPILSNKVFKYDIGYHYQK
1JPR , Knot 160 375 0.84 40 222 365 
AYTTFSQTKNDQLKEPMFFGQPVNVARYDQQKYDIFEKLIEKQLSFFWRPEEVDVSRDRIDYQALPEHEKHIFISNLKYQTLLDSIQGRSPNVALLPLISIPELETWVETWAFSETIHSRSYTHIIRNIVNDPSVVFDDIVTNEQIQKRAEGISSYYDELIEMTSYWHLLGEGTHTVNGKTVTVSLRELKKKLYLCLMSVNALEAIRFYVSFACSFAFAERELMEGNAKIIRLIARDEALHLTGTQHMLNLLRSGADDPEMAEIAEECKQECYDLFVQAAQQEKDWADYLFRDGSMIGLNKDILCQYVEYITNIRMQAVGLDLPFQTRSNPIPWINTWLVSDNVQVAPQEVEVSSYLVGQIDSEVDTDDLSNFQL
5RTZ , Knot 80 169 0.81 38 119 162 
SMVNSFSGYLKLTDNVYIKNADIVEEAKKVKPTVVVNAANVYLKHGGGVAGALNKATNNAMQVESDDYIATNGPLKVGGSCVLSGHNLAKHCLHVVGPNVNKGEDIQLLKSAYENFNQHEVLLAPLLSAGIFGADPIHSLRVCVDTVRTNVYLAVFDKNLYDKLVSSFL

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 \(c\).

\(|P_{f(3LGN_1)}(2) \setminus P_{f(1JPR_1)}(2)|=30\), \(|P_{f(1JPR_1)}(2) \setminus P_{f(3LGN_1)}(2)|=163\). 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 1:10111100010100101000100100001100101100111000100000001011011000001001100011001000101000001000111000110001100000
Pair \(Z_2\) Length of longest common subsequence
3LGN_1,1JPR_1 193 4
3LGN_1,5RTZ_1 138 3
1JPR_1,5RTZ_1 187 3

Newick tree

 
[
	1JPR_1:10.22,
	[
		3LGN_1:69,5RTZ_1:69
	]:33.22
]

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 \((0.85)(0.8)(\frac{485 }{\log_{20} 485}-\frac{110}{\log_{20}110})=112.\)
Status Protein1 Protein2 d d1/2
Query variables 3LGN_1 1JPR_1 141 89.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]