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Parikh vectors
1QAM_1 8AJJ_1 5LDK_1 Letter Amino acid
8 14 9 Q Glutamine
9 23 5 H Histidine
30 24 4 K Lycine
19 15 10 F Phenylalanine
9 18 3 Y Tyrosine
6 32 20 A Alanine
26 37 8 I Isoleucine
5 15 14 P Proline
17 24 11 S Serine
9 32 10 T Threonine
11 31 6 V Valine
13 32 5 D Aspartic acid
6 32 11 G Glycine
5 6 3 M Methionine
1 0 5 W Tryptophan
12 21 9 E Glutamic acid
26 36 4 N Asparagine
2 2 1 C Cysteine
21 40 21 L Leucine
9 12 18 R Arginine 

1QAM_1|Chain A|ERMC' METHYLTRANSFERASE|Bacillus subtilis (1423)
>8AJJ_1|Chains A, B, C, D|Dihydrolipoamide dehydrogenase|Staphylococcus aureus (1280)
>5LDK_1|Chains A, B|RNA 2',3'-cyclic phosphodiesterase|Escherichia coli (strain B / BL21-DE3) (469008)
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 \(w=f(c)\)
1QAM , Knot 110 244 0.82 40 159 234 
MNEKNIKHSQNFITSKHNIDKIMTNIRLNEHDNIFEIGSGKGHFTLELVQRCNFVTAIEIDHKLCKTTENKLVDHDNFQVLNKDILQFKFPKNQSYKIFGNIPYNISTDIIRKIVFDSIADEIYLIVEYGFAKRLLNTKRSLALFLMAEVDISILSMVPREYFHPKPKVNSSLIRLNRKKSRISHKDKQKYNYFVMKWVNKEYKKIFTKNQFNNSLKHAGIDDLNNISFEQFLSLFNSYKLFNK
8AJJ , Knot 181 446 0.82 38 218 413 
MGTYDLIVIGFGKAGKTLAKYAASTGQHVAVIEQSPKMYGGTCINIGCIPSKTLVHDGLEGKSFEASYNRKNDVVNALNNKNYHLLADDNNIDVLDFKAQFKSNTEVNLLDQHDDIVDSITAPHIIINTGATSVIPNIKGLDQAKHVFDSTGLLNISYQPKHLVIVGGGYIALEFASMFANLGSKVTVLERGESFMPREDQDVVAYGITDLENKGIALHTNVETTELSSDNHHTTVHTNVDNFEADAVLLAIGRKPNTDLALENTDIELGDRGEIKVNAHLQTTVPHIYAAGDVKGGLQFTYISLDDYRIIKSALYGNQSRTTDNRGSVPYTVFIDPPLSRVGLTSKEAAAQHYDYTEHQLLVSAIPRHKINNDPRGLFKVVINNENNMILGATLYGKQSEELINIIKLAIDQNIPYTVLRDNIYTHPTMAESFNDLFNFHHHHHH
5LDK , Knot 85 177 0.82 40 132 168 
GMSEPQRLFFAIDLPAEIREQIIHWRAKHFPPEAGRPVAADNLHLTLAFLGEVSAEKEKALSLLAGRIRQPGFTLTLDDAGQWLRSRVVWLGMRQPPRGLIQLANMLRSQAARSGCFQSNRPFHPHITLLRDASEAVTIPPPGFNWSYAVTEFTLYASSFARGRTRYTPLKRWALTQ

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 \(c\).

\(|P_{f(1QAM_1)}(2) \setminus P_{f(8AJJ_1)}(2)|=48\), \(|P_{f(8AJJ_1)}(2) \setminus P_{f(1QAM_1)}(2)|=107\). 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 1:1000010000011000001001100101000001101101010101011000011011010001000000011000010110001101011000000111011001000110011100110010111001110011000001111111010101101110001010101000110100000010000000000111011000000110000100010011100100101001101100001100
Pair \(Z_2\) Length of longest common subsequence
1QAM_1,8AJJ_1 155 4
1QAM_1,5LDK_1 181 3
8AJJ_1,5LDK_1 184 4

Newick tree

 
[
	5LDK_1:95.39,
	[
		1QAM_1:77.5,8AJJ_1:77.5
	]:17.89
]

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 \((0.85)(0.8)(\frac{690 }{\log_{20} 690}-\frac{244}{\log_{20}244})=124.\)
Status Protein1 Protein2 d d1/2
Query variables 1QAM_1 8AJJ_1 154 118
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]