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Parikh vectors
5ABA_1 6TBH_1 1KJQ_1 Letter Amino acid
9 36 18 D Aspartic acid
9 13 9 H Histidine
8 33 11 K Lycine
18 34 17 P Proline
16 27 16 T Threonine
9 52 45 A Alanine
5 23 13 F Phenylalanine
1 15 2 W Tryptophan
8 23 8 Y Tyrosine
5 15 8 M Methionine
9 29 6 N Asparagine
19 24 27 R Arginine
7 26 21 Q Glutamine
13 32 30 E Glutamic acid
15 37 35 G Glycine
6 18 21 I Isoleucine
17 47 38 L Leucine
20 23 20 S Serine
14 42 41 V Valine
11 1 5 C Cysteine 

5ABA_1|Chains A, B|CELLULAR TUMOR ANTIGEN P53|HOMO SAPIENS (9606)
>6TBH_1|Chains A, B, C, D, E, F, G, H|Beta-galactosidase, putative, bgl35A|Cellvibrio japonicus Ueda107 (498211)
>1KJQ_1|Chains A, B|phosphoribosylglycinamide formyltransferase 2|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 \(w=f(c)\)
5ABA , Knot 102 219 0.83 40 150 211 
SSSVPSQKTYQGSYGFRLGFLHSGTAKSVTCTYSPALNKLFCQLAKTCPVQLWVDSTPPPGTRVRAMAIYKQSQHMTEVVRRCPHHERCSDSDGLAPPQHLIRVEGNLRAEYLDDRNTFRHSVVVPCEPPEVGSDCTTIHYNYMCYSSCMGGMNRRPILTIITLEDSSGNLLGRDSFEVRVCACPGRDRRTEEENLRKKGEPHHELPPGSTKRALPNNT
6TBH , Knot 224 550 0.85 40 269 527 
MGSSHHHHHHAAPLPELLSNNGKHALMVDGAPYIILGSQTNNSSNYPDALKDVWPSMEKMGANTLSIPVAWEQIEPVEGQFDFSFVDVLLKEARQRKVRLVLLWFATWKNNAPHYAPAWVKLDNARFPRVVKEDGDTLNSLSPLGQNTLAADKKAFVELMKYLAKRDKDHTVIMVQVQNEVGTYGAVRDYSPMAQAVFNAAVPDDLIQKLQLKPGTWSQVFGRDADEFFHAYQIARYCDEVTVAGKAIKNLPMYVNVALRNPFNPGLPGQYSSGGGTDNVLHIWKAAAPNIDLIAPDIYFRDYKTVSKVLELYTRPDNALFVAEIGNDQPFARYLFPTLGKGGIGFSPFGMDDTDYTNYPLGAKVYNDETIEQFAQVYRLVNPMMREWARLSYQGQVWGVAEPLDSTTETQKIWNAEATPEEKEQHKKDRASALTQQLDLGLWDAEVTYGRPMFWVTPPEGNTPAAGGALIAQLDDNEYLVTAYKARVEFKPSQELAGKKFMIERVEEGRFEKGKWVMERVWNGDQTDWGLNFTDRPHLLRVKMASYSVQ
1KJQ , Knot 161 391 0.82 40 201 369 
TLLGTALRPAATRVMLLGSGELGKEVAIECQRLGVEVIAVDRYADAPAMHVAHRSHVINMLDGDALRRVVELEKPHYIVPEIEAIATDMLIQLEEEGLNVVPCARATKLTMNREGIRRLAAEELQLPTSTYRFADSESLFREAVADIGYPCIVKPVMSSSGKGQTFIRSAEQLAQAWKYAQQGGRAGAGRVIVEGVVKFDFEITLLTVSAVDGVHFCAPVGHRQEDGDYRESWQPQQMSPLALERAQEIARKVVLALGGYGLFGVELFVCGDEVIFSEVSPRPHDTGMVTLISQDLSEFALHVRAFLGLPVGGIRQYGPAASAVILPQLTSQNVTFDNVQNAVGADLQIRLFGKPEIDGSRRLGVALATAESVVDAIERAKHAAGQVKVQG

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 \(c\).

\(|P_{f(5ABA_1)}(2) \setminus P_{f(6TBH_1)}(2)|=43\), \(|P_{f(6TBH_1)}(2) \setminus P_{f(5ABA_1)}(2)|=162\). 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 1:000110000001001101111001010010000011100110011000110111000111100101111000000100110001000000000111110011010101010010000010001111001101100000100001000001111000111011010000101110001010101011000000000100010100011110000111000
Pair \(Z_2\) Length of longest common subsequence
5ABA_1,6TBH_1 205 3
5ABA_1,1KJQ_1 183 4
6TBH_1,1KJQ_1 166 4

Newick tree

 
[
	5ABA_1:10.43,
	[
		1KJQ_1:83,6TBH_1:83
	]:18.43
]

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 \((0.85)(0.8)(\frac{769 }{\log_{20} 769}-\frac{219}{\log_{20}219})=152.\)
Status Protein1 Protein2 d d1/2
Query variables 5ABA_1 6TBH_1 197 136
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]