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Parikh vectors
5EGN_1 4IDT_1 4CYN_1 Letter Amino acid
7 17 23 K Lycine
14 24 20 S Serine
11 22 22 R Arginine
16 14 11 H Histidine
21 41 34 L Leucine
15 8 21 F Phenylalanine
2 6 7 W Tryptophan
27 28 33 A Alanine
15 6 20 N Asparagine
16 8 22 I Isoleucine
6 8 11 M Methionine
15 24 30 P Proline
7 9 24 Y Tyrosine
25 25 32 V Valine
0 11 4 C Cysteine
8 15 15 Q Glutamine
26 35 16 G Glycine
9 12 16 T Threonine
16 13 23 D Aspartic acid
15 30 27 E Glutamic acid 

5EGN_1|Chains A, B, C, D, E, F, G, H|Esterase|uncultured bacterium (77133)
>4IDT_1|Chains A, B|Mitogen-activated protein kinase kinase kinase 14|Homo sapiens (9606)
>4CYN_1|Chain A|GLYCYLPEPTIDE N-TETRADECANOYLTRANSFERASE|LEISHMANIA MAJOR (5664)
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 \(w=f(c)\)
5EGN , Knot 128 271 0.88 38 177 261 
MPHVENDGVKIYYDSYGEGVPIVFLHPFSTNGGIWYFQTFPFAQTNHVIVIDHRGHGRSDKPATGYSIMEHADDVVAVLDALKVDRAVFVGNSIGGMIAMQLNLDHPQRVIGNLILSSGTGLGEGMPPEAGAAFQNDYIGAFGGLLEGAVSARSKRERPEILAVMKAHFSVPSNFPKHVFDAATADPNGVFAWNIKDRLSSIQAPTLVVAGEEDLVTTVANNQLLADNIPGAELRVINDVGHFYQLERPSEFNELLRGFVAAALEHHHHHH
4IDT , Knot 150 356 0.82 40 196 338 
GAMGSKFSVEEYLVHALQGSVSSGQAHSLTSLAKTWAARGSRSREPSPKTEDNEGVLLTEKLKPVDYEYREEVHWATHQLRLGRGSFGEVHRMEDKQTGFQCAVKKVRLEVFRAEELMACAGLTSPRIVPLYGAVREGPWVNIFMELLEGGSLGQLVKEQGCLPEDRALYYLGQALEGLEYLHSRRILHGDVKADNVLLSSDGSHAALCDFGHAVCLQPDGLGKDLLTGDYIPGTETHMAPEVVLGRSCDAKVDVWSSCCMMLHMLNGCHPWTQFFRGPLCLKIASEPPPVREIPPSCAPLTAQAIQEGLRKEPIHRVSAAELGGKVNRALQQVGGLKSPWRGEYKEPRHPPPNQA
4CYN , Knot 177 411 0.86 40 245 397 
AHAFWSTQPVPQTEDETEKIVFAGPMDEPKTVADIPEEPYPIASTFEWWTPNMEAADDIHAIYELLRDNYVEDDDSMFRFNYSEEFLQWALCPPNYIPDWHVAVRRKADKKLLAFIAGVPVTLRMGTPKYMKVKAQEKGEGEEAAKYDEPRHICEINFLCVHKQLREKRLAPILIKEATRRVNRTNVWQAVYTAGVLLPTPYASGQYFHRSLNPEKLVEIRFSGIPAQYQKFQNPMAMLKRNYQLPSAPKNSGLREMKPSDVPQVRRILMNYLDSFDVGPVFSDAEISHYLLPRDGVVFTYVVENDKKVTDFFSFYRIPSTVIGNSNYNLLNAAYVHYYAATSIPLHQLILDLLIVAHSRGFDVCNMVEILDNRSFVEQLKFGAGDGHLRYYFYNWAYPKIKPSQVALVML

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 \(c\).

\(|P_{f(5EGN_1)}(2) \setminus P_{f(4IDT_1)}(2)|=74\), \(|P_{f(4IDT_1)}(2) \setminus P_{f(5EGN_1)}(2)|=93\). 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 1:1101000110100000101111111011000111101001111000011110001010000110100110010011111011010011111001111111010100100111011100101110111101111100001111111101110100000010111110101011001100110110101011111010001001011011111000110011000111001111010110011010010010010011011111110000000
Pair \(Z_2\) Length of longest common subsequence
5EGN_1,4IDT_1 167 4
5EGN_1,4CYN_1 174 4
4IDT_1,4CYN_1 163 4

Newick tree

 
[
	5EGN_1:86.48,
	[
		4IDT_1:81.5,4CYN_1:81.5
	]:4.98
]

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 \((0.85)(0.8)(\frac{627 }{\log_{20} 627}-\frac{271}{\log_{20}271})=99.7\)
Status Protein1 Protein2 d d1/2
Query variables 5EGN_1 4IDT_1 127 111
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]