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Parikh vectors
2PNY_1 5AHM_1 3SAM_1 Letter Amino acid
16 25 56 I Isoleucine
4 15 10 M Methionine
6 9 36 Y Tyrosine
20 21 18 R Arginine
3 2 4 C Cysteine
8 17 14 Q Glutamine
15 14 14 H Histidine
7 7 26 F Phenylalanine
11 14 27 P Proline
10 54 18 A Alanine
14 16 40 D Aspartic acid
22 23 25 E Glutamic acid
27 24 33 L Leucine
16 26 55 S Serine
11 20 42 T Threonine
6 1 2 W Tryptophan
10 5 44 N Asparagine
11 51 31 G Glycine
12 18 32 K Lycine
17 38 49 V Valine 

2PNY_1|Chain A|Isopentenyl-diphosphate Delta-isomerase 2|Homo sapiens (9606)
>5AHM_1|Chains A, B|INOSINE-5'-MONOPHOSPHATE DEHYDROGENASE|PSEUDOMONAS AERUGINOSA PAO1 (208964)
>3SAM_1|Chains A, B, C|Rifampicin resistance protein|Vaccinia virus (10254)
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 \(w=f(c)\)
2PNY , Knot 111 246 0.82 40 172 235 
MGSSHHHHHHSSGLVPRGSMSDINLDWVDRRQLQRLEEMLIVVDENDKVIGADTKRNCHLNENIEKGLLHRAFSVVLFNTKNRILIQQRSDTKVTFPGYFTDSCSSHPLYNPAELEEKDAIGVRRAAQRRLQAELGIPGEQISPEDIVFMTIYHHKAKSDRIWGEHEICYLLLVRKNVTLNPDPSETKSILYLSQEELWELLEREARGEVKVTPWLRTIAERFLYRWWPHLDDVTPFVELHKIHRV
5AHM , Knot 160 400 0.80 40 196 363 
MGSSHHHHHHSSGLVPRGSHMLRISQEALTFDDVLLIPGYSEVLPKDVSLKTRLTRGIELNIPLVSAAMDTVTEARLAIAMAQEGGIGIIHKNMGIEQQAAEVRKVKKHETAKAKTYPLASKDEQGRLRVGAAVGTGADTGERVAALVAAGVDVVVVDTAHGHSKGVIERVRWVKQTFPDVQVIGGNIATAEAAKALAEAGADAVKVGIGPGSICTTRIVAGVGVPQISAIANVAAALEGTGVPLIADGGIRFSGDLAKAMVAGAYCVMMGSMFAGTEEAPGEIELFQGRSYKSYRGMGSLGAMSGSQGSSDRYFQDASAGAEKLVPEGIEGRVPYKGALSAIVHQLMGGLRAAMGYTGSADIQQMRTQPQFVRITGAGMAESHVHDVQITKEAPNYRVG
3SAM , Knot 228 576 0.83 40 254 533 
MSYYHHHHHHDYDIPTTENLYFQGAMNNTIINSLIGGDDSIKRSNVFAVDSQIPTLYMPQYISLSGVMTNDGPDNQAIASFEIRDQYITALNHLVLSLELPEVKGMGRFGYVPYVGYKCINHVSISSCNGVIWEIEGEELYNNCINNTIALKHSGYSSELNDISIGLTPNDTIKEPSTVYVYIKTPFDVEDTFSSLKLSDSKITVTVTFNPVSDIVIRDSSFDFETFNKEFVYVPELSFIGYMVKNVQIKPSFIEKPRRVIGQINQPTATVTEVHAATSLSVYTKPYYGNTDNKFISYPGYSQDEKDYIDAYVSRLLDDLVIVSDGPPTGYPESAEIVEVPEDGIVSIQDADVYVKIDNVPDNMSVYLHTNLLMFGTRKNSFIYNISKKFSAITGTYSDATKRTIFAHISHSINIIDTSIPVSLWTSQRNVYNGDNRSAESKAKDLFINDPFIKGIDFKNKTDIISRLEVRFGNDVLYSENGPISRIYNELLTKSNNGTRTLTFNFTPKIFFRPTTITANVSRGKDKLSVRVVYSTMGVNHPIYYVQKQLVVVCNDLYKVSYDQGVSITKIMGDNN

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 \(c\).

\(|P_{f(2PNY_1)}(2) \setminus P_{f(5AHM_1)}(2)|=65\), \(|P_{f(5AHM_1)}(2) \setminus P_{f(2PNY_1)}(2)|=89\). 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 1:110000000000111101010010101100001001001111100000111100000001000100111001101111000001110000000101110100000001100110100001111001100010101111100101001111010000100001110001001111000101010100000110100001101100010101010111001100110011101001011101001001
Pair \(Z_2\) Length of longest common subsequence
2PNY_1,5AHM_1 154 19
2PNY_1,3SAM_1 172 6
5AHM_1,3SAM_1 148 6

Newick tree

 
[
	2PNY_1:84.01,
	[
		5AHM_1:74,3SAM_1:74
	]:10.01
]

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 \((0.85)(0.8)(\frac{646 }{\log_{20} 646}-\frac{246}{\log_{20}246})=112.\)
Status Protein1 Protein2 d d1/2
Query variables 2PNY_1 5AHM_1 131 108
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]