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Parikh vectors
7HNW_1 4IDS_1 2AFH_1 Letter Amino acid
8 76 31 A Alanine
10 15 15 N Asparagine
10 41 30 D Aspartic acid
14 47 46 G Glycine
14 43 34 V Valine
6 9 16 M Methionine
11 22 19 F Phenylalanine
9 31 20 P Proline
6 9 9 W Tryptophan
9 37 25 R Arginine
10 22 11 H Histidine
9 28 36 I Isoleucine
12 46 28 L Leucine
7 13 38 K Lycine
10 33 20 T Threonine
7 18 21 Y Tyrosine
4 2 9 C Cysteine
10 17 11 Q Glutamine
7 23 40 E Glutamic acid
15 31 32 S Serine 

7HNW_1|Chain A[auth B]|E3 ubiquitin-protein ligase TRIM21|Mus musculus (10090)
>4IDS_1|Chain A|Delta-1-pyrroline-5-carboxylate dehydrogenase|Mycobacterium tuberculosis (83332)
>2AFH_1|Chains A, C|Nitrogenase molybdenum-iron protein|Azotobacter vinelandii (354)
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 \(w=f(c)\)
7HNW , Knot 91 188 0.84 40 148 179 
MHHHHHHMVHITLDRNTANSWLIISKDRRQVRMGDTHQNVSDNKERFSNYPMVLGAQRFSSGKMYWEVDVTQKEAWDLGVCRDSVQRKGQFSLSPENGFWTIWLWQDSYEAGTSPQTTLHIQVPPCQIGIFVDYEAGVVSFYNITDHGSLIYTFSECVFAGPLRPFFNVGFNYSGGNAAPLKLCPLKM
4IDS , Knot 226 563 0.84 40 246 518 
MGSSHHHHHHSSGLVPRGSHMDAITQVPVPANEPVHDYAPKSPERTRLRTELASLADHPIDLPHVIGGRHRMGDGERIDVVQPHRHAARLGTLTNATHADAAAAVEAAMSAKSDWAALPFDERAAVFLRAADLLAGPWREKIAAATMLGQSKSVYQAEIDAVCELIDFWRFNVAFARQILEQQPISGPGEWNRIDYRPLDGFVYAITPFNFTSIAGNLPTAPALMGNTVIWKPSITQTLAAYLTMQLLEAAGLPPGVINLVTGDGFAVSDVALADPRLAGIHFTGSTATFGHLWQWVGTNIGRYHSYPRLVGETGGKDFVVAHASARPDVLRTALIRGAFDYQGQKCSAVSRAFIAHSVWQRMGDELLAKAAELRYGDITDLSNYGGALIDQRAFVKNVDAIERAKGAAAVTVAVGGEYDDSEGYFVRPTVLLSDDPTDESFVIEYFGPLLSVHVYPDERYEQILDVIDTGSRYALTGAVIADDRQAVLTALDRLRFAAGNFYVNDKPTGAVVGRQPFGGARGSDTNDKAGSPLNLLRWTSARSIKETFVAATDHIYPHMAVD
2AFH , Knot 201 491 0.84 40 253 465 
TGMSREEVESLIQEVLEVYPEKARKDRNKHLAVNDPAVTQSKKCIISNKKSQPGLMTIRGCAYAGSKGVVWGPIKDMIHISHGPVGCGQYSRAGRRNYYIGTTGVNAFVTMNFTSDFQEKDIVFGGDKKLAKLIDEVETLFPLNKGISVQSECPIGLIGDDIESVSKVKGAELSKTIVPVRCEGFRGVSQSLGHHIANDAVRDWVLGKRDEDTTFASTPYDVAIIGDYNIGGDAWSSRILLEEMGLRCVAQWSGDGSISEIELTPKVKLNLVHCYRSMNYISRHMEEKYGIPWMEYNFFGPTKTIESLRAIAAKFDESIQKKCEEVIAKYKPEWEAVVAKYRPRLEGKRVMLYIGGLRPRHVIGAYEDLGMEVVGTGYEFAHNDDYDRTMKEMGDSTLLYDDVTGYEFEEFVKRIKPDLIGSGIKEKFIFQKMGIPFREMHSWDYSGPYHGFDGFAIFARDMDMTLNNPCWKKLQAPWEASEGAEKVAASA

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 \(c\).

\(|P_{f(7HNW_1)}(2) \setminus P_{f(4IDS_1)}(2)|=48\), \(|P_{f(4IDS_1)}(2) \setminus P_{f(7HNW_1)}(2)|=146\). 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 1:10000001101010000100111100000010110000010000001000111111001001010101010000110111000010001010101001110111100000110010001010111001111100011110100100010110010001111110111011100011011110101101
Pair \(Z_2\) Length of longest common subsequence
7HNW_1,4IDS_1 194 6
7HNW_1,2AFH_1 209 3
4IDS_1,2AFH_1 151 4

Newick tree

 
[
	7HNW_1:10.94,
	[
		4IDS_1:75.5,2AFH_1:75.5
	]:32.44
]

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 \((0.85)(0.8)(\frac{751 }{\log_{20} 751}-\frac{188}{\log_{20}188})=157.\)
Status Protein1 Protein2 d d1/2
Query variables 7HNW_1 4IDS_1 196 131
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]