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Parikh vectors
3AAQ_1 4CVM_1 7NBH_1 Letter Amino acid
15 24 20 T Threonine
27 11 13 N Asparagine
19 26 16 D Aspartic acid
5 2 6 C Cysteine
28 28 24 S Serine
19 34 36 V Valine
12 16 12 H Histidine
34 60 29 L Leucine
8 8 5 M Methionine
22 5 9 Y Tyrosine
17 24 20 E Glutamic acid
21 47 22 G Glycine
10 12 7 F Phenylalanine
7 2 3 W Tryptophan
14 13 21 K Lycine
18 17 20 P Proline
19 83 33 A Alanine
4 29 10 R Arginine
29 16 15 Q Glutamine
25 18 25 I Isoleucine 

3AAQ_1|Chain A|Ectonucleoside triphosphate diphosphohydrolase I|Legionella pneumophila (272624)
>4CVM_1|Chain A|UDP-N-ACETYLMURAMOYL-TRIPEPTIDE--D-ALANYL-D- ALANINE LIGASE|PSEUDOMONAS AERUGINOSA PAO1 (208964)
>7NBH_1|Chains A[auth AAA], B[auth BBB], C[auth CCC], D[auth DDD]|Serine racemase|Homo sapiens (9606)
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 \(w=f(c)\)
3AAQ , Knot 156 353 0.86 40 217 342 
KHSCIAVIDAGSTGSRLHIYSYDTDDTNTPIHIEEIWNKKIKPGFASIQPNSVTIDAYLTMLLADAPIHNIPVYFYATAGMRLLPQSQQKKYYDELEYWFRQQSQWQLVEAKTITGNDEALFDWLAVNYKLDTLKSVQNKSVGVMDMGGASVQIVFPMPKNAEISKHNQVELNIYGQNINLYVHSFLGLGQTEMSHQFLNSPSCFANDYPLPDGESGQGNAPSCKEEVTSLMNSVHKVNQQIQPLLALNPVNEWYSIGGISNLASSQLFHFENSELTNQSLLQQGDNQICHQQWDILNGQYPDDEYLYQYCLLSSYYYALMVDGYGINPNQTIHYIPPEQNLDWTIGVVLHRA
4CVM , Knot 186 475 0.80 40 202 421 
MLEPLRLSQLTVALDARLIGEDAVFSAVSTDSRAIGPGELFIALSGPRFDGHDYLAEVAAKGAVAALVEREVADAPLPQLLVRDTRAALGRLGALNRRKFTGPLAAMTGSSGKTTVKEMLASILRTQAGDAESVLATRGNLNNDLGVPLTLLQLAPQHRSAVIELGASRIGEIAYTVELTRPHVAIITNAGTAHVGEFGGPEKIVEAKGEILEGLAADGTAVLNLDDKAFDTWKARASGRPLLTFSLDRPQADFRAADLQRDARGCMGFRLQGVAGEAQVQLNLLGRHNVANALAAAAAAHALGVPLDGIVAGLQALQPVKGRAVAQLTASGLRVIDDSYNANPASMLAAIDILSGFSGRTVLVLGDMGELGSWAEQAHREVGAYAAGKVSALYAVGPLMAHAVQAFGATGRHFADQASLIGALATEDPTTTILIKGSRSAAMDKVVAALCGSSEESHSSVDKLAAALEHHHHHH
7NBH , Knot 146 346 0.82 40 198 323 
MDAQYDISFADVEKAHINIRDSIHLTPVLTSSILNQLTGRNLFFKCELFQKTGSFKIRGALNAVRSLVPDALERKPKAVVTHSSGNHGQALTYAAKLEGIPAYIVVPQTAPDCKKLAIQAYGASIVYCEPSDESRENVAKRVTEETEGIMVHPNQEPAVIAGQGTIALEVLNQVPLVDALVVPVGGGGMLAGIAITVKALKPSVKVYAAEPSNADDCYQSKLKGKLMPNLYPPETIADGVKSSIGLNTWPIIRDLVDDIFTVTEDEIKCATQLVWERMKLLIEPTAGVGVAAVLSQHFQTVSPEVKNICIVLSGGNVDLTSSITWVKQAERPASYQSVSVHHHHHH

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 \(c\).

\(|P_{f(3AAQ_1)}(2) \setminus P_{f(4CVM_1)}(2)|=101\), \(|P_{f(4CVM_1)}(2) \setminus P_{f(3AAQ_1)}(2)|=86\). 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 1:00001111011001001010000000000110100110001011110101001010101011110111001110101011101110000000000100110000010110100101000111011110001001001000011110111101011111100101000001010101001010100111110001000110010011000111010010101100000100110010010001011111011001001111001100011010000100001100100010000101101001000010000110000011110101101000100111000101011111001
Pair \(Z_2\) Length of longest common subsequence
3AAQ_1,4CVM_1 187 4
3AAQ_1,7NBH_1 159 3
4CVM_1,7NBH_1 160 6

Newick tree

 
[
	4CVM_1:89.37,
	[
		3AAQ_1:79.5,7NBH_1:79.5
	]:9.87
]

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 \((0.85)(0.8)(\frac{828 }{\log_{20} 828}-\frac{353}{\log_{20}353})=128.\)
Status Protein1 Protein2 d d1/2
Query variables 3AAQ_1 4CVM_1 159 142
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]