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Parikh vectors
8IAX_1 6ABJ_1 7CKY_1 Letter Amino acid
17 15 20 F Phenylalanine
14 12 14 P Proline
28 25 29 R Arginine
1 3 8 C Cysteine
39 14 24 I Isoleucine
33 44 30 L Leucine
37 4 25 K Lycine
24 12 24 N Asparagine
17 5 7 M Methionine
35 16 18 T Threonine
2 1 4 W Tryptophan
46 23 26 V Valine
49 47 26 A Alanine
35 21 29 D Aspartic acid
45 17 33 E Glutamic acid
25 14 18 S Serine
8 9 14 Y Tyrosine
11 14 21 Q Glutamine
44 27 17 G Glycine
11 22 7 H Histidine 

8IAX_1|Chains A, B, C, D, E, F, G, H|Pyruvate kinase|Streptococcus pneumoniae R6 (171101)
>6ABJ_1|Chains A, B|D-lactate dehydrogenase (Fermentative)|Pseudomonas aeruginosa (strain ATCC 15692 / DSM 22644 / CIP 104116 / JCM 14847 / LMG 12228 / 1C / PRS 101 / PAO1) (208964)
>7CKY_1|Chain A|Guanine nucleotide-binding protein G(s) subunit alpha isoforms short|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 \(w=f(c)\)
8IAX , Knot 203 521 0.81 40 229 477 
MGSSHHHHHHSSGLVPRGSHMNKRVKIVATLGPAVEIRGGKKFGEDGYWGEKLDVEASAKNIAKLIEAGANTFRFNFSHGDHQEQGERMATVKLAEKIAGKKVGFLLDTKGPEIRTELFEGEAKEYSYKTGEKIRVATKQGIKSTREVIALNVAGALDIYDDVEVGRQVLVDDGKLGLRVVAKDDATREFEVEVENDGIIAKQKGVNIPNTKIPFPALAERDNDDIRFGLEQGINFIAISFVRTAKDVNEVRAICEETGNGHVQLFAKIENQQGIDNLDEIIEAADGIMIARGDMGIEVPFEMVPVYQKMIIKKVNAAGKVVITATNMLETMTEKPRATRSEVSDVFNAVIDGTDATMLSGESANGKYPLESVTTMATIDKNAQALLNEYGRLDSDSFERNSKTEVMASAVKDATSSMDIKLVVTLTKTGHTARLISKYRPNADILALTFDELTERGLMLNWGVIPMLTDAPSSTDDMFEIAERKAVEAGLVESGDDIVIVAGVPVGEAVRTNTMRIRTVR
6ABJ , Knot 145 345 0.81 40 189 322 
MNHKVHHHHHHIEGRHMRILFFSSQAYDSESFQASNHRHGFELHFQQAHLQADTAVLAQGFEVVCAFVNDDLSRPVLERLAAGGTRLVALRSAGYNHVDLAAAEALGLPVVHVPAYSPHAVAEHAVGLILTLNRRLHRAYNRTREGDFSLHGLTGFDLHGKRVGVIGTGQIGETFARIMAGFGCELLAYDPYPNPRIQALGGRYLALDALLAESDIVSLHCPLTADTRHLIDAQRLATMKPGAMLINTGRGALVNAAALIEALKSGQLGYLGLDVYEEEADIFFEDRSDQPLQDDVLARLLSFPNVVVTAHQAFLTREALAAIADTTLDNIAAWQDGTPRNRVRA
7CKY , Knot 169 394 0.85 40 228 378 
MGCLGNSKTEDQRNEEKAQREANKKIEKQLQKDKQVYRATHRLLLLGAGESGKSTIVKQMRILHVNGFNGEGGEEDPQAARSNSDGEKATKVQDIKNNLKEAIETIVAAMSNLVPPVELANPENQFRVDYILSVMNVPDFDFPPEFYEHAKALWEDEGVRACYERSNEYQLIDCAQYFLDKIDVIKQADYVPSDQDLLRCRVLTTGIFETKFQVDKVNFHMFDVGAQRDERRKWIQCFNDVTAIIFVVASSSYNMVIREDNQTNRLQEALNLFKSIWNNRWLRTISVILFLNKQDLLAEKVLAGKSKIEDYFPEFARYTTPEDATPEPGEDPRVTRAKYFIRDEFLRISTASGDGRHYCYPHFTCSVDTENIRRVFNDCRDIIQRMHLRQYELL

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 \(c\).

\(|P_{f(8IAX_1)}(2) \setminus P_{f(6ABJ_1)}(2)|=97\), \(|P_{f(6ABJ_1)}(2) \setminus P_{f(8IAX_1)}(2)|=57\). 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 1:11000000000011110100100010111011111010110011001011001010101001101101110010101001000001001101011001110011111000110100011010100000001001011000110000011110111110100010110011100101110111000100010101000111100011011000111111100000010111001101111011001001001011000010101011101000011001001101101111101011101110111100011100101110111010011001000101000010011011101001011010010100110010011010001011100010100001000000011101100100010101110100010010110000101011110100100011110111111100110000011011000110111100100111111111101100001010010
Pair \(Z_2\) Length of longest common subsequence
8IAX_1,6ABJ_1 154 6
8IAX_1,7CKY_1 147 5
6ABJ_1,7CKY_1 169 4

Newick tree

 
[
	6ABJ_1:83.13,
	[
		8IAX_1:73.5,7CKY_1:73.5
	]:9.63
]

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 \((0.85)(0.8)(\frac{866 }{\log_{20} 866}-\frac{345}{\log_{20}345})=140.\)
Status Protein1 Protein2 d d1/2
Query variables 8IAX_1 6ABJ_1 177 144.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]