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Parikh vectors
7DYS_1 1MYI_1 8FSO_1 Letter Amino acid
63 18 39 L Leucine
34 7 17 F Phenylalanine
10 4 14 P Proline
26 2 15 Y Tyrosine
15 0 6 C Cysteine
27 14 31 E Glutamic acid
24 15 27 G Glycine
13 3 16 M Methionine
36 8 23 S Serine
33 8 24 V Valine
25 14 18 A Alanine
27 8 30 D Aspartic acid
11 9 8 H Histidine
32 5 22 T Threonine
9 2 4 W Tryptophan
22 2 22 R Arginine
22 7 18 Q Glutamine
40 6 22 I Isoleucine
21 3 13 N Asparagine
28 18 42 K Lycine 

7DYS_1|Chains A, B, C, D|Mucolipin-2|Mus musculus (10090)
>1MYI_1|Chains A, B|MYOGLOBIN|Sus scrofa (9823)
>8FSO_1|Chain A|Integrin beta-6,Talin-1|Mus musculus (10090)
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 \(w=f(c)\)
7DYS , Knot 219 518 0.88 40 265 485 
MPGDEETLDLPAWNRVPDLTWGPHHRSAMASLDSEVREECLREDLKFYFMSPCEKYRARRQIPWKLGLQILKIVMVTTQLVRFGLSNQLVVAFKEDNTVAFKHLFLKGFSGVDEDDYSCSIYTQENTYESIFFAIKQYRHLKNISLATLGYGESEDNRTGLKVCKQHYKTGAMFSSNETLNIDSDIETDCIHLDLQVLTTEPEDWAQTSFFRLDFYRLVQVDISFALKGIDLQAVHSREIPDCYLFQNTITFDNTAHSGKIKIYLNSEANIEECKNMNISGSTQRSTHYLLVFDVFVIMICLASLILCTRSIVLALRLRKRFLNFFLEKYKQRVCGADQWEFVNGWYVLVTISDLMTIIGSILKMEIKAKKLTNYDVCSILLGTSTLFVWVGVIRYLGYFQTYNVLILTMQASLPKVLRFCACAGMIYLGYTFCGWIVLGPYHEKFENLNIVAECLFSLVNGDDMFATFAQIQQKSILVWLFSRLYLYSFISLFIYMVLSLFIALITDSYHTIKKYQQ
1MYI , Knot 75 153 0.82 38 115 148 
GLSDGEWQLVLNVWGKVEADVAGHGQEVLIRLFKGHPETLEKFDSFKHLKSEDEMKASEDLKKHGNTVLTALGGILKKKGHHEAELTPLAQSHATKHKIPVKYLEFISEAIIQVLQSKHPGDFGADAQGAMSKALELFRNDMAAKYKELGFQG
8FSO , Knot 176 411 0.86 40 231 394 
MTGTNPLYRGSMVALSLKISIGNVVKTMQFEPSTMVYDACRMIRERIPEALAGPPNDFGLFLSDDDPKKGIWLEAGKALDYYMLRNGDTMEYRKKQRPLKIRMLDGTVKTIMVDDSKTVTDMLMTICARIGITNHDEYSLVRELMEEKKDELNWLDHGRTLREQGVEEHETLLLRRKFFYSDQNVDSRDPVQLNLLYVQARDDILNGSHPVSFDKACEFAGFQCQIQFGPHNEQKHKAGFLDLKDFLPKEYVKQKGERKIFQAHKNCGQMSEIEAKVRYVKLARSLKTYGVSFFLVKEKMKGKNKLVPRLLGITKECVMRVDEKTKEVIQEWSLTNIKRWAASPKSFTLDFGDYQDGYYSVQTTEGEQIAQLIAGYIDIILKKKKSKDHFGLEGDEESTMLEDSVSPKKST

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 \(c\).

\(|P_{f(7DYS_1)}(2) \setminus P_{f(1MYI_1)}(2)|=178\), \(|P_{f(1MYI_1)}(2) \setminus P_{f(7DYS_1)}(2)|=28\). 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 1:11100001011110011010111000011101000100001000101011010000010001110111011011110001101110001111100000111001110110110000000010000000001111100000100101101101000000011010000000111100000101000100001010101100010011000110101001101010111011010110000110001100010100010010101010001010000010101000000001111011111101101110000111110100011011100000010110010110110111010011011101101010100100001001111000111111110011010000111101010110110101011110110010111111100001001011100110110100111011010000111111001010011011101110111111000000100000
Pair \(Z_2\) Length of longest common subsequence
7DYS_1,1MYI_1 206 4
7DYS_1,8FSO_1 170 4
1MYI_1,8FSO_1 178 5

Newick tree

 
[
	1MYI_1:99.72,
	[
		7DYS_1:85,8FSO_1:85
	]:14.72
]

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 \((0.85)(0.8)(\frac{671 }{\log_{20} 671}-\frac{153}{\log_{20}153})=148.\)
Status Protein1 Protein2 d d1/2
Query variables 7DYS_1 1MYI_1 190 120
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]