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Parikh vectors
6FEX_1 6UHB_1 6CVN_1 Letter Amino acid
5 4 4 W Tryptophan
12 15 23 N Asparagine
21 13 26 D Aspartic acid
18 15 36 E Glutamic acid
12 6 10 H Histidine
11 20 15 K Lycine
13 10 28 S Serine
24 11 22 R Arginine
11 9 29 T Threonine
8 7 16 Y Tyrosine
24 12 29 A Alanine
15 7 22 Q Glutamine
10 4 18 M Methionine
16 4 23 F Phenylalanine
8 3 8 C Cysteine
22 9 35 G Glycine
12 11 18 I Isoleucine
37 25 32 L Leucine
20 15 20 P Proline
22 11 31 V Valine 

6FEX_1|Chain A|Epithelial discoidin domain-containing receptor 1|Homo sapiens (9606)
>6UHB_1|Chains A, B|Ryanodine receptor 3|Homo sapiens (9606)
>6CVN_1|Chains A, C|Tubulin beta chain|Sus scrofa (9823)
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 \(w=f(c)\)
6FEX , Knot 141 321 0.84 40 198 308 
PGAVGDGPPRVDFPRSRLRFKEKLGEGQFGEVHLCEVDSPQDLVSLDFPLNVRKGHPLLVAVKILRPDATKNARNDFLKEVKIMSRLKDPNIIRLLGVCVQDDPLCMITDYMENGDLNQFLSAHQLEDKAAEGAPGDPTISYPMLLHVAAQIASGMRYLATLNFVHRDLATRNCLVGENFTIKIADFGMSRNLYAGDYYRVQGRAVLPIRWMAWECILMGKFTTASDVWAFGVTLWEVLMLCRAQPFGQLTDEQVIENAGEFFRDQGRQVYLSRPPACPQGLYELMLRCWSRESEQRPPFSQLHRFLAEDALNTVHHHHHH
6UHB , Knot 100 211 0.84 40 146 198 
SNASFIPCPVDTSQVILPPHLEKIRDRLAENIHELWGMNKIELGWTFGKIRDDNKRQHPCLVEFSKLPETEKNYNLQMSTETLKTLLALGCHIAHVNPAAEEDLKKVKLPKNYMMSNGYKPAPLDLSDVKLLPPQEILVDKLAENAHNVWAKDRIKQGWTYGIQQDLKNKRNPRLVPYALLDERTKKSNRDSLREAVRTFVGYGYNIEPSD
6CVN , Knot 188 445 0.85 40 257 432 
MREIVHIQAGQCGNQIGAKFWEVISDEHGIDPTGSYHGDSDLQLERINVYYNEAAGNKYVPRAILVDLEPGTMDSVRSGPFGQIFRPDNFVFGQSGAGNNWAKGHYTEGAELVDSVLDVVRKESESCDCLQGFQLTHSLGGGTGSGMGTLLISKIREEYPDRIMNTFSVVPSPKVSDTVVEPYNATLSVHQLVENTDETYCIDNEALYDICFRTLKLTTPTYGDLNHLVSATMSGVTTCLRFPGQLNADLRKLAVNMVPFPRLHFFMPGFAPLTSRGSQQYRALTVPELTQQMFDAKNMMAACDPRHGRYLTVAAVFRGRMSMKEVDEQMLNVQNKNSSYFVEWIPNNVKTAVCDIPPRGLKMSATFIGNSTAIQELFKRISEQFTAMFRRKAFLHWYTGEGMDEMEFTEAESNMNDLVSEYQQYQDATADEQGEFEEEGEEDEA

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 \(c\).

\(|P_{f(6FEX_1)}(2) \setminus P_{f(6UHB_1)}(2)|=117\), \(|P_{f(6UHB_1)}(2) \setminus P_{f(6FEX_1)}(2)|=65\). 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 1:111110111010110001010001101011010100100100110101110100101111110110101000100011001011001001011011110100011011000100101001101001000110111101010011110111011011001101011000110000111001010110111000101100001010111110111100111101001001111110110111100101110100001100110110001001010011101011001110010000000111001001110011001000000
Pair \(Z_2\) Length of longest common subsequence
6FEX_1,6UHB_1 182 3
6FEX_1,6CVN_1 183 4
6UHB_1,6CVN_1 213 4

Newick tree

 
[
	6CVN_1:10.89,
	[
		6FEX_1:91,6UHB_1:91
	]:10.89
]

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 \((0.85)(0.8)(\frac{532 }{\log_{20} 532}-\frac{211}{\log_{20}211})=92.3\)
Status Protein1 Protein2 d d1/2
Query variables 6FEX_1 6UHB_1 121 99
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]