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Parikh vectors
8FTB_1 1QRP_1 1HUJ_1 Letter Amino acid
23 41 11 S Serine
22 25 26 I Isoleucine
16 5 2 M Methionine
31 21 24 D Aspartic acid
31 14 19 E Glutamic acid
25 34 15 G Glycine
8 1 6 H Histidine
18 15 11 F Phenylalanine
14 17 19 P Proline
21 3 7 R Arginine
15 16 16 N Asparagine
20 26 19 T Threonine
4 5 6 W Tryptophan
24 27 13 V Valine
19 17 22 A Alanine
14 16 11 Y Tyrosine
39 21 18 L Leucine
42 0 28 K Lycine
6 6 1 C Cysteine
19 16 7 Q Glutamine 

8FTB_1|Chain A[auth B]|Integrin beta-2,Talin-1|Mus musculus (10090)
>1QRP_1|Chain A[auth E]|PEPSIN 3A|Homo sapiens (9606)
>1HUJ_1|Chains A, B|INORGANIC PYROPHOSPHATASE|Saccharomyces cerevisiae (4932)
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 \(w=f(c)\)
8FTB , Knot 178 411 0.87 40 235 395 
MNNDNPLFKSAMVALSLKISIGNVVKTMQFEPSTMVYDACRMIRERIPEALAGPPNDFGLFLSDDDPKKGIWLEAGKALDYYMLRNGDTMEYRKKQRPLKIRMLDGTVKTIMVDDSKTVTDMLMTICARIGITNHDEYSLVRELMEEKKDELNWLDHGRTLREQGVEEHETLLLRRKFFYSDQNVDSRDPVQLNLLYVQARDDILNGSHPVSFDKACEFAGFQCQIQFGPHNEQKHKAGFLDLKDFLPKEYVKQKGERKIFQAHKNCGQMSEIEAKVRYVKLARSLQTYGVSFFLVKEKMKGKNKLVPRLLGITKECVMRVDEKTKEVIQEWSLTNIKRWAASPKSFTLDFGDYQDGYYSVQTTEGEQIAQLIAGYIDIILKKKKSKDHFGLEGDEESTMLEDSVSPKKST
1QRP , Knot 137 326 0.81 38 177 304 
VDEQPLENYLDMEYFGTIGIGTPAQDFTVVFDTGSSNLWVPSVYCSSLACTNHNRFNPEDSSTYQSTSETVSITYGTGSMTGILGYDTVQVGGISDTNQIFGLSETEPGSFLYYAPFDGILGLAYPSISSSGATPVFDNIWNQGLVSQDLFSVYLSADDQSGSVVIFGGIDSSYYTGSLNWVPVTVEGYWQITVDSITMNGEAIACAEGCQAIVDTGTSLLTGPTSPIANIQSDIGASENSDGDMVVSCSAISSLPDIVFTINGVQYPVPPSAYILQSEGSCISGFQGMNLPTESGELWILGDVFIRQYFTVFDRANNQVGLAPVA
1HUJ , Knot 127 281 0.85 40 180 270 
TYTTRQIGAKNTLEYKVYIEKDGKPVSAFHDIPLYADKENNIFNMVVEIPRWTNAKLEITREETLNPIIQDTKKGKLRFVRNCFPHHGYIHNYGAFPQTWEDPNVSHPETKAVGDNDPIDVLEIGETIAYTGQVKQVKALGIMALLDEGETDWKVIAIDINDPLAPKLNDIEDVEKYFPGLLRATNEWFRIYKIPDGKPENQFAFSGEAKNKKYALDIIKETHDSWKQLIAGKSSDSKGIDLTNVTLPDTPTYSKAASDAIPPASLKADAPIDKSIDKWFF

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 \(c\).

\(|P_{f(8FTB_1)}(2) \setminus P_{f(1QRP_1)}(2)|=121\), \(|P_{f(1QRP_1)}(2) \setminus P_{f(8FTB_1)}(2)|=63\). 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 1:100001110011111010101101100101010011001001100011011111100111110000100111101101100011001001000000011010110101001110000010011101010111000000011001100000010110010010001100000111000110000010000110101101010001101001101001001111000101110000000111101001110001000100011010000101001010100101100100011011110001010001110111100001101000000110010100100111010010101100001000100001001101111010111000000001110100000110001010000
Pair \(Z_2\) Length of longest common subsequence
8FTB_1,1QRP_1 184 3
8FTB_1,1HUJ_1 173 5
1QRP_1,1HUJ_1 179 3

Newick tree

 
[
	1QRP_1:92.13,
	[
		8FTB_1:86.5,1HUJ_1:86.5
	]:5.63
]

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 \((0.85)(0.8)(\frac{737 }{\log_{20} 737}-\frac{326}{\log_{20}326})=112.\)
Status Protein1 Protein2 d d1/2
Query variables 8FTB_1 1QRP_1 148 130.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]