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Parikh vectors
9IWM_1 3KAR_1 9ASA_1 Letter Amino acid
21 24 27 K Lycine
15 13 25 F Phenylalanine
9 6 14 P Proline
18 22 33 V Valine
14 31 28 N Asparagine
10 13 18 Q Glutamine
33 29 57 L Leucine
9 15 13 R Arginine
17 19 18 E Glutamic acid
12 19 18 G Glycine
7 5 15 C Cysteine
18 28 34 I Isoleucine
12 5 15 M Methionine
12 39 53 S Serine
11 27 27 T Threonine
0 2 7 W Tryptophan
7 8 14 Y Tyrosine
23 11 29 A Alanine
16 19 20 D Aspartic acid
9 11 6 H Histidine 

9IWM_1|Chain A|Peroxisome proliferator-activated receptor alpha|Homo sapiens (9606)
>3KAR_1|Chain A|KINESIN-LIKE PROTEIN KAR3|Saccharomyces cerevisiae (4932)
>9ASA_1|Chain A|5-hydroxytryptamine receptor 2A|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 \(w=f(c)\)
9IWM , Knot 120 273 0.82 38 178 260 
GSHMTADLKSLAKRIYEAYLKNFNMNKVKARVILSGKASNNPPFVIHDMETLCMAEKTLVAKLVANGIQNKEAEVRIFHCCQCTSVETVTELTEFAKAIPGFANLDLNDQVTLLKYGVYEAIFAMLSSVMNKDGMLVAYGNGFITREFLKSLRKPFCDIMEPKFDFAMKFNALELDDSDISLFVAAIICCGDRPGLLNVGHIEKMQEGIVHVLRLHLQSNHPDDIFLFPKLLQKMADLRQLVTEHAQLVQIIKKTESDAALHPLLQEIYRDMY
3KAR , Knot 150 346 0.84 40 194 331 
MGNIRVYCRIRPALKNLENSDTSLINVNEFDDNSGVQSMEVTKIQNTAQVHEFKFDKIFDQQDTNVDVFKEVGQLVQSSLDGYNVCIFAYGQTGSGKTFTMLNPGDGIIPSTISHIFNWINKLKTKGWDYKVNCEFIEIYNENIVDLLRSDNNNKEDTSIGLKHEIRHDQETKTTTITNVTSCKLESEEMVEIILKKANKLRSTASTASNEHSSRSHSIFIIHLSGSNAKTGAHSYGTLNLVDLAGSERINVSQVVGDRLRETQNINKSLSCLGDVIHALGQPDSTKRHIPFRNSKLTYLLQYSLTGDSKTLMFVNISPSSSHINETLNSLRFASKVNSTRLVSRK
9ASA , Knot 191 471 0.83 40 223 450 
MDILCEENTSLSSTTNSLMQLNDDTRLYSNDFNSGEANTSDAFNWTVDSENRTNLSCEGCLSPSCLSLLHLQEKNWSALLTAVVIILTIAGNILVIMAVSLEKKLQNATNYFLMSLAIADMLLGFLVMPVSMLTILYGYRWPLPSKLCAVWIYLDVLFSTASIMHLCAISLDRYVAIQNPIHHSRFNSRTKAFLKIIAVWTISVGISMPIPVFGLQDDSKVFKEGSCLLADDNFVLIGSFVSFFIPLTIMVITYFLTIKSLQKEATLCVSDLGTRAKLASFSFLPQSSLSSEKLFQRSIHREPGSYTGRRTMQSISNEQKACKVLGIVFFLFVVMWCPFFITNIMAVICKESCNEDVIGALLNVFVWIGYLSSAVNPLVYTLFNKTYRSAFSRYIQCQYKENKKPLQLILVNTIPALAYKSSQLQMGQKKNSKQDAKTTDNDCSMVALGKQHSEEASKDNSDGVNEKVSCV

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 \(c\).

\(|P_{f(9IWM_1)}(2) \setminus P_{f(3KAR_1)}(2)|=71\), \(|P_{f(3KAR_1)}(2) \setminus P_{f(9IWM_1)}(2)|=87\). 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 1:100101010011001001010010100101011101010001111100100101100011101110110000101011000000010010010011011111101010001011001100111111001100011111010111000110010011001101010111010110100001011111110010011110110100100111011010100001001111101100110100110001011011000000111011100100010
Pair \(Z_2\) Length of longest common subsequence
9IWM_1,3KAR_1 158 4
9IWM_1,9ASA_1 181 4
3KAR_1,9ASA_1 147 4

Newick tree

 
[
	9IWM_1:88.43,
	[
		3KAR_1:73.5,9ASA_1:73.5
	]:14.93
]

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 \((0.85)(0.8)(\frac{619 }{\log_{20} 619}-\frac{273}{\log_{20}273})=97.0\)
Status Protein1 Protein2 d d1/2
Query variables 9IWM_1 3KAR_1 122 109.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]