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Parikh vectors
8CNY_1 1WZK_1 5KFA_1 Letter Amino acid
9 24 31 Q Glutamine
12 16 8 H Histidine
8 26 25 K Lycine
1 13 7 W Tryptophan
9 49 34 A Alanine
8 25 18 P Proline
10 24 31 S Serine
10 45 28 R Arginine
9 48 28 E Glutamic acid
18 43 29 G Glycine
11 22 21 I Isoleucine
19 54 44 L Leucine
17 36 30 V Valine
9 17 13 N Asparagine
9 36 19 D Aspartic acid
2 5 11 C Cysteine
6 10 12 M Methionine
9 38 13 F Phenylalanine
13 28 25 T Threonine
3 26 8 Y Tyrosine 

8CNY_1|Chain A|Protease 3C|Enterovirus (12059)
>1WZK_1|Chains A, B|Alpha-amylase II|Thermoactinomyces vulgaris (2026)
>5KFA_1|Chain A|DNA polymerase eta|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 \(w=f(c)\)
8CNY , Knot 92 192 0.84 40 143 186 
MGPSLDFALSLLRRNIRQVQTDQGHFTMLGVRDRLAVLPRHSQPGKTIWVEHKLINILDAVELVDEQGVNLELTLVTLDTNEKFRDITKFIPENISAASDATLVINTEHMPSMFVPVGDVVQYGFLNLSGKPTHRTMMYNFPTKAGQCGGVVTSVGKVIGIHIGGNGRQGFCAGLKRSYFASEQLEHHHHHH
1WZK , Knot 232 585 0.84 40 269 550 
MLLEAIFHEAKGSYAYPISETQLRVRLRAKKGDVVRCEVLYADRYASPEEELAHALAGKAGSDERFDYFEALLECSTKRVKYVFLLTGPQGEAVYFGETGFSAERSKAGVFQYAYIHRSEVFTTPEWAKEAVIYQIFPERFANGDPSNDPPGTEQWAKDARPRHDSFYGGDLKGVIDRLPYLEELGVTALYFTPIFASPSHHKYDTADYLAIDPQFGDLPTFRRLVDEAHRRGIKIILDAVFNHAGDQFFAFRDVLQKGEQSRYKDWFFIEDFPVSKTSRTNYETFAVQVPAMPKLRTENPEVKEYLFDVARFWMEQGIDGWRLDVANEVDHAFWREFRRLVKSLNPDALIVGEIWHDASGWLMGDQFDSVMNYLFRESVIRFFATGEIHAERFDAELTRARMLYPEQAAQGLWNLLDSHDTERFLTSCGGNEAKFRLAVLFQMTYLGTPLIYYGDEIGMAGATNPDCRRPMIWEEKEQNRGLFEFYKELIRLRHRLASLTRGNVRSWHADKQANLYAFVRTVQDQHVGVVLNNRGEKQTVLLQVPESGGKTWLDCLTGEEVHGKQGQLKLTLRPYQGMILWNGR
5KFA , Knot 183 435 0.85 40 244 418 
GPHMATGQDRVVALVDMDCFFVQVEQRQNPHLRNKPCAVVQYKSWKGGGIIAVSYEARAFGVTRSMWADDAKKLCPDLLLAQVRESRGKANLTKYREASVEVMEIMSRFAVIERASIDEAYVDLTSAVQERLQKLQGQPISADLLPSTYIEGLPQGPTTAEETVQKEGMRKQGLFQWLDSLQIDNLTSPDLQLTVGAVIVEEMRAAIERETGFQCSAGISHNKVLAKLACGLNKPNRQTLVSHGSVPQLFSQMPIRKIRSLGGKLGASVIEILGIEYMGELTQFTESQLQSHFGEKNGSWLYAMCRGIEHDPVKPRQLPKTIGCSKNFPGKTALATREQVQWWLLQLAQELEERLTKDRNDNDRVATQLVVSIRVQGDKRLSSLRRCCALTRYDAHKMSHDAFTVIKNCNTSGIQTEWSPPLTMLFLCATKFSAS

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 \(c\).

\(|P_{f(8CNY_1)}(2) \setminus P_{f(1WZK_1)}(2)|=35\), \(|P_{f(1WZK_1)}(2) \setminus P_{f(8CNY_1)}(2)|=161\). 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 1:111010111011000100100001010111100011111000011001110001101101101100011010101101000001001001110010110010111000011011111101100111010101000011001100110011110011011110111010011011100001100010000000
Pair \(Z_2\) Length of longest common subsequence
8CNY_1,1WZK_1 196 5
8CNY_1,5KFA_1 185 4
1WZK_1,5KFA_1 139 4

Newick tree

 
[
	8CNY_1:10.45,
	[
		5KFA_1:69.5,1WZK_1:69.5
	]:32.95
]

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 \((0.85)(0.8)(\frac{777 }{\log_{20} 777}-\frac{192}{\log_{20}192})=163.\)
Status Protein1 Protein2 d d1/2
Query variables 8CNY_1 1WZK_1 206 134.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]