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Parikh vectors
8UWW_1 5BNN_1 7QDO_1 Letter Amino acid
12 23 11 N Asparagine
22 9 15 D Aspartic acid
28 17 39 G Glycine
30 19 17 I Isoleucine
19 16 16 K Lycine
12 16 18 F Phenylalanine
19 17 30 P Proline
18 12 17 R Arginine
9 7 9 H Histidine
16 8 7 M Methionine
24 21 47 S Serine
16 9 8 Y Tyrosine
21 23 32 V Valine
29 25 27 A Alanine
6 2 8 C Cysteine
26 31 39 T Threonine
4 3 8 W Tryptophan
11 7 24 Q Glutamine
28 15 35 E Glutamic acid
27 17 33 L Leucine 

8UWW_1|Chains A, B|Actin, alpha cardiac muscle 1|Sus scrofa (9823)
>5BNN_1|Chain A|Capsid protein VP1|Enterovirus D68 (42789)
>7QDO_1|Chains A, B|Isoform 2 of Immunoglobulin heavy constant mu|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 \(w=f(c)\)
8UWW , Knot 160 377 0.84 40 218 361 
MCDDEETTALVCDNGSGLVKAGFAGDDAPRAVFPSIVGRPRHQGVMVGMGQKDSYVGDEAQSKRGILTLKYPIEHGIITNWDDMEKIWHHTFYNELRVAPEEHPTLLTEAPLNPKANREKMTQIMFETFNVPAMYVAIQAVLSLYASGRTTGIVLDSGDGVTHNVPIYEGYALPHAIMRLDLAGRDLTDYLMKILTERGYSFVTTAEREIVRDIKEKLCYVALDFENEMATAASSSSLEKSYELPDGQVITIGNERFRCPETLFQPSFIGMESAGIHETTYNSIMKCDIDIRKDLYANNVLSGGTTMYPGIADRMQKEITALAPSTMKIKIIAPPERKYSVWIGGSILASLSTFQQMWISKQEYDEAGPSIVHRKCF
5BNN , Knot 133 297 0.85 40 198 286 
VESIIKTATDTVKSEINAELGVVPSLNAVETGATSNTEPEEAIQTRTVINQHGVSETLVENFLGRAALVSKKSFEYKNHASSSAGTHKNFFKWTINTKSFVQLRRKLELFTYLRFDAEITILTTVAVNGNNDSTYMGLPDLTLQAMFVPTGALTPKEQDSFHWQSGSNASVFFKISDPPARMTIPFMCINSAYSVFYDGFAGFEKNGLYGINPADTIGNLCVRIVNEHQPVGFTVTVRVYMKPKHIKAWAPRPPRTMPYMSIANANYKGRDTAPNTLNAIIGNRASVTTMPHNIVTT
7QDO , Knot 175 440 0.80 40 214 391 
MGILPSPGMPALLSLVSLLSVLLMGCVAETGVIAELPPKVSVFVPPRDGFFGNPRKSKLICQATGFSPRQIQVSWLREGKQVGSGVTTDQVQAEAKESGPTTYKVTSTLTIKESDWLSQSMFTCRVDHRGLTFQQNASSMCVPDQDTAIRVFAIPPSFASIFLTKSTKLTCLVTDLTTYDSVTISWTRQNGEAVKTHTNISESHPNATFSAVGEASISEDDWNSGERFTCTVTHTDLPSPLKQTISRPKGVALHRPDVYLLPPAREQLNLRESATITCLVTGFSPADVFVQWMQRGQPLSPEKYVTSAPMPEPQAPGRYFAHSILTVSEEEWNTGETYTCVVAHEALPNRVTERTVDKSTEGEVSADEEGFENEIAQLEYEISQLEQEIQALESGGGSGGGSENLYFQGGGSWSHPQFEKGGGSGGGSGGSAWSHPQFEK

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 \(c\).

\(|P_{f(8UWW_1)}(2) \setminus P_{f(5BNN_1)}(2)|=92\), \(|P_{f(5BNN_1)}(2) \setminus P_{f(8UWW_1)}(2)|=72\). 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 1:10000000111000101110111110011011110111010001111111000001100100001110100110011100100100110001000101110001011001110101000010011100101111011101110101010001111001011000111001011101110101110010001101100010011001000110010001001110100011011000010000011010110110001001001101011110011100000001100010100010100110110010111100100010111100101011111000001111101110100100111000000011101100001
Pair \(Z_2\) Length of longest common subsequence
8UWW_1,5BNN_1 164 4
8UWW_1,7QDO_1 164 4
5BNN_1,7QDO_1 150 4

Newick tree

 
[
	8UWW_1:84.20,
	[
		5BNN_1:75,7QDO_1:75
	]:9.20
]

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 \((0.85)(0.8)(\frac{674 }{\log_{20} 674}-\frac{297}{\log_{20}297})=104.\)
Status Protein1 Protein2 d d1/2
Query variables 8UWW_1 5BNN_1 134 118.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]