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Parikh vectors
8CHU_1 5ZIT_1 3SDS_1 Letter Amino acid
15 22 15 R Arginine
10 31 15 D Aspartic acid
12 8 8 Q Glutamine
12 26 22 E Glutamic acid
7 32 33 S Serine
0 6 5 W Tryptophan
6 22 5 Y Tyrosine
9 32 22 K Lycine
2 14 10 M Methionine
7 23 18 P Proline
8 27 31 V Valine
13 27 35 A Alanine
6 19 13 N Asparagine
7 30 19 I Isoleucine
17 40 26 L Leucine
7 23 13 F Phenylalanine
5 26 25 T Threonine
0 7 3 C Cysteine
14 30 23 G Glycine
1 12 12 H Histidine 

8CHU_1|Chains A, B|Transcriptional activator protein Pur-alpha|Homo sapiens (9606)
>5ZIT_1|Chain A|RdRp|Enterovirus D68 (42789)
>3SDS_1|Chains A, B, C|Ornithine carbamoyltransferase, mitochondrial|Coccidioides immitis (246410)
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 \(w=f(c)\)
8CHU , Knot 75 158 0.80 36 118 152 
GPETQELASKRVDIQNKRFYLDVKQNAKGRFLKIAEVGAGGNESRLTLSMSVAVEFRDYLGDFIEHYAQLGPSQPPDLAQAQDEPRRALKSEFLVRENRKYYMDLKENQRGRFLRIRQTVNRGPGLGSTQGQTIALPAQGLIEFRDALAKLIDDYGVE
5ZIT , Knot 190 457 0.84 40 252 441 
GEIVSNEKSGVCINAPAKTKLQPSVFHQVFEGSKEPAVLNSKDPRLKTDFEEAIFSKYTGNKIMLMDEYMEEAVDHYVGCLEPLDISIDPIPLESAMYGMDGLEALDLTTSAGFPYLLQGKKKRDIFNRQTRDTTEMTRMLEKYGVDLPFVTFVKDELRSREKVEKGKSRLIEASSLNDSVAMRVAFGNLYATFHNNPGTATGSAVGCDPDVFWSKIPILLNGEIFAFDYTGYDASLSPVWFACLKKVLIKLGYTHQTSFIDYLCHSVHLYKDRKYIVNGGMPSGSSGTSIFNTMINNIIIRTLLIRVYKGIDLDQFKMIAYGDDVIASYPHKIDPGLLAEAGKHYGLIMTPADKGTSFVDTNWENVTFLKRYFRADDQYPFLIHPVMPMKEIHESIRWTKDPRNTQDHVRSLCYLAWHNGEEAYNEFCRKIRSVPVGRALTLPAYSSLRRKWLDSF
3SDS , Knot 154 353 0.85 40 207 340 
GPGSMAKNQWRPYTNGSHAPSTPRHLLSIADLTPTEFATLVRNASSYKKTIKSDSMPERLTGALSGKTVAMMFSKRSTRTRVSTEGAVVKMGGHPMFLGKDDIQLGVNESLYDTSVVISSMVSCIVARVGPHSDIANLAKHSSVPVINALCDTFHPLQAIADFLTIHESFASQSATHGTHPSSLGLEGLKIAWVGDANNVLFDLAIAATKMGVNVAVATPRGYEIPSHIVELIQKAREGVQSPGNLTQTTVPEVAVKDADVIVTDTWISMGQETEKIKRLEAFKDFKVTSELAKRGGAKENWKFMHCLPRHPEEVSDEVFYSERSLVFPEAENRLWAAISALEAFVVNKGKIA

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 \(c\).

\(|P_{f(8CHU_1)}(2) \setminus P_{f(5ZIT_1)}(2)|=30\), \(|P_{f(5ZIT_1)}(2) \setminus P_{f(8CHU_1)}(2)|=164\). 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 1:11000011000101000010101000101011011011111000010101011101000110110001011100110110100010011000111000000010100000101101000100111110001001111101110100111011000110
Pair \(Z_2\) Length of longest common subsequence
8CHU_1,5ZIT_1 194 3
8CHU_1,3SDS_1 171 4
5ZIT_1,3SDS_1 163 4

Newick tree

 
[
	8CHU_1:94.50,
	[
		3SDS_1:81.5,5ZIT_1:81.5
	]:13.00
]

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 \((0.85)(0.8)(\frac{615 }{\log_{20} 615}-\frac{158}{\log_{20}158})=131.\)
Status Protein1 Protein2 d d1/2
Query variables 8CHU_1 5ZIT_1 169 112
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]