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Parikh vectors
7WSE_1 9HAF_1 2EYU_1 Letter Amino acid
7 0 1 C Cysteine
62 16 19 E Glutamic acid
37 13 19 G Glycine
46 9 19 T Threonine
23 0 0 W Tryptophan
48 15 15 A Alanine
35 7 15 R Arginine
40 19 12 D Aspartic acid
40 16 14 V Valine
37 3 7 Y Tyrosine
14 14 12 H Histidine
32 20 25 I Isoleucine
38 7 11 P Proline
45 8 11 S Serine
41 14 15 K Lycine
24 6 9 M Methionine
33 8 7 F Phenylalanine
35 7 8 N Asparagine
32 6 14 Q Glutamine
70 16 28 L Leucine 

7WSE_1|Chain A|Angiotensin-converting enzyme|Mammalia (40674)
>9HAF_1|Chains A, C, E|Mite allergen Der f 7|Dermatophagoides farinae (6954)
>2EYU_1|Chains A, B|twitching motility protein PilT|Aquifex aeolicus (63363)
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 \(w=f(c)\)
7WSE , Knot 281 739 0.83 40 308 684 
MSGSFWLLLSLVAVTAAQSTTEEQAKTFLQKFDHEAEDLSYRSSLASWNYNTNITDENVQKMNAARAKWSAFYEEQSRIAKTYPLEEIQNLTLKRQLQALQQSGTSVLSADKSKRLNTILNTMSTIYSSGKVLDPNTQEYLVLEPGLDDIMENSEDYNRRLWAWEGWRAEVGKQLRPFYEEYVVLENEMARANNYEDYGDYWRGDYEVTGADGYDYSRNQLIADVERTFAEIKPLYEQLHAYVRAKLMDAYPSRISPTGCLPAHLLGDMWGRFWTNLYPLTVPFGEKPSIDVTKEMQNQSWDAKRIFKEAEKFFVSIGLPNMTQEFWVNSMLTEPGDGRKVVCHPTAWDLGKGDFRIKMCTKVTMDDFLTAHHEMGHIQYDMAYATQPFLLRNGANEGFHEAVGEIMSLSAATPHYLKALGLLPPDFYEDNVTEINFLLKQALQIVGTLPFTYMLEKWRWMVFKGEIPKEQWMQKWWEMKREIVGVVEPLPHDETYCDPACLFHVAEDYSFIRYYTRTIYQFQFHEALCQTAKHEGPLYKCDISNSTEAGQRLLQMLHLGKSEPWTLALENIVGVKTMDVKPLLNYFEPLLTWLKEQNRNSPVGWSTDWTPYSDQSIKVRISLKSALGEKAYEWNDNEMYLFQSSVAYAMREYFSKVRNETIPFGEKDVWVSDLKPRISFNFFVTTPKNVSDIIPRTEVEEAIRMSRGRINDAFRLDDNSLEFLGIQPTLGPPYEPPVT
9HAF , Knot 90 204 0.78 36 142 195 
HHHHHHGSDPIHYDKITEEINKAIDDAIAAIEQSETIDPMKVPDHADKFERHVGILDFKGELAMRNIEARGLKQMKRQGDANVKGEEGIVKAHLLIGVHDDIVSMEYDLAYKLGDLHPTTHVISDIQDFVVALSLEIPDEGNITMTSFEVRQFANVVNHIGGLSILDPIFGVLSDVLTAIFQDTVRKEMTKVLAPAFKRELEKN
2EYU , Knot 112 261 0.79 38 163 240 
MPAEIPEFKKLGLPDKVLELCHRKMGLILVTGPTGSGKSTTIASMIDYINQTKSYHIITIEDPIEYVFKHKKSIVNQREVGEDTKSFADALRAALREDPDVIFVGEMRDLETVETALRAAETGHLVFGTLHTNTAIDTIHRIVDIFPLNQQEQVRIVLSFILQGIISQRLLPKIGGGRVLAYGLLIPNTAIRNLIRENKLQQVYSLMQSGQAETGMQTMNQTLYKLYKQGLITLEDAMEASPDPKELERMIRGGRHHHHHH

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 \(c\).

\(|P_{f(7WSE_1)}(2) \setminus P_{f(9HAF_1)}(2)|=185\), \(|P_{f(9HAF_1)}(2) \setminus P_{f(7WSE_1)}(2)|=19\). 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 1:1010111110111101100000001001100100010010000011010000010000100101101010110000001100011001001010001011000100110100000100110010010001011010000011101110011000000000111101101011001011000011100011010000001001010001011010000000111010001101011000101010101101010010101011101110111011001011011110010101000100001010011001001110111101000111001100110100110010110110101010100010100110100011010001101001111001100110011101101011010010111111101000010010111001101110111001100101111010110001100110100011111011100000001101101100001100000010010100110001000111000010000011001101101100011011100111100101011100101110110000000111100010100000101010100111001001000010110001101100010010000111100011100101010101110010010011100010011010010100110100001011110101111001110
Pair \(Z_2\) Length of longest common subsequence
7WSE_1,9HAF_1 204 4
7WSE_1,2EYU_1 201 4
9HAF_1,2EYU_1 129 6

Newick tree

 
[
	7WSE_1:11.82,
	[
		2EYU_1:64.5,9HAF_1:64.5
	]:46.32
]

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 \((0.85)(0.8)(\frac{943 }{\log_{20} 943}-\frac{204}{\log_{20}204})=202.\)
Status Protein1 Protein2 d d1/2
Query variables 7WSE_1 9HAF_1 259 161
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]