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Parikh vectors
4QHY_1 6TTO_1 4ERP_1 Letter Amino acid
14 13 35 N Asparagine
15 17 51 D Aspartic acid
14 16 33 Q Glutamine
21 24 31 P Proline
41 19 39 V Valine
25 15 62 A Alanine
18 8 40 R Arginine
26 13 49 S Serine
22 13 39 T Threonine
7 3 5 W Tryptophan
4 6 11 C Cysteine
34 35 75 L Leucine
33 23 42 K Lycine
9 5 16 M Methionine
12 12 28 F Phenylalanine
32 28 46 E Glutamic acid
34 18 48 G Glycine
5 7 20 H Histidine
41 24 50 I Isoleucine
8 18 41 Y Tyrosine 

4QHY_1|Chain A|Translation initiation factor 2 subunit gamma|Sulfolobus solfataricus (273057)
>6TTO_1|Chain A|Hyoscyamine 6 beta-hydroxylase|Datura metel (35625)
>4ERP_1|Chains A, B, C, D|Ribonucleoside-diphosphate reductase 1 subunit alpha|Escherichia coli K-12 (83333)
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 \(w=f(c)\)
4QHY , Knot 171 415 0.82 40 225 391 
MAWPKVQPEVNIGVVGHVDFGKTTLVQAITGIWTSKHSEELKRGMTIKLGYAETNIGVCESCKKPEAYVTEPSCKSCGSDDEPKFLRRISFIDAPGHEVLMATMLSGAALMDGAILVVAANEPFPQPQTREHFVALGIIGVKNLIIVQNKVAVVSKEEALSQYRQIKQFTKGTWAENVPIIPVSALHKINIDSLIEGIEEYIKTPYRDLSQKPVMLVIRSFDVNKPGTQFNELKGGVIGGSIIQGLFKVDQEIKVLPGLRVEKQGKVSYEPIFTKISSIRFGDEEFKEAKPGGLVAIGTYLDPSLTKADNLLGSIITLADAEVPVLWNIRIKYNLLERVVGAKEMLKVDPIRAKETLMLSVGSSTTLGIVTSVKKDEIEVELRRPVAVWSNNIRTVISRQIAGRWRMIGWGLVEI
6TTO , Knot 141 317 0.85 40 200 305 
SNADVPIIDLQQDHLLIVQQITKACQDFGLFQVINHGVPEKLMVEAMEVYKEFFALPAEEKEKFQPKGEPAKFELPLEQKAKLYVEGERRCNEEFLYWKDTLAHGCYPLHEELLNSWPEKPPTYRDVIAKYSVEVRKLTMRILDYICEGLGLKLGYFDNELTQIQMLLANYYPSCPDPSTTIGSGGHYDGNLITLLQQDLVGLQQLIVKDDKWIAVEPIPTAFVVNLGLTLKVMSNEKFEGSIHRVVTHPIRNRISIGTLIGPDYSCTIEPIKELISQENPPLYKPYPYAEFAEIYLSDKSDYDAGVKPYKINQFPN
4ERP , Knot 292 761 0.84 40 294 703 
MNQNLLVTKRDGSTERINLDKIHRVLDWAAEGLHNVSISQVELRSHIQFYDGIKTSDIHETIIKAAADLISRDAPDYQYLAARLAIFHLRKKAYGQFEPPALYDHVVKMVEMGKYDNHLLEDYTEEEFKQMDTFIDHDRDMTFSYAAVKQLEGKYLVQNRVTGEIYESAQFLYILVAACLFSNYPRETRLQYVKRFYDAVSTFKISLPTPIMSGVRTPTRQFSSCVLIECGDSLDSINATSSAIVKYVSQRAGIGINAGRIRALGSPIRGGEAFHTGCIPFYKHFQTAVKSCSQGGVRGGAATLFYPMWHLEVESLLVLKNNRGVEGNRVRHMDYGVQINKLMYTRLLKGEDITLFSPSDVPGLYDAFFADQEEFERLYTKYEKDDSIRKQRVKAVELFSLMMQERASTGRIYIQNVDHCNTHSPFDPAIAPVRQSNLCLEIALPTKPLNDVNDENGEIALCTLSAFNLGAINNLDELEELAILAVRALDALLDYQDYPIPAAKRGAMGRRTLGIGVINFAYYLAKHGKRYSDGSANNLTHKTFEAIQYYLLKASNELAKEQGACPWFNETTYAKGILPIDTYKKDLDTIANEPLHYDWEALRESIKTHGLRNSTLSALMPSETSSQISNATNGIEPPRGYVSIKASKDGILRQVVPDYEHLHDAYELLWEMPGNDGYLQLVGIMQKFIDQSISANTNYDPSRFPSGKVPMQQLLKDLLTAYKFGVKTLYYQNTRDGAEDAQDDLVPSIQDDGCESGACKI

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 \(c\).

\(|P_{f(4QHY_1)}(2) \setminus P_{f(6TTO_1)}(2)|=98\), \(|P_{f(6TTO_1)}(2) \setminus P_{f(4QHY_1)}(2)|=73\). 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 1:1111010101011111010110001101101110000000100110101101000111000000101010010000010000101100101101110011110110111110111111110011101000001111111110011110001111000011000001001001011001111110110010100110110001001000100011111100101001100100101111110110111010001011111010001010001110010010110001001011111111001010100100111011011010111110101000110011110011010110100011101100001111001000010101001111100010011000111010111111101
Pair \(Z_2\) Length of longest common subsequence
4QHY_1,6TTO_1 171 4
4QHY_1,4ERP_1 149 4
6TTO_1,4ERP_1 160 4

Newick tree

 
[
	6TTO_1:85.38,
	[
		4QHY_1:74.5,4ERP_1:74.5
	]:10.88
]

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 \((0.85)(0.8)(\frac{732 }{\log_{20} 732}-\frac{317}{\log_{20}317})=113.\)
Status Protein1 Protein2 d d1/2
Query variables 4QHY_1 6TTO_1 143 126.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]