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Parikh vectors
8JHZ_1 1NGI_1 2IMF_1 Letter Amino acid
34 7 6 M Methionine
66 11 7 P Proline
46 22 13 R Arginine
152 29 13 G Glycine
207 25 22 L Leucine
37 6 2 H Histidine
161 27 4 T Threonine
160 10 11 Y Tyrosine
111 37 22 A Alanine
305 18 12 N Asparagine
8 2 1 C Cysteine
95 31 13 V Valine
131 24 10 E Glutamic acid
272 26 9 I Isoleucine
160 18 12 F Phenylalanine
196 22 14 S Serine
27 1 5 W Tryptophan
176 29 13 D Aspartic acid
67 12 7 Q Glutamine
215 29 7 K Lycine 

8JHZ_1|Chain A|Hemorrhagic toxin|Paeniclostridium sordellii (1505)
>1NGI_1|Chain A|HEAT-SHOCK COGNATE 70 kD PROTEIN|Bos taurus (9913)
>2IMF_1|Chain A|2-hydroxychromene-2-carboxylate isomerase|Pseudomonas putida (303)
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 \(w=f(c)\)
8JHZ , Knot 731 2626 0.73 40 333 1552 
MSLISKDELIKLAYSIKPREDEYKTILTNLDEYNKLVTINNKDKYLQLKKLNDSIDIFINKYKKSSRNRALFNLKKDISKEVILIKNSNISPVEKNLHFVWIGGEVSDTALEYINQWADINRDYNIRVWYDGEAFLVNTLKKAIVEHSTTDTLQLFEEDIKNPQFDNMKFYKKRMEFIYERQNRFINYYKSEINKPIKPTIDDIIKSHLVSEYNKNSESLELYRRTSFEKISNNNGVDIRNNNLFTEQELLNIYNQELLDRENLAAASDLVRLLALKDFGGVYLDVDMLPGIQPDLFKTISRPSSIGVDSWEMIKLEAIMKYKKYIKNYTSKNFDKLDQQLKDDFQITLESKSEKSEIFSKLGNLDVSDLEIKIAFALGSVINQALISKKGSYLTNLVIEQVKNRYKFLNQHLNPAIELGGNFSDTTKNFHDSLFNSATSENSMFLTKIASYLQVGFMPEARATISLSGPGAYSSAYYDFINLQDNTIEKTLNASDLMEFKFPESNLSQFTEQEINSLWSFDQASAKYQFEKYVRDYTQESLSEDSELDFNKNTVLDKNYLLNNKIPSNNVEETGSKNYVHYIIQLQGDDISYESACNLFSKNPKNSVIIQRNMNESAKSYFLSDNGESISELNKYRIPQRLKNKEKIKVTFIGHGKDEFNTSEFAKLSVDSLSNEISSFLDIMKLDISPKNVEINLLGCNMFSYNVNVEETYPGKLLLNNIDKIISTLPNVNKDNITIGANQYEVRINNEGRKELLDHSGQWLNKEEAIMNDLSSKEYIFFDSIENRPKAKSKNLIELASISDNIKTLLLDTNIDPETKFILNNLKLNIESSIDNNIYYEKLEPVKNIIHNSIDNLTNEFNLIENVSDELYELKKINNLDDNYLISFEDISKNDSTYTIRFINKNSGESVHIQTEKEIFLKYGEHITQEINTIKNNIIIDVNGNLIGNIELEHAPQVNTLNAAFFIQSLIDYSNNKDVLNNLRTSVKVQLYAQLFSTGLNTIYDSIQLVNLISNAINDAINVLPTLTEGVPILATILDGISLGAAIKELSETNDPLLKKELEAKVGIIAINMSLSLASMISTVIGVGSELAIFLLPIAGISAGIPSLVNNELILHDKATSVVNYFTHLSESKKYGPFKLEDDKILSPIDDLVISEIDFNTDSIKLGTCDILSMEGGSGYTVTNDIDHFFSSPYINSNIPPLSIYPVMNIQTTNLDFSKDLMMLPNAPSRLLWWETGAVPGLRSLETDGTRLLDSIRDFYPGKFYWRFYAWFDFAITTLKPVYENTNIKIKLDKNTRNFIMPTITTDVIRNNLSYSFTGSGGTYSLLLSSHPISTNINLSKDDLWIFNIDNKVREISIQNGTIKKGNLVRNALSNLDINKNKLTIDNQIINFSGDVDNKYRYIFLNCSLDDEISLMIEINLFAKSYNLILSGNKDYLISNLSTIINKINNLGLNSKNISYNYTDEFNNKYFGVISKTNQKSIICYKKGSKNILELYNGNMLLFDSKDFIADDINIFMKDDINTITGKYYIDNNLDISVDFSISLISKNKVKVNGLYLNEYGYASFLEFIKNSDGHHNTSNFINLFLDNIGFWKLFGFNNIEFVIDKYFAITGKTDMGYIEFICDNNKNIDIYFGEWKTSSTKNTIFSGNGRNLIVEPIYDINAGDNISTSIDFSYEYISGIDIYINKILIVPNLYTELVNINTDYSSNKYFPEIIVLNTDTFHDKVNINLDSSSLDYEWAIDGSDFILSRYLEENNRKILQKIRIKDILSNTKSFNKMMIDFKDIKNISLDHIMNNFKSFNSESELDRDHFGFKTIDSKTYYYNEVGKLVKGSININDSLFYFDTIESNLVTGWKTINGKTYYFDINNGVAYIGYKTIDGKNFYFDGNGIMQIGVFKVPDGFKYFAPANTYNNNEEGQTIVYQNKFLTINGKKYYFDNNSKAVTGWQIINGNKYYFDTNTGIAAVGLQIINNNKYYFNPHTAIAATGWQSINNAKYYFDINTYIGTTGYKTIDSKNFYFDSNCIMQIGVFKVSDGFKYFAPANTYNNNEEGQAIVYQNKFLTINGKRYYFDNNSKAVTGWHTIDGKKYYFNPNTAIAATGWHTIDDKKYYFNPDTAIAATGWHTIDDKKYYFNPNTGITSTGETTINNKSFYFNDKGIMQIGVFKVPDGFKYFAPANTHNNNLEGQAIVYQNKFLTINGKKYYFDNNSKAITGWQVIDDKKYYFNSNTAVADTNLCTINNEKYYFSYDGILQNGYITIGRLNFYFDSNNDSKMTTGVFKGPNGFEYFAPANTYNNNLEGQAIVYQNKFLTINGKKYYFDNKSKAVTGWQTIDGKKYYFNPNTAIAAMGWQAIDGKKYYFNPNTAIATTGWQTIDGKKYYFNPNTAIAATGWQAIDGKKYYFNPNTATTSIGYTTINSKNFYFNNDGIMQLGVFKGPDGFEYFAPANTHNNNEEGQSITYQNKFLIFNEDVYYFDSSSKAVTGWRTIDDHRFYFEPNTGIGANGYKTLDGKNFYFRNGLPQFGVFKGPDGFEYFAPANTHNNNEEGQSITYQNKFLVFLGNRYYFDSSSKAVTGWQTINGNTYYFMPDTAIAAAGGFFTIDGAIYFFGIDGVKQPGIYGHHHHHHHH
1NGI , Knot 165 386 0.84 40 214 368 
MSKGPAVGIDLGTTYSCVGVFQHGKVEIIANDQGNRTTPSYVAFTDTERLIGDAAKNQVAMNPTNTVFDAKRLIGRRFDDAVVQSDMKHWPFMVVNDAGRPKVQVEYKGETKSFYPEEVSSMVLTKMKEIAEAYLGKTVTNAVVTVPAYFNDSQRQATKDAGTIAGLNVLRIINEPTAAAIAYGLDKKVGAERNVLIFDLGGGTFDVSILTIEDGIFEVKSTAGDTHLGGEDFDNRMVNHFIAEFKRKHKKDISENKRAVRRLRTACERAKRTLSSSTQASIEIDSLYEGIDFYTSITRARFEELNADLFRGTLDPVEKALRDAKLDKSQIHDIVLVGGSTRIPKIQKLLQDFFNGKELNKSINPDEAVAYGAAVQAAILSGDKSE
2IMF , Knot 94 203 0.82 40 144 197 
MIVDFYFDFLSPFSYLANQRLSKLAQDYGLTIRYNAIDLARVKIAIGNVGPSNRDLKVKLDYLKVDLQRWAQLYGIPLVFPANYNSRRMNIGFYYSGAEAQAAAYVNVVFNAVWGEGIAPDLESLPALVSEKLGWDRSAFEHFLSSNAATERYDEQTHAAIERKVFGVPTMFLGDEMWWGNDRLFMLESAMGRLCRQNADLSS

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 \(c\).

\(|P_{f(8JHZ_1)}(2) \setminus P_{f(1NGI_1)}(2)|=131\), \(|P_{f(1NGI_1)}(2) \setminus P_{f(8JHZ_1)}(2)|=12\). 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 1:1011000011011001010000000110010000011010000001010010001011100000000001110100010001111000010110001011111101000110010011010000010110010111100100111000000010110001001010010100001011000000110000001001101010011000110000000001010000010010000110100001100001101000011000011110011011110011110101011111010110010010011100101101011100000100000001001000100010101000000001100110101001010111111011001110001001001110010000011000101110111010000001000110010000011100110010111110101010101111000100011010000100010100110101100010010000100110100101000100010000000100000101000011000011000110001000100001001101010010000100110001000111000100010001100010010010000110010000010101110100010000110101001000100110110101010010101110011000101000011011100100110011010000101110000101000100011000101100001110010000011100100010100001101101000100111000101000111001010100010001000010110011000100100010110010001001001001000011010010000000010110000100101000001110010010001001000111010101110101001101001011111001100000001100100010101010110011001000101101100110011011101001111110110110111110010000011100010101111110101011011001111100111111111110111101100011100010011001001000000111010000110110011100101000010110001101011010010001001100101000111101011101000010100011111011001111001111110010001001100100101101010101110111001011000001010100000011110100011000100010101100011100011000101000011110100010010100101001011001100101000010100011010101000000111000100010111010111000011101000011001001100100111000010000000100001111000000011000010001101001011110000111001011100010010100010001010101010110000101011010001010110110000100000011011100111101111001011100011101000110101100000010101101000000001101010011101100101100100010100001011010100111110100011010000000001101111000010001010100001000111010011100010000001100101001100000100111010010010100110010010000010000111001000000000110110101010001101001000110110010100001010011101100010100101010111011110110110011110000000010011000011010100001000001101101101000010000111111101100000010100111101100100100010100011001000100001010000110111101001100111100000000101110000110101000010000011011001010000101001111011001000000101001111011001000000101001100010001000010100011101111011011001111000000101011100001101010000100000110110110000001000011100010010000001000111001010110101010000000100111011011001111000000101011100001101010000100000110110010100001010011111110110100001010011100110010100001010011110110110100001010010001100010000101000111011110110110011110000000010010000011110001001000001101100100001010100111101000101001010011101111011011001111000000001001000001111110000100000110110010100001110011111111101011101111011001110100000000
Pair \(Z_2\) Length of longest common subsequence
8JHZ_1,1NGI_1 143 5
8JHZ_1,2IMF_1 203 4
1NGI_1,2IMF_1 174 4

Newick tree

 
[
	2IMF_1:10.04,
	[
		8JHZ_1:71.5,1NGI_1:71.5
	]:29.54
]

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 \((0.85)(0.8)(\frac{3012 }{\log_{20} 3012}-\frac{386}{\log_{20}386})=633.\)
Status Protein1 Protein2 d d1/2
Query variables 8JHZ_1 1NGI_1 693 404.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]