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Parikh vectors
3SPB_1 9CJP_1 8QJA_1 Letter Amino acid
48 18 118 A Alanine
8 7 58 H Histidine
30 23 130 T Threonine
37 28 119 V Valine
13 21 96 N Asparagine
31 8 119 E Glutamic acid
32 12 84 I Isoleucine
38 28 139 L Leucine
16 15 125 S Serine
3 3 29 W Tryptophan
5 11 97 Y Tyrosine
24 11 124 R Arginine
21 17 150 D Aspartic acid
41 26 154 G Glycine
20 11 73 K Lycine
10 10 30 M Methionine
11 18 68 F Phenylalanine
16 13 68 P Proline
4 12 11 C Cysteine
11 14 87 Q Glutamine 

3SPB_1|Chains A, B, C, D|UDP-N-acetylglucosamine 1-carboxyvinyltransferase|Enterobacter cloacae (716541)
>9CJP_1|Chains A, B|3C-like proteinase nsp5|Severe acute respiratory syndrome coronavirus 2 (2697049)
>8QJA_1|Chains A, B, C, D|Putative type VI secretion system YD repeat-containing Rhs element Vgr protein|Advenella mimigardefordensis DPN7 (1247726)
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 \(w=f(c)\)
3SPB , Knot 171 419 0.82 40 203 380 
MDKFRVQGPTRLQGEVTISGAKNAALPILFAALLAEEPVEIQNVPKLKDIDTTMKLLTQLGTKVERDGSVWIDASNVNNFSAPYDLVKTMRASIWALGPLVARFGQGQVSLPGGCAIGARPVDLHIFGLEKLGAEIKLEEGYVKASVNGRLKGAHIVMDKVSVGATVTIMSAATLAEGTTIIENAAREPEIVDTANFLVALGAKISGQGTDRITIEGVERLGGGVYRVLPDRIETGTFLVAAAISGGKIVCRNAQPDTLDAVLAKLREAGADIETGEDWISLDMHGKRPKAVTVRTAPHPAFPTDMQAQFTLLNLVAEGTGVITETIFENRFMHVPELIRMGAHAEIESNTVICHGVEKLSGAQVMATDLRASASLVLAGCIAEGTTVVDRIYHIDRGYERIEDKLRALGANIERVKGE
9CJP , Knot 133 306 0.83 40 194 290 
SGFRKMAFPSGKVEGCMVQVICGTTTLNGLWLDDVVYCPRHVICTSEDMFNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGACGSVGFNIDYDCVSFCYMHHMVLPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ
8QJA , Knot 633 1879 0.84 40 355 1521 
GSGSMNRTVNATTPLGDKLKFKSLKGHEQLSDLFEWTVEFVSDSPGLVLEDLLGKTISLEVETSAAPRYMHGVITAFKLVHRETQTRRYYIYEATVRPWLWYSTQVTDNRIFQDQTAVQIITQVLQAYDYPIENRLVGQYRKWGYSVQFQESDFNFISRLMEHEGIYYWFRHEKDQHVLVLMDDAHSHAPLPVMPDIPFYPDDTRSVPLEEYIRDWQIAGELTPTTYSTMDYDFQKPQAEMSARRWVKNQNTQGLDLDWYDPMGGYVDSADSDHYARVHLESMQCLQEQAWAVSNVRNLAPGYTFSLKYYPNSEENKSYLILRAEYDFRDPSYASAGGVQENATFVIRSQHIPASVQFRAPRSTPAPRMSGPQTATVVGPDGQEIWTDKYGRIKVQFHWDRQGAMDENSSCWLRVSSPWAGGGFGGVQIPRVREEVVVDFINGDVDRPIAVGRVYNASNMPPVSLPENATQSGFLTRTKNGTPENANKMLFEDSQGNELLSMVAEKDMNTHVKNNQQHDVVGNAVSSIGGLRSHTAHSTSSITMASGAVKSYQSNHNRTVQASLDDSVGGNLEQTLSDGVDETITGAHSHTVSGAASHQLLGLHVNTTTQDVETVNGTVTETVSADETTNVTSSSELEAADITLETPSLYKESTKQDININAGGSLDIQSTGPGIIQTPTDIEKESPANLEAAVMLDNNTVNREDKYILKLNLDAMSDVMVSGAKTDRNIADLAFYGMGGTMQLANVGIYGASIQMGSSDQKMGLASLSITGLELDKGFKIKTMGAGGRSGYKGKHRAAGGGRGPRGGKGGRHRAGNKGDPKKPHADCKECRGRVGGSIGLDVGDERFSHLDFVLPGVFPLEWNRTYRSNMTAKDGQGELGPRWITAFTLVVLPLETGGYEYISEVGRAVSVIELEPGQEWYDRTEELIWRRPDVDTLEVSQKYQRTETFERSGDVFRLKTIADRAGNQAQLIYDAENGFLVRVETAIQTVHLDHDAHGRITTIWHEVTGDDGTPLRRTLARYQYDEAHDLVAAVDQYERTHTYAYQNHLITRYSDKTGRGINLEWDGDHPQAKCVREYRDDGSNLLRFRWDESESKTYVTDALSATTVYTFDAHNYIIHIDFADGTRQSRIRDEFHNIVEVRYPDDSFEKYEYDEFDNILKLTRADHTTVGWEYDQYSQITKIVDPGGNIWTRDYDQFGNMIRETDPKGHTSEFQYTAEGLLIQSVNPAGGVSTLSYNPAGLLIRYSDCSNKTRRWEYDLLGRTVKQTDPCGNAESYEYNRYGFINEVRRPDGSVLTLDFDEEGRLLSFIDPIDNLTVYAYDGAGRLARKTDPLKQDFHYNYDKKGRLASLQDENGAQFTFQFDPVDRLIRTVGFDGKVKKYNYDKPTGLLFSMEDADRETHFEYDVMGQLLKRRAGHLVDSFEYDISNRIIRAHNEYCDQHFEYDVLNNPIRETHIYNAFGQNREYVWENEFDELSNRLTTRRPSGEKVSWLRYGAGHVHGILLDDQEVLSFERDNSHRAVRKRQSNSLLAVTNYDVMGRVKEQKLNLAQGNRTNLRSRQYAYGLDGSLVAIEDSRHGTTTYRYDALDRLVSATAFNETELFAFDPASNLVDRDKTERSKPANTFPKNVSKVLGNILKRCAGMHFEYDAQGNLIRKRKPDSLQEFEWDEFGRLRKTVNTDLNSKHVSEAEYIYDTFDRRIGKVNQQETSGAAVTFYGWDGHHLAFEESTVGDAVDKTHYLYEENSFVPLIQYQYSADNTIAEGRNHLSVSHYQCDHIGTPQLLTDDNGHIVWEGRYSALGKQLDSIGGDAGVAGGQNNLCYQGQYYDRESGLHYNRFRYYDPDIGRFIQQDPIGLFGDSNFYTYAPNTANWIDPFGL

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 \(c\).

\(|P_{f(3SPB_1)}(2) \setminus P_{f(9CJP_1)}(2)|=86\), \(|P_{f(9CJP_1)}(2) \setminus P_{f(3SPB_1)}(2)|=77\). 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 1:10010101100101010101100111111111111001101001101001000101100110010001011101001001011001100101011111111101101010111101111011010111100111010100101010101010110111001011101011011011010011001100101100101111111010101000101011001111100111001001011111110110110001010010111101001110100100110101010010110100110111100101010110111010111000110001101101101110101000011001100101101110010101011111011010011001001001000100010111101001010
Pair \(Z_2\) Length of longest common subsequence
3SPB_1,9CJP_1 163 5
3SPB_1,8QJA_1 178 4
9CJP_1,8QJA_1 191 4

Newick tree

 
[
	8QJA_1:95.63,
	[
		3SPB_1:81.5,9CJP_1:81.5
	]:14.13
]

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 \((0.85)(0.8)(\frac{725 }{\log_{20} 725}-\frac{306}{\log_{20}306})=115.\)
Status Protein1 Protein2 d d1/2
Query variables 3SPB_1 9CJP_1 141 123.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]