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Parikh vectors
6VNH_1 4PJW_1 6TSB_1 Letter Amino acid
16 40 24 N Asparagine
13 45 16 F Phenylalanine
13 19 20 P Proline
17 51 18 V Valine
15 59 22 S Serine
17 67 30 D Aspartic acid
15 49 12 Q Glutamine
18 59 29 I Isoleucine
30 118 34 L Leucine
9 39 7 M Methionine
10 53 19 T Threonine
8 63 29 A Alanine
5 20 2 C Cysteine
26 82 18 E Glutamic acid
16 23 1 H Histidine
16 35 20 Y Tyrosine
19 50 17 R Arginine
19 36 30 G Glycine
23 64 15 K Lycine
3 9 4 W Tryptophan 

6VNH_1|Chains A, B|Tyrosine-protein kinase JAK2|Homo sapiens (9606)
>4PJW_1|Chain A|Cohesin subunit SA-2|Homo sapiens (9606)
>6TSB_1|Chain A[auth AAA]|Peroxiredoxin|Clostridioides difficile (strain 630) (272563)
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 \(w=f(c)\)
6VNH , Knot 135 308 0.83 40 197 295 
HHHHHHHHENLYFQGDPTQFEERHLKFLQQLGKGNFGSVEMCRYDPLQDNTGEVVAVKKLQHSTEEHLRDFEREIEILKSLQHDNIVKYKGVCYSAGRRNLKLIMEYLPYGSLRDYLQKHKERIDHIKLLQYTSQICKGMEYLGTKRYIHRDLATRNILVENENRVKIGDFGLTKVLPQDKEYYKVKEPGESPIFWYAPESLTESKFSVASDVWSFGVVLYELFTYIEKSKSPPAEFMRMIGNDKQGQMIVFHLIELLKNNGRLPRPDGCPDEIYMIMTECWNNNVNQRPSFRDLALRVDQIRDNMAG
4PJW , Knot 361 981 0.84 40 318 885 
ENMMLFEVVKMGKSAMQSVVDDWIESYKHDRDIALLDLINFFIQCSGCKGVVTAEMFRHMQNSEIIRKMTEEFDEDSGDYPLTMAGPQWKKFKSSFCEFIGVLVRQCQYSIIYDEYMMDTVISLLTGLSDSQVRAFRHTSTLAAMKLMTALVNVALNLSINMDNTQRQYEAERNKMIGKRANERLELLLQKRKELQENQDEIENMMNAIFKGVFVHRYRDAIAEIRAICIEEIGIWMKMYSDAFLNDSYLKYVGWTMHDKQGEVRLKCLTALQGLYYNKELNSKLELFTSRFKDRIVSMTLDKEYDVAVQAIKLLTLVLQSSEEVLTAEDCENVYHLVYSAHRPVAVAAGEFLYKKLFSRRDPEEDGMMKRRGRQGPNANLVKTLVFFFLESELHEHAAYLVDSMWDCATELLKDWECMNSLLLEEPLSGEEALTDRQESALIEIMLCTIRQAAECHPPVGRGTGKRVLTAKEKKTQLDDRTKITELFAVALPQLLAKYSVDAEKVTNLLQLPQYFDLEIYTTGRLEKHLDALLRQIRNIVEKHTDTDVLEACSKTYHALCNEEFTIFNRVDISRSQLIDELADKFNRLLEDFLQEGEEPDEDDAYQVLSTLKRITAFHNAHDLSKWDLFACNYKLLKTGIENGDMPEQIVIHALQCTHYVILWQLAKITESSSTKEDLLRLKKQMRVFCQICQHYLTNVNTTVKEQAFTILCDILMIFSHQIMSGGRDMLEPLVYTPDSSLQSELLSFILDHVFIEQDDDNNSADGQQEDEASKIEALHKRRNLLAAFCKLIVYTVVEMNTAADIFKQYMKYYNDYGDIIKETMSKTRQIDKIQCAKTLILSLQQLFNEMIQENGYNFDRSSSTFSGIKELARRFALTFGLDQLKTREAIAMLHKDGIEFAFKEPNPQGESHPPLNLAFLDILSEFSSKLLRQDKRTVYVYLEKFMTFQMSLRREDVWLPLMSYRNSLLAGGDDDTMSVI
6TSB , Knot 156 367 0.83 40 207 347 
GPAMKTLKDSKKLVRPQITDPYNPIVENANCPDINPIVAEYVLGNPTNVDAQLLDAVIFAFAEIDQSGNLFIPYPRFLNQLLALKGEKPSLKVIVAIGGWGAEGFSDAALTPTSRYNFARQVNQMINEYALDGIDIDWEYPGSSASGITSRPQDRENFTLLLTAIRDVIGDDKWLSVAGTGDRGYINSSAEIDKIAPIIDYFNLMSYDFTAGETGPNGRKHQANLFDSDLSLPGYSVDAMVRNLENAGMPSEKILLGIPFYGRLGATITRTYDELRRDYINKNGYEYRFDNTAQVPYLVKDGDFAMSYDDALSIFLKTQYVLRNCLGGVFSWTSTYDQANILARTMSIGINDPEVLKEELEGIYGQF

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 \(c\).

\(|P_{f(6VNH_1)}(2) \setminus P_{f(4PJW_1)}(2)|=31\), \(|P_{f(4PJW_1)}(2) \setminus P_{f(6VNH_1)}(2)|=152\). 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 1:00000000001010101001000010110011010110101000011000010111100100000001001000101100100001100011000110001011100110101000100000010010110000010011001100001000110001110000010110111001110000000100110011110110010000101100110111110011001000001110110111000010111101101100010110101010010111000100010001010011101001000111
Pair \(Z_2\) Length of longest common subsequence
6VNH_1,4PJW_1 183 4
6VNH_1,6TSB_1 164 3
4PJW_1,6TSB_1 167 5

Newick tree

 
[
	4PJW_1:89.37,
	[
		6VNH_1:82,6TSB_1:82
	]:7.37
]

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 \((0.85)(0.8)(\frac{1289 }{\log_{20} 1289}-\frac{308}{\log_{20}308})=257.\)
Status Protein1 Protein2 d d1/2
Query variables 6VNH_1 4PJW_1 329 213.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]