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Parikh vectors
5MMN_1 6TNY_1 8FHP_1 Letter Amino acid
18 78 52 E Glutamic acid
17 66 78 S Serine
22 82 76 V Valine
12 88 48 R Arginine
7 19 43 N Asparagine
15 45 85 I Isoleucine
6 78 59 P Proline
16 60 51 D Aspartic acid
1 26 18 C Cysteine
1 10 20 W Tryptophan
25 82 88 G Glycine
12 46 50 K Lycine
18 21 21 H Histidine
16 128 102 L Leucine
6 23 24 M Methionine
7 39 57 F Phenylalanine
12 47 57 T Threonine
6 33 36 Y Tyrosine
12 81 93 A Alanine
5 55 34 Q Glutamine 

5MMN_1|Chain A|DNA gyrase subunit B|Escherichia coli O157:H7 (83334)
>6TNY_1|Chain A|DNA polymerase delta catalytic subunit|Homo sapiens (9606)
>8FHP_1|Chains A, B|Solute carrier family 12 member 2,Solute carrier family 12 member 3 chimera|Danio rerio (7955)
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 \(w=f(c)\)
5MMN , Knot 105 234 0.81 40 150 220 
MRGSHHHHHHGIDHMSNSYDSSSIKVLKGLDAVRKRPGMYIGDTDDGTGLHHMVFEVVDNAIDEALAGHCKEIIVTIHADNSVSVQDDGRGIPTGIHPEEGVSAAEVIMTVLHAGGKFDDNSYKVSGGLHGVGVSVVNALSQKLELVIQREGKIHRQIYEHGVPQAPLAVTGETEKTGTMVRFWPSLETFTNVTEFEYEILAKRLRELSFLNSGVSIRLRDKRDGKEDHFHYEG
6TNY , Knot 395 1107 0.83 40 320 952 
MDGKRRPGPGPGVPPKRARGGLWDDDDAPRPSQFEEDLALMEEMEAEHRLQEQEEEELQSVLEGVADGQVPPSAIDPRWLRPTPPALDPQTEPLIFQQLEIDHYVGPAQPVPGGPPPSRGSVPVLRAFGVTDEGFSVCCHIHGFAPYFYTPAPPGFGPEHMGDLQRELNLAISRDSRGGRELTGPAVLAVELCSRESMFGYHGHGPSPFLRITVALPRLVAPARRLLEQGIRVAGLGTPSFAPYEANVDFEIRFMVDTDIVGCNWLELPAGKYALRLKEKATQCQLEADVLWSDVVSHPPEGPWQRIAPLRVLSFDIECAGRKGIFPEPERDPVIQICSLGLRWGEPEPFLRLALTLRPCAPILGAKVQSYEKEEDLLQAWSTFIRIMDPDVITGYNIQNFDLPYLISRAQTLKVQTFPFLGRVAGLCSNIRDSSFQSKQTGRRDTKVVSMVGRVQMDMLQVLLREYKLRSYTLNAVSFHFLGEQKEDVQHSIITDLQNGNDQTRRRLAVYCLKDAYLPLRLLERLMVLVNAVEMARVTGVPLSYLLSRGQQVKVVSQLLRQAMHEGLLMPVVKSEGGEDYTGATVIEPLKGYYDVPIATLDFSSLYPSIMMAHNLCYTTLLRPGTAQKLGLTEDQFIRTPTGDEFVKTSVRKGLLPQILENLLSARKRAKAELAKETDPLRRQVLDGRQLALKVSANSVYGFTGAQVGKLPCLEISQSVTGFGRQMIEKTKQLVESKYTVENGYSTSAKVVYGDTDSVMCRFGVSSVAEAMALGREAADWVSGHFPSPIRLEFEKVYFPYLLISKKRYAGLLFSSRPDAHDRMDCKGLEAVRRDNCPLVANLVTASLRRLLIDRDPEGAVAHAQDVISDLLCNRIDISQLVITKELTRAASDYAGKQAHVELAERMRKRDPGSAPSLGDRVPYVIISAAKGVAAYMKSEDPLFVLEHSLPIDTQYYLEQQLAKPLLRIFEPILGEGRAEAVLLRGDHTRCKTVLTGKVGGLLAFAKRRNCCIGCRTVLSHQGAVCEFCQPRESELYQKEVSHLNALEERFSRLWTQCQRCQGSLHEDVICTSRDCPIFYMRKKVRKDLEDQEQLLRRFGPPGPEAW
8FHP , Knot 397 1092 0.84 40 337 958 
MSASPPISAGDYLSAPEPDALKPAGPTPSQSRFQVDLVTESAGDGETTVGFDSSPPEYVAEPPPDGLRDSVSGGEKAKGRFRVVNFAASSPDAAPAETAQNGDTVMSEGSLHSSTGGQQHHHYDTHTNTYYLRTFGHNTIDAVPKIDFYRQTAAPLGEKLIRPTLSELHDELDKEPFEDGFANGEELTPAEESAAKDVSESKEPVRFGWVKGVMIRCMLNIWGVILYLRLPWITAQAGIVLTWIIILLSVTVTSITGLSISAISTNGKVKSGGTYFLISRSLGPELGGSIGLIFAFANAVGVAMHTVGFAATVRDLLQEYGAPIVDPINDIRIIGVVSVTVLLAISLAGMEWESKAQVLFFLVIMVSFANYLVGTLIPPSEDKASKGFFSYRADIFVQNLVPDWRGPDGTFFGMFSIFFPSATGILAGANISGDLKDPAIAIPKGTLMAIFWTTISYLAISATIGSCVVRDASGVLNDTVTPGWGACEGLACSYGWNFTECTQQHSCHYGLINYYQTMSMVSGFAPLITAGIFGATLSSALACLVSAAKVFQCLCEDQLYPLIGFFGKGYGKNKEPVRGYLLAYAIAVAFIIIAELNTIAPIISNFFLCSYALINFSCFHASITNSPGWRPSFQYYNKWAALFGAIISVVIMFLLTWWAALIAIGVVLFLLLYVIYKKPEVNWGSSVQAGSYNLALSYSVGLNEVEDHIKNYRPQCLVLTGPPNFRPALVDFVGTFTRNLSLMICGHVLIGPHKQRMPELQLIANGHTKWLNKRKIKAFYSDVIAEDLRRGVQILMQAAGLGRMKPNILVVGFKKNWQSAHPATVEDYIGILHDAFDFNYGVCVMRMREGLNVSKMMQAHINPVFDPAEDGKEASARVDPKALVKEEQATTIFQSEQGKKTIDIYWLFDDGGLTLLIPYLLGRKRRWSKCKIRVFVGGQINRMDQERKAIISLLSKFRLGFHEVHILPDINQNPRAEHTKRFEDMIAPFRLNDGFKDEATVNEMRRDCPWKISDEEITKNRVKSLRQVRLNEIVLDYSRDAALIVITLPIGRKGKCPSSLYMAWLETLSQDLRPPVILIRGNQENVLTFYCQ

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 \(c\).

\(|P_{f(5MMN_1)}(2) \setminus P_{f(6TNY_1)}(2)|=19\), \(|P_{f(6TNY_1)}(2) \setminus P_{f(5MMN_1)}(2)|=189\). 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 1:101000000011001000000001011011011000111011000010110011101100110011110000111010100010100010111011010011011011101101110100000010111011110110110001011100010100010001110111110100000101101110100100100100011100100101100110101000001000010001
Pair \(Z_2\) Length of longest common subsequence
5MMN_1,6TNY_1 208 4
5MMN_1,8FHP_1 219 4
6TNY_1,8FHP_1 95 4

Newick tree

 
[
	5MMN_1:12.21,
	[
		6TNY_1:47.5,8FHP_1:47.5
	]:72.71
]

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 \((0.85)(0.8)(\frac{1341 }{\log_{20} 1341}-\frac{234}{\log_{20}234})=291.\)
Status Protein1 Protein2 d d1/2
Query variables 5MMN_1 6TNY_1 366 219
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]