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Parikh vectors
2WNW_1 5EAW_1 3PDI_1 Letter Amino acid
29 72 34 R Arginine
26 82 32 E Glutamic acid
15 26 14 H Histidine
21 53 22 I Isoleucine
12 23 14 M Methionine
37 93 21 S Serine
31 64 55 A Alanine
23 38 10 Q Glutamine
45 142 43 L Leucine
18 51 26 P Proline
10 22 10 C Cysteine
13 58 23 K Lycine
10 48 24 T Threonine
29 53 33 D Aspartic acid
33 55 40 G Glycine
19 33 11 F Phenylalanine
10 10 7 W Tryptophan
20 24 14 Y Tyrosine
24 71 36 V Valine
22 38 14 N Asparagine 

2WNW_1|Chains A, B|ACTIVATED BY TRANSCRIPTION FACTOR SSRB|SALMONELLA ENTERICA SUBSP. ENTERICA SEROVAR TYPHIMURIUM STR. LT2 (99287)
>5EAW_1|Chains A, B|DNA replication ATP-dependent helicase/nuclease DNA2|Mus musculus (10090)
>3PDI_1|Chains A, C, E, G|Nitrogenase MoFe cofactor biosynthesis protein NifE|Azotobacter vinelandii (322710)
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 \(w=f(c)\)
2WNW , Knot 186 447 0.84 40 247 423 
MKGRLISSDPYRQQFLVERAVSFSHRQRDCSELISVLPRHALQQIDGFGGSFTEGAGVVFNSMSEKTKAQFLSLYFSAQEHNYTLARMPIQSCDFSLGNYAYVDSSADLQQGRLSFSRDEAHLIPLISGALRLNPHMKLMASPWSPPAFMKTNNDMNGGGKLRRECYADWADIIINYLLEYRRHGINVQALSVQNEPVAVKTWDSCLYSVEEETAFAVQYLRPRLARQGMDEMEIYIWDHDKDGLVDWAELAFADEANYKGINGLAFHWYTGDHFSQIQYLAQCLPDKKLLFSEGCVPMESDAGSQIRHWHTYLHDMIGNFKSGCSGFIDWNLLLNSEGGPNHQGNLCEAPIQYDAQNDVLRRNHSWYGIGHFCRYVRPGARVMLSSSYDNLLEEVGFVNPDGERVLVVYNRDVQERRCRVLDGDKEIALTLPPSGASTLLWRQESI
5EAW , Knot 388 1056 0.85 40 321 935 
MEPLDELDLLLLEEDGGAEAVPRVELLRKKADALFPETVLSRGVDNRYLVLAVETSQNERGAEEKRLHVTASQDREHEVLCILRNGWSSVPVEPGDIVHLEGDCTSEPWIIDDDFGYFILYPDMMISGTSVASSIRCLRRAVLSETFRGSDPATRQMLIGTILHEVFQKAISESFAPERLQELALQTLREVRHLKEMYRLNLSQDEILCEVEEYLPSFSKWAEDFMRKGPSSEFPQMQLSLPSDGSNRSSPCNIEVVKSLDIEESIWSPRFGLKGKIDVTVGVKIHRDCKMKYKVMPLELKTGKESNSIEHRSQVVLYTLLSQERREDPEAGWLLYLKTGQMYPVPANHLDKRELLKLRNWLAASLLHRVSRAAPGEEARLSALPQIIEEEKTCKYCSQIGNCALYSRAVEEQGDDASIPEAMLSKIQEETRHLQLAHLKYFSLWCLMLTLESQSKDNRKTHQSIWLTPASELEESGNCVGNLVRTEPVSRVCDGQYLHNFQRKNGPMPATNLMAGDRIILSGEERKLFALSKGYVKKMNKAAVTCLLDRNLSTLPATTVFRLDREERHGDISTPLGNLSKLMESTDPSKRLRELIIDFREPQFIAYLSSVLPHDAKDTVANILKGLNKPQRQAMKRVLLSKDYTLIVGMPGTGKTTTICALVRILSACGFSVLLTSYTHSAVDNILLKLAKFKVGFLRLGQSHKVHPDIQKFTEEEICRSRSIASLAHLEELYNSHPIVATTCMGINHPIFSRKTFDFCIVDEASQISQPVCLGPLFFSRRFVLVGDHQQLPPLVVNREARALGMSESLFKRLERNESAVVQLTVQYRMNRKIMSLSNKLTYAGKLECGSDRVANAVLALPNLKDARLSLQLYADYSDSPWLAGVLEPDNPVCFLNTDKVPAPEQVENGGVSNVTEARLIVFLTSTFIKAGCSPSDIGVIAPYRQQLRIISDLLARSSVGMVEVNTVDKYQGRDKSLILVSFVRSNEDGTLGELLKDWRRLNVALTRAKHKLILLGSVSSLKRFPPLGTLFDHLNAEQLILDLPSREHESLSHIL
3PDI , Knot 195 483 0.83 40 238 451 
MHHHHHHHMKAKDIAELLDEPACSHNKKEKSGCAKPKPGATDGGCSFDGAQIALLPVADVAHIVHGPIACAGSSWDNRGTRSSGPDLYRIGMTTDLTENDVIMGRAEKRLFHAIRQAVESYSPPAVFVYNTCVPALIGDDVDAVCKAAAERFGTPVIPVDSAGFYGTKNLGNRIAGEAMLKYVIGTREPDPLPVGSERPGIRVHDVNLIGEYNIAGEFWHVLPLLDELGLRVLCTLAGDARYREVQTMHRAEVNMMVCSKAMLNVARKLQETYGTPWFEGSFYGITDTSQALRDFARLLDDPDLTARTEALIAREEAKVRAALEPWRARLEGKRVLLYTGGVKSWSVVSALQDLGMKVVATGTKKSTEEDKARIRELMGDDVKMLDEGNARVLLKTVDEYQADILIAGGRNMYTALKGRVPFLDINQEREFGYAGYDGMLELVRQLCITLECPVWEAVRRPAPWDIPASQDAAPSAPARSANA

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 \(c\).

\(|P_{f(2WNW_1)}(2) \setminus P_{f(5EAW_1)}(2)|=31\), \(|P_{f(5EAW_1)}(2) \setminus P_{f(2WNW_1)}(2)|=105\). 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 1:101011000100001110011010000000001101110011001011110100111111001000001011010101000000110111000010110010100010100101010000101111101110101010111011011111000001011101000001011011100110000011010110100011110010001001000011110010101100110010101100000111011011110010001101111010010010010011001100011100101110001100100100010011101001001110101110001110001010011100010001100000101110100010111011100000011001111010100111100001000000110100011101110110011100001
Pair \(Z_2\) Length of longest common subsequence
2WNW_1,5EAW_1 136 5
2WNW_1,3PDI_1 159 4
5EAW_1,3PDI_1 141 5

Newick tree

 
[
	3PDI_1:77.36,
	[
		2WNW_1:68,5EAW_1:68
	]:9.36
]

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 \((0.85)(0.8)(\frac{1503 }{\log_{20} 1503}-\frac{447}{\log_{20}447})=269.\)
Status Protein1 Protein2 d d1/2
Query variables 2WNW_1 5EAW_1 340 241
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]