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Parikh vectors
4CEI_1 8URX_1 2OFV_1 Letter Amino acid
84 3 13 D Aspartic acid
5 0 3 C Cysteine
56 6 11 Q Glutamine
120 9 26 E Glutamic acid
69 7 18 I Isoleucine
49 1 15 P Proline
54 2 12 Y Tyrosine
52 5 11 F Phenylalanine
28 1 11 N Asparagine
29 4 4 H Histidine
87 10 15 K Lycine
10 1 7 W Tryptophan
104 7 19 A Alanine
74 8 17 R Arginine
66 12 16 G Glycine
138 3 27 L Leucine
24 2 9 M Methionine
65 3 10 S Serine
53 5 18 T Threonine
65 14 15 V Valine 

4CEI_1|Chain A|ATP-DEPENDENT HELICASE/NUCLEASE SUBUNIT A|BACILLUS SUBTILIS SUBSP. SUBTILIS STR. 168 (224308)
>8URX_1|Chain A[auth 0]|Ribosomal protein L21|Escherichia coli (562)
>2OFV_1|Chains A, B|Proto-oncogene tyrosine-protein kinase LCK|Homo sapiens (9606)
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 \(w=f(c)\)
4CEI , Knot 432 1232 0.83 40 319 1056 
MNIPKPADSTWTDDQWNAIVSTGQDILVAAAAGSGKTAVLVERMIRKITAEENPIDVDRLLVVTFTNASAAEMKHRIAEALEKELVQRPGSLHIRRQLSLLNRASISTLHSFCLQVLKKYYYLIDLDPGFRIADQTEGELIGDEVLDELFEDEYAKGEKAFFELVDRYTTDRHDLDLQFLVKQVYEYSRSHPNPEAWLESFVHLYDVSEKSAIEELPFYQYVKEDIAMVLNGAKEKLLRALELTKAPGGPAPRADNFLDDLAQIDELIQHQDDFSELYKRVPAVSFKRAKAVKGDEFDPALLDEATDLRNGAKKLLEKLKTDYFTRSPEQHLKSLAEMKPVIETLVQLVISYGKRFEAAKQEKSIIDFSDLEHYCLAILTAENDKGEREPSEAARFYQEQFHEVLVDEYQDTNLVQESILQLVTSGPEETGNLFMVGDVKQSIYRFRLAEPLLFLSKYKRFTESGEGTGRKIDLNKNFRSRADILDSTNFLFKQLMGGKIGEVDYDEQAELKLGAAYPDNDETETELLLIDNAEDTDASEEAEELETVQFEAKAIAKEIRKLISSPFKVYDGKKKTHRNIQYRDIVILLRSMPWAPQIMEELRAQGIPVYANLTSGYFEAVEVAVALSVLKVIDNPYQDIPLASVLRSPIVGADENELSLIRLENKKAPYYEAMKDYLAAGDRSDELYQKLNTFYGHLQKWRAFSKNHSVSELIWEVYRDTKYMDYVGGMPGGKQRQANLRVLYDRARQYESTAFRGLFRFLRFIERMQERGDDLGTARGLSEQEDVVRLMTIHSSKGLEFPVVFVAGLGRNFNMMDLNKSYLLDKELGFGTKYIHPQLRISYPTLPLIAMKKKMRRELLSEELRVLYVALTRAKEKLFLIGSCKDHQKQLAKWQASASQTDWLLPEFDRYQARTYLDFIGPALARHRDLGDLAGVPAHADISGHPARFAVQMIHSYDLLDDDLEERMEEKSERLEAIRRGEPVPGSFAFDEKAREQLSWTYPHQEVTQIRTKQSVSEIKRKREYEDEYSGRAPVKPADGSILYRRPAFMMKKGLTAAEKGTAMHTVMQHIPLSHVPSIEEAEQTVHRLYEKELLTEEQKDAIDIEEIVQFFHTEIGGQLIGAKWKDREIPFSLALPAKEIYPDAHEADEPLLVQGIIDCLYETEDGLYLLAYKSDRIEGKFQHGFEGAAPILKKRYETQIQLYTKAVEQIAKTKVKGCALYFFDGGHILTL
8URX , Knot 53 103 0.79 38 82 98 
MYAVFQSGGKQHRVSEGQTVRLEKLDIATGETVEFAEVLMIANGEEVKIGVPFVDGGVIKAEVVAHGRGEKVKIVKFRRRKHYRKQQGHRQWFTDVKITGISA
2OFV , Knot 125 277 0.84 40 190 274 
TQKPQKPWWEDEWEVPRETLKLVERLGAGQFGEVWMGYYNGHTKVAVKSLKQGSMSPDAFLAEANLMKQLQHQRLVRLYAVVTQEPIYIITEYMENGSLVDFLKTPSGIKLTINKLLDMAAQIAEGMAFIEERNYIHRDLRAANILVSDTLSCKIADFGLARLIEDNEYTAREGAKFPIKWTAPEAINYGTFTIKSDVWSFGILLTEIVTHGRIPYPGMTNPEVIQNLERGYRMVRPDNCPEELYQLMRLCWKERPEDRPTFDYLRSVLEDFFTATE

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 \(c\).

\(|P_{f(4CEI_1)}(2) \setminus P_{f(8URX_1)}(2)|=240\), \(|P_{f(8URX_1)}(2) \setminus P_{f(4CEI_1)}(2)|=3\). 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 1:10110110001000010111001001111111101001111001100101000110100111101001011010001101100011001101010001011001010010010101100000110101110110000101110011001100001010011101100000000010101110010000000101011100110100100001100111000100011111011000110110100111111101001100110100110000010010001111010010110100101111001001001100110010000100010001001101011100110111001001011000001101001000011110100001000100110100001001110000000110001101100110001011111010001001011011111000001000101010010100010001011000011100111101101000001010111101000000001111001000010001001001010101110010011001101001000000010000111110011111011001010111101010010101101111101101100100011110110011111000010110100001100011000111100000100010010101001011000001001110100000010011111110000101011000100000011011101101100100010011010110000011011010000110111111111100101101000011000111100010101010010111111000100011000101101110010001111100000000110101010000111101000010001011111110000110111111010101011011101100001100010001000000101100101111011100010001010010001001000001001000000000001011101101011000111110011011001011001100111001101001000100100001100000011010011011000111011110100001110111110010101001001111011100100000110111000001010100110111111000000010100011001100010101101101101101
Pair \(Z_2\) Length of longest common subsequence
4CEI_1,8URX_1 243 4
4CEI_1,2OFV_1 163 4
8URX_1,2OFV_1 178 3

Newick tree

 
[
	8URX_1:11.61,
	[
		4CEI_1:81.5,2OFV_1:81.5
	]:32.11
]

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 \((0.85)(0.8)(\frac{1335 }{\log_{20} 1335}-\frac{103}{\log_{20}103})=332.\)
Status Protein1 Protein2 d d1/2
Query variables 4CEI_1 8URX_1 411 221.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]