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Parikh vectors
8WIL_1 8ARR_1 8RGA_1 Letter Amino acid
35 10 15 Y Tyrosine
51 12 44 V Valine
60 24 46 A Alanine
68 14 47 G Glycine
84 24 76 L Leucine
21 5 23 F Phenylalanine
56 10 41 T Threonine
21 2 3 W Tryptophan
28 7 16 N Asparagine
41 10 47 I Isoleucine
39 19 47 S Serine
60 29 40 E Glutamic acid
56 17 32 K Lycine
32 5 18 P Proline
74 13 26 R Arginine
57 14 19 D Aspartic acid
5 2 1 C Cysteine
26 7 26 Q Glutamine
23 3 13 H Histidine
35 9 19 M Methionine 

8WIL_1|Chain A|Jingmen tick virus NSP1|Jingmen tick virus (1491393)
>8ARR_1|Chain A|14-3-3 protein sigma|Homo sapiens (9606)
>8RGA_1|Chains A, B|Multidrug resistance ABC transporter ATP-binding/permease protein BmrA|Bacillus subtilis (1423)
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 \(w=f(c)\)
8WIL , Knot 326 872 0.84 40 302 792 
MEPTLGDPTELGRQAKKEMNAMTYEEFDRMKFRGVVAEVYAGDKPSKGYDKLRTLLDLMDRPKLGTTVDLCAGRGGWSQLVGELEGPGDITAVSLWERGREEWMADPAIRRINANVKNVKPWRVDTLLFDGGETFKRDQNIMKEESYNDALLDAVDAWMMQATPPTNFVIKIQVPYTQKAMRLIEKWQVKTKKGRLVRLAGDRLSNTVMYFISDRLETQIRGRITSFLRELRERRVDRSLTSDPALQYERITPEWTAEAVIEGCAPLTPLNMTRSIAEMHMEYPPLGITRFFKELGYKIAKKKGSEGTRKNRFVGQLIEPLRRVLERHHLFGAWQLTSTTPRAVFNMFRSKVDRAPVELHSHYPGLKKMYDILADLWLERYGSMKRLTEEEMASAINRRGAMGYQMDNRNYGDLGAYWDSGDWRQDVNTFKRALLSGTPTHAVYNTTAKKEKTKNLTRQVNKGSRIIQYLPADARLYELKVLGGLHKYLEKCGWSVAGQGLYKYGDRVKKSMDATGAAISEDVAGWDTKISKGLLTLESHMFTKLAEDEEMAREIHHLYRLYADPHMVVQREIEGEVHDVLLRGRGQVSSGRQPTYAANTITNFITTTYGMAVTLGIPEADWPRLIRDLTDERGNRRLLVSGDDKVLFLRGDEARVYASSAYRISNDMGLVRKDMALEQESEIIVDVKEISFCSHRYWPVKYGNEIHYMPVRDVGEIFAKATMALGVYKDDMTQEAWARVQGLNMLVNYHHIPECRMLALAILSVTRIGLNLKGITKGWMMSTEWLRDDLAPDTIHALITEGRTSGWDQLGYVDFKDRKGILLRPDTNYKNWRRDLPEKVRQLREDGQYKDWLQKMAVFGGSSSGTHHHHHH
8ARR , Knot 108 236 0.83 40 164 229 
GAMGSMERASLIQKAKLAEQAERYEDMAAFMKGAVEKGEELSCEERNLLSVAYKNVVGGQRAAWRVLSSIEQKSNEEGSEEKGPEVREYREKVETELQGVCDTVLGLLDSHLIKEAGDAESRVFYLKMKGDYYRYLAEVATGDDKKRIIDSARSAYQEAMDISKKEMPPTNPIRLGLALNFSVFHYEIANSPEEAISLAKTTFDEAMADLHTLSEDSYKDSTLIMQLLRDNLTLWT
8RGA , Knot 226 599 0.80 40 236 551 
MSSSHHHHHHMPTKKQKSKSKLKPFFALVRRTNPSYGKLAFALALSVVTTLVSLLIPLLTKQLVDGFSMSNLSGTQIGLIALVFFVQAGLSAYATYALNYNGQKIISGLRELLWKKLIKLPVSYFDTNASGETVSRVTNDTMVVKELITTHISGFITGIISVIGSLTILFIMNWKLTLLVLVVVPLAALILVPIGRKMFSISRETQDETARFTGLLNQILPEIRLVKASNAEDVEYGRGKMGISSLFKLGVREAKVQSLVGPLISLVLMAALVAVIGYGGMQVSSGELTAGALVAFILYLFQIIMPMGQITTFFTQLQKSIGATERMIEILAEEEEDTVTGKQIENAHLPIQLDRVSFGYKPDQLILKEVSAVIEAGKVTAIVGPSGGGKTTLFKLLERFYSPTAGTIRLGDEPVDTYSLESWREHIGYVSQESPLMSGTIRENICYGLERDVTDAEIEKAAEMAYALNFIKELPNQFDTEVGERGIMLSGGQRQRIAIARALLRNPSILMLDAATSSLDSQSEKSVQQALEVLMEGRTTIVIAHRLSTVVDADQLLFVEKGEITGRGTHHELMASHGLYRDFAEQQLKMNADLENKAG

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 \(c\).

\(|P_{f(8WIL_1)}(2) \setminus P_{f(8ARR_1)}(2)|=155\), \(|P_{f(8ARR_1)}(2) \setminus P_{f(8WIL_1)}(2)|=17\). 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 1:10101101001100100010110000100101011110101100100100010011011001011001010110111001110101110101101100100011101110010101001011010011101100100000110000000111011011110101100111010110000110110010100001011011100100011011000100010101001100100001000100011100001010101011101011101101000110101001111100110011001100010010000011101101100110000111110100001011101100010011101000011100100111011100010100100001101100011110010000010111010010100010010011101010011000010000000100010010011001110101001011111000100011011101100010010001010111100011110001001110100011001100001100100100101010111000101010011101010100100100110010011000011110111101011011001000010001110100011110100101010010010001111000111000001110100101000001110010010011100110111010111110000100011101011011100001100011111110100111010110011110001100011100101110010001100110101000011110100000010001100100100010000110011111100010000000
Pair \(Z_2\) Length of longest common subsequence
8WIL_1,8ARR_1 172 4
8WIL_1,8RGA_1 124 6
8ARR_1,8RGA_1 142 5

Newick tree

 
[
	8ARR_1:83.72,
	[
		8WIL_1:62,8RGA_1:62
	]:21.72
]

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 \((0.85)(0.8)(\frac{1108 }{\log_{20} 1108}-\frac{236}{\log_{20}236})=233.\)
Status Protein1 Protein2 d d1/2
Query variables 8WIL_1 8ARR_1 293 184
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]