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Parikh vectors
3SMH_1 4BWV_1 9FDQ_1 Letter Amino acid
26 22 105 A Alanine
5 4 42 M Methionine
29 18 96 V Valine
17 13 45 P Proline
28 18 62 S Serine
37 7 39 N Asparagine
19 18 47 D Aspartic acid
1 5 2 C Cysteine
25 8 48 Q Glutamine
31 20 103 L Leucine
25 17 43 K Lycine
5 9 27 Y Tyrosine
11 17 17 H Histidine
22 15 72 I Isoleucine
21 17 55 F Phenylalanine
1 3 9 W Tryptophan
33 16 38 R Arginine
43 23 51 E Glutamic acid
27 20 89 G Glycine
12 13 67 T Threonine 

3SMH_1|Chains A, B, C, D, E, F|Allergen Ara h 1, clone P41B|Arachis hypogaea (3818)
>4BWV_1|Chains A, B|PHOSPHOADENOSINE-PHOSPHOSULPHATE REDUCTASE|PHYSCOMITRELLA PATENS (3218)
>9FDQ_1|Chain A[auth C]|Multidrug efflux pump subunit AcrB|Escherichia coli K-12 (83333)
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 \(w=f(c)\)
3SMH , Knot 172 418 0.82 40 215 387 
SRNNPFYFPSRRFSTRYGNQNGRIRVLQRFDQRSRQFQNLQNHRIVQIEAKPNTLVLPKHADADNILVIQQGQATVTVANGNNRKSFNLDEGHALRIPSGFISYILNRHDNQNLRVAKISMPVNTPGQFEDFFPASSRDQSSYLQGFSRNTLEAAFNAEFNEIRRVLLEENAGGEQEERGQRRWSTRSSENNEGVIVKVSKEHVEELTKHAKSVSKKGSEEEGDITNPINLREGEPDLSNNFGKLFEVKPDKKNPQLQDLDMMLTCVEIKEGALVLPHFNSKAMVIVVVNKGTGNLELVAVRKEQQQRGRREEEEDEDEEEEGSNREVRRYTARLKEGDVFIMPAAHPVAINASSELHLLGFGINAENNHRIFLAGDKDNVIDQIEKQAKDLAFPGSGEQVEKLIKNQKESHFVSARP
4BWV , Knot 128 283 0.85 40 192 273 
MGSSHHHHHHSSGLVPRGSHMASAVPVTTSGSSIDIKKCNEQARDARLQHLEAQALETLQKTVENFEKPAFPCALIAGDVVILDLLHRIGAFSDNKVKIIFIDTFHLFPETYKFLSEVEERYGFKAHVFHAADVNNKEAYDAKFGSDLFITDIEEYDRICKVEPFSRALKTLEVDAMINGRRRDHGAERAHLEVFEEGKMVKVQPLAYWEFRDCWDYLTKYSLPYHPLHDQGFPSIGDVQSTIPVPREKWFEYAGERSGRFQGLTNPDGSAKTECGIHVGGRT
9FDQ , Knot 377 1057 0.82 40 302 939 
MPNFFIDRPIFAWVIAIIIMLAGGLAILKLPVAQYPTIAPPAVTISASYPGADAKTVQDTVTQVIEQNMNGIDNLMYMSSNSDSTGTVQITLTFESGTDADIAQVQVQNKLQLAMPLLPQEVQQQGVSVEKSSSSFLMVVGVINTDGTMTQEDISDYVAANMKDAISRTSGVGDVQLFGSQYAMRIWMNPNELNKFQLTPVDVITAIKAQNAQVAAGQLGGTPPVKGQQLNASIIAQTRLTSTEEFGKILLKVNQDGSRVLLRDVAKIELGGENYDIIAEFNGQPASGLGIKLATGANALDTAAAIRAELAKMEPFFPSGLKIVYPYDTTPFVKISIHEVVKTLVEAIILVFLVMYLFLQNFRATLIPTIAVPVVLLGTFAVLAAFGFSINTLTMFGMVLAIGLLVDDAIVVVENVERVMAEEGLPPKEATRKSMGQIQGALVGIAMVLSAVFVPMAFFGGSTGAIYRQFSITIVSAMALSVLVALILTPALCATMLKPIAKGDHGEGKKGFFGWFNRMFEKSTHHYTDSVGGILRSTGRYLVLYLIIVVGMAYLFVRLPSSFLPDEDQGVFMTMVQLPAGATQERTQKVLNEVTHYYLTKEKNNVESVFAFNGFGFAGRGQNTGIAFVSLKDWADRPGEENKVEAITMRATRAFSQIKDAMVFAFNLPAIVELGTATGFDFELIDQAGLGHEKLTQARNQLLAEAAKHPDMLTSVRPNGLEDTPQFKIDIDQEKAQALGVSINDINTTLGAAWGGSYVNDFIDRGRVKKVYVMSEAKYRMLPDDIGDWYVRAADGQMVPFSAFSSSRWEYGSPRLERYNGLPSMEILGQAAPGKSTGEAMELMEQLASKLPTGVGYDWTGMSYQERLSGNQAPSLYAISLIVVFLCLAALYESWSIPFSVMLVVPLGVIGALLAATFRGLTNDVYFQVGLLTTIGLSAKNAILIVEFAKDLMDKEGKGLIEATLDAVRMRLRPILMTSLAFILGVMPLVISTGAGSGAQNAVGTGVMGGMVTATVLAIFFVPVFFVVVRRRFSRKNEDIEHSHTVDHHLEHHHHHH

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 \(c\).

\(|P_{f(3SMH_1)}(2) \setminus P_{f(4BWV_1)}(2)|=92\), \(|P_{f(4BWV_1)}(2) \setminus P_{f(3SMH_1)}(2)|=69\). 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 1:0000110110001000010001010110010000001001000011010101001111001010011110010101011010000010100101101101110011000000010110101110011010011110000000010110000101110101001001110001110000010001000000000111101000010010001001000100001010011010010101000110110101000010100101110010100111111010001111111001010101111000000010000000000000100001000010100101111111011110100010111111010000011111000011001000100111110100100110000000110101
Pair \(Z_2\) Length of longest common subsequence
3SMH_1,4BWV_1 161 3
3SMH_1,9FDQ_1 127 4
4BWV_1,9FDQ_1 170 6

Newick tree

 
[
	4BWV_1:88.27,
	[
		3SMH_1:63.5,9FDQ_1:63.5
	]:24.77
]

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 \((0.85)(0.8)(\frac{701 }{\log_{20} 701}-\frac{283}{\log_{20}283})=115.\)
Status Protein1 Protein2 d d1/2
Query variables 3SMH_1 4BWV_1 145 123
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]