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Parikh vectors
4CBG_1 6MOI_1 7SCN_1 Letter Amino acid
26 5 20 P Proline
2 1 7 W Tryptophan
27 3 15 Y Tyrosine
28 4 13 R Arginine
29 14 18 I Isoleucine
13 2 10 F Phenylalanine
27 3 31 S Serine
39 16 22 V Valine
28 13 15 K Lycine
21 13 26 N Asparagine
20 5 7 Q Glutamine
36 20 23 G Glycine
14 16 10 H Histidine
44 32 28 L Leucine
36 10 19 T Threonine
38 39 16 A Alanine
23 20 9 D Aspartic acid
3 0 9 C Cysteine
41 9 22 E Glutamic acid
21 6 2 M Methionine 

4CBG_1|Chains A, B, C, D|SERINE PROTEASE NS3|CLASSICAL SWINE FEVER VIRUS (11096)
>6MOI_1|Chain A|Dimeric DARPing CCR5 (C_angle_R5)|synthetic construct (32630)
>7SCN_1|Chains A, E[auth C], I[auth G]|Hemagglutinin HA1 chain|Influenza A virus (strain A/New Zealand:South Canterbury/35/2000 H1N1) (363066)
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 \(w=f(c)\)
4CBG , Knot 205 516 0.82 40 252 477 
MASHHHHHHHENLYFQGMSGIQTVSKSATDLTEMVKKITTMNRGEFRQITLATGAGKTTELPRSVIEEIGRHKRVLVLIPLRAAAESVYQYMRQKHPSIAFNLRIGEMKEGDMATGITYASYGYFCQMSQPKLRAAMVEYSFIFLDEYHCATPEQLAIMGKIHRFSENLRVVAMTATPAGTVTTTGQKHPIEEFIAPEVMKGEDLGSEYLDIAGLKIPVEEMKNNMLVFVPTRNMAVEAAKKLKAKGYNSGYYYSGEDPSNLRVVTSQSPYVVVATNAIESGVTLPDLDVVVDTGLKCEKRIRLSPKMPFIVTGLKRMAVTIGEQAQRRGRVGRVKPGRYYRSQETPVGSKDYHYDLLQAQRYGIEDGINITKSFREMNYDWSLYEEDSLMITQLEILNNLLISEELPMAVKNIMARTDHPEPIQLAYNSYETQVPVLFPKIRNGEVTDTYDNYTFLNARKLGDDVPPYVYATEDEDLAVELLGLDWPDPGNQGTVEAGRALKQVVGLSTAENALL
6MOI , Knot 73 231 0.57 38 100 152 
MGHHHHHHSDLGKKLLKAARAGQDDEVRILMANGADVNATDIWDATPLHLAALIGHLEIVEVLLKNGADVNASDITGTTPLHLAATMGHKEIVEVLLKYGADVNAYDLNGATPLHLAARMGHAEIVAVLLEYGADVNAQDAAGMTPLHLAAANGHDEIVLILLLKGADVNAKDAAGMTPLHLAAANGHKRIVLVLILAGADVNAQDKFGKTAFDISIDNGNEDLAKILQKQ
7SCN , Knot 139 322 0.83 40 197 305 
DTICIGYHANNSTDTVDTVLEKNVTVTHSVNLLEDSHNGKLCLLKGIAPLQLGNCSVAGWILGNPECELLISKESWSYIVETPNPENGTCYPGYFADYEELREQLSSVSSFERFEIFPKESSWPNHTVTGVSASCSHNGKSSFYRNLLWLTGKNGLYPNLSKSYVNNKEKEVLVLWGVHHPPNIGNQRALYHTENAYVSVVSSHYSRRFTPEIAKRPKVRDQEGRINYYWTLLEPGDTIIFEANGNLIAPWYAFALSRGFGSGIITSNAPMDECDAKCQTPQGAINSSLPFQNVHPVTIGECPKYVRSAKLRMVTGLRNIPS

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 \(c\).

\(|P_{f(4CBG_1)}(2) \setminus P_{f(6MOI_1)}(2)|=174\), \(|P_{f(6MOI_1)}(2) \setminus P_{f(4CBG_1)}(2)|=22\). 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 1:110000000000101011011001000100100110010010010100101101110000110011001100001111111011100100010000101110101101001011011001001010010010101111000111100000101001111101001000101111010111010001000110011110110100110001011110111001000111111000111011001010100010000100100101100001011110011001101101011100110000010101011111011001110110010001011010110000000011100000001101000110011010001001000101000001110010110011100011111001110000101101100000001111110100101000000001101001100111010100000111011110110110010101101100111100100111
Pair \(Z_2\) Length of longest common subsequence
4CBG_1,6MOI_1 196 6
4CBG_1,7SCN_1 171 5
6MOI_1,7SCN_1 191 4

Newick tree

 
[
	6MOI_1:10.23,
	[
		4CBG_1:85.5,7SCN_1:85.5
	]:14.73
]

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 \((0.85)(0.8)(\frac{747 }{\log_{20} 747}-\frac{231}{\log_{20}231})=143.\)
Status Protein1 Protein2 d d1/2
Query variables 4CBG_1 6MOI_1 186 121.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]