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Parikh vectors
5RHW_1 7VJS_1 3NNE_1 Letter Amino acid
34 11 35 T Threonine
32 18 45 V Valine
23 6 21 I Isoleucine
30 32 38 L Leucine
15 3 19 M Methionine
20 13 28 S Serine
10 10 11 Y Tyrosine
14 7 20 N Asparagine
31 16 42 E Glutamic acid
9 5 20 H Histidine
25 6 11 K Lycine
9 9 13 Q Glutamine
22 32 35 P Proline
28 17 46 G Glycine
15 6 15 F Phenylalanine
9 3 9 W Tryptophan
42 18 55 A Alanine
36 22 38 R Arginine
25 10 40 D Aspartic acid
6 2 5 C Cysteine 

5RHW_1|Chain A|NS3 Helicase|Zika virus (64320)
>7VJS_1|Chain A|Alpha-ketoglutarate-dependent dioxygenase alkB homolog 6|Homo sapiens (9606)
>3NNE_1|Chains A, B, C, D, E, F, G, H|Choline oxidase|Arthrobacter globiformis (1665)
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 \(w=f(c)\)
5RHW , Knot 177 435 0.82 40 234 407 
MLKKKQLTVLDLHPGAGKTRRVLPEIVREAIKKRLRTVILAPTRVVAAEMEEALRGLPVRYMTTAVNVTHSGTEIVDLMCHATFTSRLLQPIRVPNYNLNIMDEAHFTDPSSIAARGYISTRVEMGEAAAIFMTATPPGTRDAFPDSNSPIMDTEVEVPERAWSSGFDWVTDHSGKTVWFVPSVRNGNEIAACLTKAGKRVIQLSRKTFETEFQKTKNQEWDFVITTDISEMGANFKADRVIDSRRCLKPVILDGERVILAGPMPVTHASAAQRRGRIGRNPNKPGDEYMYGGGCAETDEGHAHWLEARMLLDNIYLQDGLIASLYRPEADKVAAIEGEFKLRTEQRKTFVELMKRGDLPVWLAYQVASAGITYTDRRWCFDGTTNNTIMEDSVPAEVWTKYGEKRVLKPRWMDARVCSDHAALKSFKEFAAGKR
7VJS , Knot 108 246 0.80 40 147 232 
MEEQDARVPALEPFRVEQAPPVIYYVPDFISKEEEEYLLRQVFNAPKPKWTQLSGRKLQNWGGLPHPRGMVPERLPPWLQRYVDKVSNLSLFGGLPANHVLVNQYLPGEGIMPHEDGPLYYPTVSTISLGSHTVLDFYEPRRPEDDDPTEQPRPPPRPTTSLLLEPRSLLVLRGPAYTRLLHGIAAARVDALDAASSPPNAAACPSARPGACLVRGTRVSLTIRRVPRVLRAGLLLGKGTENLYFQ
3NNE , Knot 211 546 0.81 40 254 511 
MHIDNIENLSDREFDYIVVGGGSAGAAVAARLSEDPAVSVALVEAGPDDRGVPEVLQLDRWMELLESGYDWDYPIEPQENGNSFMRHARAKVMGGCSSHNACIAFWAPREDLDEWEAKYGATGWNAEAAWPLYKRLETNEDAGPDAPHHGDSGPVHLMNVPPKDPTGVALLDACEQAGIPRAKFNTGTTVVNGANFFQINRRADGTRSSSSVSYIHPIVEQENFTLLTGLRARQLVFDADRRCTGVDIVDSAFGHTHRLTARNEVVLSTGAIDTPKLLMLSGIGPAAHLAEHGIEVLVDSPGVGEHLQDHPEGVVQFEAKQPMVAESTQWWEIGIFTPTEDGLDRPDLMMHYGSVPFDMNTLRHGYPTTENGFSLTPNVTHARSRGTVRLRSRDFRDKPMVDPRYFTDPEGHDMRVMVAGIRKAREIAAQPAMAEWTGRELSPGVEAQTDEELQDYIRKTHNTVYHPVGTVRMGAVEDEMSPLDPELRVKGVTGLRVADASVMPEHVTVNPNITVMMIGERCADLIRSARAGETTTADAELSAALA

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 \(c\).

\(|P_{f(5RHW_1)}(2) \setminus P_{f(7VJS_1)}(2)|=132\), \(|P_{f(7VJS_1)}(2) \setminus P_{f(5RHW_1)}(2)|=45\). 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 1:110000101101011110000111011001100010011111001111010011011110010011010001001101100101000110110110001011001010010011101010001011011111101011100011100001110001011001100110110000100111110100100111010011001101000010001000000010111000100111010100110000010111101001111111110010110001011001001100010111010000101011010111001010011110100101001111010101000000011011001011111100110111000000101010000011000111011000100011010110101000011100100111100
Pair \(Z_2\) Length of longest common subsequence
5RHW_1,7VJS_1 177 4
5RHW_1,3NNE_1 148 4
7VJS_1,3NNE_1 185 3

Newick tree

 
[
	7VJS_1:95.39,
	[
		5RHW_1:74,3NNE_1:74
	]:21.39
]

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 \((0.85)(0.8)(\frac{681 }{\log_{20} 681}-\frac{246}{\log_{20}246})=121.\)
Status Protein1 Protein2 d d1/2
Query variables 5RHW_1 7VJS_1 154 117.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]