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Parikh vectors
6MRG_1 4LPQ_1 2BFR_1 Letter Amino acid
30 7 21 E Glutamic acid
23 4 8 F Phenylalanine
16 12 8 Y Tyrosine
32 11 6 T Threonine
7 6 1 W Tryptophan
28 12 7 D Aspartic acid
49 23 17 G Glycine
79 15 15 L Leucine
40 13 5 P Proline
14 7 11 I Isoleucine
30 6 18 K Lycine
43 15 8 S Serine
40 21 24 A Alanine
18 5 4 N Asparagine
14 1 3 C Cysteine
29 11 3 Q Glutamine
29 15 6 R Arginine
6 3 3 H Histidine
19 6 5 M Methionine
34 14 19 V Valine 

6MRG_1|Chains A, B, C, D|Fatty-acid amide hydrolase 1|Rattus norvegicus (10116)
>4LPQ_1|Chain A|ErfK/YbiS/YcfS/YnhG family protein|Xylanimonas cellulosilytica (446471)
>2BFR_1|Chain A|HYPOTHETICAL PROTEIN AF1521|ARCHAEOGLOBUS FULGIDUS (2234)
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 \(w=f(c)\)
6MRG , Knot 231 580 0.84 40 252 538 
GSGMKETAAAKFERQHMDSPDLGTDDDDKAMETGRQKARGAATRARQKQRASLETMDKAVQRFRLQNPDLDSEALLTLPLLQLVQKLQSGELSPEAVFFTYLGKAWEVNKGTNCVTSYLTDCETQLSQAPRQGLLYGVPVSLKECFSYKGHDSTLGLSLNEGMPSESDCVVVQVLKLQGAVPFVHTNVPQSMFSYDCSNPLFGQTMNPWKSSKSPGGSSGGEGALIGSGGSPLGLGTDIGGSIRFPSAFCGICGLKPTGNRLSKSGLKGCVYGQTAVQLSLGPMARDVESLALCLKALLCEHLFTLDPTVPPLPFREEVYRSSRPLRVGYYETDNYTMPSPAMRRALIETKQRLEAAGHTLIPFLPNNIPYALEVLSTGGLFSDGGRSFLQNFKGDFVDPCLGDLILILRLPSWFKRLLSLLLKPLFPRLAAFLNNMRPRSAEKLWKLQHEIEMYRQSVIAQWKAMNLDVLLTPMLGPALDLNTPGRATGAVSYTMLYNCLDFPAGVVPVTTVTAEDDAQMELYKGYFGDIWDIILKKAMKNSVGLPVAVQCVALPWQEELCLRFMREVEQLMTPQKQPS
4LPQ , Knot 95 207 0.81 40 138 200 
SNAEPPAPQYPQRGSTGPEVLALQQRLMDLGYFILKADGDYGWATQQAVWAFQKAAGLYRDGVVGPQTQAALDAGYRPTPRSSSGKVVEIDLDKQILLAVEDGRVVRIINASSGNGETYEAKGRTYRATTPRGDFAVYMQRDGMHSSTLELGDMWRPKYFRGGYAVHGSSSIPTYPASHGCVRVSNAAMNWLWDSWGMPIGTRVLLY
2BFR , Knot 92 192 0.84 40 131 183 
MEVLFEAKVGDITLKLAQGDITQYPAKAIVNAANKRLEHGGGVAYAIAKACAGDAGLYTEISKKAMREQFGRDYIDHGEVVVTPAMNLEERGIKYVFHTVGPICSGMWSEELKEKLYKAFLGPLEKAEEMGVESIAFPAVSAGIYGCDLEKVVETFLEAVKNFKGSAVKEVALVIYDRKSAEVALKVFERSL

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 \(c\).

\(|P_{f(6MRG_1)}(2) \setminus P_{f(4LPQ_1)}(2)|=150\), \(|P_{f(4LPQ_1)}(2) \setminus P_{f(6MRG_1)}(2)|=36\). 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 1:1011000111010000100101100000011001000101110010000010100100110010100101000111011110110010010101011110011011010010001000100000010011001110111101000100010000111010011100000111011010111111000110011000000111100101100000111001101111101101111100111010110110110110101001000110101010011010111110010011101011100011010101111110001000001101100000000110111001110000010111001111110011011011001111001100110010101101011011111011011001101110111101111100101001001101000101000011101011010111011111110100110101110001100010111111110010100010101001011011011100110001111111001111100010101100100110100010
Pair \(Z_2\) Length of longest common subsequence
6MRG_1,4LPQ_1 186 3
6MRG_1,2BFR_1 187 4
4LPQ_1,2BFR_1 149 4

Newick tree

 
[
	6MRG_1:98.71,
	[
		4LPQ_1:74.5,2BFR_1:74.5
	]:24.21
]

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 \((0.85)(0.8)(\frac{787 }{\log_{20} 787}-\frac{207}{\log_{20}207})=161.\)
Status Protein1 Protein2 d d1/2
Query variables 6MRG_1 4LPQ_1 208 138.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]