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Parikh vectors
6XEU_1 2OOF_1 1ARX_1 Letter Amino acid
15 48 33 A Alanine
19 36 38 G Glycine
10 22 29 P Proline
26 34 19 V Valine
7 4 2 W Tryptophan
7 20 4 H Histidine
33 53 28 L Leucine
25 17 38 S Serine
25 22 20 T Threonine
17 22 15 R Arginine
19 10 18 F Phenylalanine
25 30 14 E Glutamic acid
29 17 17 I Isoleucine
16 13 4 K Lycine
10 11 7 M Methionine
13 8 13 N Asparagine
14 19 20 D Aspartic acid
6 6 8 C Cysteine
10 14 14 Q Glutamine
14 10 3 Y Tyrosine 

6XEU_1|Chains A, B[auth D], C[auth G], D[auth J]|G protein-activated inward rectifier potassium channel 2|Mus musculus (10090)
>2OOF_1|Chain A|4-imidazolone-5-propanoate amidohydrolase|unidentified (32644)
>1ARX_1|Chain A|PEROXIDASE|'Arthromyces ramosus' (5451)
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 \(w=f(c)\)
6XEU , Knot 148 340 0.84 40 206 328 
MAKRKIQRYVRKDGKCNVHHGNVRETYRYLTDIFTTLVDLKWRFNLLIFVMVYTVTWLFFGMIWWLIAYIRGDMDHIEDPSWTPCVTNLNGFVSAFLFSIETETTIGYGYRVITDKCPEGIILLLIQSVLGSIVNAFMVGCMFVKISQPKKRAETLVFSTHAVISMRDGKLCLMFRVGDLRNSHIVEASIRAKLIKSKQTSEGEFIPLNQTDINVGYYTGDDRLFLVSPLIISHEINQQSPFWEISKAQLPKEELEIVVILEGMVEATGMTCQARSSYITSEILWGYRFTPVLTLEDGFYEVDYNSFHETYETSTPSLSAKELAELANRAESNSLEVLFQ
2OOF , Knot 174 416 0.84 40 211 386 
MSLNCERVWLNVTPATLRSDLADYGLLEPHALGVHEGRIHALVPMQDLKGPYPAHWQDMKGKLVTPGLIDCHTHLIFAGSRAEEFELRQKGVPYAEIARKGGGIISTVRATRAASEDQLFELALPRVKSLIREGVTTVEIKSGYGLTLEDELKMLRVARRLGEALPIRVKTTLLAAHAVPPEYRDDPDSWVETICQEIIPAAAEAGLADAVDVFCEHIGFSLAQTEQVYLAADQYGLAVKGHMDQLSNLGGSTLAANFGALSVDHLEYLDPEGIQALAHRGVVATLLPTAFYFLKETKLPPVVALRKAGVPMAVSSDINPGTAPIVSLRMAMNMACTLFGLTPVEAMAGVTRHAARALGEQEQLGQLRVGMLADFLVWNCGHPAELSYLIGVDQLVSRVVNGEETLHGEGHHHHHH
1ARX , Knot 141 344 0.79 40 189 315 
QGPGGGGGSVTCPGGQSTSNSQCCVWFDVLDDLQTNFYQGSKCESPVRKILRIVFHDAIGFSPALTAAGQFGGGGADGSIIAHSNIELAFPANGGLTDTIEALRAVGINHGVSFGDLIQFATAVGMSNCPGSPRLEFLTGRSNSSQPSPPSLIPGPGNTVTAILDRMGDAGFSPDEVVDLLAAHSLASQEGLNSAIFRSPLDSTPQVFDTQFYIETLLKGTTQPGPSLGFAEELSPFPGEFRMRSDALLARDSRTACRWQSMTSSNEVMGQRYRAAMAKMSVLGFDRNALTDCSDVIPSAVSNNAAPVIPGGLTVDDIEVSCPSEPFPEIATASGPLPSLAPAP

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 \(c\).

\(|P_{f(6XEU_1)}(2) \setminus P_{f(2OOF_1)}(2)|=89\), \(|P_{f(2OOF_1)}(2) \setminus P_{f(6XEU_1)}(2)|=94\). 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 1:1100010001000100010010100000010011001101010101111111001011111111111101010100100101010100101110111101000001101001100001011111110011101101111101110100100010011100011101001010111011010000110101010110000000101111000010110001000111101111000100001110100101100010111110111010110001000010001111001011101001100100001000000001010100110110010000101110
Pair \(Z_2\) Length of longest common subsequence
6XEU_1,2OOF_1 183 5
6XEU_1,1ARX_1 159 4
2OOF_1,1ARX_1 160 5

Newick tree

 
[
	2OOF_1:87.98,
	[
		6XEU_1:79.5,1ARX_1:79.5
	]:8.48
]

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 \((0.85)(0.8)(\frac{756 }{\log_{20} 756}-\frac{340}{\log_{20}340})=113.\)
Status Protein1 Protein2 d d1/2
Query variables 6XEU_1 2OOF_1 143 131
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]