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Parikh vectors
5WCG_1 1AUS_1 6ETH_1 Letter Amino acid
15 11 11 Q Glutamine
19 22 32 P Proline
4 8 3 W Tryptophan
29 43 35 A Alanine
17 9 6 C Cysteine
42 32 28 E Glutamic acid
14 10 12 M Methionine
21 26 26 D Aspartic acid
15 16 7 N Asparagine
22 46 51 G Glycine
16 15 4 H Histidine
27 18 40 S Serine
20 19 7 Y Tyrosine
21 29 24 R Arginine
41 41 38 L Leucine
35 23 28 K Lycine
17 20 11 F Phenylalanine
13 34 30 T Threonine
25 34 48 V Valine
20 19 30 I Isoleucine 

5WCG_1|Chain A|N-lysine methyltransferase SMYD2|Homo sapiens (9606)
>1AUS_1|Chains A[auth L], C[auth M], E[auth N], G[auth O]|RIBULOSE BISPHOSPHATE CARBOXYLASE/OXYGENASE|Spinacia oleracea (3562)
>6ETH_1|Chain A|Molybdopterin biosynthesis protein CNX1|Arabidopsis thaliana (3702)
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 \(w=f(c)\)
5WCG , Knot 181 433 0.84 40 240 409 
MRAEGLGGLERFCSPGKGRGLRALQPFQVGDLLFSCPAYAYVLTVNERGNHCEYCFTRKEGLSKCGRCKQAFYCNVECQKEDWPMHKLECSPMVVFGENWNPSETVRLTARILAKQKIHPERTPSEKLLAVKEFESHLDKLDNEKKDLIQSDIAALHHFYSKHLEFPDNDSLVVLFAQVNCNGFTIEDEELSHLGSAIFPDVALMNHSCCPNVIVTYKGTLAEVRAVQEIKPGEEVFTSYIDLLYPTEDRNDRLRDSYFFTCECQECTTKDKDKAKVEIRKLSDPPKAEAIRDMVRYARNVIEEFRRAKHYKSPSELLEICELSQEKMSSVFEDSNVYMLHMMYQAMGVCLYMQDWEGALQYGQKIIKPYSKHYPLYSLNVASMWLKLGRLYMGLEHKAAGEKALKKAIAIMEVAHGKDHPYISEIKQEIESH
1AUS , Knot 192 475 0.83 40 254 450 
MSPQTETKASVGFKAGVKDYKLTYYTPEYETLDTDILAAFRVSPQPGVPPEEAGAAVAAESSTGTWTTVWTDGLTNLDRYKGRCYHIEPVAGEENQYICYVAYPLDLFEEGSVTNMFTSIVGNVFGFKALRALRLEDLRIPVAYVKTFQGPPHGIQVERDKLNKYGRPLLGCTIKPKLGLSAKNYGRAVYECLRGGLDFTKDDENVNSQPFMRWRDRFLFCAEALYKAQAETGEIKGHYLNATAGTCEDMMKRAVFARELGVPIVMHDYLTGGFTANTTLSHYCRDNGLLLHIHRAMHAVIDRQKNHGMHFRVLAKALRLSGGDHIHSGTVVGKLEGERDITLGFVDLLRDDYTEKDRSRGIYFTQSWVSTPGVLPVASGGIHVWHMPALTEIFGDDSVLQFGGGTLGHPWGNAPGAVANRVALEACVQARNEGRDLAREGNTIIREATKWSPELAAACEVWKEIKFEFPAMDTV
6ETH , Knot 192 471 0.83 40 221 436 
MSRGSMEGQGCCGGGGGKTEMIPTEEALRIVFGVSKRLPPVIVSLYEALGKVLAEDIRAPDPLPPYPASVKDGYAVVASDGPGEYPVITESRAGNDGLGVTVTPGTVAYVTTGGPIPDGADAVVQVEDTKVIGDVSTESKRVKILIQTKKGTDIRRVGCDIEKDATVLTTGERIGASEIGLLATAGVTMVKVYPMPIVAILSTGDELVEPTAGTLGRGQIRDSNRAMLVAAVMQQQCKVVDLGIVRDDRKELEKVLDEAVSSGVDIILTSGGVSMGDRDFVKPLLEEKGKVYFSKVLMKPGKPLTFAEIRAKPTESMLGKTVLAFGLPGNPVSCLVCFNIFVVPTIRQLAGWTSPHPLRVRLRLQEPIKSDPIRPEFHRAIIKWKDNDGSGTPGFVAESTGHQMSSRLLSMRSANALLELPATGNVLSAGSSVSAIIVSDISAFSIDKKASLSEPGSTSGGSAWSHPQFEK

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 \(c\).

\(|P_{f(5WCG_1)}(2) \setminus P_{f(1AUS_1)}(2)|=79\), \(|P_{f(1AUS_1)}(2) \setminus P_{f(5WCG_1)}(2)|=93\). 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 1:1010111110010011010110110110110111001101011010001000000100001100010000110001000000111001000111111001010001010101110001010001000111100100010010000001100011110010000101100001111110100011010000100110111101111000001011100010110101100101100110001011010000000100001100000000000000101010010011010110011001001100100100000100110100100001001100001011011001111010100101110010011010000011001011011101101011100011100110011111011010001010010001000
Pair \(Z_2\) Length of longest common subsequence
5WCG_1,1AUS_1 172 4
5WCG_1,6ETH_1 157 3
1AUS_1,6ETH_1 153 4

Newick tree

 
[
	5WCG_1:84.19,
	[
		6ETH_1:76.5,1AUS_1:76.5
	]:7.69
]

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 \((0.85)(0.8)(\frac{908 }{\log_{20} 908}-\frac{433}{\log_{20}433})=126.\)
Status Protein1 Protein2 d d1/2
Query variables 5WCG_1 1AUS_1 161 153
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]