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Parikh vectors
8GJN_1 1EKA_1 7VTQ_1 Letter Amino acid
2 2 48 C Cysteine
7 0 49 Q Glutamine
6 0 80 E Glutamic acid
1 0 36 H Histidine
4 0 40 I Isoleucine
11 0 157 L Leucine
7 0 72 K Lycine
3 0 36 N Asparagine
5 0 16 W Tryptophan
6 0 24 Y Tyrosine
10 0 57 V Valine
7 0 50 F Phenylalanine
19 0 80 S Serine
5 0 55 R Arginine
3 0 24 M Methionine
7 0 59 D Aspartic acid
13 3 56 G Glycine
3 0 31 P Proline
14 0 44 T Threonine
5 1 43 A Alanine 

8GJN_1|Chain A|Heavy chain of 17B10 Fab|Mus musculus (10090)
>1EKA_1|Chains A, B|RNA (5'-R(*GP*AP*GP*UP*GP*CP*UP*C)-3')|
>7VTQ_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|NACHT, LRR and PYD domains-containing protein 3|Mus musculus (10090)
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 \(w=f(c)\)
8GJN , Knot 71 138 0.84 40 110 134 
MGWSWIFLFLLSGTAGVLSEVQLQQSGPELVKPGASVRISCKTSGYTFTEYSMFWVKQSHGQSLEWIGGINPNDDSTTYKQNFKGKATLTVDKSSSTAFMELRSLTSEDSAVYYCTRDRYDGRVVDFWGQGTTLTVSS
1EKA , Knot 5 8 0.43 8 7 6 
GAGUGCUC
7VTQ , Knot 378 1057 0.83 40 335 900 
HHHHHHDYKDDDDKLEVLFQGPEFMTSVRCKLAQYLEDLEDVDLKKFKMHLEDYPPEKGCIPVPRGQMEKADHLDLATLMIDFNGEEKAWAMAVWIFAAINRRDLWEKAKKDQPEWNDTCTSHSSMVCQEDSLEEEWMGLLGYLSRISICKKKKDYCKMYRRHVRSRFYSIKDRNARLGESVDLNSRYTQLQLVKEHPSKQEREHELLTIGRTKMRDSPMSSLKLELLFEPEDGHSEPVHTVVFQGAAGIGKTILARKIMLDWALGKLFKDKFDYLFFIHCREVSLRTPRSLADLIVSCWPDPNPPVCKILRKPSRILFLMDGFDELQGAFDEHIGEVCTDWQKAVRGDILLSSLIRKKLLPKASLLITTRPVALEKLQHLLDHPRHVEILGFSEAKRKEYFFKYFSNELQAREAFRLIQENEVLFTMCFIPLVCWIVCTGLKQQMETGKSLAQTSKTTTAVYVFFLSSLLQSRGGIEEHLFSDYLQGLCSLAADGIWNQKILFEECDLRKHGLQKTDVSAFLRMNVFQKEVDCERFYSFSHMTFQEFFAAMYYLLEEEAEGETVRKGPGGCSDLLNRDVKVLLENYGKFEKGYLIFVVRFLFGLVNQERTSYLEKKLSCKISQQVRLELLKWIEVKAKAKKLQWQPSQLELFYCLYEMQEEDFVQSAMDHFPKIEINLSTRMDHVVSSFCIKNCHRVKTLSLGFFHNSPKEEEEERRGGRPLDQVQCVFPDTHVACSSRLVNCCLTSSFCRGLFSSLSTNRSLTELDLSDNTLGDPGMRVLCEALQHPGCNIQRLWLGRCGLSHQCCFDISSVLSSSQKLVELDLSDNALGDFGIRLLCVGLKHLLCNLQKLWLVSCCLTSACCQDLALVLSSNHSLTRLYIGENALGDSGVQVLCEKMKDPQCNLQKLGLVNSGLTSICCSALTSVLKTNQNFTHLYLRSNALGDTGLRLLCEGLLHPDCKLQMLELDNCSLTSHSCWNLSTILTHNHSLRKLNLGNNDLGDLCVVTLCEVLKQQGCLLQSLQLGEMYLNRETKRALEALQEEKPELTIVFEISW

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 \(c\).

\(|P_{f(8GJN_1)}(2) \setminus P_{f(1EKA_1)}(2)|=108\), \(|P_{f(1EKA_1)}(2) \setminus P_{f(8GJN_1)}(2)|=5\). 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 1:111011111110101111001010001101101110101000001001000011110000100101111101000000000010101010100000011101001000001100000000010110111010010100
Pair \(Z_2\) Length of longest common subsequence
8GJN_1,1EKA_1 113 2
8GJN_1,7VTQ_1 243 4
1EKA_1,7VTQ_1 336 2

Newick tree

 
[
	7VTQ_1:16.11,
	[
		8GJN_1:56.5,1EKA_1:56.5
	]:10.61
]

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 \((0.85)(0.8)(\frac{146 }{\log_{20} 146}-\frac{8}{\log_{20}8})=51.8\)
Status Protein1 Protein2 d d1/2
Query variables 8GJN_1 1EKA_1 69 36
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]