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Parikh vectors
2YQK_1 6PMI_1 1ZRZ_1 Letter Amino acid
0 23 15 A Alanine
4 19 13 T Threonine
4 5 11 Y Tyrosine
4 23 20 R Arginine
2 9 15 N Asparagine
1 4 5 C Cysteine
6 36 29 E Glutamic acid
0 5 12 M Methionine
2 1 4 W Tryptophan
2 30 24 V Valine
1 10 16 Q Glutamine
4 38 35 L Leucine
8 16 21 K Lycine
2 16 16 P Proline
1 21 30 D Aspartic acid
8 20 24 G Glycine
0 8 8 H Histidine
3 24 19 I Isoleucine
4 4 23 F Phenylalanine
7 17 24 S Serine 

2YQK_1|Chain A|Arginine-glutamic acid dipeptide repeats protein|Homo sapiens (9606)
>6PMI_1|Chains A, B|DNA-directed RNA polymerase subunit alpha|Escherichia coli O157:H7 (83334)
>1ZRZ_1|Chain A|Protein kinase C, iota|Homo sapiens (9606)
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 \(w=f(c)\)
2YQK , Knot 35 63 0.76 34 52 58 
GSSGSSGIEKCWTEDEVKRFVKGLRQYGKNFFRIRKELLPNKETGELITFYYYWKKTSGPSSG
6PMI , Knot 146 329 0.85 40 180 315 
MQGSVTEFLKPRLVDIEQVSSTHAKVTLEPLERGFGHTLGNALRRILLSSMPGCAVTEVEIDGVLHEYSTKEGVQEDILEILLNLKGLAVRVQGKDEVILTLNKSGIGPVTAADITHDGDVEIVKPQHVICHLTDENASISMRIKVQRGRGYVPASTRIHSEEDERPIGRLLVDACYSPVERIAYNVEAARVEQRTDLDKLVIEMETNGTIDPEEAIRRAATILAEQLEAFVDLRDVRQPEVKEEKPEFDPILLRPVDDLELTVRSANCLKAEAIHYIGDLVQRTEVELLKTPNLGKKSLTEIKDVLASRGLSLGMRLENWPPASIADE
1ZRZ , Knot 160 364 0.86 40 222 351 
EKEAMNTRESGKASSSLGLQDFDLLRVIGRGSYAKVLLVRLKKTDRIYAMKVVKKELVNDDEDIDWVQTEKHVFEQASNHPFLVGLHSCFQTESRLFFVIEYVNGGDLMFHMQRQRKLPEEHARFYSAEISLALNYLHERGIIYRDLKLDNVLLDSEGHIKLTDYGMCKEGLRPGDTTSTFCGTPNYIAPEILRGEDYGFSVDWWALGVLMFEMMAGRSPFDIVGSSDNPDQNTEDYLFQVILEKQIRIPRSMSVKAASVLKSFLNKDPKERLGCLPQTGFADIQGHPFFRNVDWDMMEQKQVVPPFKPNISGEFGLDNFDSQFTNERVQLTPDDDDIVRKIDQSEFEGFEYINPLLMSAEECV

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 \(c\).

\(|P_{f(2YQK_1)}(2) \setminus P_{f(6PMI_1)}(2)|=23\), \(|P_{f(6PMI_1)}(2) \setminus P_{f(2YQK_1)}(2)|=151\). 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 1:100100110001000010011011000100110100011100001011010001000011001
Pair \(Z_2\) Length of longest common subsequence
2YQK_1,6PMI_1 174 3
2YQK_1,1ZRZ_1 202 3
6PMI_1,1ZRZ_1 156 3

Newick tree

 
[
	2YQK_1:99.08,
	[
		6PMI_1:78,1ZRZ_1:78
	]:21.08
]

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 \((0.85)(0.8)(\frac{392 }{\log_{20} 392}-\frac{63}{\log_{20}63})=102.\)
Status Protein1 Protein2 d d1/2
Query variables 2YQK_1 6PMI_1 131 77
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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