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Parikh vectors
2IWM_1 8ZBM_1 2RKU_1 Letter Amino acid
22 12 9 N Asparagine
2 5 7 C Cysteine
26 28 17 G Glycine
4 9 10 H Histidine
24 30 20 I Isoleucine
19 19 23 K Lycine
12 12 16 F Phenylalanine
11 18 23 R Arginine
16 26 22 E Glutamic acid
16 19 18 P Proline
18 29 14 A Alanine
9 11 6 Q Glutamine
28 26 33 L Leucine
12 16 4 M Methionine
24 23 16 S Serine
3 4 1 W Tryptophan
18 16 10 Y Tyrosine
20 20 12 D Aspartic acid
30 27 13 T Threonine
24 21 20 V Valine 

2IWM_1|Chains A, B, C, D|PENICILLIN ACYLASE|BACILLUS SPHAERICUS (1421)
>8ZBM_1|Chains A[auth E], B[auth H], C[auth O], D[auth P], F[auth Q], G[auth R]|Actin, alpha skeletal muscle|Rattus norvegicus (10116)
>2RKU_1|Chain A|Serine/threonine-protein kinase PLK1|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 \(w=f(c)\)
2IWM , Knot 144 338 0.82 40 199 328 
MLGSSSLSIRTTDDKSLFARTMDFTMEPDSKVIIVPRNYGIRLLEKENVVINNSYAFVGMGSTDITSPVLYDGVNEKGLMGAMLYYATFATYADEPKKGTRGINPVYVISQVLGNCVTVDDVIEKLTSYTLLNEANIILGFAPPLHYTFTDASGESIVIEPDKTGITIHRKTIGVMTNSPGYEWHQTNLRAYIGVTPNPPQDIMMGDLDLTPFGQGAGGLGLPGDFTPSARFLRVAYWKKYTEKAKNETEGVTNLFHILSSVNIPKGVVLTNEGKTDYTIYTSAMCAQSKNYYFKLYDNSRISAVSLMAENLNSQDLITFEWDRKQDIKQLNQVNVMS
8ZBM , Knot 158 371 0.84 40 218 357 
TTALVCDNGSGLVKAGFAGDDAPRAVFPSIVGRPRHQGVMVGMGQKDSYVGDEAQSKRGILTLKYPIEHGIITNWDDMEKIWHHTFYNELRVAPEEHPTLLTEAPLNPKANREKMTQIMFETFNVPAMYVAIQAVLSLYASGRTTGIVLDSGDGVTHNVPIYEGYALPHAIMRLDLAGRDLTDYLMKILTERGYSFVTTAEREIVRDIKEKLCYVALDFENEMATAASSSSLEKSYELPDGQVITIGNERFRCPETLFQPSFIGMESAGIHETTYNSIMKCDIDIRKDLYANNVMSGGTTMYPGIADRMQKEITALAPSTMKIKIIAPPERKYSVWIGGSILASLSTFQQMWITKQEYDEAGPSIVHRKCF
2RKU , Knot 128 294 0.82 40 186 282 
AKEIPEVLVDPRSRRRYVRGRFLGKGGFAKCFEISDADTKEVFAGKIVPKSLLLKPHQREKMSMEISIHRSLAHQHVVGFHGFFEDNDFVFVVLELCRRRSLLELHKRRKALTEPEARYYLRQIVLGCQYLHRNRVIHRDLKLGNLFLNEDLEVKIGDFGLATKVEYDGERKKVLCGTPNYIAPEVLSKKGHSFEVDVWSIGCIMYTLLVGKPPFETSCLKETYLRIKKNEYSIPKHINPVAASLIQKMLQTDPTARPTINELLNDEFFTSGYIPARLPITCLTIPPRFSIAPS

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 \(c\).

\(|P_{f(2IWM_1)}(2) \setminus P_{f(8ZBM_1)}(2)|=68\), \(|P_{f(8ZBM_1)}(2) \setminus P_{f(2IWM_1)}(2)|=87\). 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 1:11100010100000001110010101010001111100011011000011100001111110001001110011000111111100101100100100100110110110011100101001100100001100101111111110001001010011101000110100001111000110010000101011101011001111010101110111111111010101011011010000001000001100110110010110111100010000010001101000000101000001011011100100001101010000010010010110
Pair \(Z_2\) Length of longest common subsequence
2IWM_1,8ZBM_1 155 4
2IWM_1,2RKU_1 163 4
8ZBM_1,2RKU_1 178 4

Newick tree

 
[
	2RKU_1:87.78,
	[
		2IWM_1:77.5,8ZBM_1:77.5
	]:10.28
]

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 \((0.85)(0.8)(\frac{709 }{\log_{20} 709}-\frac{338}{\log_{20}338})=101.\)
Status Protein1 Protein2 d d1/2
Query variables 2IWM_1 8ZBM_1 129 121
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]