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Parikh vectors
1RBD_1 8JOS_1 3UDO_1 Letter Amino acid
2 37 24 E Glutamic acid
0 58 43 L Leucine
1 19 12 F Phenylalanine
0 21 22 N Asparagine
1 28 11 Q Glutamine
0 25 32 I Isoleucine
0 6 2 W Tryptophan
0 35 24 G Glycine
0 31 11 P Proline
1 51 25 S Serine
1 35 9 T Threonine
0 33 15 V Valine
0 17 13 Y Tyrosine
4 58 39 A Alanine
1 34 10 R Arginine
1 28 23 D Aspartic acid
0 4 5 C Cysteine
1 16 4 H Histidine
2 21 29 K Lycine
0 12 8 M Methionine 

1RBD_1|Chain A[auth S]|RIBONUCLEASE S (S-PEPTIDE)|Bos taurus (9913)
>8JOS_1|Chain A|Acyltransferase|Pseudozyma tsukubaensis (5483)
>3UDO_1|Chain A|3-isopropylmalate dehydrogenase|Campylobacter jejuni (197)
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 \(w=f(c)\)
1RBD , Knot 12 16 0.69 22 14 14 
KETAAAKFERQHADSX
8JOS , Knot 226 569 0.84 40 261 531 
MGSSHHHHHHSSGLVPRGSHMLGDQVWKEVSSNTLRRPLIGVEKMINYAEYYQNGNFQLTIALSLDTNLSADKMQERLGLALWTVRGYLPELGTWTVGSSQDSTLDLDHVTFKAIQTVEEAQEWIEDTAILVDEGTTVSEMVDLLSNKRIEPVGKQFRAYLVSQPRHGKPALVMNASHTLNGHRMLFQGSAILQALVDARLTALVASSNSSREALEAVFVPEDISRLARKLPQSLNTAYADKFRPDEADVEAGFAKLGERILNSTQPTLGIPRFSTPSQNPQYTLGSANGQPLTMLNLRRQIGVTEHRMLHRAFKKRGSSLPSFTYACIVNSIDRRCKASTSEADEAPGANLVYSAHASRWFPAETFMSRSPVNMAIVPGSGYVAPEELRSKQRGRDLNESELFALAKTIRAKQEQYLESPHIISYTAQVGDDIAAAMAETANKQRQAGTDPYVALSENSPAICPPTLTSQGEVPIKNLYTPEGASFDPKPKQPEYEYIYFGEGLLGGRTTDASVCFALWSLAGILTLQAHFDSRFFDAQVIDTILDDVVLQLRRSAASAVDEAPEAKL
3UDO , Knot 153 361 0.83 40 189 345 
SNAMKTYKVAVLAGDGIGPLVMKEALKILTFIAQKYNFSFELNEAKIGGASIDAYGVALSDETLKLCEQSDAILFGSVGGPKWDNLPIDQRPERASLLPLRKHFNLFANLRPCKIYESLTHASPLKNEIIQKGVDILCVRELTGGIYFGKQDLGKESAYDTEIYTKKEIERIARIAFESARIRKKKVHLIDKANVLASSILWREVVANVAKDYQDINLEYMYVDNAAMQIVKNPSIFDVMLCSNLFGDILSDELAAINGSLGLLSSASLNDKGFGLYEPAGGSAPDIAHLNIANPIAQILSAALMLKYSFKEEQAAQDIENAISLALAQGKMTKDLNAKSYLNTDEMGDCILEILKENDNG

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 \(c\).

\(|P_{f(1RBD_1)}(2) \setminus P_{f(8JOS_1)}(2)|=3\), \(|P_{f(8JOS_1)}(2) \setminus P_{f(1RBD_1)}(2)|=250\). 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 1:0001110100001000
Pair \(Z_2\) Length of longest common subsequence
1RBD_1,8JOS_1 253 3
1RBD_1,3UDO_1 189 2
8JOS_1,3UDO_1 164 4

Newick tree

 
[
	1RBD_1:11.91,
	[
		3UDO_1:82,8JOS_1:82
	]:37.91
]

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 \((0.85)(0.8)(\frac{585 }{\log_{20} 585}-\frac{16}{\log_{20}16})=175.\)
Status Protein1 Protein2 d d1/2
Query variables 1RBD_1 8JOS_1 222 113.5
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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