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Parikh vectors
1YLS_1 2KOB_1 6KCN_1 Letter Amino acid
0 6 43 E Glutamic acid
0 3 28 T Threonine
0 1 2 W Tryptophan
0 4 21 Y Tyrosine
0 3 18 V Valine
0 6 23 R Arginine
0 6 30 D Aspartic acid
4 2 33 G Glycine
0 4 33 F Phenylalanine
0 7 20 H Histidine
0 10 47 L Leucine
0 9 41 K Lycine
0 4 29 P Proline
0 10 21 S Serine
1 10 19 A Alanine
0 8 31 N Asparagine
5 1 6 C Cysteine
0 3 14 Q Glutamine
0 10 45 I Isoleucine
0 1 12 M Methionine 

1YLS_1|Chains A, C|RNA Diels-Alder ribozyme|null
>2KOB_1|Chain A|Uncharacterized protein|Clostridium leptum DSM 753 (428125)
>6KCN_1|Chains A, B, C, D|Lysine--tRNA ligase|Plasmodium falciparum (isolate NF54) (5843)
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 \(w=f(c)\)
1YLS , Knot 7 11 0.50 8 8 9 
GGAGCUCGCCC
2KOB , Knot 55 108 0.79 40 89 102 
RDSFGDWAEKFLKSKEADGVSVSQLNSYKNYCRNHLSPLYMKSLSEILPADIQSIINETKLAKNTLKAIRNTASQIFRLAIENRAIDFNPADYVRIPKIALEHHHHHH
6KCN , Knot 209 516 0.84 40 253 483 
MEVDPRLYFENRSKFIQDQKDKGINPYPHKFERTISIPEFIEKYKDLGNGEHLEDTILNITGRIMRVSASGQKLRFFDLVGDGEKIQVLANYSFHNHEKGNFAECYDKIRRGDIVGIVGFPGKSKKGELSIFPKETILLSACLHMLPMKYGLKDTEIRYRQRYLDLLINESSRHTFVTRTKIINFLRNFLNERGFFEVETPMMNLIAGGANARPFITHHNDLDLDLYLRIATELPLKMLIVGGIDKVYEIGKVFRNEGIDNTHNPEFTSCEFYWAYADYNDLIKWSEDFFSQLVYHLFGTYKISYNKDGPENQPIEIDFTPPYPKVSIVEEIEKVTNTILEQPFDSNETIEKMINIIKEHKIELPNPPTAAKLLDQLASHFIENKYNDKPFFIVEHPQIMSPLAKYHRTKPGLTERLEMFICGKEVLNAYTELNDPFKQKECFKLQQKDREKGDTEAAQLDSAFCTSLEYGLPPTGGLGLGIDRITMFLTNKNSIKDVILFPTMRPANGGHHHHHH

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 \(c\).

\(|P_{f(1YLS_1)}(2) \setminus P_{f(2KOB_1)}(2)|=8\), \(|P_{f(2KOB_1)}(2) \setminus P_{f(1YLS_1)}(2)|=89\). 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 1:11110001000
Pair \(Z_2\) Length of longest common subsequence
1YLS_1,2KOB_1 97 1
1YLS_1,6KCN_1 253 3
2KOB_1,6KCN_1 196 6

Newick tree

 
[
	6KCN_1:12.61,
	[
		1YLS_1:48.5,2KOB_1:48.5
	]:79.11
]

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 \((0.85)(0.8)(\frac{119 }{\log_{20} 119}-\frac{11}{\log_{20}11})=41.3\)
Status Protein1 Protein2 d d1/2
Query variables 1YLS_1 2KOB_1 53 29.5
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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