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Parikh vectors
2KDA_1 3DKZ_1 3SET_1 Letter Amino acid
0 7 26 E Glutamic acid
0 9 5 H Histidine
0 9 22 I Isoleucine
0 2 34 K Lycine
0 8 23 D Aspartic acid
0 3 22 N Asparagine
0 5 7 M Methionine
0 7 15 F Phenylalanine
0 12 5 R Arginine
0 7 21 P Proline
0 8 12 S Serine
0 2 9 Q Glutamine
4 2 0 C Cysteine
2 16 29 G Glycine
0 12 30 L Leucine
2 13 19 T Threonine
0 0 8 W Tryptophan
0 0 15 Y Tyrosine
0 8 20 V Valine
2 12 50 A Alanine 

2KDA_1|Chain A|5'-D(*CP*GP*TP*AP*CP*(63H)P*CP*AP*TP*GP*C)-3'|
>3DKZ_1|Chains A, B|Thioesterase superfamily protein|Bordetella parapertussis (519)
>3SET_1|Chains A, B[auth C]|Maltose-binding periplasmic protein|Escherichia coli (83333)
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 \(w=f(c)\)
2KDA , Knot 8 11 0.58 10 10 9 
CGTACXCATGC
3DKZ , Knot 66 142 0.76 36 105 134 
MSTPHTDFFGLTIPFMQLLGVVPEHSGNGTARTRLPARADLVNSRGDIHGGTLMSVLDFTLGAAIRGDTPEVGVATIDMNTSFMSPGRGDLVIETRCLRRGASIAFCEGEIRDSAGELVAKATATFKIIQRRPGLEHHHHHH
3SET , Knot 152 372 0.80 38 193 344 
MKIEEGKLVIWINGDKGYNGLAEVGKKFEKDTGIKVTVEHPDKLEEKFPQVAATGDGPDIIFWAHDRFGGYAQSGLLAEITPDKAFQDKLYPFTWDAVRYNGKLIAYPIAVEALSLIYNKDLLPNPPKTWEEIPALDKELKAKGKSALMFNLQEPYFTWPLIAADGGYAFKYENGKYDIKDVGVDNAGAKAGLTFLVDLIKNKHMNADTDYSIAEHAFNHGETAMTINGPWAWSNIDTSKVNYGVTVLPTFKGQPSKPFVGVLSAGINAASPNKELAKEFLENYLLTDEGLEAVNKDKPLGAVALKSYEEELAKDPRIAATMENAQKGEIMPNIPQMSAFWYAVRTAVINAASGRQTVDAALAAAQTNAAAA

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 \(c\).

\(|P_{f(2KDA_1)}(2) \setminus P_{f(3DKZ_1)}(2)|=7\), \(|P_{f(3DKZ_1)}(2) \setminus P_{f(2KDA_1)}(2)|=102\). 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 1:01010001010
Pair \(Z_2\) Length of longest common subsequence
2KDA_1,3DKZ_1 109 3
2KDA_1,3SET_1 197 3
3DKZ_1,3SET_1 172 3

Newick tree

 
[
	3SET_1:10.02,
	[
		2KDA_1:54.5,3DKZ_1:54.5
	]:47.52
]

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 \((0.85)(0.8)(\frac{153 }{\log_{20} 153}-\frac{11}{\log_{20}11})=52.6\)
Status Protein1 Protein2 d d1/2
Query variables 2KDA_1 3DKZ_1 62 33.5
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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