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Parikh vectors
8HEE_1 1ZCD_1 7WZY_1 Letter Amino acid
8 8 4 H Histidine
7 11 17 K Lycine
12 15 9 P Proline
8 24 10 S Serine
14 36 17 G Glycine
11 9 12 N Asparagine
18 64 32 L Leucine
26 49 16 A Alanine
1 3 2 C Cysteine
8 6 9 Q Glutamine
8 40 13 I Isoleucine
3 11 6 M Methionine
6 18 4 F Phenylalanine
23 13 11 T Threonine
16 32 20 V Valine
8 10 13 D Aspartic acid
10 9 13 E Glutamic acid
1 8 0 W Tryptophan
8 8 7 Y Tyrosine
15 14 12 R Arginine 

8HEE_1|Chains A, D, E, F, G|VP1 of capsid protein|Foot-and-mouth disease virus (12110)
>1ZCD_1|Chains A, B|Na(+)/H(+) antiporter 1|Escherichia coli (562)
>7WZY_1|Chains A, B|ATP phosphoribosyltransferase|Acinetobacter baumannii (470)
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 \(w=f(c)\)
8HEE , Knot 102 211 0.86 40 153 207 
TTSAGESADPVTTTVENYGGETQVQRRHHTDVGFIMDRFVKINNTNPTHVIDLMQTHQHGLVGALLRAATYYFSDLEIVVRHEGNLTWVPNGAPEAALSNAGNPTAYNKAPFTRLALPYTAPHRVLATVYNGTSKYSTTGERTRGDLGALAARVATQLPASFNFGAIRATDISELLVRMKRAELYCPRPLLAVEVTAQDRHKQKIIAPAKQ
1ZCD , Knot 158 388 0.81 40 188 352 
MKHLHRFFSSDASGGIILIIAAILAMIMANSGATSGWYHDFLETPVQLRVGSLEINKNMLLWINDALMAVFFLLVGLEVKRELMQGSLASLRQAAFPVIAAIGGMIVPALLYLAFNYADPITREGWAIPAATDIAFALGVLALLGSRVPLALKIFLMALAIIDDLGAIIIIALFYTNDLSMASLGVAAVAIAVLAVLNLCGARRTGVYILVGVVLWTAVLKSGVHATLAGVIVGFFIPLKEKHGRSPAKRLEHVLHPWVAYLILPLFAFANAGVSLQGVTLDGLTSILPLGIIAGLLIGKPLGISLFCWLALRLKLAHLPEGTTYQQIMVVGILCGIGFTMSIFIASLAFGSVDPELINWAKLGILVGSISSAVIGYSWLRVRLRPSV
7WZY , Knot 102 227 0.81 38 153 218 
MNDVRNDDPNFNVMGNFDHGLTLALSKGRILKETLPLLATAGINLLEDPEKSRKLIFPTTHKQVRILILRASDVPTYVENGAADLGVAGKDVLMEHGAQHVYELLDLQIAKCKLMTAGKVGMERPKGRLKIATKYVNLTRQYYASLGEQVDVIKLYGSMELAPLVGLGDYIVDVVDTGNTLRANGLEPLEEICKVSSRLIVNKASFKRKQVLLNPIISQLEQAVQSR

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 \(c\).

\(|P_{f(8HEE_1)}(2) \setminus P_{f(1ZCD_1)}(2)|=60\), \(|P_{f(1ZCD_1)}(2) \setminus P_{f(8HEE_1)}(2)|=95\). 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 1:0001100101100010001100010000000111110011010000100110110000011111110110001001011100010101110111011100110101000111001111001100111010010000000100001011111101100111010111101001001110100101001011111010100000001111100
Pair \(Z_2\) Length of longest common subsequence
8HEE_1,1ZCD_1 155 4
8HEE_1,7WZY_1 136 4
1ZCD_1,7WZY_1 155 4

Newick tree

 
[
	1ZCD_1:80.41,
	[
		8HEE_1:68,7WZY_1:68
	]:12.41
]

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 \((0.85)(0.8)(\frac{599 }{\log_{20} 599}-\frac{211}{\log_{20}211})=110.\)
Status Protein1 Protein2 d d1/2
Query variables 8HEE_1 1ZCD_1 134 104.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]