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Parikh vectors
7LRV_1 4WCU_1 1DOH_1 Letter Amino acid
5 20 5 Q Glutamine
3 11 32 G Glycine
4 15 11 P Proline
16 21 30 A Alanine
3 15 15 R Arginine
6 17 11 N Asparagine
11 29 13 D Aspartic acid
3 3 7 C Cysteine
8 27 20 E Glutamic acid
8 16 4 H Histidine
2 21 20 I Isoleucine
10 40 12 L Leucine
11 17 17 K Lycine
2 20 15 S Serine
3 23 11 T Threonine
4 26 33 V Valine
3 11 9 M Methionine
2 14 9 F Phenylalanine
1 8 7 Y Tyrosine
1 5 2 W Tryptophan 

7LRV_1|Chains A, B[auth C]|Soluble cytochrome b562|Escherichia coli (562)
>4WCU_1|Chains A, B, C, D|cAMP-specific 3',5'-cyclic phosphodiesterase 4D|Homo sapiens (9606)
>1DOH_1|Chains A, B|TRIHYDROXYNAPHTHALENE REDUCTASE|Magnaporthe grisea (148305)
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 \(w=f(c)\)
7LRV , Knot 55 106 0.80 40 81 101 
ADLEDNMETLNDNLKVIEKADNAAQVKDALTKMRAAALDAQKATPPKLEDKSPDSPEMWHFRHGFDHLVGHIDDALKLANEGKVKEAQAAAEQLKCHCNHCHQHYR
4WCU , Knot 157 359 0.85 40 215 338 
IPRFGVKTEQEDVLAKELEDVNKWGLHVFRIAELSGNRPLTVIMHTIFQERDLLKTFKIPVDTLITYLMTLEDHYHADVAYHNNIHAADVVQSTHVLLSTPALEAVFTDLEILAAIFASAIHDVDHPGVSNQFLINTNSELALMYNDSSVLENHHLAVGFKLLQEENCDIFQNLTKKQRQSLRKMVIDIVLATDMSKHMNLLADLKTMVETKKVTSSGVLLLDNYSDRIQVLQNMVHCADLSNPTKPLQLYRQWTDRIMEEFFRQGDRERERGMEISPMCDKHNASVEKSQVGFIDYIVHPLWETWADLVHPDAQDILDTLEDNREWYQSTIPQSPSPAPDDPEEGRQGQTEKFQFELT
1DOH , Knot 122 283 0.81 40 177 272 
MPAVTQPRGESKYDAIPGPLGPQSASLEGKVALVTGAGRGIGREMAMELGRRGCKVIVNYANSTESAEEVVAAIKKNGSDAACVKANVGVVEDIVRMFEEAVKIFGKLDIVCSNSGVVSFGHVKDVTPEEFDRVFTINTRGQFFVAREAYKHLEIGGRLILMGSITGQAKAVPKHAVYSGSKGAIETFARCMAIDMADKKITVNVVAPGGIKTDMYHAVCREYIPNGENLSNEEVDEYAAVQWSPLRRVGLPIDIARVVCFLASNDGGWVTGKVIGIDGGACM

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 \(c\).

\(|P_{f(7LRV_1)}(2) \setminus P_{f(4WCU_1)}(2)|=27\), \(|P_{f(4WCU_1)}(2) \setminus P_{f(7LRV_1)}(2)|=161\). 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 1:1010001001000101100100110100110010111101001011010000100101101001100111010011011001010010111001000000000000
Pair \(Z_2\) Length of longest common subsequence
7LRV_1,4WCU_1 188 4
7LRV_1,1DOH_1 182 4
4WCU_1,1DOH_1 178 3

Newick tree

 
[
	7LRV_1:93.65,
	[
		1DOH_1:89,4WCU_1:89
	]:4.65
]

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 \((0.85)(0.8)(\frac{465 }{\log_{20} 465}-\frac{106}{\log_{20}106})=107.\)
Status Protein1 Protein2 d d1/2
Query variables 7LRV_1 4WCU_1 134 86.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]