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Parikh vectors
3DIB_1 5FOW_1 9EVY_1 Letter Amino acid
5 17 3 E Glutamic acid
0 1 3 W Tryptophan
9 13 3 V Valine
4 15 9 R Arginine
15 11 5 S Serine
6 4 7 Y Tyrosine
3 7 4 F Phenylalanine
4 9 2 H Histidine
2 17 7 I Isoleucine
2 27 10 L Leucine
4 5 1 M Methionine
10 9 9 N Asparagine
5 11 8 D Aspartic acid
8 0 0 C Cysteine
7 11 6 Q Glutamine
3 21 18 G Glycine
10 12 7 K Lycine
4 7 12 P Proline
10 13 11 T Threonine
13 21 11 A Alanine 

3DIB_1|Chain A|Ribonuclease pancreatic|Bos taurus (9913)
>5FOW_1|Chains A, B[auth C]|DNA repair and recombination protein RadA|Pyrococcus furiosus (strain ATCC 43587 / DSM 3638 / JCM 8422 / Vc1) (186497)
>9EVY_1|Chains A, B, C, D|Nucleoprotein|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
3DIB , Knot 64 124 0.83 38 107 121 
KETAAAKFERQHMDSSTSAASSSNYCNQMMKSRNLTKDRCKPVNTFVHESLADVQAVCSQKNVACKNGQTNCYQSYSTMSITDCRETGSSKYPNCAYKTTQANKHAIVACEGNPYVPVHFDASV
5FOW , Knot 102 231 0.80 38 154 223 
MATIGRISTGSKSLDKLLGGGIETQAITEVFGEFGSGKTQLAHTLAVMVQLPPEEGGLNGSVMWIDTENTFRPERIREIAQNRGLDPDEVLKHIAYARAFNSNHQMLLVQQAEDMIKELLNTDRPVKLLIVDSLTSHFRSEYIGRGALAERQQKLAKHLADLHRLANLYDIAVFVTNQVQANGGHILAHSATLRVYLRKGKGGKRIARLIDAPHLPEGEAVFSITEKGIED
9EVY , Knot 65 136 0.78 38 103 132 
GSRPQGLPNNTASWFTALTQHGKEDLKFPRGQGVPINTNSSPDDQIGYYRRATRRIRGGDGKMKDLSPRWYFYYLGTGPEAGLPYGANKDGIIWVATDGALNTPKDHIGTRNPANNAAIVLQLPQGTTLPKGFYAE

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 \(c\).

\(|P_{f(3DIB_1)}(2) \setminus P_{f(5FOW_1)}(2)|=60\), \(|P_{f(5FOW_1)}(2) \setminus P_{f(3DIB_1)}(2)|=107\). 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 1:0001110100001000001100000000110000100000011001100011010110000011000100000000001010000001000010010000010001111001010111010101
Pair \(Z_2\) Length of longest common subsequence
3DIB_1,5FOW_1 167 3
3DIB_1,9EVY_1 152 3
5FOW_1,9EVY_1 173 3

Newick tree

 
[
	5FOW_1:87.81,
	[
		3DIB_1:76,9EVY_1:76
	]:11.81
]

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 \((0.85)(0.8)(\frac{355 }{\log_{20} 355}-\frac{124}{\log_{20}124})=70.7\)
Status Protein1 Protein2 d d1/2
Query variables 3DIB_1 5FOW_1 89 69.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]