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Parikh vectors
6DWR_1 8OZE_1 2ZUQ_1 Letter Amino acid
10 0 6 Y Tyrosine
6 0 5 D Aspartic acid
14 0 28 L Leucine
3 0 13 F Phenylalanine
8 0 10 P Proline
4 0 7 W Tryptophan
14 5 19 A Alanine
2 0 9 R Arginine
4 0 6 E Glutamic acid
10 0 6 T Threonine
25 8 10 G Glycine
3 0 2 H Histidine
15 0 8 I Isoleucine
2 0 7 M Methionine
33 0 6 S Serine
16 0 1 N Asparagine
12 0 3 C Cysteine
10 0 9 Q Glutamine
14 0 6 K Lycine
17 0 15 V Valine 

6DWR_1|Chain A|Cationic trypsin|Bos taurus (9913)
>8OZE_1|Chains A, G[auth N]|RNA (5'-R(P*UP*GP*AP*GP*GP*UP*AP*GP*UP*AP*GP*GP*UP*UP*GP*UP*AP*UP*AP*G)-3')|Maribacter polysiphoniae (429344)
>2ZUQ_1|Chains A, D|Disulfide bond formation protein B|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 \(w=f(c)\)
6DWR , Knot 101 223 0.81 42 142 216 
IVGGYTCGANTVPYQVSLNSGYHFCGGSLINSQWVVSAAHCYKSGIQVRLGEDNINVVEGNEQFISASKSIVHPSYNSNTLNNDIMLIKLKSAASLNSRVASISLPTSCASAGTQCLISGWGNTKSSGTSYPDVLKCLKAPILSDSSCKSAYPGQITSNMFCAGYLEGGKDSCQGDXGGPVVCSGKLQGIVSWGSGCAQKNKPGVYTKVCNYVSWIKQTIASN
8OZE , Knot 9 20 0.45 6 8 12 
UGAGGUAGUAGGUUGUAUAG
2ZUQ , Knot 86 176 0.84 40 127 171 
MLRFLNQASQGRGAWLLMAFTALALELTALWFQHVMLLKPSVLCIYERVALFGVLGAALIGAIAPKTPLRYVAMVIWLYSAFRGVQLTYEHTMLQLYPSPFATCDFMVRFPEWLPLDKWVPQVFVASGDCAERQWDFLGLEMPQWLLGIFIAYLIVAVLVVISQPFKAKKRDLFGR

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 \(c\).

\(|P_{f(6DWR_1)}(2) \setminus P_{f(8OZE_1)}(2)|=139\), \(|P_{f(8OZE_1)}(2) \setminus P_{f(6DWR_1)}(2)|=5\). 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 1:1111000110011001010010010110110001110110000011010110001011010001101000110100000010001111010011010001101011000101100011011100000100010110010111100000001011010001101101011000001001111100101011101101010000111000100010110001100
Pair \(Z_2\) Length of longest common subsequence
6DWR_1,8OZE_1 144 2
6DWR_1,2ZUQ_1 187 3
8OZE_1,2ZUQ_1 133 2

Newick tree

 
[
	6DWR_1:88.37,
	[
		8OZE_1:66.5,2ZUQ_1:66.5
	]:21.87
]

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 \((0.85)(0.8)(\frac{243 }{\log_{20} 243}-\frac{20}{\log_{20}20})=76.5\)
Status Protein1 Protein2 d d1/2
Query variables 6DWR_1 8OZE_1 98 52.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]