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Parikh vectors
5UCE_1 8CLN_1 1PIJ_1 Letter Amino acid
0 3 4 C Cysteine
1 12 3 Q Glutamine
2 6 0 I Isoleucine
5 19 8 E Glutamic acid
1 10 0 M Methionine
4 11 2 F Phenylalanine
6 33 11 A Alanine
0 20 3 R Arginine
3 5 2 N Asparagine
5 17 6 D Aspartic acid
2 11 3 Y Tyrosine
3 28 5 V Valine
4 27 7 G Glycine
2 28 2 L Leucine
0 22 4 P Proline
1 8 3 W Tryptophan
0 18 4 H Histidine
6 3 1 K Lycine
0 23 4 S Serine
11 11 1 T Threonine 

5UCE_1|Chain A|Immunoglobulin G-binding protein G|Streptococcus sp. GX7805 (1325)
>8CLN_1|Chains A, B, C, D|Hydrolase|Streptomyces coelicoflavus (285562)
>1PIJ_1|Chain A|HIGH POTENTIAL IRON SULFUR PROTEIN|Halorhodospira halophila (1053)
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 \(w=f(c)\)
5UCE , Knot 33 56 0.79 30 46 54 
MTFKAIINGKTLKGETTTEAVDAATAEKVFKQYFNDNGLDGEWTYDDATKTFTVTE
8CLN , Knot 140 315 0.85 40 195 303 
MVTSPALRDVHVPHAYPEQQVDLGEITMNYAEAGDPDRPAVLLIPEQTGSWWSYEEAMGLLSEHFHVYAVDLRGQGRSSWTPKRYSLDNFGNDLVRFIALVVKRPVVVAGNSSGGVLAAWLSAYSMPGQLRGVLCEDPPFFASELVPAHGHSVRQGAGPVFELFRTYLGDQWSVGDWEGFCRAAGASASPMARSFVADGIPQHLQEYDPEWARVFYEGTVGLSCPHERMLGQVKTPVLLTHHMRGIDPETGNLLGALSDEQALRARRLMDSAGVTVDYESVPDASHMMHQSAPARYVEIFTRWAAALAPHHHHHH
1PIJ , Knot 43 73 0.84 36 61 71 
ASEPRAEDGHAHDYVNEAADASGHPRYQEGQLCENCAFWGEAVQDGWGRCTHPDFDEVLVKAEGWCSVYAPAS

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 \(c\).

\(|P_{f(5UCE_1)}(2) \setminus P_{f(8CLN_1)}(2)|=24\), \(|P_{f(8CLN_1)}(2) \setminus P_{f(5UCE_1)}(2)|=173\). 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 1:10101110100101000001101101001100010001101010000100010100
Pair \(Z_2\) Length of longest common subsequence
5UCE_1,8CLN_1 197 3
5UCE_1,1PIJ_1 91 3
8CLN_1,1PIJ_1 182 3

Newick tree

 
[
	8CLN_1:10.29,
	[
		5UCE_1:45.5,1PIJ_1:45.5
	]:60.79
]

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 \((0.85)(0.8)(\frac{371 }{\log_{20} 371}-\frac{56}{\log_{20}56})=99.4\)
Status Protein1 Protein2 d d1/2
Query variables 5UCE_1 8CLN_1 130 75.5
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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